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+Centralized Cooperative Exploration Policy for Continuous
+Control Tasks
+Chao Li*
+Institute of Automation, Chinese
+Academy of Sciences, China
+[email protected]
+Chen Gong∗
+Institute of Automation, Chinese
+Academy of Sciences, China
+[email protected]
+Qiang He
+University of Tubingen, Germany
+[email protected]
+Xinwen Hou†
+Institute of Automation, Chinese
+Academy of Sciences, Beijing, China
+[email protected]
+Yu Liu
+Institute of Automation, Chinese
+Academy of Sciences, Beijing, China
+[email protected]
+ABSTRACT
+The deep reinforcement learning (DRL) algorithm works brilliantly
+on solving various complex control tasks. This phenomenal success
+can be partly attributed to DRL encouraging intelligent agents to
+sufficiently explore the environment and collect diverse experiences
+during the agent training process. Therefore, exploration plays a
+significant role in accessing an optimal policy for DRL. Despite
+recent works making great progress in continuous control tasks,
+exploration in these tasks has remained insufficiently investigated.
+To explicitly encourage exploration in continuous control tasks,
+we propose CCEP (Centralized Cooperative Exploration Policy),
+which utilizes underestimation and overestimation of value func-
+tions to maintain the capacity of exploration. CCEP first keeps two
+value functions initialized with different parameters, and generates
+diverse policies with multiple exploration styles from a pair of value
+functions. In addition, a centralized policy framework ensures that
+CCEP achieves message delivery between multiple policies, fur-
+thermore contributing to exploring the environment cooperatively.
+Extensive experimental results demonstrate that CCEP achieves
+higher exploration capacity. Empirical analysis shows diverse ex-
+ploration styles in the learned policies by CCEP, reaping benefits in
+more exploration regions. And this exploration capacity of CCEP
+ensures it outperforms the current state-of-the-art methods across
+multiple continuous control tasks shown in experiments.
+KEYWORDS
+Deep Reinforcement Learning, Cooperative Exploration, Continu-
+ous Control Tasks
+1
+INTRODUCTION
+Deep reinforcement learning (DRL) [46], which utilizes deep neu-
+ral networks to learn an optimal policy, works brilliantly and has
+demonstrated beyond human-level performance in solving various
+challenging sequential decision-making tasks e.g., video games [17,
+34, 35, 52, 54], autonomous driving [18, 53], robotic control tasks [2,
+31], etc. In DRL settings, an agent needs to sufficiently explore the
+environment and collect a set of experiences to obtain an optimal
+policy. The agent aims to learn an optimal policy to maximize its ex-
+pected cumulative rewards through trial and error. Therefore, DRL
+∗These two authors contributed equally.
+†The corresponding author.
+can be regarded as learning from reward feedback from environ-
+ments. It is essential that during the training phase the agent should
+be encouraged to explore the environments and gather sufficient
+reward signals for well-training.
+In DRL, exploration has obsessed with a critical problem: submit-
+ting solutions too quickly without sufficient exploration, leading
+to getting stuck at local minima or even complete failure. DRL
+researchers adopt the neural network to yield the policy with sig-
+nificant feature extraction and expression capabilities in a range of
+continuous control tasks. Whereas this phenomenal practice has
+achieved great performance, it is still obsessed with the notorious
+insufficient exploration problem in continuous control tasks. Good
+exploration becomes extremely difficult when the environment
+is distracting or provides little feedback. Whereas existing explo-
+ration methods remain a problematic drawback – lacking diversity
+to explore. The classic exploration methods such as 𝜖-Greedy strate-
+gies [35] or Gaussian noise [31] indirectly and implicitly change
+the style of the exploration. However, in massive situations, diverse
+styles of exploration are necessary. For instance, in chess games,
+players should perform different styles of policies (e.g., radical, con-
+servative, etc.) to keep competitive when facing various situations;
+humanoid robots attempt diverse control styles and eventually learn
+to walk efficiently.
+Recent studies enrich diverse styles of policies by construct-
+ing relationships between the distribution of policy and trajecto-
+ries [1, 13, 26, 27, 43]. In Diversity is All You Need (DIAYN) [11],
+authors highlight the diversity of policies plays a significant role in
+well-training agents. It trains the policy that maximizes the mutual
+information between the latent variable and the states, then altering
+the latent variable of the policy network creates multiple policies
+performing disparately. Although this interesting viewpoint has
+attracted a spectrum of following works [1, 11, 13, 20, 43], the afore-
+mentioned methods achieve diverse policies depending on the task
+in an unsupervised way, resulting in the algorithm performing in-
+sufficient generality in different tasks. In fact, the ideal algorithm
+to implement the diverse policies should be general across a range
+of tasks, which motivates us to design a task-oriented algorithm
+working towards developing various policies. Eysenbach et al. pro-
+posed [12] that the learned skills could not construct all the state
+marginal distributions in the downstream tasks. For task relevance,
+the mutual information is added as an intrinsic reward for em-
+powerment [7, 36], but these methods change the original reward
+arXiv:2301.02375v1  [cs.LG]  6 Jan 2023
+
+function resulting in the performance being extremely sensitive to
+the trade-off between original and intrinsic rewards.
+Our method insights from an interesting phenomenon during
+the exploration. The critic aims to approximate the accumulated
+reward by bootstrapping in the actor-critic framework. However,
+the different critic functions may have great differences even if they
+approximate the same target due to the function approximation er-
+ror. For instance, Twin Delayed Deep Deterministic policy gradient
+(TD3) [16] presents that two value functions with different initial
+parameters perform quite differently with identical targets. It is
+knotty to measure whether a value function is exact or not, and
+the gap between these two value functions is termed as controversy,
+which sees a decreasing trend along with the value function updat-
+ing process. Our intuition can be ascribed that controversy in the
+value estimation will lead to sub-optimal policies, and these policies
+have a bias toward message acquisition known as the style.
+This paper highlights that controversy can be utilized to encour-
+age policies to yield multiple styles, and encourages exploration
+for a continuous control task by applying multi-styled policies.
+Our paper contributes three aspects. (1) We first describe that the
+estimation bias in double value functions can lead to various ex-
+ploration styles. (2) This paper proposes the CCEP algorithm,
+encouraging diverse exploration for environments by cooperation
+from multi-styled policies. (3) Finally, in CCEP, we design a novel
+framework, termed as the centralized value function framework,
+which is updated by experience collected from all the policies and
+accomplishes the message delivery mechanism between different
+policies. Extensive experiments are conducted on the MuJoCo plat-
+form to evaluate the effectiveness of our method. The results reveal
+that the proposed CCEP approach attains substantial improvements
+in both average return and sample efficiency on the baseline across
+selected environments, and the average return of agents trained
+using CCEP is 6.7% higher than that of the baseline. Besides, CCEP
+also allows agents to explore more states during the same train-
+ing time steps as the baseline. Additional analysis indicates that
+message delivery leading to the cooperative multi-styled policies
+further enhanced the exploration efficiency by 8.6% compared with
+that of without cooperation.
+We organize the rest of this paper as follows. Section 2 briefly
+explains the concepts of RL. We elaborate on how to perform the
+CCEP algorithm in Section 3. We introduce our experimental set-
+tings in Section 4. Section 5 presents and analyzes results to validate
+the effectiveness of CCEP, after which section 6 discusses related
+works. Finally, we conclude our paper in Section 7. The code and
+documentation are released in the link for validating reproducibil-
+ity.1
+2
+PRELIMINARIES
+Reinforcement learning (RL) aims at training an agent to tackle the
+sequential decision problems that can be formalized as a Markov
+Decision Process (MDP). This process can be defined as a tuple
+(S, A, 𝑃,𝑟,𝛾), where S is the state space, A is the action space,
+𝑃 : S × A × S ↦→ [0, 1] denotes the transition probability, 𝑟 (𝑠,𝑎) is
+the reward function 𝑟 : S ×A ↦→ R, determining the reward agents
+will receive in the state 𝑠 while executing the action 𝑎. The 𝛾 ∈
+1https://github.com/Jincate/CCEP
+(0, 1) is the discount factor. The return is defined as the discounted
+accumulated reward.
+𝑅 =
+∞
+∑︁
+𝑡=0
+𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡)
+(1)
+In the DRL community, developers usually use the neural network
+parameterized with 𝜙 to indicate the policy 𝜋(𝑎|𝑠), which inputs
+an observation and outputs an action. The goal of DRL is to solve
+this MDP process and find the optimal policy 𝜋𝜙∗ : S ↦→ A with
+parameter 𝜙∗ that maximizes the expected accumulated return.
+𝜙∗ = arg max
+𝜙
+E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 )
+� ∞
+∑︁
+𝑡=0
+𝛾𝑡𝑟 (𝑠𝑡,𝑎𝑡)
+�
+(2)
+David Silver, et al. [44] propose that solving Eq. (2) with determin-
+istic policy gradient strategy,
+∇𝜙 𝐽 (𝜙) = E𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 )
+�
+∇𝑎𝑄𝜋 (𝑠,𝑎)|𝑎=𝜋 (𝑠)∇𝜙𝜋𝜙 (𝑠)
+�
+(3)
+where 𝑄𝜋 (𝑠,𝑎) = E𝑎��� ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃 (·|𝑠𝑡,𝑎𝑡 ) [𝑅|𝑠,𝑎] is known as the
+value function, indicating how good it is for an agent to pick action
+𝑎 while being in state 𝑠. To use the gradient-based approach (e.g.,
+Stochastic Gradient Descent [41]) to solve this equation, deep Q-
+learning uses the neural network to approximate the value function.
+The value function parameterized with 𝜃 is updated by minimizing
+the temporal difference (TD) error [45] between the estimated value
+of the subsequent state 𝑠′ and the current state 𝑠.
+𝜃∗ = arg min
+𝜃
+E
+�
+𝑟 (𝑠,𝑎) + 𝛾𝑄𝜋
+𝜃 (𝑠′,𝑎′) − 𝑄𝜋
+𝜃 (𝑠,𝑎)
+�2
+(4)
+We store the trajectories of the agent exploring the environment
+in a replay buffer [32] from which sample a random mini-batch of
+samples, updating the parameters mentioned above.
+3
+CENTRALIZED COOPERATIVE
+EXPLORATION POLICY
+This section details technologies of CCEP (Centralized Cooper-
+ative Exploration Policy). We first analyze value estimation bias
+from function approximation errors and generate multi-styled value
+functions by encouraging overestimation bias and underestimation
+bias for the value functions, respectively. To achieve multi-styled
+exploration, we propose a multi-objective update method for train-
+ing policy and randomly select one policy to explore at each time
+step. These historical trajectories during exploration are stored for
+training a single policy function to achieve cooperative message
+delivery. We denote our policy as 𝜋(𝑠,𝑧), where 𝑧 is a one-hot label
+and represents different policies. In this work, we focus on gener-
+ating multiple policies with different styles to encourage diverse
+exploration. We implement our method based on TD3 [16] which
+maintains double critics and uses the minimum of the critics as the
+target estimate.
+3.1
+Function Approximation Error
+This section shows that there exist approximation errors in the value
+function optimization and can accumulate to substantial scales. The
+accumulated approximation error will lead to value estimation bias,
+which plays a significant role in policy improvement.
+
+Sample one style per step
+𝒛~𝒑(𝒛)
+𝒂𝒕~𝝅𝝓(𝒂𝒕|𝒔𝒕, 𝒛𝒕)
+STYLE
+𝒔𝒕�𝟏~𝒑 (𝒔𝒕�𝟏|𝒔𝒕, 𝒂𝒕)
+ENVIRONMENT
+𝒂𝒕
+REPLAY BUFFER
+𝒔𝒕�𝟏
+𝒛
+𝒛
+𝒔𝒕�𝟏
+Sample a batch of N transitions
+(𝒔𝒕, 𝒂𝒕, 𝒛𝒕, 𝒔𝒕�𝟏, 𝒛𝒕�𝟏, 𝒓)
+UPDATE CRITICS
+𝜽𝒊
+𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐢𝐧
+𝜽𝒊 𝑵�𝟏∑�𝒚 − 𝑸𝜽𝒊(𝒔, 𝒂)�
+𝟐
+UPDATE POLICY
+Generate critics by Eq. (9) 
+𝝓𝒕�𝟏 ←𝐚𝐫𝐠 𝐦𝐚𝐱
+𝝓
+𝟏
+𝟒 �
+𝒌=𝟏
+𝟒
+𝑸𝒌�𝒔, 𝝅(𝒔, 𝒛𝒌)�
+𝝓𝒕�𝟏
+POLICY
+    No.1
+POLICY
+    No.2
+POLICY
+    No.3
+POLICY
+   No.4
+Cooperative exploration
+Centralization
+Figure 1: The workflow of CCEP Algorithm. The agent 𝜋 interacts with the environment with diverse style cooperatively and
+produce the transition 𝑠𝑡 → 𝑠𝑡+1. The actor and critic are updated over a mini-batch of the transition samples. A centralized
+policy with four different styles is learned from the multi-styled critics.
+In value-based deep reinforcement, deep neural networks ap-
+proximate the value functions, and the function approximation
+error exists correspondingly. One major source of the function
+approximation error comes from the optimization procedure. In
+this procedure, stochastic gradient descent, which uses a batch of
+random samples for gradient update each time, is the mainstream
+method due to the consideration of computational resources and
+training efficiency. However, as [40] has indicated, a mini-batch
+gradient update may have unpredictable effects on samples outside
+the training batch, which leads to the function approximation error.
+For explanation, we use 𝑒𝑡 to represent the approximation error of
+the value function with the state-action pair(𝑠𝑡,𝑎𝑡) as input and
+approximation error 𝑒𝑡 can be modeled as follows:
+𝑄𝜃 (𝑠𝑡,𝑎𝑡) = 𝑟 (𝑠𝑡,𝑎𝑡) + 𝛾E[𝑄𝜃 (𝑠𝑡+1,𝑎𝑡+1)] − 𝑒𝑡
+(5)
+Approximation errors influence the value estimation when using
+the value function as an estimator. The estimation may be skewed
+towards an overestimation, causing a wrong estimate for a given
+state. This leads to a problem of an optimal action being chosen but
+replaced by a sub-optimal action, owing to the overestimation of
+a sub-optimal action. Thus, the overestimation bias is a common
+problem in Q-Learning with discrete actions, as we choose the
+seemly best action 𝑎𝑡+1 in the target value. Still, there is little chance
+for the optimal state-action pair to be updated.
+E[ max
+𝑎𝑡+1∈A 𝑄(𝑠𝑡+1,𝑎𝑡+1)] ≥
+max
+𝑎𝑡+1∈A E[𝑄(𝑠𝑡+1,𝑎𝑡+1)]
+(6)
+Mentioned overestimated bias can also occur in continuous con-
+trol tasks[16], since the policy approximator always provides the
+optimal action at the current state based on the value function.
+While this bias can be quite small in an individual update, the bias
+can be accumulated to a substantial overestimation. Eq.(5) can be
+expanded as follows:
+𝑄𝜃 (𝑠𝑡,𝑎𝑡) = E𝑎𝑡 ∼𝜋𝜙 (·|𝑠𝑡 ),𝑠𝑡+1∼𝑃
+� ∞
+∑︁
+𝑡=0
+𝛾𝑡 (𝑟 (𝑠𝑡,𝑎𝑡) − 𝑒𝑡)
+�
+(7)
+Previous works such as Double Q-learning [23] and Double DQN [24]
+are proposed to alleviate value functions of underestimating. The
+idea is to maintain two independent estimators in which one is
+used for estimation while the other is for selecting maximal action.
+Similarly, as an extension in dealing with continuous control tasks,
+TD3 [16] reduce the overestimation bias by using double value func-
+tions and taking the minimum between the two value functions
+for an estimation which suffers from underestimation problems as
+well [50, 51].
+Does the estimation error influence the performance? Given
+a continuous control task, we use 𝑓 to approximate the true under-
+lying value function 𝑄∗, which indicates the accumulated reward
+obtained by acting 𝑎 before taking optimal policy 𝜋∗ at state 𝑠. 𝑉
+represents the true underlying value function(which is not known
+during training). 𝑉 ∗ and 𝑉 𝜋𝑓 represent the accumulated return
+obtained by adopting the optimal policy 𝜋∗ and 𝜋𝑓 in state 𝑠 re-
+spectively in which 𝜋𝑓 is a learned policy by maximizing the value
+function approximate 𝑓 .
+Lemma 1. (Performance Gap). The performance gap of the policy
+between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓 is defined
+by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have
+∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 − 𝑄∗∥∞
+1 − 𝛾
+We provide proof detail of Lemma 1 in Supplementary A. This
+inequality indicates that the performance gap of the policy can be
+bounded by the estimation error of the value function and accurate
+value estimate can reduce the upper bound of the performance gap
+
+and enhance the performance.
+Do overestimation bias and underestimation bias affect per-
+formance in the same way? An empirical study shows that esti-
+mation bias may not always be a detrimental problem while both
+underestimation bias and overestimation bias may improve learn-
+ing performance which depends on the environment [30]. As an
+example, for an unknown area with high stochasticity, overesti-
+mation bias may help to explore the overestimated area but un-
+derestimation bias prevents this. However, if these areas of high
+stochasticity are given low values, the overestimation bias may lead
+to excess exploration in low-value regions. The fact is that we can
+not choose the environment and these different circumstances can
+always occur during exploration. Our method is designed to utilize
+the difference in exploration behavior brought by estimation bias
+to encourage multi-styled exploration.
+3.2
+Multi-Style Critics: Radical, Conservative
+As mentioned above, function approximation error exists in value
+functions and can accumulate to substantial scales which have a
+great influence on the value estimation resulting in overestimation
+or underestimation bias. Estimation bias has been researched in
+recent works [16, 25, 50, 51]. While these works focus on an accurate
+value estimation and discussed the method to control the estimation
+bias with the use of multiple value functions for auxiliary, they
+just choose one of the value functions, which seems to be the
+most accurate, for policy update neglecting other value functions.
+However, there is no accurate value function without trial and
+error. In this section, we show how to utilize the estimation bias
+and introduce our method for the generalization of multi-styled
+critics.
+Our intuition is that there are different degrees of estimation
+bias in double randomly initialized value functions when perform-
+ing function approximation. However, the estimation bias can be
+controlled by applying a maximum operator and minimum op-
+erator, namely the maximum of the two estimates is relatively
+overestimated and the minimum of the two estimates is relatively
+underestimated. Two different estimates raise a controversy about
+which critic gives the accurate estimate. The best way to resolve
+the controversy is to follow one of the critics to explore and collect
+reward messages. While controversy does not always exist because
+there is only one accurate value, the critics reach an agreement
+when the state value has been exactly estimated. And the existence
+of controversy means more exploration is needed.
+We start by maintaining double randomly initialized value func-
+tions 𝑄𝜃1 and 𝑄𝜃2 with parameters 𝜃1 and 𝜃2 respectively and up-
+date the value function with TD3 [16] which takes the minimum
+between the two value functions as the target value estimate:
+𝑦 = 𝑟 + min
+𝑖=1,2𝑄𝜃𝑖 (𝑠′,𝑎′),𝑎′ ∼ 𝜋𝜙
+(8)
+But the two randomly initialized value functions potentially have
+different value estimations for a given state-action pair due to the
+accumulated function approximation error. This difference leads to
+the result that the two critics may give two different suggestions
+for the best action choice. While these estimates are relatively
+overestimated or underestimated, these different criteria for a given
+state-action pair may lead to a different style of action choice. It
+Algorithm 1 Centralized Cooperative Exploration Policy (CCEP)
+Initialize critic networks 𝑄𝜃1,𝑄𝜃2
+Initialize actor network 𝜋𝜙 with random parameters 𝜃1,𝜃2,𝜙
+Initialize target networks 𝜃 ′
+1 ← 𝜃1,𝜃 ′
+2 ← 𝜃2,𝜙′ ← 𝜙
+Initialize replay buffer B
+Initialize number of skills K
+1: for 𝑡 = 1 to 𝑇 do
+2:
+Sample a skill 𝑧 from 𝑝(𝑧)
+3:
+Select action with noise 𝑎 ∼ 𝜋𝜙 (𝑠,𝑎) + 𝜖,𝜖 ∼ N (0, 𝜎)
+4:
+Observe a reward 𝑟 and a new state 𝑠′
+5:
+Store transition tuple (𝑠,𝑧,𝑎,𝑟,𝑠′,𝑧′) in B
+6:
+Sample mini-batch of 𝑁 transitions (𝑠,𝑧,𝑎,𝑟,𝑠′,𝑧′) from 𝐵
+7:
+𝑎′ ← 𝜋𝜙′(𝑠′,𝑧′) + 𝜖,𝜖 ∼ 𝑐𝑙𝑖𝑝(N (0, 𝜎), −𝑐,𝑐)
+8:
+𝑦 = 𝑟 + 𝛾 min𝑖=1,2 𝑄𝜃′
+𝑖 (𝑠′,𝑎′)
+9:
+Update critics: 𝜃𝑖 ← arg min𝜃𝑖 𝑁 −1 �(𝑦 − 𝑄𝜃𝑖 (𝑠,𝑎))2
+10:
+if t mod d then
+11:
+Update policy:
+12:
+∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 � ∇𝑎𝑄 𝑗 (𝑠,𝑎)|𝑎=𝜋𝜙 (𝑠,𝑧)∇𝜙𝜋𝜙 (𝑠,𝑧)
+13:
+Update target networks
+14:
+𝜃 ′
+𝑖 ← 𝜏𝜃𝑖 + (1 − 𝜏)𝜃 ′
+𝑖
+15:
+𝜙′
+𝑖 ← 𝜏𝜙𝑖 + (1 − 𝜏)𝜙′
+𝑖
+is reasonable the estimation is radical if we choose the maximum
+value of the two to estimate and the estimation is conservative if
+we choose the minimum value of the two. Thus, we consider four
+critics:
+𝑄 𝑗 (𝑠,𝑎) =
+
+
+𝑄𝜃1 (𝑠,𝑎)
+𝑗 = 0
+𝑄𝜃2 (𝑠,𝑎)
+𝑗 = 1
+max(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎))
+𝑗 = 2
+min(𝑄𝜃1 (𝑠,𝑎),𝑄𝜃2 (𝑠,𝑎))
+𝑗 = 3
+(9)
+There exists controversy among these critics, and the controversy
+can further influence the performance of the policy learned.
+3.3
+Opposite Value Functions
+This approach for generating diverse styles raises the problem that
+the value functions may not provide sufficient difference in style
+when the controversy disappear. This phenomenon is very com-
+mon when the value function converges. But we don’t want this to
+happen too soon, because we want the value functions to provide
+more exploration for the policy. The controversy exists due to the
+randomly initialized parameters of the neural networks and the
+error accumulation. But actually, there is a small probability that
+the two networks have great similarities, which will lead to double
+consistent critics. This is not what we want, because consistent
+critics mean monotonous policy. To avoid this, we try to enlarge the
+controversy. The solution in this paper is to learn two opposite tar-
+gets respectively for the two networks, where one of the networks
+approximates the positive value and another approximates the neg-
+ative one. This approach is equivalent to adding a factor -1 to the
+final layer of either network. We find that the controversy is guar-
+anteed with this simple network structure change. To provide some
+intuition, we compared the controversy changes after the double
+value functions learn the opposite target. We express the amount
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time Steps (1e6)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+Average Error
+(a) HalfCheetah-v3
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Time Steps (1e6)
+0
+1
+2
+3
+Average Error
+Target
+same target
+opposite target
+(b) Walker2d-v3
+Figure 2: Measuring the error between double critics given
+same/opposite targets in TD3 on MuJoCo environments over
+1 million time steps
+of controversy between the two value functions by the errors of
+state values in a batch of samples. Figure 2 shows the controversy
+measuring over MuJoCo [48] environments in HalfCheetah-v3 and
+Walker2d-v3. The results show that with the simple network struc-
+ture change, the controversy is enlarged. But this approach will
+not influence the value estimation because we just fine-tune the
+structure.
+3.4
+Centralized Cooperation
+With four critics, we train a centralized cooperative policy to encour-
+age multi-styled explorations through diverse value estimations.
+We model this problem as a multi-objective optimization problem.
+The target is to train multiple policies, with each policy targeting an
+individual value function. We express the policy function as 𝜋(𝑠,𝑧),
+with state 𝑠 and latent variable 𝑧 as input. The latent variable 𝑧,
+which is a one-hot label in our method, represents different policies.
+The architecture of our centralized cooperative policy is shown
+in Figure 3. This idea comes from skill discovery method [11, 43],
+which use the latent variable 𝑧 to express different skills. And in
+skill discovery, the target is to maximize the mutual information be-
+tween latent variable 𝑧 and some aspects of the trajectories, which
+is a different target for a different latent variable 𝑧. Our method
+encourages diverse styles of policies by different targets as well.
+Particularly, we sample latent variable 𝑧 from set {0, 1, 2, 3} and
+encode it in a one-hot label. For a given latent variable 𝑧, the policy
+targets 𝑧-th value functions in Eq.(9). With different latent variable
+𝑧, the policy shows diverse styles due to the multi-styled targets.
+We make an experiment showing that there exists different explo-
+ration preferences for these policies (Section 5.2) In the exploration
+procedure, we randomly sample latent variable 𝑧 and make deci-
+sions by policy 𝜋(𝑠,𝑧). This approach enables diverse styles to be
+applied at each time step. Broadly speaking, our exploration policy
+has the following characteristics: Multi-styled, Centralized, and
+Cooperative.
+Multi-styled. We train four policies to accomplish the explo-
+ration. These policies learn from the corresponding value function
+𝑄 𝑗:
+𝜋∗
+𝑗 = arg max
+𝜋
+𝑄 𝑗
+(10)
+There are four value estimators, in which two of them (𝑗 = 0, 1)
+are normal but different estimators, one (𝑗 = 2) is an overestimated
+Policy 1
+Policy 2
+Policy 3
+Policy 4
+Centralized 
+Cooperative 
+Policy
+Environment
+Label variance
+Input
+Output
+Figure 3: The architecture of our centralized cooperative pol-
+icy. The agent cooperatively explores the environments by
+selecting one of the styles at each time step. The style se-
+lection process is implemented by sampling latent variable
+𝑧. Policies with diverse styles exchange messages through a
+centralized network.
+estimator compared to the other (𝑗 = 3) and helps encourage ex-
+plorations in overestimated actions, the last remaining one (𝑗 = 3)
+is a conservative estimator and brings more exploitation as illus-
+trated in the previous section. It is appropriate for the policies to
+perform in a variety of ways given the varied estimators they use
+(e.g., conservative, radical).
+Centralized. Our policy is a centralized policy because we make
+use of all the policies learned in each episode. At each time step𝑡, we
+sample one of these policies for exploration. It allows us to generate
+a variety of trajectories adopting this exploration approach as this
+centralized policy can generate 4𝑛 types of trajectories theoretically
+for 𝑛-step exploration, compared to using a single policy that can
+only generate one. These trajectories are stored as experience and
+maintain the update for a pair of centralized value functions.
+Cooperative. We update the policy cooperatively. With multi-
+ple policies learning their respective value functions, “knowledge”
+learned by each policy cannot be shared. Our method is to learn a
+single network for policies and learn cooperatively [4]. To repre-
+sent different policies, we feed latent variables 𝑧 which are one-hot
+labels as extra input to the network. The policy which inputs latent
+variable 𝑧 and state𝑠 and outputs action𝑎 can be defined as 𝜋(𝑠,𝑧).
+We sample 𝑧 to represent the sampling of different policies. Thus,
+the policy can be updated by taking deterministic policy gradient.
+∇𝜙 𝐽 (𝜙) = 𝑁 −1K−1 ∑︁
+∇𝑎𝑄 𝑗 (𝑠,𝑎)|𝑎=𝜋𝜙 (𝑠,𝑧)∇𝜙𝜋𝜙 (𝑠,𝑧)
+(11)
+Where 𝑄 𝑗 (𝑠,𝑎) refer to the multi-styled critics in Eq.(9), K is the
+number of styles which is 4 in this algorithm. The specific algorithm
+is shown in Algorithm 1.
+4
+EXPERIMENTAL SETTINGS
+To evaluate our method, we test our algorithm on the suit of
+MujoCo [48] continuous control tasks, including HalfCheetah-v3,
+
+G(a)
+(b)
+(c)
+(d)
+Figure 4: Screenshots of MuJoCo environments. (a) Ant-v3,
+(b) HalfCheetah-v3, (c) Walker2d-v3, (d) Hopper-v3
+Hopper-v3, Walker2d-v3, Ant-v3, Pusher-v2 and Humanoid-v3 (the
+screenshots are presented in Figure 4).
+For implementation, our method builds on TD3 [16], and for com-
+parison, we also establish three-layer feedforward neural networks
+with 256 hidden nodes per hidden layer for both critics and actors.
+Particularly, the actor takes state 𝑠 and latent variable 𝑧 concate-
+nated as input, where the latent variable 𝑧 is encoded as one-hot
+label. At each time step, both networks are trained with a mini-
+batch of 256 samples. We apply soft updates for target networks as
+well.
+We compared our algorithm against some classic algorithms
+such as DDPG [30], which is an efficient off-policy reinforcement
+learning method for continuous tasks; PPO [42], the state-of-the-
+art policy gradient algorithms; TD3 [16], which is an extension
+to DDPG; SAC [22], which is an entropy-based method with high
+sample efficiency. Further, we compared our algorithm with the
+latest algorithm in solving the exploration problems in continuous
+control tasks such as OAC [8], which makes improvements on SAC
+for better exploration. We implement DDPG and PPO by OpenAI’s
+baselines repository and SAC, TD3, and OAC by the github the
+author provided. And we use the parameter the author recommend
+for implementation. The details of the implementation are shown
+in Supplementary B.
+5
+EXPERIMENTAL RESULTS AND ANALYSIS
+5.1
+Evaluation
+To validate the performance of the CCEP algorithm, we evaluate
+our algorithm in MuJoCo continuous control suites. We perform
+interactions for 1 million steps in 10 different seeds and evaluate
+the algorithm over 10 episodes every 5k steps. Our results report
+the mean scores and standard deviations in the 10 seeds. We show
+learning curves in Figure 5 and the max average return over 10 trials
+of 1 million time steps in Table 1. The learning curves in 1 million
+time steps show that our algorithm achieves a higher sample effi-
+ciency compared with the latest algorithm. Furthermore, the results
+in the Table 1 indicates that our algorithm shows superior perfor-
+mance. And in HalfCheetah-v3, Walker2d-v3, Hopper-v3, Ant-v3,
+Table 1: The highest average return over 10 trials of 1 million
+time steps. The maximum value for each task is bolded.
+Environment
+Ours
+OAC
+SAC
+TD3
+DDPG
+PPO
+HalfCheetah
+11945
+9921
+11129
+9758
+8469
+3681
+Hopper
+3636
+3364
+3357
+3479
+2709
+3365
+Walker2d
+4706
+4458
+4349
+4229
+3669
+3668
+Ant
+5630
+4519
+5084
+5142
+1808
+909
+Pusher
+-21
+-25
+-20
+-25
+-29
+-21
+Humanoid
+5325
+5747
+5523
+5356
+1728
+586
+our algorithm outperforms all the other baselines and achieve sig-
+nificant improvements. While in the Pusher-v2 task, our algorithm
+show higher stability than that of TD3. For further evaluation, we
+evaluate our algorithm in the state-based suite PyBullet [9] which
+is considered to be harder than the suite MuJoCo. Our algorithm
+still shows better performance compared to the baseline algorithms.
+The corresponding results are shown in Supplementary C.1.
+5.2
+Policy Style
+To ensure that our proposed CCEP algorithm learns diverse styles,
+we compared the distribution of explored trajectories when ex-
+ploring with a single style only. We test the algorithm in Ant-v3
+environment over 1𝑒6 time steps and use the states sampled to rep-
+resent the trajectories. Figure 6 shows the states explored by each
+style at 1𝑒5, 2𝑒5 and 3𝑒5 learning steps, and a more detailed results
+are shown in Supplementary C.3. We collect the states sampled over
+10 episodes with different seeds and apply t-SNE [49] for better
+visualization. The results show that while part of the states can be
+gathered by all styles which implies a compromise in controversy,
+there is a considerably large region of states that can only be ex-
+plored by a unique style of policy. Though different styles, diverse
+styles come to be in compromise as training process goes on. This
+phenomenon suggests that CCEP behaves in multi-styled explo-
+ration which leads to an exploration preference, and styles come
+to an agreement with sufficient exploration. Another phenomenal
+conclusion is that although the style tends to be consistent, new
+styles are emerging which brings enduring exploration capabilities.
+5.3
+Measuring Exploration Ability
+The critical problem of our proposed method is whether we achieve
+higher sample efficiency. Although the learning curves (Figure 5)
+gives considerably convincing results, a more intuitive result has
+been given in Figure 7. We compared the exploration of CCEP
+with that of TD3 and SAC (which achieve the trade-off between
+exploration and exploitation by entropy regularization.) over 10
+episodes with different seeds (Figure 7). For a fair comparison, these
+methods are trained in Ant-v3 with the same seed at half of the
+training process. In order to get reliable results, the states explored
+are gathered in 10 episodes with different seeds. We still apply the
+same t-SNE [49] transformation to the states explored by all of the
+algorithms for better visualization. While there are great differences
+between the states explored by TD3 (green) and SAC (blue), the
+result shows that our algorithm (red) explores a wider range of
+states which even covers that TD3 and SAC explored.
+
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+5000
+10000
+Average Return
+HalfCheetah-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+2000
+4000
+Walker2d-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+0
+1000
+2000
+3000
+4000
+Hopper-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+2000
+4000
+Average Return
+Humanoid-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+2000
+0
+2000
+4000
+6000
+Ant-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+80
+60
+40
+20
+Pusher-v2
+Ours
+OAC
+PPO
+DDPG
+TD3
+SAC
+Figure 5: Learning curves for 6 MuJoCo continuous control tasks.For better visualization,the curves are smoothed uniformly.
+The bolded line represents the average evaluation over 10 seeds. The shaded region represents a standard deviation of the
+average evaluation over 10 seeds.
+(1) Learning Steps = 100000
+(2) Learning Steps = 200000
+(3) Learning Steps = 300000
+POLICY No.1
+POLICY No.2
+POLICY No.3
+POLICY No.4
+Figure 6: The states visited by each style. For better visualization, the states get dimension reduction by t-SNE. The points with
+different color represents the states visited by the policy with the style. The distance between points represents the difference
+between states.
+5.4
+Ablation Study
+We perform an ablation study to understand the contribution of the
+cooperation between policies for message delivery. The results are
+shown in Table 2 where we compare the performance of training
+policies by removing policy cooperation and training them sepa-
+rately. We perform interactions for 1 million time steps for each
+method. The results show that without cooperation, the policy net-
+work not only trains 4 times more network parameters but also
+fails to reduce performance. And this performance degradation is
+even more pronounced on Walker2d. Additional learning curves
+can be found in Supplementary C.2.
+6
+RELATED WORK
+This section discusses several methods proposed recently for im-
+proving the exploration of deep reinforcement learning.
+A range of works take an effort in encouraging explorations with
+
+(1) Ours
+(2) TD3
+(3) SAC
+Figure 7: Measuring the exploration region. Comparison of exploration capabilities of TD3 (green), SAC (blue) and Ours (red).
+The points represent region explored by each method in 10 episodes. All the states get dimension reduction by the same t-SNE
+transformation for better visualization.
+Table 2: Max Average Return over 5 trials of 1 million time
+steps, comparing ablation over cooperation for message de-
+livery. The maximum value for each task is bolded.
+Method
+HCheetah
+Hopper
+Walker2d
+Ant
+CCEP
+11969
+3672
+4789
+5488
+CCEP-Cooperation
+11384
+3583
+4087
+4907
+TD3
+9792
+3531
+4190
+4810
+the use of randomness over model parameters [6]. Another preva-
+lent series of works propose to enhance exploration by simulta-
+neously maximizing the expected return and entropy of the pol-
+icy [15, 21, 22, 39, 55]. Whereas, these methods do not provide
+heuristic knowledge to guide the exploration, which can be consid-
+ered to be insufficient and time-consuming.
+To achieve effective exploration, the curiosity mechanism [19, 38]
+has been proposed in recent works, e.g., the counted-based ap-
+proaches [33] which quantify the “novelty” of a state by the times
+visited. However, these methods maintain the state-action visitation
+counts which make it challenging in solving high-dimensional or
+continuous tasks. Other works rely on errors in predicting dynam-
+ics, which have been used to address the difficulties in complex
+environments [5, 37, 38]. Though the Intrinsic Curiosity Module
+(ICM) [37] maintains a predictor on state transitions and considers
+the prediction error as an intrinsic reward, Random Network Distil-
+lation (RND) [5] utilizes the prediction errors of networks trained
+on historical trajectories to quantify the novelty of states, which is
+effective and easy to implement in real applications.
+Another direction in previous work is to study exploration in
+hierarchical reinforcement learning (HRL) [3, 47]. These methods
+are insight from the fact that developers prefer to divide the com-
+prehensive and knotty problems into several solvable sub-problems.
+There are some further studies on hierarchy in terms of tasks, rep-
+resentative of which are goal-based reinforcement learning and
+skill discovery. The similarity of these approaches is that they both
+identify different policies by utilizing latent variables. In goal-based
+RL, the latent variables are defined by the policy’s goal, which
+aims to complete several sub-goals and accomplish the whole task.
+These methods introduce prior human knowledge, causing them
+to work brilliantly on some tasks but fail when unaware of human
+knowledge. Despite our method also introducing latent variables to
+represent different styles of policies, all the policies share the same
+objective, nevertheless differing in the road to reach the destination.
+Skill discovery methods, which adopt the latent variable to repre-
+sent the skill learned from the policy, introduce mutual information
+to organize relationships between the latent variable 𝑧 and some
+aspects of the trajectories to acquire diverse skills (also known
+as style) [1, 11, 13, 20, 43]. Nevertheless, these methods train the
+policy in an unsupervised way [11, 13, 43], suggesting that the
+skills trained are unaware of task-driven, and they cannot rep-
+resent the optimal policies when adapted to downstream tasks
+illustrated in [12]. Our method avoids this issue because we train
+the policy task-oriented and demonstrate the benefit brought by
+the attention of these policies to the state value making them differ
+considerably in exploration style. For task relevance, some related
+works that learn skills by jointly learning a set of skills and a meta-
+controller [3, 10, 13, 14, 28, 29]. The options of the meta-controller
+control different attentions of each policy. However, these meth-
+ods usually choose the best option to explore and rarely execute
+sub-optimal options, leading to the drawback – the algorithm tends
+to ignore sub-optimal actions that maybe fail in most states but
+are effective in a few critical scenarios. Our proposed approach
+randomly selects different styles of policies for directed coopera-
+tive exploration, which are improved accordingly with the value
+function and produce different styles due to differences in attention.
+7
+CONCLUSION
+In the value-based method, value estimation bias has been a com-
+mon problem. While different estimation bias in double value func-
+tions lead to value function controversy, the controversy can be
+utilized to encourage policies to yield multiple styles. In this paper,
+we aim at encouraging explorations by multi-styled policies. We
+
+start by analysis on estimation bias during the value function train-
+ing process and its effect on the exploration. We then encourage
+this controversy between the value functions and generate four
+critics for producing multi-styled policies. Finally, we apply these
+policies with diverse styles for centralized cooperative exploration
+which perform superior sample efficiency in the test environment.
+Though there are a lot of works focusing on reducing the estimation
+bias for an accurate value estimation, few works try to utilize these
+inevitable errors to make improvements. Our results show that it
+is also an option to use the errors to encourage explorations. For
+future work, it is an exciting avenue for focusing on more expres-
+sive policy styles. A style that can be represented as a continuous
+distribution may be more efficient and more expressive.
+REFERENCES
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+Variational option discovery algorithms. CoRR abs/1807.10299 (2018).
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+//arxiv.org/abs/1807.10299
+[2] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong,
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+Zaremba. 2017. Hindsight experience replay. 30 (2017), 5048–5058.
+[3] Pierre-Luc Bacon, Jean Harb, and Doina Precup. 2017. The option-critic archi-
+tecture. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 31.
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+
+Supplementary Materials
+A
+PROOF OF LEMMA 1
+(Lemma 1)(Performance Gap). Let 𝑉 ∗ be the ground truth state
+value in Bellman value iterations, 𝑄∗ be the ground truth state
+action value, 𝑉 𝜋𝑓 be the state value when applying learned policy
+𝜋𝑓 , 𝑓 be the value function approximator. The performance gap of
+the policy between the optimal policy 𝜋∗ and the learned policy 𝜋𝑓
+is defined by an infinity norm ∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ and we have
+∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞ ≤ 2∥𝑓 −𝑄∗ ∥∞
+1−𝛾
+Proof. For any 𝑠 ∈ S
+𝑉 ∗(𝑠) − 𝑉 𝜋𝑓 (𝑠) =𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠))
++ 𝑄∗(𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠))
+≤𝑄∗(𝑠, 𝜋∗(𝑠)) − 𝑓 (𝑠, 𝜋∗(𝑠))
++ 𝑓 (𝑠, 𝜋𝑓 (𝑠)) − 𝑄∗(𝑠, 𝜋𝑓 (𝑠))
++ 𝛾E𝑠′∼𝑃 (𝑠,𝜋𝑓 (𝑠)) [𝑉 ∗(𝑠′) − 𝑉 𝜋𝑓 (𝑠′)]
+≤2∥𝑓 − 𝑄∗∥∞ + 𝛾∥𝑉 ∗ − 𝑉 𝜋𝑓 ∥∞
+B
+EXPERIMENTAL DETAILS
+B.1
+Environments
+We evaluate the performance of CCEP on environments from Mu-
+joCo Control Suite [48]which can be listed as HalfCheetah-v3, Ant-
+v3, Walker2d-v3, Humanoid-v3, Hopper-v3, and Pusher-v2, and the
+specific parameters of these environments are listed in Table 3. We
+use the publicly available environments without any modification.
+B.2
+Implementation and Hyper-parameters
+Here, we describe certain implementation details of CCEP. For
+our implementation of CCEP, we follows a standard actor-critic
+framework.
+B.3
+Soft Actor-Critic Implementation Details
+For implementation of SAC, we use the code the author provided
+and use the parameters the author recommended. We use a single
+Gaussian distribution and use the environment-dependent reward
+scaling as described by the authors. For a fair comparison, we
+apply the version of soft target update and train one iteration per
+time step. We use the reward scales as the author recommended
+(except for Pusher-v2 which is not mentioned by the author in
+the article). Considering that there are similar action dimensions
+between Pusher-v2 and HalfCheetah-v3, we set the same reward
+scale for Pusher-v2. The specific reward scales for each environment
+is shown in Table 4.
+Table 3: Environment Specific Parameters
+Environment
+State Dimensions
+Action Dimensions
+Ant-v3
+111
+8
+HalfCheetah-v3
+17
+6
+Hopper-v3
+11
+3
+Humanoid-v3
+376
+17
+Pusher-v2
+23
+7
+Walker2d-v3
+17
+6
+Table 4: SAC Environment Specific Parameters
+Environment
+Reward Scale
+Ant-v3
+5
+HalfCheetah-v3
+5
+Hopper-v3
+5
+Humanoid-v3
+20
+Pusher-v2
+5
+Walker2d
+5
+B.4
+Optimistic Actor-Critic Implementation
+Details
+The implementation of OAC is mainly based on the open source
+code. We set the hyper-parameters the same as OAC used in MuJoCo
+which is listed in Table 5. And for fair comparison, we train with 1
+training gradient per environment step. We use the same reward
+scales as SAC, listed in Table 4.
+Table 5: SAC Environment Specific Parameters
+Parameter
+Value
+shift multiplier
+√
+2𝛿
+6.86
+𝛽𝑈 𝐵
+4.66
+𝛽𝐿𝐵
+-3.65
+B.5
+Reproducing Other Baselines
+For reproduction of TD3, we use the official implementation (
+https://github.com/sfujim/TD3). For reproduction of DDPG and
+PPO we use OpenAI’s baselines repository and apply default hyper-
+parameters.
+Table 6: CCEP Parameters settings
+Parameter
+Value
+Exploration policy
+N (0, 0.1),𝑧 ∼ 𝑝(𝑧)
+Number of policy
+4
+Variance of exploration noise
+0.2
+Random starting exploration time steps
+2.5 × 104
+Optimizer
+Adam[30]
+Learning rate for actor
+3 × 10−4
+Learning rate for critic
+3 × 10−4
+Replay buffer size
+1 × 106
+Batch size
+256
+Discount (𝛾)
+0.99
+Number of hidden layers
+2
+Number of hidden units per layer
+256
+Activation function
+ReLU
+Iterations per time step
+1
+Target smoothing coefficient (𝜂)
+5 × 10−3
+Variance of target policy smoothing
+0.2
+Noise clip range
+[−0.5, 0.5]
+Target critic update interval
+2
+
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+2000
+4000
+6000
+8000
+10000
+12000
+Average Return
+(a) HalfCheetah-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+1000
+2000
+3000
+4000
+5000
+(b) Walker2d-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+1000
+2000
+3000
+(c) Hopper-v3
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Average Return
+(d) Ant-v3
+CCEP
+CCEP-Cooperation
+TD3
+Figure 8: Ablation over the use of cooperation. Comparison of CCEP, TD3 and the subtraction of cooperation (CCEP-
+cooperation).
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+500
+1000
+1500
+2000
+Average Return
+(a) Walker2D
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+1000
+2000
+3000
+(b) Ant
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+0
+500
+1000
+1500
+2000
+2500
+(c) Hopper
+0.00
+0.25
+0.50
+0.75
+1.00
+Time Steps (1e6)
+1000
+0
+1000
+2000
+3000
+Average Return
+(d) HalfCheetah
+TD3
+Ours
+SAC
+DDPG
+PPO
+Figure 9: Learning curves for 4 PyBullet continuous control tasks. For better visualization, the curves are smoothed uniformly.
+The bolded line represents the average evaluation over 10 seeds. The shaded region represents the standard deviation of the
+average evaluation over 10 seeds.
+Table 7: Evaluation in PyBullet control suite. The highest average return over 10 trials of 1 million time steps. The maximum
+value for each task is bolded.
+Pybullet Environment
+Ours
+SAC
+TD3
+DDPG
+PPO
+HalfCheetah
+2670 ± 275
+2494 ± 266
+2415 ± 236
+1120 ± 373
+465 ± 30
+Hopper
+2254 ± 186
+2167 ± 323
+1860 ± 288
+1762 ± 368
+623 ± 131
+Walker2d
+1829 ± 418
+1369 ± 408
+1676 ± 342
+929 ± 345
+509 ± 106
+Ant
+3175 ± 184
+2423 ± 680
+2711 ± 253
+483 ± 70
+578 ± 19
+C
+ADDITIONAL EXPERIMENTS
+C.1
+Additional Evaluation
+For an additional Evaluation, We conduct experiments on the state-
+based PyBullet [9] suite which is based on the well-known open-
+source physics engine bullet and is packaged as a Python module for
+robot simulation and learning. The suite of Pybullet is considered
+to be a harder environment than MuJoCo [48]. We choose TD3
+[16], SAC [22], PPO [42], DDPG [31] as our baselines due to their
+superior performance. We perform interactions for 1 million steps
+in 10 different seeds and evaluate the algorithm over 10 episodes
+every 5k steps. We evaluate our algorithm in HalfCheetah, Hopper,
+Walker2d and ant in the suite of pybullet. Our results report the
+mean scores and standard deviations in the 10 seeds. We show the
+learning curves in Figure 9 and the max average return over 10
+trials in Table 7.
+C.2
+Additional Ablation Results
+We compare the learning curves of CCEP, TD3 and the subtraction
+of cooperation (CCEP-cooperation) for better understanding the
+contribution of policy cooperation (Section 5.4). We perform inter-
+actions for 1 million steps in 10 different seeds and evaluate over
+10 episodes every 5k steps. Our results report the mean scores and
+standard deviations in the 10 seeds. We show the learning curves
+in Figure 8
+C.3
+Supplementary Results
+We provide supplementary results for Section 5.2. Figure 10 shows
+the states visited by each style over 1M time steps with intervals
+of 100k. The results show that different styles get consistent but
+
+new styles emerges as well, which brings enduring exploration
+capabilities.
+
+(1) Learning Steps = 100000
+(2) Learning Steps = 200000
+(3) Learning Steps = 300000
+(4) Learning Steps = 400000
+(5) Learning Steps = 500000
+(6) Learning Steps = 600000
+(7) Learning Steps = 700000
+(8) Learning Steps = 800000
+(9) Learning Steps = 900000
+POLICY No.1
+POLICY No.2
+POLICY No.3
+POLICY No.4
+Figure 10: The states visited by each style. For better visualization, the states get dimension reduction by t-SNE. The points with
+different color represents the states visited by the policy with the style. The distance between points represents the difference
+between states.
+

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+MNRAS 000, 1–11 (2022)
+Preprint 1 February 2023
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+Stacey N. Bright3 , P. D. Klaassen1 , O. D. Fox3
+3, Danny Gasman6
+V. C. Geers1 ,
+Adrian M. Glauser7 , Pierre Guillard10,11 , Omnarayani Nayak3 , A. Noriega-Crespo3 ,
+Michael E. Ressler12 , B. Sargent3,13 , T. Temim14 , B. Vandenbussche6 ,
+Macarena García Marín3
+1 UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK
+2 Centro de Astrobiología (CSIC-INTA), Carretera de Ajalvir, 28850 Torrejón de Ardoz, Madrid, Spain
+3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+4Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA
+5Dublin Institute for Advanced Studies, School of Cosmic Physics, Astronomy & Astrophysics Section, 31 Fitzwilliam Place, Dublin 2, Ireland
+6Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
+7Telespazio UK for the European Space Agency (ESA), ESAC, Spain
+8ETH Zurich, Institute for Particle Physics and Astrophysics, Wolfgang-Paulistr. 27, CH-8093 Zurich, Switzerland
+9Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
+10Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis bd Arago,
+75014 Paris, France
+11Institut Universitaire de France, Ministére de l’Enseignement Supérieur et de la Recherche, 1 rue Descartes, 75231 Paris Cedex 05, France
+12Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, CA 91109
+13Center for Astrophysical Sciences, The William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
+14Princeton University, 4 Ivy Ln, Princeton, NJ 08544, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+During the commissioning of JWST, the Medium-Resolution Spectrometer (MRS) on the
+Mid-Infrared Instrument (MIRI) observed the planetary nebula SMP LMC 058 in the Large
+Magellanic Cloud. The MRS was designed to provide medium resolution (R = 𝜆/Δ𝜆) 3D
+spectroscopy in the whole MIRI range. SMP LMC 058 is the only source observed in JWST
+commissioning that is both spatially and spectrally unresolved by the MRS and is a good test
+of JWST’s capabilities. The new MRS spectra reveal a wealth of emission lines not previously
+detected in this metal-poor planetary nebula. From these lines, the spectral resolving power
+(𝜆/Δ𝜆) of the MRS is confirmed to be in the range R = 4000 to 1500, depending on the MRS
+spectral sub-band. In addition, the spectra confirm that the carbon-rich dust emission is from
+SiC grains and that there is little to no time evolution of the SiC dust and emission line strengths
+over a 16-year epoch. These commissioning data reveal the great potential of the MIRI MRS.
+Key words: instrumentation: spectrographs; infrared: general; Astrophysics - Instrumentation
+and Methods for Astrophysics
+1
+INTRODUCTION
+The succession of increasingly powerful mid-infrared spectrographs
+(e.g., the Short Wavelength Spectrometer (SWS) and the Infrared
+Spectrograph (IRS) on board the Infrared Space Observatory and
+★ E-mail: [email protected]
+the Spitzer Space Telescope) launched into space has revolutionised
+our knowledge of the cool universe (e.g., Waters et al. 1996).
+The Mid-Infrared Instrument (MIRI; Wright et al. (submitted))
+on the James Webb Space Telescope (JWST) includes, in addition
+to the imager and coronographs, both a low-resolution spectrometer
+(LRS) covering wavelengths from 5 to 14 𝜇m (Kendrew et al. 2015)
+and a medium-resolution spectrometer (MRS; Wells et al. 2015,
+Argyriou et al. (in prep.)), which is an Integral Field Unit (IFU),
+© 2022 The Authors
+arXiv:2301.13233v1  [astro-ph.IM]  30 Jan 2023
+
+2
+O. C. Jones et al.
+that has a field of view ranging from 3.2′′ × 3.7′′ to 6.6′′ × 7.7′′
+(Law et al. (in prep.)) and can spatially resolve spectroscopic data
+between 4.9 and 27.9 𝜇m. This is the first time a mid-IR IFU has
+been deployed outside our atmosphere and will enable resolved
+spectroscopic studies of individual stars at the beginning and end
+of their evolution, diffuse structure in galaxies and planets.
+The Large Magellanic Cloud (LMC) is a gas-rich, metal-poor,
+star-forming, irregular galaxy, which is a satellite of the Milky Way,
+hosts ∼103 planetary nebulae (PNe) (Reid & Parker 2010; Reid
+2014), and is at a uniform distance of ∼50 kpc (Pietrzyński et al.
+2013). In lower metallicity environments like the LMC, which has
+about half the metallicity of the Milky Way (Westerlund 1997;
+Choudhury et al. 2016), significant dust production is expected to
+occur in the outflows of asymptotic giant branch (AGB) stars with
+further processing as these objects become planetary nebula (PNe).
+The gas and dust ejected into the interstellar medium (ISM) by a
+strong stellar wind from this phase of evolution contains elements
+synthesised in the stellar interior and dredged up to the surface by
+convection (e.g., Karakas & Lattanzio 2007, 2014). Their chemical
+composition is expected to primarily depend upon the initial stel-
+lar mass and the interstellar elemental abundance at the time the
+progenitor stars were formed (Kwok 2000; Gonçalves et al. 2014;
+Kwitter & Henry 2022).
+As such, the infrared spectra of PNe host a rich variety of
+features; forbidden emission lines arising from ionisation from the
+hot central star (e.g., Stanghellini et al. 2007), complex organic
+molecules (e.g., Ziurys 2006), polycyclic aromatic hydrocarbons
+(PAHs), and inorganic and organic solids (e.g., Stanghellini et al.
+2007; Bernard-Salas et al. 2009; Guzman-Ramirez et al. 2011;
+García-Hernández & Górny 2014) with the frequency of carbona-
+ceous features higher in the LMC than in Galactic PNe. This is likely
+due to the increased efficiency of third dredge-up (TDU) and the
+increased C/O ratio at low metallicities (Karakas et al. 2002). Pro-
+cessing by an external ambient UV radiation field which is stronger
+in the LMC (Gordon et al. 2008) may also affect the circumstellar
+chemistry. Detailed examination of PNe at low-metallicity, there-
+fore, provides a unique insight into chemical abundances and their
+effect on late-stage stellar evolution, dust production, and the for-
+mation of PNe in conditions comparable to those during the epoch
+of peak star formation in the Universe (Madau et al. 1996). Fur-
+thermore, due to their compact nature and brightness over a broad
+wavelength range, PNe are also useful calibration sources (e.g.,
+Swinyard et al. 1996; Feuchtgruber et al. 1997; Perley & Butler
+2013; Brown et al. 2014).
+SMP LMC 058 was observed by JWST as part of commission-
+ing and calibration activities for MIRI. First identified by Sanduleak
+et al. (1978), SMP LMC 058 is a carbon-rich planetary nebula (PN)
+in the LMC, with a heliocentric radial velocity of 278±7 km s−1
+(Margon et al. 2020). The central star of SMP 058 is a likely C ii
+emitter (Margon et al. 2020), consistent with a very early Wolf-
+Rayet type star on the carbon sequence (WC). Several dozen very
+strong, common emission lines of PNe were also detected in its
+optical spectra (Margon et al. 2020). SMP LMC 058 has also been
+observed with the Spitzer Infrared Spectrograph (IRS) at both low-
+resolution (R∼60–127) and high-resolution (R∼600). The Spitzer
+spectra show SMP LMC 058 has unusual dust chemistry with a
+strong SiC feature at ∼11.3 𝜇m (Bernard-Salas et al. 2009) and
+other associated features, including emission from PAHs at 6–9
+𝜇m, and a shoulder at 18 𝜇m from an unidentified carrier. However,
+the Spitzer-IRS data show no clear evidence of fullerenes (Sloan
+et al. 2014). SiC is rarely seen in Galactic PNe, in spite of the
+higher Si abundance in the Milky Way compared to the Magellanic
+Clouds (Jones et al. 2017). Its strength may be due to photoexci-
+tation, or because at a high C/O ratio SiC forms on the surface of
+carbon grains (Sloan et al. 2014).
+In this paper, we describe the observations and calibration of
+JWST MIRI MRS commissioning data of SMP LMC 058 (Sec-
+tion 2). We then present its MRS spectra in Section 3 and determine
+the resolving power of the MRS in Section 4. In Section 5 we
+identify and analyse the new emission lines and solid-state features
+detected in this carbon-rich planetary nebula and compare this with
+Spitzer IRS data. The potential of the MRS and our conclusions are
+discussed in Section 6.
+2
+OBSERVATIONS AND CALIBRATIONS
+The observations were taken as part of the MIRI MRS commis-
+sioning program, program ID 1049 (the commissioning purpose of
+these observations was PSF characterization). They use the standard
+MRS observing template, with 4-point dither patterns optimized for
+channels 2, 3, and 4 respectively. Each dither pattern was used twice,
+in the ‘positive’ and ‘negative’ direction. Target acquisition was ac-
+tivated, with the science target itself serving as an acquisition target.
+All three bands (SHORT, MEDIUM, LONG) in all channels were
+observed in all dithers. Simultaneous MIRI imaging in filter F770W
+was taken in the dither optimized for channel 2.
+A dedicated background observation was taken, employing
+a 2-point dither optimised for all channels, on a field roughly 3
+arcmin away. The background field was chosen to be relatively
+clear of astronomical sources based on archival WISE imaging data
+(Wright et al. 2010).
+A total of 45 FASTR1 frames were taken per integration. In
+target observations, a single integration was taken per dither point.
+The background observation had two integrations (to match the total
+integration time on source, accounting for the use of only a two-
+point dither on the background). The integration time per MRS sub-
+band and complete dither were therefore 499.5s or roughly 1,500s
+to cover the entire wavelength range (bands SHORT, MEDIUM,
+and LONG). Between the six dithers on-target and the single back-
+ground, the total integration time was approximately 2.9 hours (6.9
+hr including all overheads).
+The MRS observations were processed with version 1.7.3 of
+the JWST calibration pipeline and context 0995 of the Calibra-
+tion Reference Data System (CRDS). In general, we follow the
+standard MRS pipeline procedure (Labiano et al. 2016; Bushouse
+et al. 2022; and see Álvarez-Márquez et al. 2022 for an in-flight
+example of MRS data calibration). The background subtraction
+has been performed using the dedicated background observation.
+We have generated twelve 3D spectral cubes, one for each of
+the MRS channels and bands, with a spatial and spectral sam-
+pling of 0.13" × 0.13" × 0.001 𝜇m, 0.17" × 0.17" × 0.002 𝜇m,
+0.20" × 0.20" × 0.003 𝜇m, and 0.35" × 0.35" × 0.006 𝜇m for chan-
+nels 1, 2, 3, and 4, respectively. We have performed 1D spectral ex-
+tractions individually in each of the MRS cubes using a circular aper-
+ture of radius equal to 1.5 × 𝐹𝑊𝐻𝑀(𝜆), where 𝐹𝑊𝐻𝑀(𝜆) = 0.3
+arcsec for 𝜆 < 8𝜇m and 𝐹𝑊𝐻𝑀(𝜆) = 0.31 × 𝜆[𝜇𝑚]/8 arcsec for
+𝜆 > 8𝜇m. The selected FWHM (𝜆) values follow the MRS PSF
+Full Width at Half Maximum (FWHM). NIRCam observation (see
+Figure 1), and MRS observations suggest that SMP LMC 058 is an
+unresolved source. We use the MRS PSF models (Patapis et al. in
+prep.) to correct the aperture losses in the 1D spectra. The percent-
+age of flux that is lost out of the selected aperture is 17% for channel
+1 and increases to 30% in channel 4.
+MNRAS 000, 1–11 (2022)
+
+JWST MRS observations of SMP LMC 058
+3
+5h24m21.5s
+21.0s
+20.5s
+20.0s
+70 04′57′′
+05′00′′
+03′′
+06′′
+RA (ICRS)
+Dec (ICRS)
+F356W
+0.5 pc
+Figure 1. NIRCam F356W image of SMP LMC 058 shown in an Asinh
+stretch. At this spatial resolution (0.063′′) SMP LMC 058 is an unresolved
+point source.
+The 12 spectral segments extracted from these cubes were
+corrected for residual fringing using a post-pipeline spectral-level
+correction which is a modified version of the detector-level correc-
+tion available in the JWST calibration pipeline. The residual fringe
+contrasts are reduced by employing an empirical multi-component
+sine fitting method (e.g. Kester et al. 2003), under the assumption
+that the pipeline fringe flat correction has reduced fringe contrasts
+to the point where this multi-component sine approximation is valid
+(Kavanagh et al., in prep.).
+Finally, each of the 12 individual spectral segments was
+stitched together to remove minor flux discontinuities. This was
+done by determining a scaling factor between the median flux (ex-
+cluding spectral lines) in the overlapping MRS segments; then ap-
+plying this multiplicative factor to the longer wavelength segments,
+in turn, to effectively shift the spectrum to match the flux of its
+neighbouring shorter wavelength segment. This factor was typi-
+cally on the order of <5 per cent. The flux data in the overlapping
+spectral regions were then averaged. The final stitched spectrum was
+inspected to ensure there were no remaining discontinuities which
+may affect the continuum and model fitting.
+3
+SMP LMC 058 SPECTRUM
+Figure 2 shows the extracted spectrum of SMP LMC 058 which
+exhibits a rich variety of atomic, molecular and solid state features,
+including PAHs and silicon carbide, characteristic of carbon-rich
+material, and a strong continuum which rises towards the longest
+wavelengths. Due to the superior sensitivity and spectral resolution
+(see Section 4) of the MRS, the MIRI spectrum of SMP LMC
+058 shows features that are not seen in the Spitzer IRS data (see
+Section 5), notably in the number of emission lines detected.
+In the spectrum presented here, there is a large amount of fine-
+structure line emission present, from the strong nebular forbidden
+lines of [Ar ii], [Ar iii], [S iv], [Ne ii], [Ne iii], [S iii] to weak
+H recombination lines (Hi) from the Pfund and Humphreys series,
+and beyond. To ensure we measure and identify all the emission
+lines in the spectra we fit a pseudo-continuum to the broadband
+spectral features using a piece-wise spline model. Obvious narrow
+band features were identified and masked in the fitting based on
+their amplitudes exceeding a threshold value. We used an outlier
+rejection fitter to flag and ignore any weaker narrow-band features
+that may compromise the continuum fit. After visual inspection of
+the fit, it was subtracted to isolate any narrow-band features present.
+Figure 3 shows the spectrum of SMP LMC 058 after subtraction
+of the pseudocontinuum from the total spectrum. The spectrum is
+extremely rich in emission lines. In total 51 lines were detected.
+Using the 12 original MRS segments, we identified and ana-
+lyzed all detected emission lines with a signal-to-noise ratio (SNR)
+greater than 3 in the SMP LMC 058 spectra. Depending on the
+line profiles (see Figure 4), we performed one-component and two-
+component Gaussian fits, plus a second-order polynomial to simul-
+taneously fit the continuum and emission line.1 The uncertainties
+on the derived emission line parameters, like the line FWHM, flux,
+central wavelength, etc, were estimated using a Monte Carlo simu-
+lation (following the same methodology as Álvarez-Márquez et al.
+2021, 2022). Systemic velocity shifts were removed using a he-
+liocentric radial velocity of 278 ± 7 km/s (Reid & Parker 2006;
+Margon et al. 2020). Table 1 presents the measured wavelengths
+and fluxes together with the identification of the mid-IR emission
+lines in SMP LMC 058 spectra. Small wavelength offsets are con-
+sistent with known errors in the MRS FLT-4 wavelength solution
+(see discussion by Argyriou et al. (in prep.)) and should be reduced
+further by ongoing calibration efforts later in Cycle 1. Weak lines are
+more prevalent at shorter wavelengths in channels 1 and 2 where the
+MRS sensitivity is higher and the uncertainties in the flux are better
+constrained. Furthermore, we find that the current MRS wavelength
+calibration is <40 km/s for all spectral sub-bands, better than the
+FWHM of the MRS line spread function (75-200 km/s, Labiano
+et al. 2021, Argyriou et al. (in prep.)).
+4
+MRS RESOLVING POWER
+The resolving power (R) is defined as 𝜆/Δ𝜆, where Δ𝜆 is the mini-
+mum distance to distinguish two features in a spectrum. We define
+the Δ𝜆 as the FWHM of an unresolved emission line. SMP LMC
+058 is the only source observed in the JWST commissioning datasets
+that is considered both spatially and spectrally unresolved with the
+MRS, this makes it an excellent target for determining the inflight
+MRS resolving power. Here, we assume the intrinsic width of the
+emission lines in SMP LMC 058 to be negligible, as we do not have
+high-resolution spectroscopy to characterize its intrinsic velocity
+dispersion. Nearby planetary nebula eject gas with typical velocity
+dispersions of about 10–51 kms−1 (Reid & Parker 2006). If this is
+the case for SMP LMC 058, then assuming a velocity of 25 kms−1
+we might underestimate the MRS resolving power by up to 5% for
+channel 1, and up to 1% for channel 4 (see e.g., Law et al. 2021).
+The pre-launch MRS resolving power has been established
+from MIRI ground-based test and calibration campaigns, using a
+set of etalons which provided lines in all MRS bands. It was de-
+termined to be in the range of about 4000 to 1500 (Labiano et al.
+2021). Figure 5 shows the comparison between the ground-based
+MRS resolving power estimates and the inflight estimates derived
+from the SMP LMC 058 spectra. The inflight MRS resolving power
+has been determined using only emission lines with SNR higher
+than 6, and following the FWHM results obtained in the one- and
+1 We used the mpfit (Markwardt 2009) Python routine to perform the fits,
+the code is publicly available here.
+MNRAS 000, 1–11 (2022)
+
+4
+O. C. Jones et al.
+5
+10
+15
+20
+25
+Wavelength (microns)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+Flux (Jy)
+MIRI MRS
+Figure 2. The MIRI MRS spectrum of SMP LMC 058. Numerous emission lines, PAH features and dust features are clearly seen on a rising continuum. These
+features are much better resolved in the MRS spectra due to the higher spectral resolution.
+two-component Gaussian fits (see Section 3). In the case of Hi
+emission lines, we used the FWHM of the narrow gaussian compo-
+nent. The errors in the resolving power are, on average, larger for
+the Hi emission lines due to the uncertainty in the two-component
+Gaussian fit. Given the uncertainties, the inflight MRS resolving
+power agrees with the ground-based estimates, and it presents a
+trend followed by the equation 4401 − 112 × 𝜆[𝜇𝑚] + 10−9×𝜆[𝜇𝑚].
+The ground-based estimates consider the width of the etalon
+emission lines to be negligible, which could imply an underesti-
+mation of the MRS resolving power by a factor of 10%. A similar
+situation is potentially happening with the inflight estimations due
+to the lack of the intrinsic velocity dispersion of SMP LMC 058. We
+conclude that the estimations of the ground-based MRS resolving
+power (Labiano et al. 2021) are valid, within a 10% of uncertainty,
+for the MRS inflight performance. As JWST observes more sources
+with spatially and spectrally unresolved spectral lines, their char-
+acterisation will provide a more comprehensive understanding of
+the inflight variations of the resolving power within each of the 12
+spectral bands. As of now, the continuous "trend" curve in Figure 5
+presents the state of knowledge of the MRS resolving power.
+5
+DISCUSSION
+5.1
+Emission Lines
+Short-High and Long-High Spitzer spectroscopic data of SMP LMC
+058 were published in Bernard-Salas et al. (2008). A comparison of
+the MRS line fluxes with those found by Bernard-Salas et al. (2008)
+is given in Table 2. In general, there is good agreement between our
+measurements of the forbidden emission line strengths of [S iv],
+[Ne ii], [Ne iii], [S iii]. Furthermore, the high-excitation lines of
+[Ar v] at 13.10 𝜇m and [O iv] with ionisation potentials of 60 and
+55 eV, respectively, are not detected in either spectrum. These lines
+are excited by high-temperature stars with Teff between 140,000
+– 180,000 K. The highest ionisation species in the MRS spectra
+are [K iv] (46 eV) and [Na iii] (47 eV), these lines have not been
+previously detected by Spitzer. Thus, we consider SMP LMC 058
+to be a low-excitation source.
+Given the superior sensitivity of JWST (Rigby et al. 2022) and
+the MRS, we detect a line at 14.38 𝜇m an order of magnitude below
+the upper-limit of [Ne v] reported by Bernard-Salas et al. (2008).
+The ionisation potential of [Ne v] is 97 eV, thus it is unlikely given
+the absence of other high-excitation lines in the MRS spectra of SMP
+LMC 058, that this emission is from [Ne v], instead, we attribute
+this line to [Cl ii] which has an ionisation potential of 13 eV.
+As seen in Table 1 higher ionisation potential species expand
+at a lower velocity than the lower ionisation potential species (e.g.,
+Reid & Parker 2006). This is due to ionisation occurring at a greater
+distance from the centre of the PN where velocities are greater, and
+can cause lower excitation species to expand to larger radii in the
+PN.
+Hydrogen recombination lines are abundant in the spectrum
+of SMP LMC 058, all are new detections. Hi emission lines more
+closely trace the ionized regions, compared to molecular hydrogen.
+As shown in Figure 4, the line profiles of the bright Hi emission lines
+are asymmetric, exhibiting a blue tail, whereas the forbidden emis-
+sion lines present symmetric unresolved profiles. Hi emission lines
+are composed of a spectrally unresolved main component contain-
+MNRAS 000, 1–11 (2022)
+
+JWST MRS observations of SMP LMC 058
+5
+5
+6
+7
+8
+9
+Wavelength (microns)
+0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Flux (Jy)
+H Hu 
+H Hu 
+[ArII]
+H Pf 
+H Hu 
+H10-7
+[ArIII]
+[NaIII]
+H16-7
+H15-7
+H13-7
+H12-7
+H15-8
+H13-8
+10
+12
+14
+16
+18
+20
+22
+Wavelength (microns)
+0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Flux (Jy)
+[SIV]
+H9-7
+H Hu 
+[NeII]
+[NeV]
+[NeIII]
+H10-8
+[SIII]
+H8-7
+Figure 3. Continuum-subtracted MRS spectrum of SMP LMC 058 (where the continuum includes dust and PAH features), highlighting the atomic emission
+lines. The identification of key species are marked on the spectrum. The top panel shows lines in channels 1 and 2 of the MRS, the lower panels show channels
+3 and 4. The flux axis is truncated to highlight lower contrast lines.
+MNRAS 000, 1–11 (2022)
+
+6
+O. C. Jones et al.
+Table 1. Measured central wavelengths, line flux, line widths, and line identification for SMP LMC 058. The systemic velocity was removed prior to calculating
+the velocity shift of a line. If a line is present in multiple MRS bands, measurements are provided for each individual MRS segment. Uncertain line identifications
+are denoted by a ‘?’.
+Band
+Line
+𝜆lab
+𝜆observed
+𝜎𝜆observed
+𝜆offset
+FWHM
+Flux (×10−15)
+𝜎 (×10−15)
+Identification
+𝜇m
+𝜇m
+𝜇m
+km s−1
+nm
+erg s−1 cm−2
+erg s−1 cm−2
+1S
+Hi 23−7
+4.924
+4.92903
+0.00004
+-46.025
+50.005
+0.11
+0.02
+1S
+Hi 22−7
+4.971
+4.97588
+0.00004
+-23.351
+49.997
+0.09
+0.01
+1S
+Hi 21−7
+5.026
+5.03086
+0.00007
+-6.428
+63.362
+0.07
+0.01
+1S
+Hi 20−7
+5.091
+5.09600
+0.00003
+2.418
+50.000
+0.13
+0.01
+1S
+Hi 10−6
+5.129
+5.13342
+0.00001
+-0.165
+94.556
+1.67
+0.02
+1S
+Hi 19−7
+5.169
+5.17401
+0.00002
+4.175
+51.268
+0.16
+0.02
+1S
+Hi 18−7
+5.264
+5.26856
+0.00006
+0.558
+77.922
+0.16
+0.01
+1S
+[Fe ii]
+5.340
+5.34504
+0.00011
+5.043
+92.348
+0.06
+0.01
+1S
+Hi 17−7
+5.380
+5.38499
+0.00002
+-12.237
+77.919
+0.18
+0.01
+1S
+Hi 16−7
+5.525
+5.53072
+0.00006
+-21.881
+92.047
+0.27
+0.02
+1S
+Hi 15−7
+5.711
+5.71692
+0.00002
+-8.044
+50.000
+0.28
+0.02
+1M
+Hi 15−7
+5.711
+5.71726
+0.00005
+-26.023
+89.264
+0.28
+0.02
+1M
+Hi 9−6
+5.908
+5.91340
+0.00001
+15.046
+99.683
+2.31
+0.03
+1M
+Hi 14−7
+5.957
+5.96199
+0.00002
+19.244
+73.871
+0.33
+0.02
+1M
+[K iv]
+5.982
+5.98750
+0.0001
+2.606
+110.497
+0.12
+0.01
+1M
+Hi 13−7
+6.292
+6.29807
+0.00005
+-14.725
+74.449
+0.43
+0.04
+1L
+Hi 12−7
+6.772
+6.77852
+0.00002
+-10.975
+85.586
+0.58
+0.02
+1L
+Hi 21−8
+6.826
+6.83241
+0.00013
+-8.356
+77.331
+0.06
+0.02
+1L
+H2(0,0) S(5)
+6.910
+6.91588
+0.00013
+2.424
+95.459
+0.13
+0.02
+1L
+Hi 20−8
+6.947
+6.95306
+0.00009
+6.515
+75.574
+0.08
+0.02
+1L
+[Ar ii]
+6.985
+6.99181
+0.00001
+-2.238
+89.682
+1.17
+0.01
+1L
+Hi 19−8
+7.093
+7.09935
+0.00017
+-2.424
+76.633
+0.11
+0.01
+1L
+Hi 18−8
+7.272
+7.27877
+0.00011
+-14.968
+69.767
+0.10
+0.02
+1L
+[Na iii]
+7.318
+7.32485
+0.00006
+-14.817
+136.625
+0.46
+0.02
+1L
+Hi 6−5
+7.460
+7.46690
+0
+-4.607
+79.653
+11.51
+0.04
+1L
+Hi 8−6
+7.502
+7.50949
+0.00001
+-1.396
+79.577
+12.33
+0.07
+1L
+Hi 11−7
+7.508
+7.51515
+0.00003
+-2.922
+74.553
+0.69
+0.02
+2S
+Hi 15−8
+8.155
+8.16375
+0.00053
+-47.407
+49.995
+0.15
+0.05
+2S
+Hi 14−8
+8.665
+8.67220
+0.00019
+11.971
+53.004
+0.28
+0.03
+2M
+Hi 10−7
+8.760
+8.76815
+0.00006
+1.510
+110.130
+1.04
+0.05
+2M
+[Ar iii]
+8.991
+8.99859
+0
+37.731
+101.834
+20.61
+0.07
+2M
+Hi 13−8
+9.392
+9.40091
+0.00012
+-5.539
+84.050
+0.25
+0.04
+2M
+H2(0,0) S(3)
+9.665
+9.67502
+0.00023
+-35.450
+151.677
+0.23
+0.04
+2M
+Hi 18−9
+9.847
+9.85887
+0.00037
+-82.099
+75.445
+0.06
+0.01
+2L
+[S iv]
+10.511
+10.52014
+0.00001
+3.428
+96.022
+30.07
+0.12
+2L
+Hi 16−9
+10.804
+10.81253
+0.00068
+30.378
+140.073
+0.13
+0.03
+2L
+Hi 9−7
+11.309
+11.31929
+0.00025
+-2.583
+82.684
+1.41
+0.17
+3S
+Hi 7−6
+12.372
+12.38265
+0.00003
+17.586
+94.410
+2.90
+0.03
+3S
+Hi 11−8
+12.387
+12.39758
+0.00005
+26.303
+107.676
+0.68
+0.02
+3S
+Hi 14−9
+12.587
+12.59863
+0.00023
+3.125
+93.158
+0.15
+0.03
+3S
+[Ne ii]
+12.814
+12.82631
+0
+-20.232
+98.379
+17.63
+0.02
+3M
+Hi 13−9
+14.183
+14.19616
+0.00035
+1.955
+69.993
+0.13
+0.05
+3M
+[Cl ii]?
+14.368
+14.38034
+0.00019
+16.544
+82.249
+0.34
+0.02
+3M
+Hi 16−10
+14.962
+14.97556
+0.00064
+11.773
+50.002
+0.04
+0.02
+3L
+[Ne iii]
+15.555
+15.56957
+0.00001
+-0.617
+130.542
+179.26
+0.55
+3L
+Hi 10−8
+16.209
+16.22334
+0.00014
+14.869
+97.309
+0.60
+0.08
+3L
+Hi 12−9
+16.881
+16.89664
+0.0002
+-6.106
+91.592
+0.29
+0.04
+4S
+[S iii]
+18.713
+18.72914
+0.00002
+19.709
+136.931
+11.34
+0.07
+4S
+Hi 8−7
+19.062
+19.07898
+0.00005
+9.654
+148.238
+2.40
+0.05
+4M
+[Ar iii]
+21.830
+21.85033
+0.00032
+1.852
+155.363
+0.82
+0.06
+4M
+Hi 13−10+Hi 11−9
+22.340
+22.35612
+0.00055
+68.040
+194.363
+0.64
+0.05
+ing the majority of the line flux (>95%)), and a spectrally resolved
+blue-shifted component possibly due to thermal broadening (Chu
+et al. 1984) or from condensation outside the main core which may
+be evident as a marginally resolved envelope like structure in the
+MRS cube at 7.466𝜇m.
+Two molecular hydrogen lines (H2) have been detected in the
+MRS data, the ortho-H2 𝑣 = 0–0 S(3) and S(5) lines. The S(1) line at
+17.055 𝜇m may also be present, although this is not easily discerned
+above the continuum and we do not measure its flux. The S(3) and
+S(5) rotational line emission probably originate from irradiated,
+and perhaps also shocked, dense molecular clumps, torus structures
+(e.g., Kastner et al. 1996; Hora et al. 1999; Akras et al. 2017;
+MNRAS 000, 1–11 (2022)
+
+JWST MRS observations of SMP LMC 058
+7
+7.455
+7.460
+7.465
+7.470
+7.475
+Wavelength (microns)
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Flux (Jy)
+Pf 
+15.54
+15.56
+15.58
+15.60
+Wavelength (microns)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Flux (Jy)
+[Ne III]
+Figure 4. Top: The Pf 𝛼 H i emission line profile shows a spatially unre-
+solved main component and a weaker spectrally resolved blue-shifted wing.
+Bottom: The [Ne iii] line profile is spectrally unresolved and symmetric.
+This shape is typical of all the forbidden emission lines in SMP LMC 058.
+The dashed line marks the lines observed central wavelength.
+Table 2. Comparison of SMP LMC 058 MRS line fluxes with those of
+Bernard-Salas et al. (2008) taken with the high-resolution modules on the
+Spitzer IRS. All line strengths reported by Bernard-Salas et al. (2008) have
+a 10% error except for [S iv] which has a 10–20% error. Errors in the MRS
+flux are <1% and are provided for each line in Table 1.
+Line
+Wavelength
+MRS Flux
+Spitzer Flux
+Ionisation
+(Rest)
+×10−15
+×10−15
+potential
+𝜇m
+erg s−1 cm−2
+erg s−1 cm−2
+(eV)
+[S iv]
+10.511
+30.07
+29.2
+35
+[Ne ii]
+12.814
+17.63
+20.6
+22
+[Ar v]
+13.099
+<0.029
+<2.4
+60
+[Ne v]
+14.323
+<0.005
+<3.8
+97
+[Ne iii]
+15.555
+179.26
+200.6
+41
+[S iii]
+18.713
+11.34
+11.0
+23
+[O iv]
+25.883
+<0.23
+<21.6
+55
+Fang et al. 2018), or from the outer regions of the PNe where the
+temperature is about 1000K (Aleman & Gruenwald 2004; Matsuura
+et al. 2007b).
+5.2
+Dust and PAH Features
+The dust in SMP LMC 058 is carbon-rich. Amongst the most promi-
+nent features is the strong silicon carbide (SiC) emission at 11 𝜇m
+and the rising continuum due to the thermal emission of warm dust.
+Strong emission features from PAHs also appear in the spectrum at
+5.2, 5.7, 6.2, 7.7, 8.6, 11.2 and 12.7 𝜇m.
+At sub-solar metallicities (∼ 0.2 − 0.5 Z⊙), SiC is commonly
+observed in PNe, yet it is rarely seen in Galactic PNe or indeed
+during the earlier AGB evolutionary phase of metal-poor carbon
+stars (Casassus et al. 2001; Zijlstra et al. 2006; Matsuura et al.
+2007a; Stanghellini et al. 2007; Bernard-Salas et al. 2008; Woods
+et al. 2011, 2012; Sloan et al. 2014; Ruffle et al. 2015; Jones et al.
+2017). The strength of the SiC flux in metal-poor PNe is highly
+sensitive to the radiation field (Bernard-Salas et al. 2009). This is
+likely due to a lower abundance of Si affecting the carbonaceous dust
+condensation sequence on the AGB. In this case, rather than SiC
+forming first, it instead forms in a mantle surrounding an amorphous
+carbon core (Lagadec et al. 2007; Leisenring et al. 2008). Then as
+the PNe dust becomes heated and photo-processed, the amorphous
+carbon evaporates increasing the SiC surface area and consequently
+its feature strength, until a critical ionisation potential of >55 eV
+occurs at which point the SiC features disappear (Bernard-Salas
+et al. 2009; Sloan et al. 2014).
+Following Bernard-Salas et al. (2009), we measure the strength
+of the SiC feature by integrating the flux above a continuum-
+subtracted spectrum from 9 to 13.2 𝜇m and then subtracting the flux
+contributions from the 11.2 𝜇m PAH feature and the [Ne ii] line.
+Due to the resolution of the MRS compared to the Spitzer spectra of
+SMP LMC 058, we detect several additional lines which contribute
+to the integrated flux in the SiC region; these lines include [S iv]
+and H i. Thus to obtain a reliable measurement of the SiC feature
+strength we also subtract the flux contribution from all emission
+lines in the 9 – 13.2 𝜇m region listed in Table 1. A PAH feature at
+∼12.6 𝜇m likely contributes a small amount of flux to the measured
+SiC feature, however isolating and subtracting this emission contri-
+bution from the wing of the SiC feature is challenging even with
+the MRS spectral resolution. Additionally, an artefact at ∼12.2 𝜇m
+due to a spectral leak (e.g., Gasman et al. 2022) may also affect the
+integrated flux. Table 3 gives the measured SiC centroid and cor-
+rected feature strength. The latter agrees exceptionally well with the
+value of 29.72 ± 0.31 ×10−16 W m−2 measured by Bernard-Salas
+et al. (2009) in the Spitzer data of SMP LMC 058. This suggests
+there is little to no evolution in the SiC dust on the 16-year time
+scales between the observations. Furthermore, the agreement be-
+tween the measurements verifies the overall flux calibration of the
+MRS instrument (Gasman et al. 2022).
+In astronomical sources, the structure, wavelengths and relative
+strength of the PAHs can differ strongly between objects, with PNe
+showing the most pronounced variations in PAH profiles due to
+photoprocessing altering the ratio of aliphatics to aromatics (Peeters
+et al. 2002; Pino et al. 2008; Matsuura et al. 2014; Sloan et al. 2014;
+Jensen et al. 2022). Figure 6 shows the PAHs in SMP LMC 058. The
+PAHs in SMP LMC 058 are considered to have a class B profile by
+Bernard-Salas et al. (2009) and Sloan et al. (2014). In this schema
+devised by Peeters et al. (2002) and van Diedenhoven et al. (2004)
+the 6.2 PAH feature for class B objects has a peak between 6.24
+and 6.28 𝜇m; the dominant 7.7 PAH feature peaks between 7.8 to
+MNRAS 000, 1–11 (2022)
+
+8
+O. C. Jones et al.
+4.5
+5
+6
+7
+8
+9
+10
+12
+15
+20
+25
+30
+Wavelength [ m]
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+MRS Resolving Power
+Ground (Labiano+21)
+Fit to inflight lines
+inflight forbidden lines
+inflight HI lines
+Figure 5. Comparison between the ground-based and inflight MRS resolving power. Gray filled area and black line: ground-based MRS resolving power
+estimates (Labiano et al. 2021). Filled red circles: inflight MRS resolving power calculation using forbidden emission lines identified in this paper. Open red
+circles: inflight MRS resolving power calculation using Hi emission lines.
+6
+8
+10
+12
+14
+Wavelength (microns)
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+Flux (Jy)
+PAH 5.2 m
+PAH 5.7 m
+PAH 6.2 m
+PAH 7.7 m
+PAH 8.6 m
+PAH 11.2 m
+SiC
+Local continuum
+Figure 6. The SiC and PAH features are highlighted in the spectra of SMP LMC 058. A local continuum fit to the 11.3 𝜇m feature which is superimposed
+on the broad SiC emission feature is also shown. The colours highlight the spectral region for each feature, to which a local continuum was fit and the flux
+measured over.
+MNRAS 000, 1–11 (2022)
+
+JWST MRS observations of SMP LMC 058
+9
+Table 3. PAH and SiC Fluxes and Centroids.
+Centroid
+Integrated Flux
+Integrated Flux Error
+𝜇m
+W m−2
+W m−2
+PAH
+5.262
+1.64×10−18
+7.0×10−20
+PAH
+5.698
+5.62×10−18
+1.1×10−19
+PAH
+6.274
+9.259×10−17
+2.9×10−19
+PAH
+7.834
+3.427×10−16
+1.2×10−18
+PAH
+8.665
+6.231×10−17
+8.5×10−19
+PAH
+11.298
+1.047×10−16
+2.0×10−18
+SiC
+11.097
+3.067×10−15
+4.5×10−18
+8.0 𝜇m; and the 8.6 PAH band is red-shifted. These values agree
+well with our measured centroids listed in Table 3. Furthermore, the
+PAHs observed in SMP LMC 058 closely resemble those observed
+in the ISO SWS spectrum of the Galactic post-AGB star, HD 44179
+(the Red Rectangle) which also shows strong aromatic features on
+top of a continuum (Waters et al. 1998).
+The relative strength of the PAH features depends on a number
+of factors including the degree of ionisation of the radiation field
+(e.g., Allamandola et al. 1999). The strength of the PAH features
+in SMP LMC 058 was determined by integrating the flux of the
+feature above an adopted local continuum, fit to each side of the
+feature and measured using specutils line_flux. Particular care
+was taken in fitting a continuum, too, and then measuring the 11.25
+𝜇m band (produced by the out-of-plane solo C–H bending mode)
+as this is superimposed on top of the broad SiC feature. Table 3
+presents the central wavelength of the features and the integrated
+flux. The ratio of the PAH strengths correlates with the source type
+and hence its physical conditions (Hony et al. 2001); ionized PAHs
+have strong features at 6.2, 7.7 and 8.6 𝜇m whilst the 11.2 𝜇m PAH
+feature is stronger for neutral PAHs. From the PAH line strengths
+given in Table 3 it is evident that the 7.7𝜇m feature dominates the
+total PAH emission, and thus the dust around SMP LMC 058 is
+likely experiencing a high degree of ionisation.
+Carbon-rich PNe can show a rich variety of solid-state material
+in their spectra in addition to PAHs. The C60 fullerene molecule
+typically exhibits features at ∼7.0, 8.5, 17.4 and 18.9 𝜇m, and all
+four were first identified in the spectrum of the Galactic PN TC-1
+(Cami et al. 2010). Fullerenes have since been detected in several
+other PNe (e.g., García-Hernández et al. 2010, 2011; Sloan et al.
+2014). The still-unidentified 21 𝜇m emission feature, first detected
+by Kwok et al. 1989, can also appear in carbon-rich PNe, often
+associated with unusual PAH emission and aliphatic hydrocarbons
+(Cerrigone et al. 2011; Matsuura et al. 2014; Sloan et al. 2014; Volk
+et al. 2020).
+The spectra of SMP LMC 058 from the IRS on Spitzer did
+not show any of these unusual hydrocarbon-related features, but the
+improved spectral resolution of the MRS allows for a much more
+careful examination. Nonetheless, these additional features remain
+too weak to be detected. SMP LMC 058 presents a classic Class
+B PAH spectrum, as expected for objects which have evolved to
+the young PN stage (Sloan et al. 2014). Younger objects which
+could still be described as post-AGB objects would show the 21 𝜇m
+feature and/or aliphatics. Sloan et al. (2014) identified SMP LMC
+058 as a member of the Big-11 group because of the combination
+of a strong SiC emission feature and the 11.2 𝜇m PAH feature and
+the absence of fullerenes. This group is actually related to the PNe
+that show fullerenes, and the presence or absence of fullerenes may
+be due to something as simple as which have a clear line of sight to
+the interior of the dust shells where the fullerenes are expected to
+be present.
+6
+SUMMARY AND CONCLUSIONS
+We have presented MIRI/MRS spectra of the carbon-rich planetary
+nebula SMP LMC 058 located in the metal-poor Large Magellanic
+Cloud. SMP LMC 058 is a point source in the MRS data and
+its spectrum contains the only spatially and spectrally unresolved
+emission lines observed during the commissioning of the JWST
+Medium-Resolution Spectrometer. In the MRS spectrum, we de-
+tected 51 emission lines, of which 47 were previously undetected
+in this source. The strongest emission lines were used to determine
+the spectral resolutions of the MIRI MRS instrument. The resolving
+power is R > 3960 in channel 1, R > 3530 in channel 2, R > 3200
+in channel 3, and R > 1920 in channel 4. This on-sky performance
+is comparable to the resolution determined from the ground cali-
+bration of the MRS which provides resolving powers from 4000 at
+channel 1 to 1500 at channel 4. Furthermore, a comparison of the
+line strengths and spectral continuum to previous observations of
+SMP LMC 058 with the IRS on the Spitzer was used to verify the
+absolute flux calibration of the MRS instrument. The MRS spectra
+confirm that the carbon-rich dust emission is from grains and not
+isolated molecules and that there is little to no time evolution of the
+SiC dust and emission line strengths in the 16 years between the
+observations. The PAH emission is dominated by the 7.7𝜇m feature.
+The strong PAHs and SiC in the spectra are consistent with the lack
+of high-excitation lines detected in the spectra, which if present,
+would indicate a hard radiation field that would likely destroy these
+grains. These commissioning data reveal the great potential and
+resolving power of the MIRI MRS to study line, molecular and
+solid-state features in individual sources in nearby galaxies.
+ACKNOWLEDGEMENTS
+We thank Kay Justtanont for her insights, comments and dis-
+cussions. This work is based on observations made with the
+NASA/ESA/CSA James Webb Space Telescope. The data were ob-
+tained from the Mikulski Archive for Space Telescopes at the Space
+Telescope Science Institute, which is operated by the Association of
+Universities for Research in Astronomy, Inc., under NASA contract
+NAS 5-03127 for JWST. These observations are associated with
+program #1049. This work is based in part on observations made
+with the Spitzer Space Telescope, which was operated by the Jet
+Propulsion Laboratory, California Institute of Technology under a
+contract with NASA
+O.C.J acknowledge support from an STFC Webb fellowship.
+J.A.M. and A.L acknowledge support by grant PIB2021-127718NB-
+100 by the Spanish Ministry of Science and Innovation/State Agency
+of Research (MCIN/AEI). P.J.K acknowledges financial support
+from the Science Foundation Ireland/Irish Research Council Path-
+way programme under Grant Number 21/PATH-S/9360. I.A., D.G.,
+and B.V. thank the European Space Agency (ESA) and the Belgian
+Federal Science Policy Office (BELSPO) for their support in the
+framework of the PRODEX Programme. PG would like to thank
+the University Pierre and Marie Curie, the Institut Universitaire de
+France, the Centre National d’Etudes Spatiales (CNES), the "Pro-
+gramme National de Cosmologie and Galaxies" (PNCG) and the
+"Physique Chimie du Milieu Interstellaire" (PCMI) programs of
+MNRAS 000, 1–11 (2022)
+
+10
+O. C. Jones et al.
+CNRS/INSU, with INC/INP co-funded by CEA and CNES, for
+there financial supports.
+MIRI draws on the scientific and technical expertise of the
+following organisations: Ames Research Center, USA; Airbus De-
+fence and Space, UK; CEA-Irfu, Saclay, France; Centre Spatial
+de Liége, Belgium; Consejo Superior de Investigaciones Científi-
+cas, Spain; Carl Zeiss Optronics, Germany; Chalmers University of
+Technology, Sweden; Danish Space Research Institute, Denmark;
+Dublin Institute for Advanced Studies, Ireland; European Space
+Agency, Netherlands; ETCA, Belgium; ETH Zurich, Switzerland;
+Goddard Space Flight Center, USA; Institute d’Astrophysique Spa-
+tiale, France; Instituto Nacional de Técnica Aeroespacial, Spain; In-
+stitute for Astronomy, Edinburgh, UK; Jet Propulsion Laboratory,
+USA; Laboratoire d’Astrophysique de Marseille (LAM), France;
+Leiden University, Netherlands; Lockheed Advanced Technology
+Center (USA); NOVA Opt-IR group at Dwingeloo, Netherlands;
+Northrop Grumman, USA; Max-Planck Institut für Astronomie
+(MPIA), Heidelberg, Germany; Laboratoire d’Etudes Spatiales et
+d’Instrumentation en Astrophysique (LESIA), France; Paul Scher-
+rer Institut, Switzerland; Raytheon Vision Systems, USA; RUAG
+Aerospace, Switzerland; Rutherford Appleton Laboratory (RAL
+Space), UK; Space Telescope Science Institute, USA; Toegepast-
+Natuurwetenschappelijk Onderzoek (TNO-TPD), Netherlands; UK
+Astronomy Technology Centre, UK; University College London,
+UK; University of Amsterdam, Netherlands; University of Arizona,
+USA; University of Bern, Switzerland; University of Cardiff, UK;
+University of Cologne, Germany; University of Ghent; University
+of Groningen, Netherlands; University of Leicester, UK; University
+of Leuven, Belgium; University of Stockholm, Sweden; Utah State
+University, USA. A portion of this work was carried out at the Jet
+Propulsion Laboratory, California Institute of Technology, under a
+contract with the National Aeronautics and Space Administration.
+The following National and International Funding Agencies
+funded and supported the MIRI development: NASA; ESA; Bel-
+gian Science Policy Office (BELSPO); Centre Nationale d’Etudes
+Spatiales (CNES); Danish National Space Centre; Deutsches Zen-
+trum fur Luftund Raumfahrt (DLR); Enterprise Ireland; Ministerio
+De Economia y Competividad; Netherlands Research School for As-
+tronomy (NOVA); Netherlands Organisation for Scientific Research
+(NWO); Science and Technology Facilities Council; Swiss Space
+Office; Swedish National Space Agency; and UK Space Agency.
+Facilities: JWST (MIRI/MRS) - James Webb Space Telescope.
+DATA AVAILABILITY
+JWST data were obtained from the Mikulski Archive for
+Space Telescopes at the Space Telescope Science Institute
+(https://archive.stsci.edu/).
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+

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+Communications in Mathematics n (2023), no. m, 00–12
+DOI: https://doi.org/10.46298/cm.ABCD
+©2023 G´abor Rom´an
+This is an open access article licensed under the CC BY-SA 4.0
+1
+On square-free numbers generated from given sets of primes
+II
+G´abor Rom´an
+Abstract. We progress with the investigation started in article [7], namely the anal-
+ysis of the asymptotic behaviour of QP(x) for different sets P, where QP(x) is the
+element count of the set containing those positive square-free integers, which are
+smaller than-, or equal to x, and which are only divisible by the elements of P. We
+study how QP(x) behaves when we require that χ(p) = 1 must hold for every p ∈ P,
+where χ is a Dirichlet character.
+1
+Introduction
+Let’s take a set of prime numbers P, and denote with QP(x) the element count of the
+set of all those positive square-free integers, which are smaller than-, or equal to x; and
+which are only divisible by the elements of P.
+In article [7] we examined how QP(x) behaves asymptotically based on the structure
+of P in two scenarios. During the first scenario, P contained those primes which are not
+greater than an x dependent bound λ(x), see [7, Prop. 1]. In the second scenario, P
+contained those primes which are not greater than an x dependent bound λ(x), and which
+fall into certain congruence classes modulo q, see [7, Prop. 2].
+In this article, we go further, and render the structure of P more complex. We would
+want to restrict ourselves to primes p, for which a certain integer a is quadratic residue
+modulo p, but to generalise, we are going to require that χ(p) = 1 holds, where χ is a
+Dirichlet character. This covers our goal, as the only real valued primitive non-principal
+χ(n) are given for positive n by the Kronecker symbol (D|n), where D is a fundamental
+discriminant, see [9].
+MSC 2020: 11M06, 11M20, 11N36, 11N37, 11N69
+Keywords: Square-free numbers, Combinatorial sieve, Dirichlet character, Square-free numbers in
+arithmetic progressions, L-functions, Euler product
+Affiliation:
+G´abor Rom´an – E¨otv¨os Lor´and University, Budapest, Hungary
+E-mail: [email protected]
+arXiv:2301.02377v1  [math.NT]  6 Jan 2023
+
+=P sciences2
+G´abor Rom´an
+The results will be of course similar to the results in article [7], heavily depending on
+the conductor q(χ) of the Dirichlet character χ in context.
+Proposition 1.1. Let χ be a real valued non-principal Dirichlet character. Furthermore, let
+λ : R → [1, +∞) be a monotone increasing function which is in o(x1/2), and let P contain
+all the primes p which are not greater than λ(x), and for which χ(p) = 1.
+Then for every ε > 0, there exit real constants a1 and a2 such that
+ea1
+ln q(χ)
+ln λ(x) x
+ln x
+�
+ln λ(x)
+�
+q(χ)ε ≪ QP(x) ≪ ea2
+ln q(χ)
+ln λ(x) x
+ln x
+�
+ln q(χ)
+�
+ln λ(x)
+(1)
+as x → +∞. In addition, if χ is primitive, q(χ) is big enough, and L(s, χ) has no real
+zero in the interval (1 − cχ, 1), then there exists a real constant a3 such that
+ea3
+ln q(χ)
+ln λ(x) x
+ln x
+�
+ln λ(x)
+�
+ln q(χ)
+≪ QP(x)
+(2)
+where cχ ≤ 1 is a χ dependant positive constant.
+We can see that for primitive χ, when q(χ) is big enough, and L(s, χ) doesn’t have a
+real zero close to 1, then we can cancel the ln q(χ) terms in the lower bounds by choosing
+a λ(x) which contains q(χ)ε, for ε > 0.
+Assuming the Riemann hypothesis for L(s, χ) we can drop the primitiveness, and the
+bounds are perturbed only by expressions containing ln ln q(χ) instead of ln q(χ).
+Proposition 1.2. Let χ be a real valued non-principal Dirichlet character. Then choose a
+function λ, and with it define the set P as in the case of proposition 1.1.
+If the Riemann hypothesis holds for L(s, χ), then there exist real constants a4 and a5
+such that
+ea4
+ln ln q(χ)
+ln λ(x)
+x
+ln x
+�
+ln λ(x)
+�
+ln ln q(χ)
+≪ QP(x) ≪ ea5
+ln ln q(χ)
+ln λ(x)
+x
+ln x
+�
+ln ln q(χ)
+�
+ln λ(x)
+(3)
+as x → +∞ and q(χ) → +∞.
+A natural extension of these results would be to only allow P to contain primes which
+are congruent to some mi modulo q, where q > 0 is an integer, and the m1, . . . , mk naturals
+are pairwise distinct relative primes to q. We are going to restrict ourselves to the case
+when q = 4; and either m = 1, or m = 3. Concerning the technicalities of this restriction
+see section 3.
+Proposition 1.3. Let χ be a real valued primitive non-principal Dirichlet character; and
+either let m = 1, or m = 3. Furthermore, let λ : R → [1, +∞) be a monotone increasing
+function which is in o(x1/2), and let P contain all those primes p which are not greater
+than λ(x), for which χ(p) = 1, and for which p ≡ m (mod 4) holds.
+
+On square-free numbers generated from given sets of primes II
+3
+When m = 1, then for every ε > 0, there exit real constants b1 and b2 such that
+eb1
+ln q(χ)
+ln λ(x) x
+ln x
+4�
+ln λ(x)
+�
+q(χ)ε ≪ QP(x) ≪ eb2
+ln q(χ)
+ln λ(x) x
+ln x
+�
+ln q(χ)
+4�
+ln λ(x)
+(4)
+as x → +∞. In addition, if q(χ) is big enough, and L(s, χ) has no real zero in the interval
+(1 − cχ, 1), then there exists a real constant b3 such that
+eb3
+ln q(χ)
+ln λ(x) x
+ln x
+4�
+ln λ(x)
+�
+ln q(χ)
+≪ QP(x)
+(5)
+where cχ ≤ 1 is a χ dependant positive constant.
+When m = 3, then for every ε > 0, there exit real constants b4 and b5 such that
+eb4
+ln q(χ)
+ln λ(x) x
+ln x
+4�
+ln λ(x)
+4�
+q(χ)ε ln q(χ)
+≪ QP(x) ≪ eb5
+ln q(χ)
+ln λ(x) x
+ln x
+4�
+q(χ)ε ln q(χ)
+4�
+ln λ(x)
+(6)
+as x → +∞. In addition, if q(χ) is big enough, L(s, χ) has no real zero in the interval
+(1 − cχ, 1), then there exists a real constant b6 such that
+QP(x) ≍ eb6
+ln q(χ)
+ln λ(x) x
+ln x
+4�
+ln λ(x)
+(7)
+where cχ ≤ 1 is a χ dependant positive constant.
+As in the previous case, we can get much better results assuming that the Riemann
+hypothesis holds for L(s, χ).
+Proposition 1.4. Let χ be a real valued primitive non-principal Dirichlet character; and
+either let m = 1, or m = 3. Then choose a function λ, and with it define the set P as in
+the case of proposition 1.3.
+If the Riemann hypothesis holds for L(s, χ), and when m = 1, then there exist real
+constants b7, and b8 such that
+eb7
+ln ln q(χ)
+ln λ(x)
+x
+ln x
+4�
+ln λ(x)
+�
+ln ln q(χ)
+≪ QP(x) ≪ eb8
+ln ln q(χ)
+ln λ(x)
+x
+ln x
+�
+ln ln q(χ)
+4�
+ln λ(x)
+(8)
+as x → +∞ and q(χ) → +∞.
+In the same setting, but with m = 3, there exists a real constant b9 such that
+QP(x) ≍ eb9
+ln ln q(χ)
+ln λ(x)
+x
+ln x
+4�
+ln λ(x)
+(9)
+as x → +∞ and q(χ) → +∞.
+
+4
+G´abor Rom´an
+2
+Proofs
+Throughout the proofs, when the index of a summation is p, or the index of a product
+is p, then p takes its values from the set of primes.
+Lemma 2.1. Let χ be a real valued non-principal Dirichlet character.
+Then we have
+�
+p≤y
+χ(p) ln
+�
+1 − 1
+p
+�
+= − ln L(1, χ) + O
+� 1
+ln y
+L′
+L (1, χ)
+�
++ O(1)
+as y → +∞.
+Proof. Fix a real valued non-principal Dirichlet character χ. We begin the proof by showing
+that
+�
+p≤y
+χ(p)
+p
+= ln L(1, χ) + O
+� 1
+ln y
+L′
+L (1, χ)
+�
++ O(1).
+(10)
+holds as y → +∞.
+• First we show that the equality
+�
+p
+χ(p)
+p
+= ln L(1, χ) + O(1)
+(11)
+holds. Based on [5, Sec. 5.9], when χ is a non-principal (or in their case non-trivial)
+character, then L(s, χ) is entire. By this, there is no pole at s = 1, so we have that
+the Euler product form
+L(1, χ) =
+�
+p
+�
+1 − χ(p)
+p
+�−1
+(12)
+see [5, Sec. 5.1, Sec. 5.9], or [2, Sec. 11.5], converges. For every non-principal
+Dirichlet character, L(1, χ) ̸= 0, see [2, Thm. 6.20, Lem. 7.7], and as χ is real
+valued, L(1, χ) is a positive real number, so we can take the logarithm of both sides
+of (12) to get
+ln L(1, χ) = −
+�
+p
+ln
+�
+1 − χ(p)
+p
+�
+=
+�
+p
+χ(p)
+p
++
+�
+p
+∞
+�
+k=2
+χ(p)k
+kpk
+where we could use the Mercator series, see [1, 4.1.24], as |χ(p)/p| < 1. Based on
+the sum of the geometric series, see [1, 3.6.10], we have
+�
+p
+∞
+�
+k=2
+����
+χ(p)k
+kpk
+���� ≤
+�
+p
+∞
+�
+k=2
+1
+pk =
+�
+p
+1
+p(p − 1) <
+�
+p
+1
+p2
+(13)
+which is finite.
+
+On square-free numbers generated from given sets of primes II
+5
+• Next we show that
+�
+y<p
+χ(p)
+p
+≪
+1
+ln y
+L′
+L (1, χ) + O(1)
+(14)
+holds as y → +∞. We show this by examining the expression
+�
+y<n≤z
+χ(n)Λ(n)
+n ln n
+(15)
+in two different ways. Here Λ is the Mangoldt function, see [2, Sec. 2.8].
+– Relying on the definition of the Mangoldt function, there exists a natural m
+such that expression (15) is equal to
+�
+y<p≤z
+χ(p)
+p
++
+m
+�
+k=2
+�
+y<pk≤z
+χ(pk)
+kpk
+=
+�
+y<p≤z
+χ(p)
+p
++ O(1)
+(16)
+where we can apply similar reasoning as in expression (13). Note that we can
+keep to positive infinity with z, the implied constant can be bounded.
+– Define
+S(t) :=
+�
+1≤n≤t
+χ(n)Λ(n)
+n
+.
+Using Abel’s identity, see [2, Thm. 4.2], we get that expression (15) is equal to
+S(z)
+ln z − S(y)
+ln y +
+� z
+y
+S(t)
+t(ln t)2 dt.
+(17)
+Based on [2, Sec. 7.5] we have
+S(t) =
+�
+p≤t
+χ(p) ln(p)
+p
++ O(1) ≪ L′
+L (1, χ) + O(1)
+as t → +∞, where the bound is due to [2, Lem. 7.5, Lem. 7.6]. Because of this,
+there exists a real threshold τ; furthermore there exist real constants η1 and η2
+such that for every t ≥ τ we have
+|S(t)| ≤ η1
+L′
+L (1, χ) + η2.
+Assuming that y ≥ τ holds, we can bound the integral in expression (17) by
+η1
+�L′
+L (1, χ) + η2
+� � z
+y
+1
+t(ln t)2 dt = η1
+�L′
+L (1, χ) + η2
+�� 1
+ln t
+�z
+y
+(18)
+
+6
+G´abor Rom´an
+furthermore there exist real constants η3 and η4 such that the remaining terms
+in expression (17) can be bounded by
+η3
+ln z
+L′
+L (1, χ) + η4
+ln y
+L′
+L (1, χ) + O(1).
+(19)
+Keeping to positive infinity with z in expression (18), and expression (19), we
+get that expression (17) can be bounded by some real constant times
+1
+ln y
+L′
+L (1, χ) + O(1).
+(20)
+Using the right hand side of equality (16) and expression (20) we get expression (14).
+Combining (11) and (14) we get equality (10). Because 0 < 1/p ≤ 1/2 for every prime p,
+we can use the Mercator series to write the sum in lemma 2.1 as
+−
+�
+p≤y
+χ(p)
+∞
+�
+k=1
+1
+kpk = −
+�
+p≤y
+χ(p)
+p
+−
+�
+p≤y
+∞
+�
+k=2
+χ(p)
+kpk .
+We can use equality (10) on the first sum; and relying on expression (13) the value of the
+double sum on the right hand side is in O(1).
+To prove proposition 1.1 and proposition 1.2 we introduce the following function.
+α(y) :=
+�
+p≤y
+χ(p)=1
+�
+1 −
+1
+p + 1
+�
+Lemma 2.2. Let χ be a real valued non-principal Dirichlet character.
+Then we have
+α(y) ≍
+1
+�
+L(1, χ)
+1
+√ln yeO
+�
+1
+ln y
+L′
+L (1,χ)
+�
+as y → +∞.
+Proof. Fix a real valued non-principal Dirichlet character χ. Observe that we can rewrite
+α(y) as
+�
+p≤y
+χ(p)=1
+�
+1 −
+1
+p + 1
+�
+=
+�
+p≤y
+χ(p)=1
+�
+1 − 1
+p2
+�−1 �
+p≤y
+χ(p)=1
+�
+1 − 1
+p
+�
+where the first product on the right hand side can be bounded by a small positive constant.
+Taking the logarithm of the second product on the right hand side we get
+�
+p≤y
+χ(p)=1
+ln
+�
+1 − 1
+p
+�
+= 1
+2
+�
+p≤y
+(1 + χ(p)) ln
+�
+1 − 1
+p
+�
+
+On square-free numbers generated from given sets of primes II
+7
+where we can split the finite sum on the right hand side, and use lemma 2.1 to get
+1
+2
+�
+p≤y
+ln
+�
+1 − 1
+p
+�
+− ln
+�
+L(1, χ) + O
+� 1
+ln y
+L′
+L (1, χ)
+�
++ O(1)
+as y → +∞. Via exponentiation, we get
+1
+�
+L(1, χ)
+1
+√ln yeO
+�
+1
+ln y
+L′
+L (1,χ)
+�
++O(1)
+where we have used [8, Thm. 7, Col.] stating that
+�
+p≤y
+�
+1 − 1
+p
+�
+≍
+1
+ln y
+for every y > 1.
+Now we prove proposition 1.1.
+Proof. Fix a real valued non-principal Dirichlet character χ, and select a function λ sat-
+isfying the requirements of proposition 1.1. According article [7], we have to bound the
+product
+�
+p≤x1/2
+�
+1 −
+1
+p + 1
+�
+α(λ(x))−1 ≍
+1
+ln x
+�
+L(1, χ)
+�
+ln λ(x)eO
+�
+1
+ln λ(x)
+L′
+L (1,χ)
+�
+(21)
+as x → +∞, where we have used [7, Lem. 2], and lemma 2.2. When χ is a non-principal
+character, then based on article [3] we have
+cεq(χ)−ε < L(1, χ) < ln q(χ)
+(22)
+where ε is any positive number and cε is a positive number depending on ε. Also, based
+on [5, Prop. 5.7] we have
+L′
+L (1, χ) ≪ ln q(χ)
+(23)
+where the implied constant being absolute. Using these bounds in expression (21) and the
+method from article [7] we can get the bounds in expression (1).
+Assuming that χ is primitive, q(χ) is big enough, and that there exists a positive
+constant cχ ≤ 1 such that L(s, χ) has no real zero in the interval (1 − cχ, 1), we have
+1
+ln q(χ) ≪ L(1, χ)
+(24)
+where the implied constant being positive and absolute, see article [4]. Using this bound
+in expression (21) and the method from article [7] we get the bound in expression (2).
+
+8
+G´abor Rom´an
+The proof of proposition 1.2 is the following.
+Proof. Fix a real valued non-principal Dirichlet character χ, and select a function λ sat-
+isfying the requirements of proposition 1.2. We are going to bound expression (21), but
+with bounds based on the Riemann hypothesis. If χ is a real valued non-principal Dirichlet
+character, and we assume that the Riemann hypothesis holds for L(s, χ), then based on
+[6, Thm. 1] we have
+1 + o(1)
+ε1 ln ln q(χ) < L(1, χ) < (1 + o(1))ε1 ln ln q(χ)
+(25)
+as q(χ) → ∞, where ε1 and ε2 are real constants.
+(As a side note, infinitely many
+real primitive characters χ satisfy these inequalities without assuming that the Riemann
+hypothesis holds for L(s, χ), see article [3].) In the same setting, based on [5, Thm. 5.17]
+we have
+L′
+L (1, χ) ≪ ln ln q(χ)
+(26)
+where the implied constant being absolute. Using these bounds in expression (21) and the
+method from article [7] we get the bounds in expression (3).
+To prove proposition 1.3 and proposition 1.4 we introduce the following function.
+βm(y) :=
+�
+p≤y
+p≡m(4)
+χ(p)=1
+�
+1 −
+1
+p + 1
+�
+Lemma 2.3. Let χ be a real valued non-principal Dirichlet character.
+Then we have
+β1(y) ≍
+1
+4�
+L(1, χ)L(1, χχ4,3)
+1
+4√ln yeO
+�
+1
+ln y
+L′
+L (1,χ)
+�
++O
+�
+1
+ln y
+L′
+L (1,χχ4,3)
+�
+(27)
+and
+β3(y) ≍
+4
+�
+L(1, χχ4,3)
+L(1, χ)
+1
+4√ln yeO
+�
+1
+ln y
+L′
+L (1,χ)
+�
++O
+�
+1
+ln y
+L′
+L (1,χχ4,3)
+�
+(28)
+as y → +∞, where χ4,3 is the non-principal Dirichlet character modulo 4.
+Proof. Fix a real valued non-principal Dirichlet character χ, and let either m = 1, or
+m = 3. We can rewrite βm(y) as
+�
+p≤y
+p≡m(4)
+χ(p)=1
+�
+1 −
+1
+p + 1
+�
+=
+�
+p≤y
+p≡m(4)
+χ(p)=1
+�
+1 − 1
+p2
+�−1 �
+p≤y
+p≡m(4)
+χ(p)=1
+�
+1 − 1
+p
+�
+
+On square-free numbers generated from given sets of primes II
+9
+where the first product on the right hand side can be bounded by a small positive constant.
+Taking the logarithm of the second product on the right hand side we get
+�
+p≤y
+p≡m(4)
+χ(p)=1
+ln
+�
+1 − 1
+p
+�
+= 1
+2
+�
+p≤y
+p≡m(4)
+(1 + χ(p)) ln
+�
+1 − 1
+p
+�
+where we can split the finite sum on the right hand side as
+1
+2
+�
+p≤y
+p≡m(4)
+ln
+�
+1 − 1
+p
+�
++ 1
+2
+�
+p≤y
+p≡m(4)
+χ(p) ln
+�
+1 − 1
+p
+�
+.
+(29)
+Based on [2, Thm. 6.16] we can write the second sum in expression (29) as
+1
+2
+�
+p≤y
+χ(p) ln
+�
+1 − 1
+p
+�
+1
+ϕ(4)
+�
+χ4
+χ4(p)χ4(m)
+(30)
+where the internal sum iterates through the ϕ(4) Dirichlet characters modulo 4. There
+are two Dirichlet characters modulo 4; we are going to denote them as χ4,1 (the principal
+character), and as χ4,3 (the non-principal character). Splitting the internal sum, we get
+χ4,1(m)
+4
+�
+p≤y
+χ(p)χ4,1(p) ln
+�
+1 − 1
+p
+�
++ χ4,3(m)
+4
+�
+p≤y
+χ(p)χ4,3(p) ln
+�
+1 − 1
+p
+�
+.
+(31)
+Concerning the sum on the left hand side of expression (31), as χ4,1(m) = 1; and as
+χ4,1(p) = 1 when (p, 4) = 1, otherwise χ4,1(p) = 0, we have
+1
+4
+�
+p≤y
+(p,4)=1
+χ(p) ln
+�
+1 − 1
+p
+�
+= 1
+4
+�
+p≤y
+χ(p) ln
+�
+1 − 1
+p
+�
++ O(1)
+where we can use lemma 2.1 to get
+−1
+4 ln L(1, χ) + O
+� 1
+ln y
+L′
+L (1, χ)
+�
++ O(1).
+Concerning the sum on the right hand side of expression (31), χχ4,3 is a real valued non-
+principal character, so we can use lemma 2.1 again to get
+−χ4,3(m)
+4
+ln L(1, χχ4,3) + O
+� 1
+ln y
+L′
+L (1, χχ4,3)
+�
++ O(1).
+Substituting these result in expression (29), and exponentiating, in the case when m = 1,
+we get expression (27) as χ4,3(1) = 1, and because
+�
+p≤y
+p≡m(4)
+�
+1 − 1
+p
+�
+≍
+1
+√ln y
+
+10
+G´abor Rom´an
+based on the article of Williams [11]. Similarly in the case when m = 3 we get expression
+(28) as χ4,3(3) = −1.
+Now we proof proposition 1.3.
+Proof. Fix a real valued primitive non-principal Dirichlet character χ; and either let m = 1,
+or m = 3. Furthermore select a function λ satisfying the requirements of proposition 1.3.
+Based on article [7], we have to bound the product
+�
+p≤x1/2
+�
+1 −
+1
+p + 1
+�
+βm(λ(x))−1
+(32)
+when m = 1, and separately when m = 3.
+When m = 1, then expression (32) is asymptotic to
+1
+ln x
+4�
+L(1, χ)L(1, χχ4,3)
+4�
+ln λ(x)eO
+�
+1
+ln y
+L′
+L (1,χ)
+�
++O
+�
+1
+ln y
+L′
+L (1,χχ4,3)
+�
+(33)
+as x → +∞, where we have used [7, Lem. 2] and lemma 2.3. Using the bounds from
+expression (22) and expression (23) we can bound expression (33) from below as
+1
+ln x
+4�
+ln λ(x)
+4�
+q(χ)εq(χχ4,3)εeO
+�
+1
+ln y (ln q(χ)+ln q(χχ4,3))
+�
+and as
+1
+ln x
+4�
+ln q(χ)
+4�
+ln q(χχ4,3)
+4�
+ln λ(x)eO
+�
+1
+ln y (ln q(χ)+ln q(χχ4,3))
+�
+from above. As χ and χ4,3 are both primitive, their product χχ4,3 is primitive too, see [10,
+Ch. 3]. But then q(χχ4,3) ∈ O(q(χ)), see [5, Sec. 3.3]. Based on this and on the method
+in article [7] we get the bounds in expression (4).
+Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such
+that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression
+(33) to get
+1
+ln x
+4�
+ln λ(x)
+4�
+ln q(χ) 4�
+ln q(χχ4,3)
+eO
+�
+1
+ln y (ln q(χ)+ln q(χχ4,3))
+�
+from where we can get bound (5) based on the previous train of thoughts.
+When m = 3, then expression (32) is asymptotic to
+1
+ln x
+4
+�
+L(1, χ)
+L(1, χχ4,3)
+4�
+ln λ(x)eO
+�
+1
+ln y
+L′
+L (1,χ)
+�
++O
+�
+1
+ln y
+L′
+L (1,χχ4,3)
+�
+(34)
+as x → +∞, where we have used [7, Lem. 2] and lemma 2.3 again. As in the previous
+case, we can bound expression (34) from below as
+1
+ln x
+4
+�
+q(χ)−ε
+ln q(χχ4,3)
+4�
+ln λ(x)eO
+�
+1
+ln y (ln q(χ)+ln q(χχ4,3))
+�
+
+On square-free numbers generated from given sets of primes II
+11
+and from above as
+1
+ln x
+4
+�
+ln q(χ)
+q(χχ4,3)−ε
+4�
+ln λ(x)eO
+�
+1
+ln y (ln q(χ)+ln q(χχ4,3))
+�
+from where we get the bounds in expression (6).
+Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such
+that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression
+(34).
+The logarithmic contribution of the terms L(1, χ) and L(1, χχ4,3) “cancel” each
+other, and we get asymptotic (7).
+And finally, the proof of proposition 1.4 is the following.
+Proof. Fix a real valued primitive non-principal Dirichlet character χ; and either let m = 1,
+or m = 3. We use the same method as in the proof of proposition 1.3, but this time we
+assume that the Riemann hypothesis holds for L(s, χ).
+When m = 1, then we can use the bounds from expression (25) and expression (26) to
+bound expression (33) from below as
+1
+ln x
+4�
+ln λ(x)
+4�
+ln ln q(χ) 4�
+ln ln q(χχ4,3)
+eO
+�
+1
+ln y (ln ln q(χ)+ln ln q(χχ4,3))
+�
+and from above as
+1
+ln x
+4�
+ln ln q(χ)
+4�
+ln ln q(χχ4,3)
+4�
+ln λ(x)eO
+�
+1
+ln y (ln ln q(χ)+ln ln q(χχ4,3))
+�
+as x → +∞ and as q(χ) → +∞. Using the same train of thought as in the proof of
+proposition 1.3, we get the bounds in expression (8).
+When m = 3, then using the above applied bounds (25) and (26) we get the asymptotic
+(9) from asymptotic (34) via “cancellation” again.
+3
+Remarks
+As we have already mentioned in section 1, before proposition 1.3, a natural extension
+of proposition 1.1 and proposition 1.2 would be to only allow P to contain primes which
+are congruent to some mi modulo q, where q > 0 is an integer, and the m1, . . . , mk naturals
+are pairwise distinct relative primes to q. In the case of modulo 4, the results were already
+different for distinct m, so we can expect a similar outcome for larger moduli. However a
+general strategy for the generalisation could go along the following train of thoughts. We
+would have to supply an asymptotic for the product
+�
+p≤y
+p≡m(q)
+χ(p)=1
+�
+1 −
+1
+p + 1
+�
+
+12
+G´abor Rom´an
+which could be done by the refinement of lemma 2.3, and its proof. If we follow this path,
+then expression (30) will turn into
+1
+2
+�
+p≤y
+χ(p) ln
+�
+1 − 1
+p
+� 1
+ϕ(q)
+�
+χq
+χq(p)χq(m)
+where we can separate the principal character from the non-principal ones as
+χq,1(m)
+ϕ(q)
+�
+p≤y
+χ(p)χq,1(p) ln
+�
+1 − 1
+p
+�
+and what remains is
+1
+ϕ(q)
+�
+χq̸=χq,1
+χq(m)
+�
+p≤y
+χ(p)χq(p) ln
+�
+1 − 1
+p
+�
+.
+Due to the fact that χq,1 is the principal character, the first sum can be handled with our
+already presented techniques. The double sum is more problematic. On the one hand, for
+the internal sum we would have to refine lemma 2.1 and its proof. We would have to make
+sure that when χ is complex valued, then we can take the logarithm of L(1, χ) and its
+product form; furthermore that the values of these two logarithms match. On the other
+hand, we would have to obtain a good estimation for the external sum.
+References
+[1] Abramowitz M. and Stegun I. A.: Handbook of Mathematical Functions with Formulas, Graphs,
+and Mathematical Tables. Dover publications (1972).
+[2] Apostol T. M.: Introduction to Analytic Number Theory. Springer-Verlag (1976).
+[3] Bateman P. T. and Chowla S. and Erd˝os P.: Remarks on the size of L(1, χ). Publ. Math.
+Debrecen 1 (2-4) (1950) 165–182.
+[4] Hoffstein J.: On the Siegel–Tatuzawa theorem. Acta. Arith. 38 (2) (1980) 168–174.
+[5] Iwaniec H. and Kowalski E.: Analytic Number Theory. A.M.S. Colloquium Publications (2004).
+[6] Littlewood J. E.: On the class-number of the corpus P(
+√
+−k). Proc. London Math. Soc. 27 (1)
+(1928) 358–372.
+[7] Rom´an G.: On square-free numbers generated from given sets of primes. Comm. Math. 30 (1)
+(2022) 229–237.
+[8] Rosser J. B. and Schoenfeld L.: Approximate formulas for some functions of prime numbers.
+Illinois Journal of Mathematics 6 (1) (1962) 64–94.
+[9] Walfisz A.: Zur additiven Zahlentheorie II.. Math. Z. 40 (1) (1936) 592–607.
+[10] Washington L. C.: Introduction to Cyclotomic Fields. Springer-Verlag (1982).
+[11] Williams K.S.: Mertens’ Theorem for Arithmetic Progressions. J. Number Theory 6 (5) (1974)
+353–359.
+Received: Received date
+Accepted for publication: Accepted date
+Communicated by: Handling Editor
+

1dE0T4oBgHgl3EQfdgBe/content/tmp_files/load_file.txt

@@ -0,0 +1,304 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf,len=303
+page_content='Communications in Mathematics n (2023), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' m, 00–12  DOI: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='46298/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='ABCD  ©2023 G´abor Rom´an  This is an open access article licensed under the CC BY-SA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='0  1  On square-free numbers generated from given sets of primes  II  G´abor Rom´an  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We progress with the investigation started in article [7], namely the anal-  ysis of the asymptotic behaviour of QP(x) for different sets P, where QP(x) is the  element count of the set containing those positive square-free integers, which are  smaller than-, or equal to x, and which are only divisible by the elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We  study how QP(x) behaves when we require that χ(p) = 1 must hold for every p ∈ P,  where χ is a Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  1  Introduction  Let’s take a set of prime numbers P, and denote with QP(x) the element count of the  set of all those positive square-free integers, which are smaller than-, or equal to x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and  which are only divisible by the elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  In article [7] we examined how QP(x) behaves asymptotically based on the structure  of P in two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' During the first scenario, P contained those primes which are not  greater than an x dependent bound λ(x), see [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' In the second scenario, P  contained those primes which are not greater than an x dependent bound λ(x), and which  fall into certain congruence classes modulo q, see [7, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  In this article, we go further, and render the structure of P more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We would  want to restrict ourselves to primes p, for which a certain integer a is quadratic residue  modulo p, but to generalise, we are going to require that χ(p) = 1 holds, where χ is a  Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' This covers our goal, as the only real valued primitive non-principal  χ(n) are given for positive n by the Kronecker symbol (D|n), where D is a fundamental  discriminant, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  MSC 2020: 11M06, 11M20, 11N36, 11N37, 11N69  Keywords: Square-free numbers, Combinatorial sieve, Dirichlet character, Square-free numbers in  arithmetic progressions, L-functions, Euler product  Affiliation:  G´abor Rom´an – E¨otv¨os Lor´and University, Budapest, Hungary  E-mail: rogpaai@inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='elte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='hu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='02377v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='NT]  6 Jan 2023  =P sciences2  G´abor Rom´an  The results will be of course similar to the results in article [7], heavily depending on  the conductor q(χ) of the Dirichlet character χ in context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Furthermore, let  λ : R → [1, +∞) be a monotone increasing function which is in o(x1/2), and let P contain  all the primes p which are not greater than λ(x), and for which χ(p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Then for every ε > 0, there exit real constants a1 and a2 such that  ea1  ln q(χ)  ln λ(x) x  ln x  �  ln λ(x)  �  q(χ)ε ≪ QP(x) ≪ ea2  ln q(χ)  ln λ(x) x  ln x  �  ln q(χ)  �  ln λ(x)  (1)  as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' In addition, if χ is primitive, q(χ) is big enough, and L(s, χ) has no real  zero in the interval (1 − cχ, 1), then there exists a real constant a3 such that  ea3  ln q(χ)  ln λ(x) x  ln x  �  ln λ(x)  �  ln q(χ)  ≪ QP(x)  (2)  where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  We can see that for primitive χ, when q(χ) is big enough, and L(s, χ) doesn’t have a  real zero close to 1, then we can cancel the ln q(χ) terms in the lower bounds by choosing  a λ(x) which contains q(χ)ε, for ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Assuming the Riemann hypothesis for L(s, χ) we can drop the primitiveness, and the  bounds are perturbed only by expressions containing ln ln q(χ) instead of ln q(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Then choose a  function λ, and with it define the set P as in the case of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  If the Riemann hypothesis holds for L(s, χ), then there exist real constants a4 and a5  such that  ea4  ln ln q(χ)  ln λ(x)  x  ln x  �  ln λ(x)  �  ln ln q(χ)  ≪ QP(x) ≪ ea5  ln ln q(χ)  ln λ(x)  x  ln x  �  ln ln q(χ)  �  ln λ(x)  (3)  as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  A natural extension of these results would be to only allow P to contain primes which  are congruent to some mi modulo q, where q > 0 is an integer, and the m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' , mk naturals  are pairwise distinct relative primes to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We are going to restrict ourselves to the case  when q = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and either m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Concerning the technicalities of this restriction  see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued primitive non-principal Dirichlet character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and  either let m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Furthermore, let λ : R → [1, +∞) be a monotone increasing  function which is in o(x1/2), and let P contain all those primes p which are not greater  than λ(x), for which χ(p) = 1, and for which p ≡ m (mod 4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  On square-free numbers generated from given sets of primes II  3  When m = 1, then for every ε > 0, there exit real constants b1 and b2 such that  eb1  ln q(χ)  ln λ(x) x  ln x  4�  ln λ(x)  �  q(χ)ε ≪ QP(x) ≪ eb2  ln q(χ)  ln λ(x) x  ln x  �  ln q(χ)  4�  ln λ(x)  (4)  as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' In addition, if q(χ) is big enough, and L(s, χ) has no real zero in the interval  (1 − cχ, 1), then there exists a real constant b3 such that  eb3  ln q(χ)  ln λ(x) x  ln x  4�  ln λ(x)  �  ln q(χ)  ≪ QP(x)  (5)  where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  When m = 3, then for every ε > 0, there exit real constants b4 and b5 such that  eb4  ln q(χ)  ln λ(x) x  ln x  4�  ln λ(x)  4�  q(χ)ε ln q(χ)  ≪ QP(x) ≪ eb5  ln q(χ)  ln λ(x) x  ln x  4�  q(χ)ε ln q(χ)  4�  ln λ(x)  (6)  as x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' In addition, if q(χ) is big enough, L(s, χ) has no real zero in the interval  (1 − cχ, 1), then there exists a real constant b6 such that  QP(x) ≍ eb6  ln q(χ)  ln λ(x) x  ln x  4�  ln λ(x)  (7)  where cχ ≤ 1 is a χ dependant positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  As in the previous case, we can get much better results assuming that the Riemann  hypothesis holds for L(s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued primitive non-principal Dirichlet character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and  either let m = 1, or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Then choose a function λ, and with it define the set P as in  the case of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  If the Riemann hypothesis holds for L(s, χ), and when m = 1, then there exist real  constants b7, and b8 such that  eb7  ln ln q(χ)  ln λ(x)  x  ln x  4�  ln λ(x)  �  ln ln q(χ)  ≪ QP(x) ≪ eb8  ln ln q(χ)  ln λ(x)  x  ln x  �  ln ln q(χ)  4�  ln λ(x)  (8)  as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  In the same setting, but with m = 3, there exists a real constant b9 such that  QP(x) ≍ eb9  ln ln q(χ)  ln λ(x)  x  ln x  4�  ln λ(x)  (9)  as x → +∞ and q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  4  G´abor Rom´an  2  Proofs  Throughout the proofs, when the index of a summation is p, or the index of a product  is p, then p takes its values from the set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Then we have  �  p≤y  χ(p) ln  �  1 − 1  p  �  = − ln L(1, χ) + O  � 1  ln y  L′  L (1, χ)  �  + O(1)  as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued non-principal Dirichlet character χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We begin the proof by showing  that  �  p≤y  χ(p)  p  = ln L(1, χ) + O  � 1  ln y  L′  L (1, χ)  �  + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (10)  holds as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  First we show that the equality  �  p  χ(p)  p  = ln L(1, χ) + O(1)  (11)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Based on [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='9], when χ is a non-principal (or in their case non-trivial)  character, then L(s, χ) is entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' By this, there is no pole at s = 1, so we have that  the Euler product form  L(1, χ) =  �  p  �  1 − χ(p)  p  �−1  (12)  see [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='9], or [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='5], converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' For every non-principal  Dirichlet character, L(1, χ) ̸= 0, see [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='20, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='7], and as χ is real  valued, L(1, χ) is a positive real number, so we can take the logarithm of both sides  of (12) to get  ln L(1, χ) = −  �  p  ln  �  1 − χ(p)  p  �  =  �  p  χ(p)  p  +  �  p  ∞  �  k=2  χ(p)k  kpk  where we could use the Mercator series, see [1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='24], as |χ(p)/p| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Based on  the sum of the geometric series, see [1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='10], we have  �  p  ∞  �  k=2  ����  χ(p)k  kpk  ���� ≤  �  p  ∞  �  k=2  1  pk =  �  p  1  p(p − 1) <  �  p  1  p2  (13)  which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  On square-free numbers generated from given sets of primes II  5  Next we show that  �  y<p  χ(p)  p  ≪  1  ln y  L′  L (1, χ) + O(1)  (14)  holds as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We show this by examining the expression  �  y<n≤z  χ(n)Λ(n)  n ln n  (15)  in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Here Λ is the Mangoldt function, see [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  – Relying on the definition of the Mangoldt function, there exists a natural m  such that expression (15) is equal to  �  y<p≤z  χ(p)  p  +  m  �  k=2  �  y<pk≤z  χ(pk)  kpk  =  �  y<p≤z  χ(p)  p  + O(1)  (16)  where we can apply similar reasoning as in expression (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Note that we can  keep to positive infinity with z, the implied constant can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  – Define  S(t) :=  �  1≤n≤t  χ(n)Λ(n)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Using Abel’s identity, see [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2], we get that expression (15) is equal to  S(z)  ln z − S(y)  ln y +  � z  y  S(t)  t(ln t)2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (17)  Based on [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='5] we have  S(t) =  �  p≤t  χ(p) ln(p)  p  + O(1) ≪ L′  L (1, χ) + O(1)  as t → +∞, where the bound is due to [2, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='5, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Because of this,  there exists a real threshold τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' furthermore there exist real constants η1 and η2  such that for every t ≥ τ we have  |S(t)| ≤ η1  L′  L (1, χ) + η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Assuming that y ≥ τ holds, we can bound the integral in expression (17) by  η1  �L′  L (1, χ) + η2  � � z  y  1  t(ln t)2 dt = η1  �L′  L (1, χ) + η2  �� 1  ln t  �z  y  (18)  6  G´abor Rom´an  furthermore there exist real constants η3 and η4 such that the remaining terms  in expression (17) can be bounded by  η3  ln z  L′  L (1, χ) + η4  ln y  L′  L (1, χ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (19)  Keeping to positive infinity with z in expression (18), and expression (19), we  get that expression (17) can be bounded by some real constant times  1  ln y  L′  L (1, χ) + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (20)  Using the right hand side of equality (16) and expression (20) we get expression (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Combining (11) and (14) we get equality (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Because 0 < 1/p ≤ 1/2 for every prime p,  we can use the Mercator series to write the sum in lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 as  −  �  p≤y  χ(p)  ∞  �  k=1  1  kpk = −  �  p≤y  χ(p)  p  −  �  p≤y  ∞  �  k=2  χ(p)  kpk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  We can use equality (10) on the first sum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and relying on expression (13) the value of the  double sum on the right hand side is in O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  To prove proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 and proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2 we introduce the following function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  α(y) :=  �  p≤y  χ(p)=1  �  1 −  1  p + 1  �  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Then we have  α(y) ≍  1  �  L(1, χ)  1  √ln yeO  �  1  ln y  L′  L (1,χ)  �  as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued non-principal Dirichlet character χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Observe that we can rewrite  α(y) as  �  p≤y  χ(p)=1  �  1 −  1  p + 1  �  =  �  p≤y  χ(p)=1  �  1 − 1  p2  �−1 �  p≤y  χ(p)=1  �  1 − 1  p  �  where the first product on the right hand side can be bounded by a small positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Taking the logarithm of the second product on the right hand side we get  �  p≤y  χ(p)=1  ln  �  1 − 1  p  �  = 1  2  �  p≤y  (1 + χ(p)) ln  �  1 − 1  p  �  On square-free numbers generated from given sets of primes II  7  where we can split the finite sum on the right hand side, and use lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 to get  1  2  �  p≤y  ln  �  1 − 1  p  �  − ln  �  L(1, χ) + O  � 1  ln y  L′  L (1, χ)  �  + O(1)  as y → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Via exponentiation, we get  1  �  L(1, χ)  1  √ln yeO  �  1  ln y  L′  L (1,χ)  �  +O(1)  where we have used [8, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 7, Col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='] stating that  �  p≤y  �  1 − 1  p  �  ≍  1  ln y  for every y > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Now we prove proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued non-principal Dirichlet character χ, and select a function λ sat-  isfying the requirements of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' According article [7], we have to bound the  product  �  p≤x1/2  �  1 −  1  p + 1  �  α(λ(x))−1 ≍  1  ln x  �  L(1, χ)  �  ln λ(x)eO  �  1  ln λ(x)  L′  L (1,χ)  �  (21)  as x → +∞, where we have used [7, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 2], and lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' When χ is a non-principal  character, then based on article [3] we have  cεq(χ)−ε < L(1, χ) < ln q(χ)  (22)  where ε is any positive number and cε is a positive number depending on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Also, based  on [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='7] we have  L′  L (1, χ) ≪ ln q(χ)  (23)  where the implied constant being absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Using these bounds in expression (21) and the  method from article [7] we can get the bounds in expression (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Assuming that χ is primitive, q(χ) is big enough, and that there exists a positive  constant cχ ≤ 1 such that L(s, χ) has no real zero in the interval (1 − cχ, 1), we have  1  ln q(χ) ≪ L(1, χ)  (24)  where the implied constant being positive and absolute, see article [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Using this bound  in expression (21) and the method from article [7] we get the bound in expression (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  8  G´abor Rom´an  The proof of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued non-principal Dirichlet character χ, and select a function λ sat-  isfying the requirements of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We are going to bound expression (21), but  with bounds based on the Riemann hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' If χ is a real valued non-principal Dirichlet  character, and we assume that the Riemann hypothesis holds for L(s, χ), then based on  [6, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 1] we have  1 + o(1)  ε1 ln ln q(χ) < L(1, χ) < (1 + o(1))ε1 ln ln q(χ)  (25)  as q(χ) → ∞, where ε1 and ε2 are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (As a side note, infinitely many  real primitive characters χ satisfy these inequalities without assuming that the Riemann  hypothesis holds for L(s, χ), see article [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=') In the same setting, based on [5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='17]  we have  L′  L (1, χ) ≪ ln ln q(χ)  (26)  where the implied constant being absolute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Using these bounds in expression (21) and the  method from article [7] we get the bounds in expression (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  To prove proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3 and proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='4 we introduce the following function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  βm(y) :=  �  p≤y  p≡m(4)  χ(p)=1  �  1 −  1  p + 1  �  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Let χ be a real valued non-principal Dirichlet character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Then we have  β1(y) ≍  1  4�  L(1, χ)L(1, χχ4,3)  1  4√ln yeO  �  1  ln y  L′  L (1,χ)  �  +O  �  1  ln y  L′  L (1,χχ4,3)  �  (27)  and  β3(y) ≍  4  �  L(1, χχ4,3)  L(1, χ)  1  4√ln yeO  �  1  ln y  L′  L (1,χ)  �  +O  �  1  ln y  L′  L (1,χχ4,3)  �  (28)  as y → +∞, where χ4,3 is the non-principal Dirichlet character modulo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued non-principal Dirichlet character χ, and let either m = 1, or  m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We can rewrite βm(y) as  �  p≤y  p≡m(4)  χ(p)=1  �  1 −  1  p + 1  �  =  �  p≤y  p≡m(4)  χ(p)=1  �  1 − 1  p2  �−1 �  p≤y  p≡m(4)  χ(p)=1  �  1 − 1  p  �  On square-free numbers generated from given sets of primes II  9  where the first product on the right hand side can be bounded by a small positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Taking the logarithm of the second product on the right hand side we get  �  p≤y  p≡m(4)  χ(p)=1  ln  �  1 − 1  p  �  = 1  2  �  p≤y  p≡m(4)  (1 + χ(p)) ln  �  1 − 1  p  �  where we can split the finite sum on the right hand side as  1  2  �  p≤y  p≡m(4)  ln  �  1 − 1  p  �  + 1  2  �  p≤y  p≡m(4)  χ(p) ln  �  1 − 1  p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (29)  Based on [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='16] we can write the second sum in expression (29) as  1  2  �  p≤y  χ(p) ln  �  1 − 1  p  �  1  ϕ(4)  �  χ4  χ4(p)χ4(m)  (30)  where the internal sum iterates through the ϕ(4) Dirichlet characters modulo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' There  are two Dirichlet characters modulo 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' we are going to denote them as χ4,1 (the principal  character), and as χ4,3 (the non-principal character).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Splitting the internal sum, we get  χ4,1(m)  4  �  p≤y  χ(p)χ4,1(p) ln  �  1 − 1  p  �  + χ4,3(m)  4  �  p≤y  χ(p)χ4,3(p) ln  �  1 − 1  p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  (31)  Concerning the sum on the left hand side of expression (31), as χ4,1(m) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and as  χ4,1(p) = 1 when (p, 4) = 1, otherwise χ4,1(p) = 0, we have  1  4  �  p≤y  (p,4)=1  χ(p) ln  �  1 − 1  p  �  = 1  4  �  p≤y  χ(p) ln  �  1 − 1  p  �  + O(1)  where we can use lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 to get  −1  4 ln L(1, χ) + O  � 1  ln y  L′  L (1, χ)  �  + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Concerning the sum on the right hand side of expression (31), χχ4,3 is a real valued non-  principal character, so we can use lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 again to get  −χ4,3(m)  4  ln L(1, χχ4,3) + O  � 1  ln y  L′  L (1, χχ4,3)  �  + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Substituting these result in expression (29), and exponentiating, in the case when m = 1,  we get expression (27) as χ4,3(1) = 1, and because  �  p≤y  p≡m(4)  �  1 − 1  p  �  ≍  1  √ln y  10  G´abor Rom´an  based on the article of Williams [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Similarly in the case when m = 3 we get expression  (28) as χ4,3(3) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Now we proof proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued primitive non-principal Dirichlet character χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and either let m = 1,  or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Furthermore select a function λ satisfying the requirements of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Based on article [7], we have to bound the product  �  p≤x1/2  �  1 −  1  p + 1  �  βm(λ(x))−1  (32)  when m = 1, and separately when m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  When m = 1, then expression (32) is asymptotic to  1  ln x  4�  L(1, χ)L(1, χχ4,3)  4�  ln λ(x)eO  �  1  ln y  L′  L (1,χ)  �  +O  �  1  ln y  L′  L (1,χχ4,3)  �  (33)  as x → +∞, where we have used [7, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 2] and lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Using the bounds from  expression (22) and expression (23) we can bound expression (33) from below as  1  ln x  4�  ln λ(x)  4�  q(χ)εq(χχ4,3)εeO  �  1  ln y (ln q(χ)+ln q(χχ4,3))  �  and as  1  ln x  4�  ln q(χ)  4�  ln q(χχ4,3)  4�  ln λ(x)eO  �  1  ln y (ln q(χ)+ln q(χχ4,3))  �  from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' As χ and χ4,3 are both primitive, their product χχ4,3 is primitive too, see [10,  Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' But then q(χχ4,3) ∈ O(q(χ)), see [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Based on this and on the method  in article [7] we get the bounds in expression (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such  that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression  (33) to get  1  ln x  4�  ln λ(x)  4�  ln q(χ) 4�  ln q(χχ4,3)  eO  �  1  ln y (ln q(χ)+ln q(χχ4,3))  �  from where we can get bound (5) based on the previous train of thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  When m = 3, then expression (32) is asymptotic to  1  ln x  4  �  L(1, χ)  L(1, χχ4,3)  4�  ln λ(x)eO  �  1  ln y  L′  L (1,χ)  �  +O  �  1  ln y  L′  L (1,χχ4,3)  �  (34)  as x → +∞, where we have used [7, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 2] and lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' As in the previous  case, we can bound expression (34) from below as  1  ln x  4  �  q(χ)−ε  ln q(χχ4,3)  4�  ln λ(x)eO  �  1  ln y (ln q(χ)+ln q(χχ4,3))  �  On square-free numbers generated from given sets of primes II  11  and from above as  1  ln x  4  �  ln q(χ)  q(χχ4,3)−ε  4�  ln λ(x)eO  �  1  ln y (ln q(χ)+ln q(χχ4,3))  �  from where we get the bounds in expression (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Assuming that q(χ) is big enough, and that there exists a positive constant cχ ≤ 1 such  that L(s, χ) has no real zero in the interval (1−cχ, 1), we can use bound (24) in expression  (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  The logarithmic contribution of the terms L(1, χ) and L(1, χχ4,3) “cancel” each  other, and we get asymptotic (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  And finally, the proof of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='4 is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Fix a real valued primitive non-principal Dirichlet character χ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and either let m = 1,  or m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We use the same method as in the proof of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3, but this time we  assume that the Riemann hypothesis holds for L(s, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  When m = 1, then we can use the bounds from expression (25) and expression (26) to  bound expression (33) from below as  1  ln x  4�  ln λ(x)  4�  ln ln q(χ) 4�  ln ln q(χχ4,3)  eO  �  1  ln y (ln ln q(χ)+ln ln q(χχ4,3))  �  and from above as  1  ln x  4�  ln ln q(χ)  4�  ln ln q(χχ4,3)  4�  ln λ(x)eO  �  1  ln y (ln ln q(χ)+ln ln q(χχ4,3))  �  as x → +∞ and as q(χ) → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Using the same train of thought as in the proof of  proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3, we get the bounds in expression (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  When m = 3, then using the above applied bounds (25) and (26) we get the asymptotic  (9) from asymptotic (34) via “cancellation” again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  3  Remarks  As we have already mentioned in section 1, before proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3, a natural extension  of proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 and proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='2 would be to only allow P to contain primes which  are congruent to some mi modulo q, where q > 0 is an integer, and the m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' , mk naturals  are pairwise distinct relative primes to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' In the case of modulo 4, the results were already  different for distinct m, so we can expect a similar outcome for larger moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' However a  general strategy for the generalisation could go along the following train of thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We  would have to supply an asymptotic for the product  �  p≤y  p≡m(q)  χ(p)=1  �  1 −  1  p + 1  �  12  G´abor Rom´an  which could be done by the refinement of lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='3, and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' If we follow this path,  then expression (30) will turn into  1  2  �  p≤y  χ(p) ln  �  1 − 1  p  � 1  ϕ(q)  �  χq  χq(p)χq(m)  where we can separate the principal character from the non-principal ones as  χq,1(m)  ϕ(q)  �  p≤y  χ(p)χq,1(p) ln  �  1 − 1  p  �  and what remains is  1  ϕ(q)  �  χq̸=χq,1  χq(m)  �  p≤y  χ(p)χq(p) ln  �  1 − 1  p  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Due to the fact that χq,1 is the principal character, the first sum can be handled with our  already presented techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' The double sum is more problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' On the one hand, for  the internal sum we would have to refine lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='1 and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' We would have to make  sure that when χ is complex valued, then we can take the logarithm of L(1, χ) and its  product form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' furthermore that the values of these two logarithms match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' On the other  hand, we would have to obtain a good estimation for the external sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  References  [1] Abramowitz M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and Stegun I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': Handbook of Mathematical Functions with Formulas, Graphs,  and Mathematical Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Dover publications (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=': Introduction to Analytic Number Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Springer-Verlag (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=' and Erd˝os P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': Remarks on the size of L(1, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Debrecen 1 (2-4) (1950) 165–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  [4] Hoffstein J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': On the Siegel–Tatuzawa theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Acta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Arith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content='  [5] Iwaniec H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' and Kowalski E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': Analytic Number Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=' Colloquium Publications (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=': On the class-number of the corpus P(  √  −k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content='  [7] Rom´an G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': On square-free numbers generated from given sets of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content='  [8] Rosser J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=': Approximate formulas for some functions of prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Illinois Journal of Mathematics 6 (1) (1962) 64–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
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+page_content=': Zur additiven Zahlentheorie II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='. Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' 40 (1) (1936) 592–607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  [10] Washington L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=': Introduction to Cyclotomic Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Springer-Verlag (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  [11] Williams K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' : Mertens’ Theorem for Arithmetic Progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content=' Number Theory 6 (5) (1974)  353–359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}
+page_content='  Received: Received date  Accepted for publication: Accepted date  Communicated by: Handling Editor' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE0T4oBgHgl3EQfdgBe/content/2301.02377v1.pdf'}

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+A Global Inventory of Feedback 
+Timothy M. Heckman 
+The William H. Miller III Department of Physics and Astronomy, The Johns Hopkins University, 
+Baltimore, MD 21218, USA 
+ 
+Philip N. Best 
+Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, 
+EH9 3HJ, UK 
+ 
+Abstract 
+Feedback from both supermassive black holes and massive stars plays a fundamental role in the 
+evolution of galaxies and the inter-galactic medium. In this paper we use available data to 
+estimate the total amount of kinetic energy and momentum created per co-moving volume 
+element over the history of the universe from three sources: massive stars and supernovae, 
+radiation pressure and winds driven by supermassive black holes, and radio jets driven by 
+supermassive black holes. Kinetic energy and momentum injection from jets peaks at z ≈ 1, 
+while the other two sources peak at z ≈ 2. Massive stars are the dominant global source of 
+momentum injection. For supermassive black holes, we find that the amount of kinetic energy 
+from jets is about an order-of-magnitude larger than that from winds. We also find that amount 
+of kinetic energy created by massive stars is about 2.5 εstar times that carried by jets (where εstar 
+is the fraction of injected energy not lost to radiative cooling). We discuss the implications of 
+these results for the evolution of galaxies and the IGM. Because the ratio of black hole mass to 
+galaxy mass is a steeply increasing function of mass, we show that the relative importance of 
+black hole feedback to stellar feedback likewise increases with mass. We show that there is a 
+trend in the present-day universe which, in the simplest picture, is consistent with galaxies that 
+have been dominated by black hole feedback being generally quenched, while galaxies that 
+have been dominated by stellar feedback are star-forming. We also note that the amount of 
+kinetic energy carried by jets and winds appears sufficient to explain the properties of hot gas 
+in massive halos (> 1013 Mʘ). 
+1. Introduction 
+The basic properties of galaxies, supermassive black holes, and the intra-group/intra-cluster 
+medium cannot be understood without considering the impact of the return of mass, metals, 
+energy, and momentum from both populations of massive stars (stellar winds and supernovae) 
+and supermassive black holes (winds and jets). Examples include the shape of the stellar mass 
+function, the quenching and subsequent suppression of star-formation in massive galaxies, the 
+mass-metallicity and mass-radius relations, the Kennicutt-Schmidt law of star-formation, and 
+the group/cluster X-ray luminosity-temperature relation (see reviews by Somerville & Davé 
+2015, Naab & Ostriker 2017, Donahue & Voit 2022).  
+
+This input of energy and momentum from massive stars and black holes is generically referred 
+to as feedback. Even the highest resolution numerical simulations cannot fully include all the 
+relevant physics ab-initio, and must rely on “sub-grid physics” (essentially, recipes for processes 
+that cannot be spatially resolved). The same is true of semi-analytic models. This underscores 
+the importance of using observations to inform the choices that are made in simulations and 
+models. While there is now a considerable body of data on feedback from both massive stars 
+and supermassive back holes (e.g. Veilleux et al. 2020; McNamara & Nulsen 2007; Thompson & 
+Heckman 2023), we still have a very incomplete understanding of the impact of this feedback 
+on the surrounding gas. 
+In this paper, we will take a different approach from previous investigations of feedback, and 
+try to compile a global inventory (that is, integrated over cosmic time) of the amount of kinetic 
+energy and momentum per co-moving volume element injected by massive stars and 
+supermassive black holes. We will then compare the respective importance of these feedback 
+sources as a function of time and of galaxy and black hole mass. 
+2. Methodology 
+2.1 Massive Stars 
+To compute the total amount of kinetic energy injected by massive stars (stellar winds and 
+supernovae) per unit volume, we start with the present-day amount of stellar mass per unit 
+volume. We use the compilation in Madau & Dickinson (2016), adjusted to a standard Chabrier 
+Initial Mass Function (IMF - Chabrier, 2003). This value is 3.4 x 108 Mʘ Mpc-3. To compute the 
+corresponding amount of kinetic energy we need to first correct this to account for stars that 
+were formed but are no longer present. This requires multiplication by 1/(1-R), where R is the 
+so-called Returned Fraction, which is 0.3 for a Chabrier IMF.  Thus, the total mass of stars 
+formed per unit volume is 4.9 x 108 Mʘ Mpc-3. Starburst99 (Leitherer et al. 1999) models for a 
+Chabrier IMF yield a total kinetic energy in stellar winds and supernova ejecta of 6.9 x 1015 erg 
+gm-1. This then gives a value for the kinetic energy density due to stars of Ustar = 6.8 x 1057 erg 
+Mpc-3.  
+How much of this kinetic energy is available to supply feedback? The stellar ejecta initially 
+carrying the energy collide and their kinetic energy is converted to thermal energy. This hot gas  
+can then expand and flow outward with the thermal energy being converted back into kinetic 
+energy (e.g. Chevalier & Clegg 1985). Some of the initial thermal energy can be lost through 
+radiative cooling, so that only a fraction εstar remains to provide feedback. Numerical 
+simulations that represent typical conditions in low-z star-forming galaxies yield εstar ≈ 0.1 (Kim 
+et al. 2020), with a value that increases with the star-formation rate per unit area (SFR/A). At 
+the much higher values of SFR/A seen in starbursts (e.g. Kennicutt & Evans 2012), simulations 
+and models predict far greater efficiency, with εstar ≈ 0.3 to 1.0 (Schneider et al. 2020; Fielding & 
+Bryan 2022). This is consistent with X-ray observations of the H-like and He-like Fe Kα emission-
+lines in starburst galaxies from the very hot (108 K) gas created as the stellar ejecta are 
+thermalized through shocks (Thompson & Heckman 2023). These results imply that rather little 
+
+of the initial kinetic energy is lost through radiative cooling, and this is substantiated by 
+estimates of the rate of PΔV work done by the wind on the ambient gas (Thompson & Heckman 
+2023). While there are no such constraints on galaxies at high (z > 1) redshift, we do know that 
+these galaxies have values of SFR/A similar to those seen in low-z starbursts (e.g. Forster-
+Schreiber & Wuyts 2020), and that galactic winds driven by massive stars at this epoch are both 
+ubiquitous and very similar to those seen in low-z starburst galaxies (see Thompson & Heckman 
+2023). Note that roughly 60% of the total present-day stellar mass was formed at z > 1, during 
+this “windy” epoch (Madau & Dickinson 2016). 
+The situation for momentum injection is less uncertain because momentum will be conserved 
+even in the face of significant radiative losses. We can simply use the methodology above but 
+use Starburst99 to compute the specific injection rate of momentum by massive stars 
+(supernovae, stellar winds, and radiation pressure). The value is 7.4 x 107 cm s-1, and for a total 
+stellar mass density of 4.8 x 108 Mʘ Mpc-3, this yields 7.1 x 1049 gm cm s-1 Mpc-3.  
+2.2 Black-Hole Driven Winds and Radiation Pressure 
+Winds driven by supermassive black holes are multi-phase and have been measured in a 
+number of different ways. Molecular outflows have been detected in both emission and 
+absorption (see the review by Veilleux et al. 2020). Calculating kinetic energy outflow rates is 
+conceptually straightforward. The luminosity of a CO transition can be converted into a total 
+molecular gas mass, albeit with uncertainties (Tacconi et al. 2020). The measured outflow 
+velocity and the radius of the outflow then yields a kinetic energy flux given as ½ Mgas vout3 rout-1. 
+For absorption, the OH column density and outflow velocity yields an outflow rate (for an 
+assumed outflow size and OH/H2 conversion factor). The first compilation of molecular outflows 
+by Fiore et al. (2017) implied typical kinetic energy fluxes of dEwind/dt ≈ 3% LBol, however more 
+recent compilations of measurements (Lutz et al. 2020, Lamperti et al. 2022 and private 
+communication) have yielded much smaller values (median of 0.1%). 
+Similarly, the outflow rates of warm ionized gas can be measured using the Hα or [OIII]5007 
+luminosity and measured electron density to derive the total mass of ionized gas and then 
+measuring the outflow velocity and radius of the outflow to determine dEwind/dt. Different 
+recent measurements have come to drastically different results, with median values ranging 
+from as high as 1% of Lbol (Kakkad et al. 2022) to 0.3 % (Fiore et al. 2017), to 0.1% (Revalski et al. 
+2021), to 0.01% (Dall’Agnol de Oliveira 2021), to 0.0003% (Trindade Falcao et al. 2021). 
+An independent measurement of the outflow rate in the warm ionized gas comes from 
+observations of BAL QSOs. Here, the absorption-lines can provide a column density and outflow 
+velocity. Direct measurements of the electron densities can be made using the ratio of column 
+densities in lines arising from an excited state vs. the ground state. Photoionization models 
+using the observed ionizing luminosity (Q) and the inferred value of the ionization parameter 
+(U) then yield a size for the outflow: rout = (Q/4π ne c U)1/2 (Miller et al. 2020). With a velocity, 
+radius, and column density, the kinetic energy flux can be estimated. The results span a huge 
+
+range, from 0.001% to 10% Lbol (median value of 0.3%). Highly ionized outflows are also 
+detected in about 40% of AGN (Tombesi et al. 2011) based on X-ray absorption-lines. However, 
+because the size scales of these outflows are so uncertain, the kinetic energy outflow rates are 
+also uncertain (by about two orders-of-magnitude, typically ranging between 0.01 and 1% of 
+LBol – Tombesi et al. 2012). 
+It is clear from the above that assigning a value for the ratio of dEwind/dt to Lbol is difficult. If we 
+take the median values of 0.1%, 0.3%, and 0.1% LBol for the molecular, warm-ionized, and 
+highly-ionized phases, we get a total value of 0.5% LBol. Multiplying this by the total bolometric 
+energy density per co-moving volume element volume produced by supermassive black holes 
+of Urad = 8.6 x 1058 erg Mpc-3 (Hopkins et al. 2007), yields Uwind = 4.3 x 1056 ergs Mpc-3.  This is 
+6% as large as the value derived for massive stars. Using the present-day mass per unit volume 
+in supermassive black holes of ρBH = 5 x 105 Mʘ Mpc-3 (Hopkins et al. 2007) this wind energy 
+density can also be expressed as Uwind = 5 x 10-4 ρBH c2. 
+We can also consider the amount of momentum provided by AGN. An initial estimate is  
+implied by the momentum carried by radiation (Urad/c) where Urad is the total amount of radiant 
+energy per unit volume produced over cosmic time by AGN. This yields an amount of 
+momentum per unit volume of 2.9 x 1048 gm cm s-1 Mpc-3 (about 4% of the value for massive 
+stars). Since the momentum flux (in the non-relativistic case) is just twice the kinetic energy flux 
+divided by the outflow velocity, we need only consider the momentum carried by the molecular 
+and warm ionized flows (since the BAL QSO and X-ray outflows are over an order-of-magnitude 
+faster, but carry similar kinetic energy fluxes). 
+For the molecular outflows, the data in Lutz (2020) and Lamperti et al. (2022 and private 
+communication) yield median values of dpwind/dt = 1.0 and 0.7 LBol/c respectively. The near 
+equality is consistent with the idea that the molecular outflows are driven by radiation 
+pressure. If so, then combining radiation pressure and the molecular outflows would be double-
+counting in the inventory of momentum. 
+As noted above, there is a very wide range in the ratio between the kinetic energy flux in the 
+warm ionized gas and the AGN bolometric luminosity, and this translates directly into 
+uncertainties in the ratio of momentum flux and radiation pressure for this gas phase. 
+Estimated median values of this ratio range from ≈10 (Kakkad et al. 2022), to ≈1 (Fiore et al. 
+2017; Revalski et al. 2021), to ≈0.1 (Dall’Agnol de Oliveira et al. 2021), to ≈0.01 (Trindade Falcao 
+et al. 2021). It appears that the momentum flux in the warm ionized outflows is not likely to be 
+significantly larger than those in the molecular gas or to that carried by radiation. 
+This represents a total injected momentum per unit volume of at most ≈1049 gm cm s-1 Mpc-3, 
+even if we simply add the three sources (radiation, molecular gas, ionized gas) together. This is 
+still an order of magnitude below the value for massive stars. 
+ 
+
+2.3 Black Hole-Driven Jets 
+The earliest evidence for the outflow of kinetic energy driven by supermassive black holes came 
+from observations of “double lobes” of synchrotron radio emission that straddled massive 
+elliptical galaxies (Baade & Minkowski 1954). Subsequent radio observations at high angular 
+resolution showed narrow collimated features (“jets”) linking the two lobes to the galactic 
+nucleus (see Miley 1980). 
+It is now possible to quantify the amount of kinetic energy carried by jets as a function of the 
+luminosity of the radio source that they power. This can be done by joint observations of the 
+radio and X-ray emission. The expanding radio sources inflate lobes of relativistic plasma, which 
+in X-rays can be observed as cavities in the surrounding hot gas. Bırzan et al. (2004,2008), Dunn 
+et al. (2005), Rafferty et al. (2006), and Cavagnolo et al. (2010) derived the pΔV work (energy) 
+associated with the cavities in a sample of massive galaxies, groups, and clusters, and used the 
+buoyancy timescale (e.g. Churazov et al. 2001) to estimate their ages. They combined these 
+cavity powers with the monochromatic 1.4 GHz radio luminosities to show that the two were 
+well-correlated. The largest uncertainty in this method is the determination of the cavity energy 
+from the measured pressure and volume: Ecav = fcavpΔV . For the relativistic plasma of the radio 
+lobes the enthalpy of the cavity is 4pΔV. Taking fcav = 4, Heckman & Best (2014) derived the 
+following best-fit relation from the cavity data: 
+1) dEjet/dt  =  1.3 × 1038 (L1.4GHz/1026 W Hz−1)0.68  W 
+This empirical relation is very similar to predictions from theoretical models of radio jets. 
+Willett et al. (1999) used synchrotron properties to derive the relation: 
+2) dEjet/dt = 2.8 × 1036 (fW)3/2 (L1.4GHz/1026 W Hz−1)0.84 W 
+Here fW is a dimensionless factor (in the range 1 to 20) accounting for the uncertainties in the 
+extrapolation from the population of relativistic electrons that produce the observed radio 
+synchrotron emission to the total energy. Agreement with the X-ray cavity data implies fW ≈10 
+to 20 (see Heckman & Best 2014). 
+We adopt the theoretical relation (equation 2), but calibrated by the cavity data (i.e. taking fW = 
+15) and use this to convert the radio luminosity function of AGN between z = 0.1 and 3 (Yuan et 
+al. 2017) into a measure of the evolution in the rate of kinetic energy injection per unit volume 
+by radio jets.  
+The results are shown in Figure 1, and show that the peak rate of kinetic energy injection by jets 
+occurs at a significantly lower redshift (z ≈ 1) than the peak rate due to massive stars and black-
+hole-driven winds (z ≈ 2, as also shown in Figure 1). We then integrate the energy injection rate 
+by interpolating the values at z = 0.1, 0.5, 1.0, 2.0, and 3.0 and extrapolating from z = 0.1 to 0 
+and from z = 3.0 to infinity (this extrapolation does not add significantly to the total - see Figure 
+1). This then gives a value for the time-integrated total kinetic energy per unit volume due to 
+
+jets of Ujet = 2.6 x 1057 erg Mpc-3. These results are broadly in line with similar estimates derived 
+from low-frequency radio luminosity functions (Kondapally et al., private communication). 
+The time-integrated kinetic energy input from jets is ≈6 times larger than the value estimated 
+about for black-hole-driven winds, and 40% (400%) the total amount of kinetic energy 
+generated by massive stars for εstar = 1 (0.1). Alternatively, using the present-day mass per unit 
+volume in supermassive black holes of ρBH = 5 x 105 Mʘ (Hopkins et al. 2007) this jet energy 
+density can also be expressed as Ujet = 2.9 x 10-3 ρBH c2.  
+The kinetic energy carried by jets is in the form of relativistic bulk motion. In this case, the 
+momentum can be taken as p ≈ KE/c. The above value of Ujet then implies a momentum density 
+of 8.7 x 1046 gm cm s-1 Mpc-3. This is much less than the momentum carried by radiation and 
+winds from supermassive black holes, and the momentum produced by massive stars. Jets are 
+therefore far more important feedback sources in terms of kinetic energy than momentum. 
+2.4 The Bottom Line 
+For total kinetic energy inventory, the largest single source is either massive stars (for εstar > 0.4) 
+or jets (for εstar < 0.4). AGN winds are only important at the <10% level. For the total 
+momentum inventory, massive stars dominate (AGN contribute at the ≈10% level). The peak 
+rate of kinetic energy injection by jets occurs at a substantially lower redshift than that from 
+stars or AGN winds (z ≈ 1 and 2, respectively). These results are summarized in Table 1 and 
+Figure 1. 
+______________________________________________________________________________ 
+Table 1 – Summary of Feedback Inventory 
+1 
+2 
+3 
+4 
+5 
+6 
+Sample 
+Log ρ 
+Log sKE 
+Log ρKE 
+Log sp 
+ρp 
+Massive Stars 
+8.69 
+-5.11 
+57.83 
+7.87 
+49.85 
+BH Winds 
+5.70 
+-3.30 
+56.63 
+10.00 
+49.00 
+BH Jets 
+5.70 
+-2.54 
+57.43 
+7.94 
+46.94 
+______________________________________________________________________________ 
+Notes: 
+Column 2 – The log of the present-day mass density of stars (row 3) and supermassive black holes (rows 
+4 and 5) formed over cosmic time in units of Mʘ Mpc-3. 
+Column 3 – The log of the specific kinetic energy released: energy per unit mass in stars (row 3 and black 
+holes (rows 4 and 5). Given in units of c2, and assuming εstar = 1.0. 
+Column 4 – The log of the amount of kinetic energy created per unit volume (in ergs Mpc-3). 
+Column 5 – The specific momentum created (momentum per unit mass in stars (row 3) and black holes 
+(rows 4 and 5). In units of cm s-1. 
+Column 6 – The log of the amount of momentum created per unit volume (gm cm s-1 Mpc-3). 
+______________________________________________________________________________ 
+
+ 
+Figure 1 – A plot of the amount of kinetic energy injected per Gyr and co-moving cubic Mpc as a function 
+of lookback time for massive stars (supernovae and stellar winds; black) and black-hole-driven jets (blue) 
+and winds (red). For massive stars we show the cases in which 100% (solid line) and 10% (dashed line) of 
+the kinetic energy created is delivered to the surroundings (i.e. not lost to radiative cooling). Note that 
+for momentum injection, massive stars dominate at all epochs, with the same time dependence as for 
+kinetic energy injection (i.e. as given by the solid black line).  
+3. Implications 
+3.1 For Galaxies 
+To assess the implications of these results for galaxy evolution, it is essential to consider the 
+dependences of feedback on the masses of both galaxies and supermassive black holes. We can 
+go beyond these simple global values and examine the relative importance of feedback (both 
+kinetic energy and momentum) as a function of the ratio of supermassive black hole mass to 
+galaxy stellar mass. In Figure 2 we show a plot of black hole vs. galaxy mass that is similar to 
+that in Heckman & Best (2014) for the z ≈ 0.1 universe (based on SDSS). In this case, these are 
+present day stellar masses, and would need to be increased by a factor 1/(1-R) = 1.42 to 
+represent the total mass of stars ever formed. The masses for the black holes were estimated 
+from the M-σ relation from McConnell & Ma (2013). In figure 2, we have color-coded the plot 
+by the fraction of galaxies in which star-formation has been quenched, which we define to be 
+
+Redshift
+0
+0.5
+1
+2
+346
+57.5
+57.0
+56.5
+56.0
+55.5
+55.0
+Supernovae (8star = 1.0)
+ - - Supernovae (star = 0.1)
+54.5
+-AGNjets
+AGNwinds
+54.0
+0
+2
+4
+6
+8
+10
+12
+Lookback time / GyrSFR/Mstar < 10-11 yr-1. It is clear that the quenched fraction depends strongly on both the stellar 
+and black hole masses.  
+The mean relation between stellar and black hole mass in Figure 2 can be approximated as log 
+MBH = 2.0 log Mstar -14.0, implying MBH/Mstar α Mstar1.0 α MBH0.50. Thus, the relative importance of 
+feedback integrated over cosmic time from massive stars and black holes should be a strong 
+function of mass. Let us quantify this for kinetic energy and then for momentum. For kinetic 
+energy, in a given galaxy (and assuming that global averages can be applied to individual 
+galaxies; see below) the inventories above imply that the ratio KEBH/KEstar = 315 εstar-1 MBH/Mstar 
+(where Mstar is the present-day stellar mass). For momentum, the corresponding ratio is 
+pBH/pstar = 100 MBH/Mstar. We can then plot these relations in Figure 2 to see the regimes in 
+which feedback from supermassive black holes exceeds that from stars. For kinetic energy, we 
+show this separately for values of εstar = 0.1 and 1.0.  
+ 
+Figure 2 – A plot of the distribution of SDSS galaxies in the plane of galaxy stellar mass vs. supermassive 
+black hole mass. The latter were estimated using the MBH vs. σ relation in McConnell & Ma (2013). The 
+relative numbers of galaxies in each bin are indicated by the green contours (increasing by factors of 2) 
+and the color-coding represents the fraction of galaxies that are quenched (SFR/Mstar < 10-11 yr-1). The 
+dark blue dashed line indicates where the momentum injected by black holes equals that from massive 
+stars. The two light blue dashed lines indicate where the kinetic energy from black holes equals that from 
+massive stars for values of εstar = 0.1 and 1.0 (see text). The transition from predominantly star forming 
+to predominantly quenched galaxies occurs near the relationship for εstar = 0.1. 
+
+10
+1.0
+0.8
+Quenched fraction
+0.6
+8
+0.4
+0.1
+KEBH
+0.2
+6
+0.0
+10.0
+10.5
+11.0
+11.5
+12.0
+log1o(Stellar mass / Msun)In terms of momentum input, stars dominate over black holes in almost all cases. However, 
+considering kinetic energy, we find that the transition from galaxies that are mostly quenched 
+to those that are mostly star-forming occurs very near the dividing line between jet-dominated 
+feedback and stellar dominated feedback for a value of εstar ≈ 0.1. This is suggestive evidence 
+that quenching is driven by the feedback of kinetic energy from jets driven by supermassive 
+black holes. However, we caution that the transition from star forming to quiescent galaxies 
+also occurs at the transition from disk dominated to bulge dominated galaxies, so the causal 
+connections between galaxy structure, star formation, and black hole feedback are not entirely 
+clear.  
+We emphasize that the relations plotted in Figure 2 explicitly assume that the global relations 
+can be applied to individual galaxies, namely that the amount of feedback from massive stars in 
+a given galaxy is proportional to stellar mass and that the amount of feedback from jets is 
+proportional to the black hole mass. The former seems like a safe assumption, but the 
+dependence of the production of radio jets on black hole mass may be complex. We know that 
+there are essentially two populations of radio galaxies (e.g.  Heckman & Best 2014). In one case 
+(“radiative mode”) the jets are launched by star-forming galaxies and are accompanied by 
+strong nuclear radiation (QSO-like). In the other class (“jet-mode”) the jets are launched by 
+quenched galaxies, with little accompanying nuclear radiation. The radiative mode becomes 
+more important at higher luminosities and at higher redshifts. For the jet-mode galaxies, 
+Sabater et al. (2019) find that the probability of producing a jet with a given luminosity depends 
+on both the stellar and black hole mass (and more strongly on the former).  
+The situation for radiative-mode radio galaxies is less clear, although the indications are that 
+any dependence of the ratio of KE/MBH on MBH or Mstar is weaker (e.g. Janssen et al. 2012, 
+Kondapally et al. 2022). In the context of Figure 2, it may be that the jet-mode is not the 
+dominant population in terms of actively quenching, since the jet-mode galaxies are already 
+quenched (instead, these may just ‘maintain’ a quenched state).  If quenching is due to jets in 
+radiative-mode galaxies, the dividing line between quenched and star-forming galaxies in Figure 
+2 would imply that time-integrated amount of jet energy contributed by a radiative mode 
+galaxy is proportional to its black hole mass (i.e. the integrated ratio of jet kinetic energy and 
+energy carried by radiation is independent of black hole mass in these galaxies). 
+Another way to consider this is to ask how the amount of energy supplied by stars and by black 
+holes scales with the binding energy of the galaxy. We take Ebind ≈ Mstar vc2 where vc is the galaxy 
+circular velocity. The Tully-Fischer relation for disk galaxies (McGaugh et al. 2000) and the 
+Faber-Jackson relation for ellipticals (Bernardi et al. 2003) both imply vc α Mstar1/4. Thus, we 
+have Ebind α Mstar3/2. Given that KEstar α Mstar and KEBH α MBH α Mstar2, this implies that KEstar/Ebind 
+α Mstar-1/2, while KEBH/Ebind α Mstar1/2 α MBH1/4. This again underscores the fundamental 
+difference in the mass-dependence of feedback from massive stars and supermassive black 
+holes: feedback from stars becomes increasingly impactful on the galaxy as the mass decreases, 
+while feedback from black holes has greater impact as the mass increases. 
+
+3.2 For the Intra-Group and Intra-Cluster Media 
+It has long been known that the basic observed properties of the hot gas in groups and clusters 
+of galaxies (Mhalo > 1013 Mʘ) are not consistent with simple models of purely gravitational 
+processes operating during the formation of these systems (see Donahue & Voit 2022 and 
+references therein). A particularly simple example of this is the observed relationship between 
+the X-ray temperature (a proxy for halo mass) and X-ray luminosity. As the halo masses 
+decrease, the observed X-ray luminosities fall further below the relationship expected simply 
+from gravitational infall and heating. These lower luminosities arise because the hot gas in 
+these less-massive halos is more spatially-extended than the dark matter, with the resulting 
+drop in gas density leading to lower X-ray luminosities. 
+This could be due to the feedback of energy injected into the hot gas, which “lifts” the gas 
+outward. As described above, there is direct observational evidence in the local universe of 
+radio jets delivering energy to the hot gas in groups and clusters. As discussed in Donahue & 
+Voit (2022), for this to be responsible for lifting the hot gas, an amount of kinetic energy equal 
+to ≈0.5% MBH/c2 must be delivered. This is close to the value for jets and AGN winds that we 
+estimated above of ≈0.34%. Note that this could be supplemented by the kinetic energy from 
+massive stars (which would be 0.25 to 2.5 the value for jets for εstar = 0.1 and 1.0 respectively). 
+4. Summary 
+Based on a global inventory of the amount of kinetic energy and momentum injected by 
+massive stars (stellar winds and supernovae), and by winds and jets driven by supermassive 
+black holes, we draw the following conclusions: 
+i) 
+The major sources of kinetic energy are massive stars and jets. Winds driven by 
+supermassive black holes provide <10% of the total. The global ratio of the kinetic 
+energy injected by massive stars to that injected by jets is 2.5 εstar (where εstar is the 
+fraction of injected energy from stars that is not lost to radiative cooling). 
+ii) 
+Massive stars are the dominant source of momentum injection (90% of the total). 
+AGN winds provide 10%, and radio jets are negligible. 
+iii) 
+The peak in the feedback from jets occurs at z ≈ 1, considerably later than the 
+contributions of AGN-winds and massive stars (peaking at z ≈ 2). 
+iv) 
+Since the ratio of the mass of the supermassive black hole to the galaxy stellar mass 
+increases steeply with mass, there will be a mass-dependence in the relative 
+importance of feedback from the two sources. 
+v) 
+For the assumptions that the total amount of kinetic energy from massive stars is 
+proportional to the galaxy’s stellar mass, and that the total amount of kinetic energy 
+from a supermassive black hole is proportional to its mass, we find that the 
+populations of quenched and star-forming galaxies occur in the regimes where 
+supermassive black hole feedback and massive star feedback dominate, respectively 
+(for a value of εstar ≈ 0.1).  
+
+vi) 
+By comparing the amount of kinetic energy injected as a function of the binding 
+energy of a galaxy, we show that feedback becomes more impactful as galaxy mass 
+decreases for massive stars, but more impactful as galaxy mass increases for black 
+holes. 
+vii) 
+The global amount of kinetic energy injected by radio jets and AGN winds per unit 
+volume, combined with the supermassive black hole mass function, yields an 
+efficiency for producing kinetic energy in jets of 0.34% c2. This is very close to the 
+amount of energy needed to explain X-ray luminosity-temperature relation in groups 
+and clusters (0.5% c2). 
+ 
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+

8NFLT4oBgHgl3EQfBC4o/content/tmp_files/2301.11968v1.pdf.txt

@@ -0,0 +1,792 @@
+arXiv:2301.11968v1  [hep-th]  27 Jan 2023
+Strong Cosmic Censorship in light of Weak Gravity
+Conjecture for Charged Black Holes
+Jafar Sadeghi ⋆1,
+Mohammad Reza Alipour ⋆2,
+Saeed Noori Gashti⋆3
+⋆Department of Physics, Faculty of Basic Sciences,
+University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
+Abstract
+In this paper, we investigate the strong cosmic censorship conjecture (SCC) for charged
+black holes in the de Sitter space by considering the weak gravity conjecture (WGC). Us-
+ing analytical methods, we find that the SCC is preserved for dS-charged black holes with
+respect to some restriction qQ ≫ 1 and r+ ≥ Q with the help of the WGC condition
+viz
+q
+m ≥ 1 for scalar fields. Where q, m are the charge and mass of the scalar field, and
+r+, Q determine the radius of the outer event horizon and the charge of the black hole,
+respectively. In that case, when the (WGC) is valid, SCC will definitely be satisfied for
+the dS-charged black holes. On the other hand, the SCC is violated when the WGC is
+not satisfied. Also, we examined the RN-dS charged black hole in the extremality state
+and found that SCC can be violated with the condition Λr2
++ = 1.
+Keywords: Strong cosmic censorship conjecture; Weak gravity conjecture; RN-dS charged
+black hole
+Contents
+1
+Introduction
+2
+2
+Weak Gravity Conjecture
+4
+3
+Charged Black Holes in dS Space
+6
+4
+The Quasinormal Resonant Frequency Spectrum
+7
+5
+Discussion and Result
+10
+1Email:
+[email protected]
+2Email:
+[email protected]
+3Email:
+[email protected]
+1
+
+1
+Introduction
+One of the studies with a long history in general relativity is the study of the collapse of small
+perturbations. We need more information on how these oscillations decay to understand better
+the gravity concept, use gravitational wave data, and study and investigate the valuable features
+of general relativity. One of the signs of the failure of determinism in general relativity can
+be the emergence of an interesting phenomenon known as Cauchy horizons that appear in the
+astrophysical solutions of Einstein’s equations. These horizons are such that it is impossible
+to specify the history of the future of an observer that passes through such horizons using
+Einstein’s equations and initial data. With these descriptions in the black holes’ space-time
+background, it is an expected possibility that the perturbations of the outer region are infinitely
+amplified by a mechanism known as the blue shift. They lead to a singularity boundary beyond
+the Cauchy horizon in the interior of black holes, where field equations cease to make sense. The
+Penrose strong cosmic censorship (SCC) confirms such an expectation. Of course, another point
+is that astrophysical black holes are stable due to a special mechanism called the perturbation-
+damping mechanism, which is applied in the outer region.
+Therefore, considering whether
+SCC retains real hinges or not depends on the very subtle competition between the collapse
+of perturbations in the outer region and their amplification (blue shift) in the inner space-
+time of black holes. In general, the fate of Cauchy horizons is related to the collapse of small
+perturbations outside the event horizon. Hence, the validity of SCC is tied to the extent of
+external damps fluctuation. In connection with various structures and conditions, SCC and
+its satisfaction and violation have been investigated in various theories. The violation of this
+conjecture near the extremal region studied in the investigation of higher curvature gravity [1].
+Also, this conjecture has been challenged in investigating many charged black holes. In [2,3], this
+conjecture was checked for a charged AdS black hole. It was shown that for a specific interval
+for the parameter (β), this conjecture is satisfied and violated in other areas as well.
+The
+strong cosmic censorship conjecture has also been investigated in two dimensions. There have
+been interesting outcomes regarding the violation of this conjecture near the extremal region
+at specific points [4]. The study of this conjecture in the structure of three-dimensional black
+strings has also carried interesting results, which you can see [5] for a deeper study. Studying
+the validity and violation of this conjecture in many recent studies in different conditions and
+frameworks has led to exciting results that you can see [6–9] for further study. Therefore, in
+this article, we want to study a different structure of this conjecture. According to the above
+explanation, we consider the general configuration of charged black holes. Then, using the
+weak gravity conjecture, we will prove that SCC is valid for specific values for all charged black
+holes. We will use the weak gravity conjecture to prove a general relation for all charged black
+holes about SCC. In connection with SCC, we need to pay attention to more concepts, which
+we will mention here for further study. The effectiveness of mass-inflation systems, which are
+involved in the transformations of the inner Cauchy horizon associated with the space-time of
+2
+
+black holes that are approximately flat, which is pathological in the estimation of SCC, into
+a series of hypersurfaces which is singular non-extendable. Those that are in an indivisible
+form are related to two different types of physical mechanisms [10–16]. First, the events in
+the exterior space-time regions of dynamic black holes formed viz the collapse of the remnant
+perturbation fields and second amplification of exponential blue shift related to the fields falling
+into the inside of black holes. We can manage these two introduced different systems through
+parameters such as (g) and (k−). It can be stated that the dimensionless physical ratio with
+the help of these two parameters can determine the fate of the inner Cauchy horizons inside
+such space-times of non-asymptotic flat black holes [8,17,18],
+β ≡ g
+k−
+.
+Of course, a certain range of parameters of black holes, such as mass and charge, etc., as
+indicated in [8,17,18],
+β > 1
+2.
+So, space-time of the corresponding black holes can be physically expanded beyond their Cauchy
+horizon which includes a pathological fact and a sign of algebraic failure or a violation of the
+Penrose SCC in classical general relativity. For the dynamics of Einstein’s equations as well
+as the destiny of the observers, the explosive structure of the curvature that is related to
+(β < 1) does not have per se much physical significance: it indicates two theorems, not the
+failure of the field equations mentioned in [19] and of course not the destruction of macroscopic
+observers which is discussed in [13]. Therefore, the physical and mathematical formulation
+of the conjecture of a SCC in such conditions leads to ignoring physical phenomena such as
+impulsive gravitational waves or the formation of shocks in relativistic fluids.
+Due to the
+aforementioned reasons, the modern form of the CC conjecture was introduced that requires
+a stronger constraint (β < 1
+2 ) and many works have been done to fit such constraints. For
+example, by studying massless scalar fields in linear form and examining the entire parametric
+space of a charged black hole, areas beyond mentioned range were obtained, which it seems
+cannot be allowed. According to the above explanations, we organize the article in the following
+form.
+In section 2, we will give basic explanations about the weak gravity conjecture and also the
+motivation to use it. In section 3, we will introduce charged black holes in dS space, and then
+we will introduce the quasinormal resonant frequency spectrum in section 4. We will check the
+conditions of compatibility and violation of (SCC) with respect to (WGC) for RNdS charged
+black holes. Finally, we describe the results in section 5.
+3
+
+2
+Weak Gravity Conjecture
+As it is known in the literature, a new idea has been put forward as a swampland program to
+check theories coupled to gravity, to check the consistency of quantum gravity, and finally, a
+proof for string theory. Recently, ones have done lots of work on this field [20–30]. Due to the
+special conditions of string theory and the fact that its testing and experimental investigations
+seem a bit difficult, this idea has been proposed to test and investigate various concepts of
+cosmology. The swampland program is challenged from two sides. From an up-bottom view
+for introducing principles and limitations to introduce conjectures, as well as mathematical
+formulations to examine cosmological concepts. A second look from the bottom-up in order
+to test each of these conjectures with various concepts of cosmology including inflation and
+matching with observable data, which is both a proof for this new idea and a proof for string
+theory. So far, many conjectures have been proposed from this theory, and now, according to
+the structure and further investigations, new conjectures will be added to this program. Some
+of these conjectures face challenges and as a result, corrections are made to the conjectures.
+We face some limitations in quantum gravity (QG). At the point when gravity is considered at
+the quantum level, the hypothesis will be incompatible. Generally having a reliable quantum
+hypothesis of gravity isn’t really straightforward and can in any case hold many surprises and
+can be interesting for physical science at low energies. The objective of the swampland program
+is to decide the limitations that an effective field theory(EFT) should fulfill to be viable with the
+consideration of ultraviolet completion(UV) in QG. They are called swampland limitations, and
+different suggestions are figured out as far as swampland conjectures(SC). The objective is to
+recognize these limitations, accumulate proof to demonstrate or refute them inside the structure
+of QG, give reasoning to make sense of them in a model-free manner, and comprehend their
+phenomenological suggestions for low-energy EFTs. Albeit the swampland idea isn’t restricted
+to string theory on a fundamental level, SC are frequently examined by string theory backdrops.
+Without a doubt, the string theory gives an ideal structure to thorough quantitative testing
+of conjectures and works on how we might interpret potential compressions of string theory.
+Strangely, it has as of late been uncovered that a large number of these conjectures are to be
+sure related, recommending that they may essentially be various countenances of some yet-to-
+be-found crucial standard of QG. As far as possible have significant ramifications for cosmology
+and particle physics. They can give new core values to building conjectures past the standard
+models in high-energy physics. They may likewise prompt UV/IR blending, which breaks the
+assumption for scale detachment and possibly gives new bits of knowledge into the regular issues
+seen in our universe. Consequently, the presence of swampland is extraordinary information
+for phenomenology. For a total rundown of references connected with swampland that might
+be valuable, we allude in [20] the swampland program (SP) has likewise been surveyed and
+presented. The shortfall of global symmetry (GS) and the completeness of charge spectra are
+at the center of the SP. Nonetheless, they need phenomenological suggestions except if we can
+4
+
+restrict the global symmetries [21,22] and whether there is any limitations point on the mass
+of charged states. In any case, they just bound the complete hypothesis but not the low-energy
+EFTs. Specifically, it is phenomenologically important whether all charged particles can be
+really super heavy and even compare to black holes(BHs), or whether there is some thought of
+completeness of the range that gets by at low energies. A large portion of the SCs examined
+address exactly these inquiries. They want to profoundly explore these assertions and measure
+them so we can draw nearer to the recuperation of a few global symmetries. For instance, we
+can deduce recuperate a global symmetry (GS) U(1) by sending the gauge coupling(GC) to
+nothing, which ought not to be permitted in QG. Attempting to comprehend string theory
+for the study of this issue, it might turn out that if one somehow managed to try to do such
+work, can give data about the imperatives that an EFT can fulfill to be viable with QG.
+Likewise, WGC forbids this cycle by flagging the presence of new charged states that denies
+the depiction of the EFTs. Thusly, it gives an upper bound on the mass of these charged states.
+The WGC comprises of some parts: the electric and the magnetic electric-WGC: As indicated
+by a quantum hypothesis, we have the following condition [20–30],
+Q
+m ≥ Q
+M |ext = O(1),
+(1)
+and
+Q = gq,
+(2)
+where, g and q are the gauge coupling and the quantized charge. The electric-WGC needs
+the presence of an electrically charged condition of a higher charge-to-mass proportion than
+extremal BH in that hypothesis, which is regularly a variable of the order one. One more
+understanding of this conjecture is that the limitations region shows that scale force determines
+stronger than the gravity on this mode — so subsequently is called WGC. This is an identical
+equation since it expects that electromagnetic force is stronger gravitational force [20–30],
+FGrav ≤ FEM
+(3)
+It implies that the charge is more prominent than the mass, so we get a similar condition as
+above. This is as of now false within the sight of massless scalar fields. The motivations of
+WGC are twofold. To begin with, it makes a QG boundary to reestablish the GS of U(1) by
+sending g → 0. If a GC goes to zero as indicated by WGC, this conducts new light particles
+and the cutoff the hypothesis arrives at nothing and nullifies the EFT. Because of the littleness
+of the scale coupling, it relies upon how much energy the interaction with which you need to
+portray the viable EFT. The smallness of the cycle energy leads to the smallness of the scale
+coupling. On the other hand, if you need to keep the EFT substantial up to an extremely high
+cut-off, the GC can’t be excessively small. This is an illustration of swampland limitations that
+5
+
+becomes more grounded for higher energies. Obviously, a hypothesis with disappearing measure
+coupling i.e., GS is incompatible because the cutoff of the viable EFT is likewise zero. One more
+fundamental inspiration for WGC is that a kinematic prerequisite permits extremality BH to
+have decomposed. Charged BHs should fulfill an extremality breaking point to stay away from
+the presence of exposed singularities, as shown by the weak cosmic censorship (WCC). For a
+given charge Q, this super bound shows that the this super bound shows that the mass M of
+the BHs should be more noteworthy than the charge [20–30],
+M ≥ Q
+(4)
+For the BHs to have a regular horizon.
+Here, we set the extremal factor O(1) to one for
+simplicity. The primary condition for starting the decay to the small black hole and particle
+is the existence of the extremal BHs (M = Q). So, one can consider the decay of an extremal
+BHs which one of the rot items has a charge more modest than its mass as far as possible, so
+M1 ≥ Q1. Then the rot item can never again have a charge more modest than the mass, that
+is m2 ≤ Q2. It is just a kinematic necessity. Since the second rot item violates the WCC, it
+can’t be a BH, so it should be a particle. The above kinematic necessity can be acquired by
+applying preservation of mass/energy and protection of charge as follows. The initial mass of
+the BH should be more prominent than the amount of the mass of the rot items Mi and the
+charge of the initial BH.
+3
+Charged Black Holes in dS Space
+The metric of charged black hole in spherical symmetric space is defined as follows,
+dS2 = f(r)dt2 − f −1(r)dr2 − r2dΩ2,
+dΩ2 = (dθ2 + sin2(θ)dϕ2).
+(5)
+Here, we consider f(r) = H(M, Q) − Λr2
+3
+in general; where Q, M, Λ > 0 are electric charge, the
+mass of the black hole and the cosmological constant respectively. In this case, we can obtain
+its event horizons as follows,
+f(r⋆) = 0
+→
+⋆ ∈ (−, +, ..., c).
+(6)
+Considering the metric in general terms, we have different event horizons, where (r−) is the
+Cauchy horizon, (r+) is the outer event horizon, and (rc) is the cosmological horizons. Using
+Klein-Gordon’s differential equation, we can determine the dynamics of a massive charged
+particle near a charged black hole [31–34],
+1
+√−g∂µ(gµν√−g∂νΦ) − 2iqgµνAµ∂νΦ − q2gµνAµAνΦ − m2Φ = 0,
+(7)
+6
+
+where m and q are the mass and charge of the particle, respectively also, Aµ =
+� Q
+r , 0, 0, 0
+�
+. We
+can define the scalar field Φ according to relation (7) as follows [36],
+Φ(t, r, θ, φ) =
+�
+m
+�
+ℓ
+e−iωtYℓm(θ, ϕ)Φ(r).
+(8)
+The integer parameters ℓ and m are the spherical and the azimuthal harmonic indices of the
+resonant eigenmodes which characterize the charged massive scalar fields in the charged black-
+hole spacetime. By putting Eq.(8) in Eq.(7) and using dx =
+dr
+f(r), we get the Schr¨odinger-like
+differential equation ,
+d2φ(r)
+dx2
++ V (r)φ(r) = 0.
+(9)
+The effective radial potential due to a massive charged particle near a charged black hole is
+defined as [8],
+V (r) = qm
+r2
+� q
+mα(r) − m
+q β(r)
+�
+,
+(10)
+where
+α(r) = Q2
+�
+1 − ωr
+qQ
+�2
+,
+β(r) = r2f(r)H(r),
+H(r) =
+�ℓ(ℓ + 1)
+m2r2
++ f ′(r)
+m2r + 1
+�
+.
+(11)
+Also, we can consider the boundary conditions for the special radial function near the outer
+event horizon as an incoming wave and at the largest event horizon as an outgoing wave [34,35]:
+φ(x) ∼
+�
+e
+−i(ω− qQ
+r+ )x,
+for
+r → r+ (x → −∞);
+e−i(ω− qQ
+rc )x,
+for
+r → rc (x → ∞).
+(12)
+According to the above boundary conditions, we can obtain the discrete spectrum of ω, defined
+as the resonance frequency of the imaginary quasi-normal state.
+4
+The Quasinormal Resonant Frequency Spectrum
+In this section, we need to obtain the imaginary part of the resonance frequency to investigate
+the linear dynamics of a massive charged particle near a general charged black hole. Also, we
+need to do this in a dimensionless regime to do this analytically. Since, the q2
+¯h ≃
+1
+137 relationship
+exists in our universe, we can consider it for black holes, even slightly charged, and get qQ ≫ 1.
+In addition, the mechanism of the Schwinger-type pair-production in space-time of charged
+black hole creates a limit to the black hole electric field with the
+Q
+r2
++ ≪ m2
+q relationship [37–40].
+7
+
+Therefore, according to the above statement, we can consider SCC and define our constraint
+regime following ansans,
+m2r2
++ ≫ ℓ(ℓ + 1)
+and
+m2r2
++ ≫ 2k+r+,
+(13)
+where k+ = f ′(r+)/2 is the gravitational acceleration of the black hole at the outer event
+horizon. In this area, we try to obtain the imaginary part of the resonance frequency in the
+background of the general charged black hole near the event horizon.
+Now, we use radial
+potential (10) to determine the linear dynamics of the massive charged particle near the event
+horizon of the black hole. We can consider this potential in region (13) as an effective potential
+and obtain the quasinormal resonance modes analytically using standard WKB techniques
+[41, 42]. In this region, we consider the maximum effective potential near the event horizon
+of the black hole at point r = r0. In the following, we use the relationship (10), (11), and
+V ′(r0) = 0 to obtain the point where the effective potential is maximum as follows,
+r0 =
+q2Q2
+qQω − m2r2
++k+
+(14)
+According to the Schr¨odinger-like differential equation (9) and [41–43], we use the WKB
+method to obtain the quasinormal mode frequencies through the following,
+iK − (n + 1
+2) − Λ(n) = Ω(n)
+(15)
+where
+K =
+V0
+�
+2V (2)
+0
+Λ(n) =
+1
+�
+2V (2)
+0
+
+
+�
+α2 + 1
+4
+�
+8
+V (4)
+0
+V (2)
+0
+− (60α2 + 7)
+288
+�
+V (3)
+0
+V (2)
+0
+�2
+
+Ω(n) = n + 1
+2
+2V (2)
+0
+
+5 (188α2 + 77)
+6912
+�
+V (3)
+0
+V (2)
+0
+�4
+− (100α2 + 51)
+384
+�
+V (3)
+0
+�2
+V (4)
+0
+�
+V (2)
+0
+�3
+
+
++ n + 1
+2
+2V (2)
+0
+
+(68α2 + 67)
+2304
+�
+V (4)
+0
+V (2)
+0
+�2
++ (28α2 + 19)
+288
+�
+V (3)
+0
+V (5)
+0
+�
+�
+V (2)
+0
+�2
+− (4α2 + 5)
+288
+V (6)
+0
+V (2)
+0
+
+
+(16)
+Here, V (k)
+0
+≡ dkV
+dxk |r=r0 is the spatial derivative of the effective potential of equation (10), and
+its scattering peak is evaluated at the point r = r0. Using relations (10), (11), (14) and (16),
+8
+
+we will have the following relation in the region of (13),
+K ≃
+k2
++m4r4
++qQ
+2f0 (k+m2r2
++ − qQω)2
+Λ(n) ≃
+k2
++m4 �
+17 − 60
+�
+n + 1
+2
+�2�
+r4
++ + 2k+m2 �
+36
+�
+n + 1
+2
+�2 − 7
+�
+qQr2
++ω
+16qQ (qQω − 3k+m2r2
++)2
+× f0
+A = 15k4
++m8
+�
+148(n + 1
+2)2 − 41
+�
+r8
++ + 12k3
++m6
+�
+121 − 420(n + 1
+2)2
+�
+qQr6
++ω
+B = 64q5Q5 �
+k+m2r2
++ − qQω
+�4
+Ω(n) ≃ −(n + 1
+2)q3Q3f 2
+0 × A
+B
+(17)
+where f0 = f(r0). In the next step, to determine the study of SCC, we need to obtain the
+minimum value of the fundamental imaginary resonance mode of the system. For this purpose,
+using equations (15) and (17), we can calculate the Im(ω0),
+ω ≃ qQ
+r+
+− 2k+m2r2
++
+qQ
+�
+1 − 14400
+11644
+�(n + 1/2)f0
+qQ
+�4�
+− i
+�
+4f0k+(n + 1
+2)m2r2
++
+q2Q2
+�
+1 − 34qQf 4
+0
+11664
+�
++ O(f 2
+0)
+�
+(18)
+Since we consider r0 near the event horizon (r+), we have f0 ≪ 1. For investigation the SCC,
+it is necessary to find the minimum value of the resonance mode and evaluate its ratio to the
+surface gravity of the event horizon,
+β = −Im(ω0)
+k+
+≃ 2f0
+m2r2
++
+q2Q2
+�
+1 − 34qQf 4
+0
+11664
+�
+.
+(19)
+Since it is f0 ≪ 1, it is sufficient to have the conditions q2Q2 > m2r2
++ in the relation above
+concepts so that −Im(ω0)
+k+
+< 1
+2 is established. Therefore, we have the following condition for the
+study of SCC,
+q
+m ≥ r+
+Q .
+(20)
+from equation (20) determine that when r+ ≥ Q, we have the weak gravity conjecture condition.
+We know that k− > k+, so the relationship of (19) and (20) is also established for β = −Im(ω0)
+k−
+<
+1
+2. Also, according to relation (19), when qQ < 2√f0mr+, SCC can be violated. Since qQ ≫ 1
+and f0 ≪ 1, the mass of the scalar field and the radius of the event horizon must be very
+massive and very large respectively. In the following, we obtain the extremality state of the
+9
+
+RN-dS black hole. We will have the following relation for the RN-dS black hole with respect
+to equation(5),
+f(r) = 1 − 2M
+r
++ Q2
+r2 − Λr2
+3 .
+(21)
+When k+ = k− = 0, we can obtain the black hole extremality state,
+Qexe = r+
+�
+1 − Λr2
++,
+Mexe = r+(1 − 2
+3Λr2
++).
+(22)
+We substitute Eq.(22) in Eq.(19) to obtain β in the extremality state of the RN black hole,
+β ≃ 2f0
+m2
+q2(1 − Λr2
++)
+�
+1 − 34qf 4
+0
+�
+1 − Λr2
++r+
+11664
+�
+(23)
+According the above relationship, when the condition
+q
+m ≥
+1
+1−Λr2
++ is satisfied, the SCC will
+definitely be preserved, and since Λr2
++ < 1, the weak gravity conjecture will also be satisfied.
+On the other hand, when Λr2
++ ≪ 1, we will have the SCC condition in light of the WGC,
+q
+m ≥ 1 + Λr2
++,
+(24)
+from the above relation WGC is clearly obtained. In relation (23) when Λr2
++ = 1, we have
+β → ∞ and the SCC is violated. Also, these result and conditions are completely compatible
+with [44,45].
+5
+Discussion and Result
+One of the indications of the failure of determinism GR can be the rise of a fascinating pecu-
+liarity known as the Cauchy horizon that shows up in the astrophysical solutions of Einstein’s
+equations. These horizons are such that it is difficult to indicate the history of the future of
+an observer that passes come of such horizons utilizing Einstein’s conditions and initial infor-
+mation. With these descriptions in the black holes’ background space-time, it is a predicted
+possibility that the perturbations of the external area are infinitely enhanced by a system
+known as the blue shift. They lead to a singularity beyond the Cauchy horizon the inside
+of BHs, where field conditions fail to seem good. The Penrose cosmic censorship conjecture
+(SCC) affirms such an assumption. Obviously, another point is that astrophysical BHs are
+stable because of an exceptional component called the perturbation-damping system, which is
+applied in the outer region. Also, the SCC resolves the issue of the idea of the singularities
+tracked down in many answers to Einstein’s gravitational field equations: Are such singular-
+ities conventionally described by unbounded curvature? Is the presence of a Cauchy horizon,
+10
+
+an unsteady characteristic element of answers of Einstein’s equations? Recently researchers,
+remarking on the historical backdrop of the SCC conjecture, overview a portion of the headway
+made in research coordinated either toward satisfying SCC or toward revealing a portion of its
+shortcomings. They specifically around model adaptations of SCC which have been demon-
+strated for constrained groups of spacetimes viz the Gowdy spacetimes and the role played by
+the conventional presence of Asymptotically speed term dominated conduct in these answers.
+Also additionally note some work on spacetimes containing weak null singularities, and their
+importance for the SCC [44, 45, 47]. SCC conjecture has been one of the main acts of pure
+confidence with regard to GR, confirming the deterministic idea of the related field relations.
+However, it holds well for asymptotically level spacetimes, an expected disappointment of the
+SCC conjecture could emerge for spacetimes acquiring Cauchy horizon alongside a positive
+cosmological constant viz its potential failure about this issue. Researchers have unequivocally
+exhibited that infringement of the restriction SCC turns out as expected within the sight of
+a Maxwell field even with the presence of higher spacetime aspects. Specifically, for higher
+dimensions of the RN black holes, the infringement of SCC is at a bigger scope compared with
+the 4D case, for specific of the cosmological constant. Then again, for a brane world BH, the
+impact of an additional dimension is to make the infringement of cosmic censorship weaker.
+For rotating BHs, intriguingly, the SCC is constantly holding even in the presence of higher
+dimensions. A comparable situation is likewise noticed for rotating BHs on the brane [47].
+In this paper, we investigated dynamically formed charged black holes. Also, to satisfy the
+SCC, the inner Cauchy horizons of the black hole must be unstable. Here, to check the SCC,
+it is necessary to get two −Im(ω0) and k− parameters to demonstrate the decay rate of the
+remaining perturbation fields in the outer regions of the black hole and the blue-shift growth
+rate of the in-falling fields of the black hole, respectively. Therefore, if β = −Im(ω0)
+k−
+< 1/2, SCC
+will be maintained. We found that for the dS charged black hole with respect to r+ ≥ Q in
+light of the WGC, viz q/m ≥ 1, SCC will definitely be satisfied. We also found that there will
+be a possibility of violation of SCC for the massive scalar field as well as when the radius of the
+event horizon of the charged black hole is very large. We also found SCC will be violated in
+the extremality state for the charged RN-dS black hole when Λr2
++ = 1 which is also mentioned
+in [44,45]. Also, these results and conditions are completely compatible with [44,45]. On the
+other hand, in [8,46], when the scalar field is uncharged, the SCC is violated, which is consistent
+with (19) in this paper. Because can be obtained β > 1/2 if assume the charge of the scalar
+field is zero viz q = 0. The above study also raises some questions as follows.
+Is the relationship researched in this article also valid for black holes in higher dimensions? Do
+other black holes in different frames satisfy the SCC and WGC simultaneously? Is it possible to
+consider the SCC relation with WGC monitoring for all black holes? Is it may such a structure
+also be established for black holes on the brane? We leave these questions for future work.
+11
+
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+

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+Instability of the cosmological DBI-Galileon in the
+non-relativistic limit
+C. Leloup1,2, L. Heitz3 and J. Neveu3,4
+1 Universit´e Paris-Cit´e, CNRS, Astroparticule et Cosmologie, 75013 Paris, France
+2 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU,
+WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
+3 Universit´e Paris-Saclay, CNRS, IJCLab, 91405, Orsay, France
+4 Sorbonne Universit´e, CNRS, Universit´e de Paris, LPNHE, 75252 Paris Cedex 05,
+France
+Abstract.
+The DBI-Galileon model is a tensor-scalar theory of gravity which finds its
+foundation as the most general theory of the dynamics of a 4D brane embedded in
+a 5D bulk.
+It is of particular interest as it provides a few free parameters with a
+physical meaning, such as the cosmological constant which is there related to the
+brane tension. Most studies of this model have been performed assuming a maximally
+symmetric geometry for the 5D bulk, in which it has been shown that the theory
+reduces to various types of Galileon. In contrast, the general case for the geometry of
+the bulk provides a different covariantization of the Galileon model than the covariant
+Galileon: the DBI-Galileon. From the tight constraints on the gravitational waves
+speed, we are naturally led to consider the non-relativistic limit of the model where
+the kinetic energy of the brane is small compared to its tension, that we study in
+the context of late-time cosmology.
+The DBI-Galileon in the non-relativistic limit
+is simply an expansion around General Relativity (GR) which can be expressed as
+a shift-symmetric Horndeski theory. We developed the description of this theory at
+the background and perturbation level. However, by studying the scalar and tensor
+perturbations around a flat FLRW background, we found that they contain a ghost
+degree of freedom leading to fatal instability of the vacuum for every combination of the
+free parameters. As a lesson, we emphasized which of the Horndeski terms competes
+to avoid this instability in more general cases.
+1. Introduction
+Dark energy has been modelled by a large variety of theories since decades. Among
+these, many rely on the introduction of additional scalar fields whose dynamics, at the
+origin of the late-time acceleration of the expansion of the Universe, is determined by
+arbitrary parametric functions, potentials and/or coupling (see e.g. [1]). These are the
+so-called scalar-tensor theories of modified gravity. In particular, the class of Horndeski
+theories is of great interest as it contains all models of modified gravity with a single
+arXiv:2301.01723v1  [hep-th]  4 Jan 2023
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+2
+additional scalar field leading to second-order equations of motion [2, 3]. Extensions of
+Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of
+higher orders have also been explored [4, 5]. Particular Horndeski theories are described
+by the specification of four arbitrary functions of the scalar field and its kinetic energy,
+leading to a huge variety of models and phenomenological behaviours.
+Among these wide classes of models, some can be built from first physical principles
+or arguments of symmetry.
+For instance, the Galileon model [6] and its covariant
+extension [7] was built by imposing a galilean symmetry for the scalar field, leaving
+only five free numerical parameters. We can also cite, among many others, the pure
+kinetic gravity theory [8], massive gravity in the non-relativistic limit [9, 10] and the
+DBI-Galileon [11] which is the main object of this paper.
+The DBI-Galileon is a model that falls into the class of Brane-world scenarios of
+extra-dimension theories, where the matter fields are confined on a 4D brane while
+gravity can propagate into the additional spatial dimensions. Of most interest for the
+DBI-Galileon is the case of a single extra-dimension as it has been shown that theories
+with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter
+solution [12]. The action include a volume term for the 4D brane in the 5D bulk which
+leads to the well-known Dirac-Born-Infeld (DBI) action.
+This action, and DBI-like
+extensions, can lead to a self-accelerating solution and has been thoroughly studied as
+a candidate model in the early Universe cosmic inflation paradigm [13, 14]. In addition,
+the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit
+[11] but giving a physical meaning to their free parameters: the Planck mass in the
+brane, the Planck mass in the bulk, etc. In particular, the brane tension here plays the
+role of the cosmological constant which brings a possible interpretation of its nature.
+The original probe brane construction has been revisited in [15] where the matter
+metric is disformally related to a standard gravitational metric, or in [16] in the
+framework of spontaneous symmetry breaking for the 5D space-time symmetries broken
+by the presence of the brane, bridging the gap with Brane-world scenarios developed
+in the context of quantum field theory and an interpretation of the scalar field as a
+Nambu-Goldstone boson [17, 18]. The DBI-Galileon model has been studied extensively
+in special cases of the maximally symmetric bulk geometry [19, 20]. However, to our
+knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential
+candidate for Dark Energy has been performed so far.
+In this paper we develop the DBI-Galileon theory in the non relativistic limit
+(Section 2) and study its dynamics in the Friedmann-Lemaˆıtre-Robertson-Walker flat
+metric (Section 3). The perturbation stability is explored in Section 4 and then discussed
+in Section 5.
+2. DBI-Galileon in the non-relativistic limit
+DBI-Galileon
+We are interested in the description of a four dimensional brane universe embedded in
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+3
+a five dimensional bulk from the cosmological perspective. In this context, it has been
+shown in [11] that the most general action on the brane is given by the 4D Lovelock
+terms [21] inside the brane and the boundary terms associated to the 5D Lovelock terms
+in the bulk:
+S =
+�
+dx4√−g
+�
+−Λ − M 3
+5K + M 2
+P
+2 R − β M 3
+5
+m2 KGB + Lm (˜qµν, ψm)
+�
+,
+(1)
+where g is the induced metric on the brane, Λ the brane cosmological constant, K is the
+extrinsic curvature of the brane, R is the Ricci scalar on the brane, KGB is the boundary
+term on the brane of the Gauss-Bonnet scalar in the bulk and Lm is the Lagrangian
+density of matter that lives confined in the brane. In the above action has been defined
+the 4D Planck mass MP, its 5D counterpart M5 and their ratio m = M 3
+5/M 2
+P, and β is
+an arbitrary constant parameter.
+Because the action is defined on the 4-dimensional brane, the above quantities are
+expressed in terms of the induced metric g, as opposed to the 5-dimensional bulk metric
+G. Assuming we have a coordinate system (xµ, y) in the bulk, Greek letters being defined
+to span from 0 to 3, such that the length element in this frame is ds2 = qµνdxµdxν+(dy)2.
+The brane position in the bulk is defined by y = π (xµ) such that we can express the
+induced metric from the bulk metric:
+gµν = qµν + ∂µπ∂νπ
+and
+gµν = qµν − γ2∂µπ∂νπ
+(2)
+with the Lorentz factor γ =
+�
+1 + (∂π)2�−1/2. From this expression of the induced metric,
+we can explicitly write the action (1) using the bulk metric q and the scalar field π. As
+an illustration, see how the cosmological constant part of the action on the brane can
+be expressed:
+SΛ = −Λ
+�
+d4x√−g = −Λ
+�
+d4x√−q
+�
+1 + (∂π)2
+(3)
+As was pointed out in the original paper by de Rham and Tolley [11], we recover a DBI
+term in the action. This leads us to associate the cosmological constant Λ with a brane
+tension f, following Λ = f 4 for unit convenience.
+Non-relativistic limit
+If the derivatives of the field vanish, ∂µπ = 0, then we recover standard GR. Because
+we are interested in the cosmological setting, where predictions from the ΛCDM model
+based on GR match precisely a large range of observations, we consider only small
+corrections to GR. Therefore, we consider the DBI-Galileon in the so-called non-
+relativistic limit where (∂π)2 ≪ 1. In particular, the DBI part of the action in the
+non-relativistic limit becomes:
+Sf = −f 4
+�
+d4x√−q
+�
+1 + ∇µπ∇µπ
+2
+− (∇µπ∇µπ)2
+8
++ . . .
+�
+(4)
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+4
+We see that, to have a canonically normalized field, we can proceed to the field
+redefinition π → ϕ = f 2π. Up to degree 5 in ∂ϕ/f 2, we find the following Lagrangian
+operators for the DBI-Galileon in the non-relativistic limit:
+Lf = 1 + L2
+2f 4 − X2
+2 + . . .
+(5)
+LK = − L3
+2f 6 − 2X
+f 6 [ψ] + . . .
+(6)
+LR = ¯R + 1
+f 4
+�
+[Φ]2 −
+�
+Φ2�
+− f 4X
+2
+¯R − 2 ¯Rµν∇µϕ∇νϕ
+�
++ L4
+4f 8 + . . .
+(7)
+LKGB = 2
+f 6
+��
+− ¯Rµν [Φ] + 2 ¯RµρΦρ
+ν + ¯RµρνλΦρλ�
+∇µϕ∇νϕ + 1
+3 [Φ]3 − [Φ]
+�
+Φ2�
++ 2
+3
+�
+Φ3�
+−1
+2
+¯R [ψ]
+�
++ L5
+3f 10 + . . .
+(8)
+We defined on the above expressions the tensor Φµν = ∇µ∇νϕ, and the three scalars
+[Φ] = Φµ
+µ, [ψ] = ∂µϕ·Φµν·∂νϕ and X = − (∂ϕ)2 /2f 4. Furthermore, the L2...5 Lagrangian
+operators are the covariant Galileon model Lagrangian operators as defined in [6, 7].
+Therefore, the theory described here is a generalization of the Galileon model. We can
+note that the additional terms in LR and LKGB compared to the Galileon Lagrangian
+operators vanish in the particular case of a flat geometry. We conclude that the DBI-
+Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon
+than the covariant Galileon [7] or dRGT massive gravity [9, 10], with an additional
+terms in Lf and another one in LK. That point noted, we will stay at leading order
+in X in the following. Being a theory of a scalar field interacting with a metric, with
+equations of motion of at most second order, it can be described as a Horndeski theory
+[2, 3] with the following Horndeski functions:
+G2 = A (ϕ) − f 4 (1 − X + . . .)
+(9)
+G3 = M 3
+5
+f 2 (X + . . .)
+(10)
+G4 = M 2
+P
+2
+�
+1 − X − X2/2 . . .
+�
+(11)
+G5 = −2β M 3
+5
+m2f 2
+�
+X + X2 + . . .
+�
+(12)
+3. Cosmological background evolution
+We expand the dynamics of the fields around a flat FLRW background on the 4D slice
+of the bulk along xµ where the properties of the brane, gµν and ϕ, are defined. On this
+slice, the length element is:
+ds2 = qµνdxµdxν = −dt2 + a2 (t) δijdxidxj
+(13)
+It might appear more natural to expand around a flat FLRW background on the
+brane with its induced metric gµν. However, the two metrics qµν and gµν are related
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+5
+by a disformal transformation involving the scalar field which, at the background level,
+depends only on the physical time:
+¯gµν = ¯qµν + ( ˙¯ϕ (t))
+2
+f 4
+δµ0δν0
+(14)
+where barred quantities are taken at the background level. Therefore, the geometry on
+the brane is also of the FLRW type, but with a different definition for the physical time,
+leading to a different scale factor and expansion history. Because in our case ˙¯ϕ ≪ f 2,
+the expansion history on the 4D slice and on the brane will, then, be approximately the
+same. In addition, equations (9)-(12) determine a self-consistent Horndeski theory of
+gravity with the metric qµν and the scalar field ϕ without referring to the induced metric
+gµν. Thus, in the following we apply the well-known techniques used in the context of
+Horndeski theories.
+In the non-relativistic limit, action (1) reads:
+S =
+�
+dx4√−q (Lf + LK + LR + LKGB + Lm (qµν, ψm)) .
+(15)
+We define Ω0
+m and Ω0
+r the standard present energy density parameters for pressureless
+matter and radiation respectively, and ¯H the normalized Hubble rate H/H0 with H0
+the present Hubble constant. Prime symbol denotes the derivative with respect to ln a.
+We set:
+˜x = ϕ′H0
+f 2 ,
+Ω0
+Λ =
+Λ
+3H2
+0M 2
+P
+=
+f 4
+3H2
+0M 2
+P
+,
+η =
+M 3
+5
+M 2
+PH0
+,
+ξ = β
+η ,
+κ = MPH0
+f 2
+.
+(16)
+Then, the two Friedmann equations derived from action (15) are:
+¯H2 = −3
+2
+¯H4˜x2 − 15
+8
+¯H6˜x4 + η ¯H4˜x3 − 10
+3 ξ ¯H6˜x3 − 14
+3 ξ ¯H8˜x5
++ Ω0
+m
+a3 + Ω0
+r
+a4 + Ω0
+Λ
+�
+1 + 1
+2
+¯H2˜x2
+�
+(17)
+¯H2 + 2
+3
+¯H ¯H′ = −2
+3ξ
+�
+2 ¯H6˜x3 + 5 ¯H5˜x3 ¯H′ + 3 ¯H6˜x2˜x′ + 2 ¯H8˜x5 + 5 ¯H8˜x4˜x′ + 7 ¯H7˜x5 ¯H′�
+− 1
+2
+¯H4˜x2 − ¯H3˜x2 ¯H′ − 2
+3
+¯H4˜x˜x′ + 1
+3η ¯H3˜x2( ¯H˜x)′ − 3
+8
+¯H6˜x4 − 5
+4
+¯H5˜x4 ¯H′ − ¯H6˜x3˜x′
+− Ω0
+r
+3a4 + Ω0
+Λ
+�
+1 − 1
+2
+¯H2˜x2
+�
+(18)
+We see that we recover the ΛCDM equations when setting ˜x to zero, but with a physical
+interpretation of the cosmological constant as the brane tension.
+The DBI model
+proposed here is then an extension of the standard model of cosmology, and as such
+follows a late accelerated expansion but with a physical interpretation of the origin of
+Λ as a brane tension. Using the same methodology and similar notations as in [22], we
+derive the field ϕ equation of motion from action (15). This leads to a system of two
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+6
+coupled equations
+˜x′
+= −˜x + αλ − σγ
+σβ − αω
+¯H′
+= ωγ − βλ
+σβ − αω
+(19)
+with
+β = 2 ¯H4 + 9
+2
+¯H6˜x2 − Ω0
+Λ ¯H2 − 2η ¯H4˜x + 4ξ ¯H6˜x + 40
+3 ξ ¯H8˜x3
+α = 6 ¯H3˜x − Ω0
+Λ ¯H˜x + 15
+2
+¯H5˜x3 − 3η ¯H3˜x2 + 10ξ ¯H5˜x2 + 70
+3 ξ ¯H7˜x4
+γ = 4 ¯H4˜x − 2Ω0
+Λ ¯H2˜x − η ¯H4˜x2 + 2ξ ¯H6˜x2 − 10
+3 ξ ¯H8˜x4
+ω = 4
+3
+¯H4˜x + ¯H6˜x3 − 1
+3η ¯H4x2 + 2ξ ¯H6˜x2 + 10
+3 ξ ¯H8˜x4
+σ = 2
+3
+¯H + 2 ¯H3˜x2 + 15
+12
+¯H5˜x4 − 1
+3η ¯H3˜x3 + 10
+3 ξ ¯H5˜x3 + 14
+3 ξ ¯H7˜x5
+λ = ¯H2 + Ω0
+r
+3a4 − Ω0
+Λ + 1
+2Ω0
+Λ ¯H2˜x2 − 1
+3
+¯H4˜x2 − 5
+8
+¯H6˜x4 + 1
+3η ¯H4˜x3 − 2
+3ξ ¯H6˜x3 − 2ξ ¯H8˜x5
+Given values for the parameters Ω0
+m, η, ξ and κ, and initial conditions ˜x0 and H0,
+this system can be integrated to compute background cosmology observables like the
+distance moduli of type Ia supernovae. In Figure 1 we illustrate this with a Hubble
+diagram prediction compared with recent type Ia supernova data [23]. As an initial
+condition for ˜x, we chose to set ˜x0 = 6 × 10−8 today.
+This is the maximum value
+allowed by the constraint on gravitational wave speed (see Section 4) coming from the
+quasi simultaneous observation of photons and gravitational waves after neutron star
+merger event GW170817A [24]. Nevertheless, before discussing more the cosmological
+scenarios proposed by the DBI-Galileon model, stability conditions much be computed
+first to assess the viability of the models for any set of parameters at the perturbation
+level.
+4. Stability conditions
+To be viable as a description of our Universe, the model has to fulfill stability
+conditions.
+These requirements apply to degrees of freedom propagating around
+the fixed background, i.e.
+to the cosmological perturbations, that are capable of
+undermining the stability of the Universe.
+In the determination of the stability
+conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism
+described in [25] for Horndeski theories. In our case, these stability conditions are defined
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+7
+14
+16
+18
+20
+22
+24
+26
+Distance moduli  [mag]
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+Redshift z
+0.2
+0.0
+Residuals [mag]
+Figure 1.
+Hubble diagram prediction for the non-relativistic DBI-Galileon model for Ω0
+Λ = 0.7,
+˜x0 = 6 × 10−8, η = ξ = 0 (blue) compared with binned Pantheon data (black points) [23]. We used
+M = 23.81 for the offset magnitude of the diagram. Residuals to the fit are presented in the bottom
+panel.
+from the following quantities derived from the particular Horndeski functions (9) to (12):
+ω1 = 1 + 1
+2
+¯H2˜x2 + 3
+8
+¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5
+(20)
+ω2 = 2 ¯H + 3 ¯H3˜x2 + 15
+4
+¯H5˜x4 − η ¯H3˜x3 + 10ξ ¯H5˜x3 + 14ξ ¯H7˜x5
+(21)
+ω3 = 9
+�
+− ¯H2 + 1
+2Ω0
+Λ ¯H2˜x2 − 3 ¯H4˜x2 + 2η ¯H4˜x3 − 10ξ ¯H6˜x3 − 45
+8
+¯H6˜x4 − 56
+3 ξ ¯H8˜x5
+�
+(22)
+ω4 = 1 − 1
+2
+¯H2˜x2 − 1
+8
+¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′
+(23)
+Tensorial stability conditions
+In order to avoid ghosts and Laplacian instabilities, we impose the following constraints
+on the sign of the kinetic term and on the sign of the gravitational waves speed squared:
+Qt ≡ ω1
+4 = 1
+4 + 1
+8
+¯H2˜x2 + 3
+32
+¯H4˜x4 + 1
+2ξ ¯H4˜x3 + 1
+2ξ ¯H6˜x5 > 0
+(24)
+c2
+t ≡ ω4
+ω1
+= 1 − 1
+2 ¯H2˜x2 − 1
+8 ¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′
+1 + 1
+2 ¯H2˜x2 + 3
+8 ¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5
+≥ 0
+(25)
+In particular, we see that the gravitational wave speed depends on ˜x, and tends to
+1 when ˜x → 0 :
+4Qt ≃ 1 + 2( ¯H˜x)2
+(26)
+ct ≃ 1 − 1
+2( ¯H˜x)2
+(27)
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+8
+Given the very tight constraint on the speed of gravitational waves, equal to the speed
+of light up to a ∼ 10−15 difference [24, 26, 27], this justifies a posteriori the relevance
+of the non-relativistic limit where ˜x ≪ 1. Moreover, we see that tensorial perturbations
+are stable in this limit since Qt > 0.
+Scalar stability conditions
+Similar stability conditions apply to the scalar degrees of freedom, here including the
+scalar perturbations of matter components:
+Qs ≡ ω1 (4ω1ω3 + 9ω2
+2)
+3ω2
+2
+> 0
+(28)
+c2
+s ≡ 3 (2ω2
+1ω2H − ω2
+2ω4 + 4ω1ω2 ˙ω1 − 2ω2
+1 ˙ω2) − 6ω2
+1
+� (1 + wi) ρi
+ω1 (4ω1ω3 + 9ω2
+2)
+≥ 0
+(29)
+where wi and ρi are respectively the equation of state parameter and the energy density
+of the fluid i, and the sum runs over all the components of the Universe (here only
+pressureless matter and radiation). At the lowest order in ˜x, we get:
+Qs ≃ 3
+2(Ω0
+Λ − ¯H2)˜x2
+(30)
+c2
+s ≃ 1 + 2
+�
+η ¯H2˜x′ − ¯H ¯H′ − 2ξ ¯H4˜x′�
+3
+�
+Ω0
+Λ − ¯H2�
+(31)
+With ˜x ≪ 1, a fit of the DBI-Galileon model to data leads to cosmological
+parameters close to the standard model ones: Ω0
+m ≈ 0.3 and Ω0
+Λ ≈ 0.7 [28]. Therefore,
+from the first Friedmann equation, we get Ω0
+Λ < ¯H2 for all relevant models in agreement
+with cosmological observations. As Qs ≤ 0, the DBI-Galileon model contains scalar
+instabilities unless it reduces to GR. One way to avoid this would be to add a spatial
+curvature to the metric, but with a strong energy density (at least ∼ 0.3) which is also
+excluded by observations [28].
+5. Discussion
+Physical interpretation
+From the definition (28), we see that the dominant terms come G2 (giving the Ω0
+Λ term)
+and G4 (giving the ¯H2 term). The competition between the two terms leads to the
+ghost-like behaviour in a cosmological setting: Qs ≤ 0. In other words, it is the result
+of the competition between the DBI and the Einstein-Hilbert terms. The DBI action
+will have the effect of stretching the brane towards an extremal surface, whereas the
+Einstein-Hilbert term on the brane will tend to make the brane contract on itself from
+the effect of curvature. However, in the non-relativistic limit of the DBI-Galileon, the
+Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect
+from the cosmological constant is not strong enough to counterbalance, leading to an
+instantaneous decay of the vacuum state.
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+9
+Because the DBI-Galileon action is the most general one can find of a 4D probe
+brane in a 5D bulk, we expect this statement to be quite general for all such theories
+studied in the current context. Indeed, this is true in a standard cosmological setting
+which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW
+background on the brane). Therefore, the only way to evade this ghostly behaviour in
+cosmology is to include the full relativistic dynamics of the theory (˜x ∼ 1). We have
+seen that, in this case, we expect significant deviations of the speed of gravitational
+waves ct from c. This is not a definitive impossibility though if the full DBI-Galileon
+is viewed as an effective theory valid only at cosmological scales for which the speed
+of gravitational waves has not been probed [29, 30].
+Indeed, the constraint on the
+gravitational speed from the observation of GW170817 in coincidence with GRB170817A
+[24] is only valid on small scales probed by LIGO and Virgo. A modification of the
+dispersion relation of gravitational waves at small scales from operators present in the
+UV complete theory could allow ct ̸= c on cosmological scales while being compatible
+with current astrophysical observations. Waiting for the next generation of gravitational
+wave interferometers, in particular LISA, which will be able to probe this relation at
+larger scales [31], this possibility remains open.
+Direct coupling to matter
+In the context of cosmology, where standard model matter is present, there might be
+direct coupling to the scalar field. In that case, the metric ˜q to which matter is sensitive
+is different than the space-time metric q:
+S =
+�
+dx4√−q (Lf + LK + LR + LKGB) +
+�
+dx4�
+−˜qLm (˜qµν, ψm) .
+(32)
+It has been shown in [32] that the two metrics are related by a disformal transformation
+of the following form:
+qµν = A
+�
+ϕ, ˜X
+�
+˜qµν + B
+�
+ϕ, ˜X
+� ∂µϕ∂νϕ
+f 4
+(33)
+where A and B are arbitrary functions of the scalar field and ˜X = −˜qµν∂µϕ∂νϕ/2f 4.
+For simplicity and following the treatment of the covariant Galileon [22], we assume
+that A and B are constant parameters. This can be further justified by the fact that,
+a dependency on X would introduce, in general, higher order terms which would go
+beyond the framework of Horndeski theories [33], and a dependency on ϕ would, in
+general, break the shift symmetry followed by the scalar field ϕ in the probe brane
+context. Note that, when A = −B, matter is coupled to the induced metric on the
+brane.
+Contrary to the covariant Galileon, the DBI-Galileon action is not invariant by
+such a change of reference frame. However, new terms that can not be absorbed into
+a redefinition of the parameters arise only at higher order in ˜X. Therefore, the non-
+relativistic dynamics is not change by the introduction of a direct coupling between the
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+10
+scalar field and matter of the form (33) with constant parameters. In particular, this
+does not prevent the perturbations around the FLRW background from showing ghost
+instabilities.
+Generalization
+The DBI-Galileon is a particular example of the more general class of shift-symmetric
+Horndeski theories. These are subclass of Horndeski theories which are invariant under
+a shift symmetry of the scalar field ϕ → ϕ + c [34, 35]. In these theories, the arbitrary
+Horndeski functions are restricted to be functions of X alone. In order to make the
+non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR:
+G2 ≡ Λ +
++∞
+�
+n=1
+g(n)
+2 Xn
+(34)
+G3 ≡
++∞
+�
+n=1
+g(n)
+3 Xn
+(35)
+G4 ≡ M 2
+P
+2
++
++∞
+�
+n=1
+g(n)
+4 Xn
+(36)
+G5 ≡
++∞
+�
+n=1
+g(n)
+5 Xn
+(37)
+The constant terms in G3 and G5 do not appear in the expansion as they lead
+to total derivative terms. Because the Horndeski functions depend only on X, the ω
+functions that determine the stability conditions reduce to:
+ω1 ≡ 2G4 − 2X
+�
+2G4,X + ˙φHG5,X
+�
+(38)
+ω2 ≡ 4HG4 − 2X
+�
+˙φG3,X + 8HG4,X + 5 ˙φH2G5,X
+�
+− 4X2H
+�
+4G4,XX + ˙φHG5,XX
+�
+(39)
+ω3 ≡ − 18H2G4 + 3X
+�
+G2,X + 12 ˙φHG3,X + 42H2G4,X + 30 ˙φH3G5,X
+�
++ 6X2 �
+G2,XX + 3 ˙φHG3,XX + 48H2G4,XX + 13H3 ˙φG5,XX
+�
++ 12X3H2 �
+6G4,XXX + H ˙φG5,XXX
+�
+(40)
+ω4 ≡ 2G4 − 2X ¨φG5,X
+(41)
+From these, we can compute the quantity Qs up to first order in X:
+Qs = X
+H2
+�
+g(1)
+2
++ 6H2g(1)
+4
+�
++ O
+�
+X
+3
+2
+�
+(42)
+This leads to a very simple formulation of the no-ghost condition, independent of
+X, in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit:
+g(1)
+2
++ 6H2g(1)
+4
+> 0
+(43)
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+11
+In the context of the brane galileon, where g(1)
+4
+= −M 2
+P/2 and g(1)
+2
+= Λ, this is
+equivalent to the inequality which is never fulfilled in flat space:
+Λ − 3M 2
+PH2 > 0
+⇔
+Ω0
+Λ > ¯H2
+(44)
+Other stability conditions are given by:
+c2
+s ≃ 1 + 2¨φg(1)
+3
++ 4 ˙Hg(1)
+4
++ 2¨φH2g(1)
+5
+g(1)
+2
++ 6H2g(1)
+4
+> 0
+(45)
+Qt ≃ M 2
+P
+4
+(46)
+c2
+t ≃ 1
+(47)
+where we expressed these quantities at the lowest order. The two tensorial conditions
+are, thus, automatically satisfied in this context.
+On the other hand, the stability
+conditions for scalar perturbations at the lowest order give a simple inequality involving
+the parameters of the Taylor expansion, that can be easily checked at the background
+level.
+6. Conclusion
+We described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk
+space-time in the non-relativistic limit where its local kinetic energy is small compared
+to its tension. This model belongs to the class of shift-symmetric Horndeski theories,
+themselves being a subclass of the more general family of Horndeski theories. From
+the construction of the DBI-Galileon model, the free parameters of the model acquire a
+physical meaning. In particular, the interpretation of the cosmological constant is linked
+to the brane tension energy density. We derived the equations driving the evolution
+of the late-time Universe around a spatially flat FLRW cosmological background and
+studied the stability of scalar and tensorial perturbations.
+This model reduces to
+an expansion around standard GR, and therefore around standard ΛCDM in the
+cosmological context. As such, it is naturally compatible with data in the non-relativistic
+limit provided the effect of the scalar field is small enough, even considering the speed
+of the gravitational waves.
+However, it revealed fatal ghostly behaviour for scalar
+perturbations around the FLRW background. From there, we derived the corresponding
+stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit
+in the cosmological context and found very simple formulations for these conditions.
+Acknowledgements
+We would like to thank Marc Besan¸con, Arnaud de Mattia and Vanina Ruhlmann-
+Kleider for their comments on the present paper. We also want to thank David Langlois
+for useful and interesting comments and suggestions.
+
+Instability of the cosmological DBI-Galileon in the non-relativistic limit
+12
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+

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+page_content=' Leloup1,2, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Heitz3 and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Neveu3,4  1 Universit´e Paris-Cit´e, CNRS, Astroparticule et Cosmologie, 75013 Paris, France  2 Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU,  WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan  3 Universit´e Paris-Saclay, CNRS, IJCLab, 91405, Orsay, France  4 Sorbonne Universit´e, CNRS, Universit´e de Paris, LPNHE, 75252 Paris Cedex 05,  France  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The DBI-Galileon model is a tensor-scalar theory of gravity which finds its  foundation as the most general theory of the dynamics of a 4D brane embedded in  a 5D bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  It is of particular interest as it provides a few free parameters with a  physical meaning, such as the cosmological constant which is there related to the  brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Most studies of this model have been performed assuming a maximally  symmetric geometry for the 5D bulk, in which it has been shown that the theory  reduces to various types of Galileon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In contrast, the general case for the geometry of  the bulk provides a different covariantization of the Galileon model than the covariant  Galileon: the DBI-Galileon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' From the tight constraints on the gravitational waves  speed, we are naturally led to consider the non-relativistic limit of the model where  the kinetic energy of the brane is small compared to its tension, that we study in  the context of late-time cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The DBI-Galileon in the non-relativistic limit  is simply an expansion around General Relativity (GR) which can be expressed as  a shift-symmetric Horndeski theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We developed the description of this theory at  the background and perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' However, by studying the scalar and tensor  perturbations around a flat FLRW background, we found that they contain a ghost  degree of freedom leading to fatal instability of the vacuum for every combination of the  free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' As a lesson, we emphasized which of the Horndeski terms competes  to avoid this instability in more general cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Introduction  Dark energy has been modelled by a large variety of theories since decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Among  these, many rely on the introduction of additional scalar fields whose dynamics, at the  origin of the late-time acceleration of the expansion of the Universe, is determined by  arbitrary parametric functions, potentials and/or coupling (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' These are the  so-called scalar-tensor theories of modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In particular, the class of Horndeski  theories is of great interest as it contains all models of modified gravity with a single  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='01723v1  [hep-th]  4 Jan 2023  Instability of the cosmological DBI-Galileon in the non-relativistic limit  2  additional scalar field leading to second-order equations of motion [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Extensions of  Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of  higher orders have also been explored [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Particular Horndeski theories are described  by the specification of four arbitrary functions of the scalar field and its kinetic energy,  leading to a huge variety of models and phenomenological behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Among these wide classes of models, some can be built from first physical principles  or arguments of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  For instance, the Galileon model [6] and its covariant  extension [7] was built by imposing a galilean symmetry for the scalar field, leaving  only five free numerical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We can also cite, among many others, the pure  kinetic gravity theory [8], massive gravity in the non-relativistic limit [9, 10] and the  DBI-Galileon [11] which is the main object of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The DBI-Galileon is a model that falls into the class of Brane-world scenarios of  extra-dimension theories, where the matter fields are confined on a 4D brane while  gravity can propagate into the additional spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Of most interest for the  DBI-Galileon is the case of a single extra-dimension as it has been shown that theories  with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter  solution [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The action include a volume term for the 4D brane in the 5D bulk which  leads to the well-known Dirac-Born-Infeld (DBI) action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  This action, and DBI-like  extensions, can lead to a self-accelerating solution and has been thoroughly studied as  a candidate model in the early Universe cosmic inflation paradigm [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In addition,  the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit  [11] but giving a physical meaning to their free parameters: the Planck mass in the  brane, the Planck mass in the bulk, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In particular, the brane tension here plays the  role of the cosmological constant which brings a possible interpretation of its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The original probe brane construction has been revisited in [15] where the matter  metric is disformally related to a standard gravitational metric, or in [16] in the  framework of spontaneous symmetry breaking for the 5D space-time symmetries broken  by the presence of the brane, bridging the gap with Brane-world scenarios developed  in the context of quantum field theory and an interpretation of the scalar field as a  Nambu-Goldstone boson [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The DBI-Galileon model has been studied extensively  in special cases of the maximally symmetric bulk geometry [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' However, to our  knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential  candidate for Dark Energy has been performed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  In this paper we develop the DBI-Galileon theory in the non relativistic limit  (Section 2) and study its dynamics in the Friedmann-Lemaˆıtre-Robertson-Walker flat  metric (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The perturbation stability is explored in Section 4 and then discussed  in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' DBI-Galileon in the non-relativistic limit  DBI-Galileon  We are interested in the description of a four dimensional brane universe embedded in  Instability of the cosmological DBI-Galileon in the non-relativistic limit  3  a five dimensional bulk from the cosmological perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In this context,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' it has been  shown in [11] that the most general action on the brane is given by the 4D Lovelock  terms [21] inside the brane and the boundary terms associated to the 5D Lovelock terms  in the bulk:  S =  �  dx4√−g  �  −Λ − M 3  5K + M 2  P  2 R − β M 3  5  m2 KGB + Lm (˜qµν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' ψm)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (1)  where g is the induced metric on the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Λ the brane cosmological constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' K is the  extrinsic curvature of the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' R is the Ricci scalar on the brane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' KGB is the boundary  term on the brane of the Gauss-Bonnet scalar in the bulk and Lm is the Lagrangian  density of matter that lives confined in the brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In the above action has been defined  the 4D Planck mass MP, its 5D counterpart M5 and their ratio m = M 3  5/M 2  P, and β is  an arbitrary constant parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Because the action is defined on the 4-dimensional brane, the above quantities are  expressed in terms of the induced metric g, as opposed to the 5-dimensional bulk metric  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Assuming we have a coordinate system (xµ, y) in the bulk, Greek letters being defined  to span from 0 to 3, such that the length element in this frame is ds2 = qµνdxµdxν+(dy)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The brane position in the bulk is defined by y = π (xµ) such that we can express the  induced metric from the bulk metric:  gµν = qµν + ∂µπ∂νπ  and  gµν = qµν − γ2∂µπ∂νπ  (2)  with the Lorentz factor γ =  �  1 + (∂π)2�−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' From this expression of the induced metric,  we can explicitly write the action (1) using the bulk metric q and the scalar field π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' As  an illustration, see how the cosmological constant part of the action on the brane can  be expressed:  SΛ = −Λ  �  d4x√−g = −Λ  �  d4x√−q  �  1 + (∂π)2  (3)  As was pointed out in the original paper by de Rham and Tolley [11], we recover a DBI  term in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' This leads us to associate the cosmological constant Λ with a brane  tension f, following Λ = f 4 for unit convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Non-relativistic limit  If the derivatives of the field vanish, ∂µπ = 0, then we recover standard GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Because  we are interested in the cosmological setting, where predictions from the ΛCDM model  based on GR match precisely a large range of observations, we consider only small  corrections to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Therefore, we consider the DBI-Galileon in the so-called non-  relativistic limit where (∂π)2 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In particular, the DBI part of the action in the  non-relativistic limit becomes:  Sf = −f 4  �  d4x√−q  �  1 + ∇µπ∇µπ  2  − (∇µπ∇µπ)2  8  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  �  (4)  Instability of the cosmological DBI-Galileon in the non-relativistic limit  4  We see that, to have a canonically normalized field, we can proceed to the field  redefinition π → ϕ = f 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Up to degree 5 in ∂ϕ/f 2, we find the following Lagrangian  operators for the DBI-Galileon in the non-relativistic limit:  Lf = 1 + L2  2f 4 − X2  2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (5)  LK = − L3  2f 6 − 2X  f 6 [ψ] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (6)  LR = ¯R + 1  f 4  �  [Φ]2 −  �  Φ2�  − f 4X  2  ¯R − 2 ¯Rµν∇µϕ∇νϕ  �  + L4  4f 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (7)  LKGB = 2  f 6  ��  − ¯Rµν [Φ] + 2 ¯RµρΦρ  ν + ¯RµρνλΦρλ�  ∇µϕ∇νϕ + 1  3 [Φ]3 − [Φ]  �  Φ2�  + 2  3  �  Φ3�  −1  2  ¯R [ψ]  �  + L5  3f 10 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (8)  We defined on the above expressions the tensor Φµν = ∇µ∇νϕ, and the three scalars  [Φ] = Φµ  µ, [ψ] = ∂µϕ·Φµν·∂νϕ and X = − (∂ϕ)2 /2f 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Furthermore, the L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='5 Lagrangian  operators are the covariant Galileon model Lagrangian operators as defined in [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Therefore, the theory described here is a generalization of the Galileon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We can  note that the additional terms in LR and LKGB compared to the Galileon Lagrangian  operators vanish in the particular case of a flat geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We conclude that the DBI-  Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon  than the covariant Galileon [7] or dRGT massive gravity [9, 10], with an additional  terms in Lf and another one in LK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' That point noted, we will stay at leading order  in X in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Being a theory of a scalar field interacting with a metric, with  equations of motion of at most second order, it can be described as a Horndeski theory  [2, 3] with the following Horndeski functions:  G2 = A (ϕ) − f 4 (1 − X + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=')  (9)  G3 = M 3  5  f 2 (X + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=')  (10)  G4 = M 2  P  2  �  1 − X − X2/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  �  (11)  G5 = −2β M 3  5  m2f 2  �  X + X2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  �  (12)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Cosmological background evolution  We expand the dynamics of the fields around a flat FLRW background on the 4D slice  of the bulk along xµ where the properties of the brane, gµν and ϕ, are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' On this  slice, the length element is:  ds2 = qµνdxµdxν = −dt2 + a2 (t) δijdxidxj  (13)  It might appear more natural to expand around a flat FLRW background on the  brane with its induced metric gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' However, the two metrics qµν and gµν are related  Instability of the cosmological DBI-Galileon in the non-relativistic limit  5  by a disformal transformation involving the scalar field which, at the background level,  depends only on the physical time:  ¯gµν = ¯qµν + ( ˙¯ϕ (t))  2  f 4  δµ0δν0  (14)  where barred quantities are taken at the background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Therefore, the geometry on  the brane is also of the FLRW type, but with a different definition for the physical time,  leading to a different scale factor and expansion history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Because in our case ˙¯ϕ ≪ f 2,  the expansion history on the 4D slice and on the brane will, then, be approximately the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In addition, equations (9)-(12) determine a self-consistent Horndeski theory of  gravity with the metric qµν and the scalar field ϕ without referring to the induced metric  gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Thus, in the following we apply the well-known techniques used in the context of  Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  In the non-relativistic limit, action (1) reads:  S =  �  dx4√−q (Lf + LK + LR + LKGB + Lm (qµν, ψm)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (15)  We define Ω0  m and Ω0  r the standard present energy density parameters for pressureless  matter and radiation respectively, and ¯H the normalized Hubble rate H/H0 with H0  the present Hubble constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Prime symbol denotes the derivative with respect to ln a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  We set:  ˜x = ϕ′H0  f 2 ,  Ω0  Λ =  Λ  3H2  0M 2  P  =  f 4  3H2  0M 2  P  ,  η =  M 3  5  M 2  PH0  ,  ξ = β  η ,  κ = MPH0  f 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (16)  Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' the two Friedmann equations derived from action (15) are:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2 = −3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x2 − 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H6˜x4 + η ¯H4˜x3 − 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H6˜x3 − 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H8˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='+ Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='a3 + Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='a4 + Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='1 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2˜x2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(17)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2 + 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H ¯H′ = −2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3ξ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2 ¯H6˜x3 + 5 ¯H5˜x3 ¯H′ + 3 ¯H6˜x2˜x′ + 2 ¯H8˜x5 + 5 ¯H8˜x4˜x′ + 7 ¯H7˜x5 ¯H′�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x2 − ¯H3˜x2 ¯H′ − 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x˜x′ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3η ¯H3˜x2( ¯H˜x)′ − 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H6˜x4 − 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H5˜x4 ¯H′ − ¯H6˜x3˜x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='− Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3a4 + Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='1 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2˜x2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(18)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='We see that we recover the ΛCDM equations when setting ˜x to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' but with a physical  interpretation of the cosmological constant as the brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  The DBI model  proposed here is then an extension of the standard model of cosmology, and as such  follows a late accelerated expansion but with a physical interpretation of the origin of  Λ as a brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Using the same methodology and similar notations as in [22], we  derive the field ϕ equation of motion from action (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' This leads to a system of two  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Instability of the cosmological DBI-Galileon in the non-relativistic limit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='coupled equations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='˜x′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='= −˜x + αλ − σγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='σβ − αω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='= ωγ − βλ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='σβ − αω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(19)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='with  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='β = 2 ¯H4 + 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H6˜x2 − Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ ¯H2 − 2η ¯H4˜x + 4ξ ¯H6˜x + 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H8˜x3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='α = 6 ¯H3˜x − Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ ¯H˜x + 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H5˜x3 − 3η ¯H3˜x2 + 10ξ ¯H5˜x2 + 70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H7˜x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='γ = 4 ¯H4˜x − 2Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ ¯H2˜x − η ¯H4˜x2 + 2ξ ¯H6˜x2 − 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H8˜x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='ω = 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x + ¯H6˜x3 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3η ¯H4x2 + 2ξ ¯H6˜x2 + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H8˜x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='σ = 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H + 2 ¯H3˜x2 + 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H5˜x4 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3η ¯H3˜x3 + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H5˜x3 + 14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H7˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='λ = ¯H2 + Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3a4 − Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ ¯H2˜x2 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x2 − 5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H6˜x4 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3η ¯H4˜x3 − 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3ξ ¯H6˜x3 − 2ξ ¯H8˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Given values for the parameters Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' ξ and κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' and initial conditions ˜x0 and H0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  this system can be integrated to compute background cosmology observables like the  distance moduli of type Ia supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In Figure 1 we illustrate this with a Hubble  diagram prediction compared with recent type Ia supernova data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' As an initial  condition for ˜x, we chose to set ˜x0 = 6 × 10−8 today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  This is the maximum value  allowed by the constraint on gravitational wave speed (see Section 4) coming from the  quasi simultaneous observation of photons and gravitational waves after neutron star  merger event GW170817A [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Nevertheless, before discussing more the cosmological  scenarios proposed by the DBI-Galileon model, stability conditions much be computed  first to assess the viability of the models for any set of parameters at the perturbation  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Stability conditions  To be viable as a description of our Universe, the model has to fulfill stability  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  These requirements apply to degrees of freedom propagating around  the fixed background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  to the cosmological perturbations, that are capable of  undermining the stability of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  In the determination of the stability  conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism  described in [25] for Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In our case, these stability conditions are defined  Instability of the cosmological DBI-Galileon in the non-relativistic limit  7  14  16  18  20  22  24  26  Distance moduli  [mag]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='75  Redshift z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='0  Residuals [mag]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Hubble diagram prediction for the non-relativistic DBI-Galileon model for Ω0  Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='7,  ˜x0 = 6 × 10−8, η = ξ = 0 (blue) compared with binned Pantheon data (black points) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We used  M = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='81 for the offset magnitude of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Residuals to the fit are presented in the bottom  panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='from the following quantities derived from the particular Horndeski functions (9) to (12):  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='ω1 = 1 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2˜x2 + 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(20)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='ω2 = 2 ¯H + 3 ¯H3˜x2 + 15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H5˜x4 − η ¯H3˜x3 + 10ξ ¯H5˜x3 + 14ξ ¯H7˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(21)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='ω3 = 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='− ¯H2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2Ω0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Λ ¯H2˜x2 − 3 ¯H4˜x2 + 2η ¯H4˜x3 − 10ξ ¯H6˜x3 − 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H6˜x4 − 56  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 ξ ¯H8˜x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(22)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='ω4 = 1 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H2˜x2 − 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='(23)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='Tensorial stability conditions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='In order to avoid ghosts and Laplacian instabilities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' we impose the following constraints  on the sign of the kinetic term and on the sign of the gravitational waves speed squared:  Qt ≡ ω1  4 = 1  4 + 1  8  ¯H2˜x2 + 3  32  ¯H4˜x4 + 1  2ξ ¯H4˜x3 + 1  2ξ ¯H6˜x5 > 0  (24)  c2  t ≡ ω4  ω1  = 1 − 1  2 ¯H2˜x2 − 1  8 ¯H4˜x4 + 2ξ ¯H3˜x2( ¯H˜x)′ + 2ξ ¯H5˜x4( ¯H˜x)′  1 + 1  2 ¯H2˜x2 + 3  8 ¯H4˜x4 + 2ξ ¯H4˜x3 + 2ξ ¯H6˜x5  ≥ 0  (25)  In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' we see that the gravitational wave speed depends on ˜x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' and tends to  1 when ˜x → 0 :  4Qt ≃ 1 + 2( ¯H˜x)2  (26)  ct ≃ 1 − 1  2( ¯H˜x)2  (27)  Instability of the cosmological DBI-Galileon in the non-relativistic limit  8  Given the very tight constraint on the speed of gravitational waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' equal to the speed  of light up to a ∼ 10−15 difference [24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' 26,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' 27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' this justifies a posteriori the relevance  of the non-relativistic limit where ˜x ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Moreover, we see that tensorial perturbations  are stable in this limit since Qt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Scalar stability conditions  Similar stability conditions apply to the scalar degrees of freedom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' here including the  scalar perturbations of matter components:  Qs ≡ ω1 (4ω1ω3 + 9ω2  2)  3ω2  2  > 0  (28)  c2  s ≡ 3 (2ω2  1ω2H − ω2  2ω4 + 4ω1ω2 ˙ω1 − 2ω2  1 ˙ω2) − 6ω2  1  � (1 + wi) ρi  ω1 (4ω1ω3 + 9ω2  2)  ≥ 0  (29)  where wi and ρi are respectively the equation of state parameter and the energy density  of the fluid i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' and the sum runs over all the components of the Universe (here only  pressureless matter and radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' At the lowest order in ˜x, we get:  Qs ≃ 3  2(Ω0  Λ − ¯H2)˜x2  (30)  c2  s ≃ 1 + 2  �  η ¯H2˜x′ − ¯H ¯H′ − 2ξ ¯H4˜x′�  3  �  Ω0  Λ − ¯H2�  (31)  With ˜x ≪ 1, a fit of the DBI-Galileon model to data leads to cosmological  parameters close to the standard model ones: Ω0  m ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3 and Ω0  Λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='7 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Therefore,  from the first Friedmann equation, we get Ω0  Λ < ¯H2 for all relevant models in agreement  with cosmological observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' As Qs ≤ 0, the DBI-Galileon model contains scalar  instabilities unless it reduces to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' One way to avoid this would be to add a spatial  curvature to the metric, but with a strong energy density (at least ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='3) which is also  excluded by observations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Discussion  Physical interpretation  From the definition (28), we see that the dominant terms come G2 (giving the Ω0  Λ term)  and G4 (giving the ¯H2 term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The competition between the two terms leads to the  ghost-like behaviour in a cosmological setting: Qs ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In other words, it is the result  of the competition between the DBI and the Einstein-Hilbert terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The DBI action  will have the effect of stretching the brane towards an extremal surface, whereas the  Einstein-Hilbert term on the brane will tend to make the brane contract on itself from  the effect of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' However, in the non-relativistic limit of the DBI-Galileon, the  Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect  from the cosmological constant is not strong enough to counterbalance, leading to an  instantaneous decay of the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Instability of the cosmological DBI-Galileon in the non-relativistic limit  9  Because the DBI-Galileon action is the most general one can find of a 4D probe  brane in a 5D bulk, we expect this statement to be quite general for all such theories  studied in the current context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Indeed, this is true in a standard cosmological setting  which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW  background on the brane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Therefore, the only way to evade this ghostly behaviour in  cosmology is to include the full relativistic dynamics of the theory (˜x ∼ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We have  seen that, in this case, we expect significant deviations of the speed of gravitational  waves ct from c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' This is not a definitive impossibility though if the full DBI-Galileon  is viewed as an effective theory valid only at cosmological scales for which the speed  of gravitational waves has not been probed [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Indeed, the constraint on the  gravitational speed from the observation of GW170817 in coincidence with GRB170817A  [24] is only valid on small scales probed by LIGO and Virgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' A modification of the  dispersion relation of gravitational waves at small scales from operators present in the  UV complete theory could allow ct ̸= c on cosmological scales while being compatible  with current astrophysical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Waiting for the next generation of gravitational  wave interferometers, in particular LISA, which will be able to probe this relation at  larger scales [31], this possibility remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Direct coupling to matter  In the context of cosmology, where standard model matter is present, there might be  direct coupling to the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In that case, the metric ˜q to which matter is sensitive  is different than the space-time metric q:  S =  �  dx4√−q (Lf + LK + LR + LKGB) +  �  dx4�  −˜qLm (˜qµν, ψm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  (32)  It has been shown in [32] that the two metrics are related by a disformal transformation  of the following form:  qµν = A  �  ϕ, ˜X  �  ˜qµν + B  �  ϕ, ˜X  � ∂µϕ∂νϕ  f 4  (33)  where A and B are arbitrary functions of the scalar field and ˜X = −˜qµν∂µϕ∂νϕ/2f 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  For simplicity and following the treatment of the covariant Galileon [22], we assume  that A and B are constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' This can be further justified by the fact that,  a dependency on X would introduce, in general, higher order terms which would go  beyond the framework of Horndeski theories [33], and a dependency on ϕ would, in  general, break the shift symmetry followed by the scalar field ϕ in the probe brane  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Note that, when A = −B, matter is coupled to the induced metric on the  brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Contrary to the covariant Galileon, the DBI-Galileon action is not invariant by  such a change of reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' However, new terms that can not be absorbed into  a redefinition of the parameters arise only at higher order in ˜X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Therefore, the non-  relativistic dynamics is not change by the introduction of a direct coupling between the  Instability of the cosmological DBI-Galileon in the non-relativistic limit  10  scalar field and matter of the form (33) with constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In particular, this  does not prevent the perturbations around the FLRW background from showing ghost  instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Generalization  The DBI-Galileon is a particular example of the more general class of shift-symmetric  Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' These are subclass of Horndeski theories which are invariant under  a shift symmetry of the scalar field ϕ → ϕ + c [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In these theories, the arbitrary  Horndeski functions are restricted to be functions of X alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In order to make the  non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR:  G2 ≡ Λ +  +∞  �  n=1  g(n)  2 Xn  (34)  G3 ≡  +∞  �  n=1  g(n)  3 Xn  (35)  G4 ≡ M 2  P  2  +  +∞  �  n=1  g(n)  4 Xn  (36)  G5 ≡  +∞  �  n=1  g(n)  5 Xn  (37)  The constant terms in G3 and G5 do not appear in the expansion as they lead  to total derivative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Because the Horndeski functions depend only on X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' the ω  functions that determine the stability conditions reduce to:  ω1 ≡ 2G4 − 2X  �  2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + ˙φHG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X  �  (38)  ω2 ≡ 4HG4 − 2X  �  ˙φG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + 8HG4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + 5 ˙φH2G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X  �  − 4X2H  �  4G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX + ˙φHG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX  �  (39)  ω3 ≡ − 18H2G4 + 3X  �  G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + 12 ˙φHG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + 42H2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X + 30 ˙φH3G5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X  �  + 6X2 �  G2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX + 3 ˙φHG3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX + 48H2G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX + 13H3 ˙φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XX  �  + 12X3H2 �  6G4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XXX + H ˙φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='XXX  �  (40)  ω4 ≡ 2G4 − 2X ¨φG5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='X  (41)  From these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' we can compute the quantity Qs up to first order in X:  Qs = X  H2  �  g(1)  2  + 6H2g(1)  4  �  + O  �  X  3  2  �  (42)  This leads to a very simple formulation of the no-ghost condition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' independent of  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit:  g(1)  2  + 6H2g(1)  4  > 0  (43)  Instability of the cosmological DBI-Galileon in the non-relativistic limit  11  In the context of the brane galileon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' where g(1)  4  = −M 2  P/2 and g(1)  2  = Λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' this is  equivalent to the inequality which is never fulfilled in flat space:  Λ − 3M 2  PH2 > 0  ⇔  Ω0  Λ > ¯H2  (44)  Other stability conditions are given by:  c2  s ≃ 1 + 2¨φg(1)  3  + 4 ˙Hg(1)  4  + 2¨φH2g(1)  5  g(1)  2  + 6H2g(1)  4  > 0  (45)  Qt ≃ M 2  P  4  (46)  c2  t ≃ 1  (47)  where we expressed these quantities at the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' The two tensorial conditions  are, thus, automatically satisfied in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  On the other hand, the stability  conditions for scalar perturbations at the lowest order give a simple inequality involving  the parameters of the Taylor expansion, that can be easily checked at the background  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' Conclusion  We described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk  space-time in the non-relativistic limit where its local kinetic energy is small compared  to its tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' This model belongs to the class of shift-symmetric Horndeski theories,  themselves being a subclass of the more general family of Horndeski theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' From  the construction of the DBI-Galileon model, the free parameters of the model acquire a  physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' In particular, the interpretation of the cosmological constant is linked  to the brane tension energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We derived the equations driving the evolution  of the late-time Universe around a spatially flat FLRW cosmological background and  studied the stability of scalar and tensorial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  This model reduces to  an expansion around standard GR, and therefore around standard ΛCDM in the  cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' As such, it is naturally compatible with data in the non-relativistic  limit provided the effect of the scalar field is small enough, even considering the speed  of the gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  However, it revealed fatal ghostly behaviour for scalar  perturbations around the FLRW background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' From there, we derived the corresponding  stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit  in the cosmological context and found very simple formulations for these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content='  Acknowledgements  We would like to thank Marc Besan¸con, Arnaud de Mattia and Vanina Ruhlmann-  Kleider for their comments on the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
+page_content=' We also want to thank David Langlois  for useful and interesting comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9AzT4oBgHgl3EQfwf5z/content/2301.01723v1.pdf'}
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+The younger flagellum coordinates the beating in
+C. reinhardtii
+Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.W. Tam2∗
+1Department of Bionanoscience, Delft University of Technology,
+2628CJ Delft, Netherlands.
+2Laboratory for Aero and Hydrodynamics, Delft University of Technology,
+2628CD Delft, Netherlands.
+3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,
+Chinese Academy of Sciences; Beijing 100190, China.
+†Corresponding author. Email: [email protected];
+∗Corresponding author. Email: [email protected].
+1
+arXiv:2301.13278v1  [physics.bio-ph]  30 Jan 2023
+
+Abstract
+Eukaryotes swim with coordinated flagellar (ciliary) beating and steer by fine-tuning
+the coordination. The model organism for studying flagellate motility, C. reinhardtii (CR),
+employs synchronous, breast-stroke-like flagellar beating to swim, and it modulates the
+beating amplitudes differentially to steer. This strategy hinges on both inherent flagellar
+asymmetries (e.g. different response to chemical messengers) and such asymmetries be-
+ing effectively coordinated in the synchronous beating. In CR, the synchrony of beating is
+known to be supported by a mechanical connection between flagella, however, how flagellar
+asymmetries persist in the synchrony remains elusive. For example, it has been speculated
+for decades that one flagellum leads the beating, as its dynamic properties (i.e. frequency,
+waveform, etc.) appear to be copied by the other one. In this study, we combine experi-
+ments, computations, and modeling efforts to elucidate the roles played by each flagellum
+in synchronous beating. With a non-invasive technique to selectively load each flagellum,
+we show that the coordinated beating essentially responds to only load exerted on the cis
+flagellum; and that such asymmetry in response derives from a unilateral coupling between
+the two flagella. Our results highlight a distinct role for each flagellum in coordination and
+have implication for biflagellates’ tactic behaviors.
+One-Sentence Summary: The younger flagellum of C. reinhardtii coordinates the synchronous
+beating and couples to external forces.
+2
+
+Introduction
+The ability to swim towards desirable environments and away from hazardous ones is funda-
+mental to the survival of many microorganisms. These so-called tactic behaviors are exhib-
+ited by many motile microorganisms ranging from bacteria (1, 2) to larger flagellates and cili-
+ates (3–5). Different microorganisms have developed specific strategies for steering, depending
+on the tactic behavior and on their specific sensory and motility repertoire. For example, bacte-
+ria modulate the tumbling rate (1) while flagellates and ciliates modulate the waveform (6–9),
+amplitude (10, 11) and frequency of their flagellar/ciliary (4, 12) beating. The goal of these
+active modulations of the motility is to achieve a spatially asymmetric generation of propulsive
+force to steer the organism.
+C. reinhardtii (CR), the model organism for studies of flagellar motility, achieves tactic nav-
+igation by a fine-tuned differential modulation on its two flagella. Studying this organism offers
+great opportunities to look into how flagella coordinate with each other and how such coordi-
+nation helps facilitate targeted steering. CR has a symmetric cell body and two near-identical
+flagella inherited from the common ancestors of land plants and animals (13). It swims by
+beating its two flagella synchronously and is capable of photo- and chemotaxis (10, 14). For
+this biflagellated organism, effective steering hinges on both flagellar asymmetry and flagellar
+coordination. On the one hand, the two flagella must be asymmetric to respond differentially
+to stimuli (10,15); on the other hand, the differential responses must be coordinated by the cell
+such that the beating would remain synchronized to guarantee effective swimming. Understand-
+ing this remarkable feat requires knowledge about both flagellar asymmetry and coordination.
+The two flagella are known to be asymmetric in several, possibly associated, aspects. First
+of all, they differ in developmental age (16, 17). The flagellum closer to the eyespot, the cis(-
+eyespot) flagellum, is always younger than the other one, the trans(-eyespot) flagellum. This
+is because the cis is organized by a basal body (BB) that develops from a pre-matured one in
+the mother cell; and this younger BB also organizes the flagellar root (D4 rootlet) that dictates
+the eyespot formation (18). Second, the two flagella have asymmetric protein composition (19–
+21). For example, the trans flagellum is richer in CAH6, a protein possibly involved in CO2
+sensing (14,20). Finally, the flagella have different dynamic properties (22–24). Their beating is
+modulated differentially by second messengers such as calcium (22,23) and cAMP (25). When
+beating alone, the trans beats at a frequency 30%-40% higher than the cis (23,26–28); the trans
+also displays an attenuated waveform (29) and a much stronger noise (29,30).
+3
+
+Remarkably, despite these inherent asymmetries, CR cells establish robust synchronization
+between the flagella. Such coordination enables efficient swimming and steering of the cells
+and takes basis on the fibrous connections between flagellar bases (31,32). Intriguingly, in the
+coordinated beating, both flagella display dynamic properties, i.e., flagellar waveform, beating
+frequency (∼50 Hz), and frequency fluctuation, that are more similar to those of the cis flag-
+ellum (26, 28–30, 33). This has led to a long-standing hypothesis that ”the cis somehow tunes
+the trans flagellum” (26). This implies that the symmetric flagellar beating (”breast-stroke”)
+observed is the result of interactions between two flagella playing differential roles in coordi-
+nation. How does the basal coupling make this possible? Recent theoretical efforts show that
+the basal coupling can give rise to different synchronization modes (34–36); and that flagellar
+dynamics, such as beating frequency, may simply emerge from the interplay between mechan-
+ics of basal coupling and bio-activity (36). Yet, most theoretical efforts examining flagellar
+synchronization have assumed two identical flagella, limiting the results’ implication for the
+realistic case. Moreover, little experiments directly probe the flagella’s differential roles during
+synchronous beating (37). Therefore, flagellar coordination in this model organism remains un-
+clear. To clarify the picture experimentally, one needs to selectively force each flagellum, and
+characterize the dynamics of the flagellar response.
+In this study, we address this challenge and devise a non-invasive approach to apply external
+forces selectively on the cis- or the trans-flagella. Oscillatory background flows are imposed
+along an angle with respect to the cell’s symmetry axis. Such flows result in controlled hydro-
+dynamic forces, which are markedly different on the two flagella. With experiments, hydrody-
+namic computations, and modeling, we show definitively that the two flagella are unilaterally
+coupled, such that the younger flagellum (cis) coordinates the beating, whereas the elder one
+simply copies the dynamic properties of the younger. This also means that only external forces
+on the cis may mechanically fine-tune the coordination. We also study the effect of calcium
+in the cis’ leading role as calcium is deeply involved in flagellar asymmetry and hence photo-
+tactic steering. In addition, a well-known mutant that lacks flagellar dominance (ptx1) (23,38)
+is examined. Results show that the coordinating role of cis does not need environmental free
+calcium, whereas it does require the genes lost in ptx1. Our results discern the differential roles
+of CR’s flagella, highlight an advanced function of the inter-flagellar mechanical coupling, and
+have implications for biflagellates’ tactic motility.
+4
+
+Experimental scheme for selective loading
+We set out to establish a non-invasive experimental technique that exerts differential loads on the
+flagella of CR. Following the study by Quaranta et al. (31), we induce oscillatory background
+flows to exert hydrodynamic forcing to flagella of captured cells. With programmed oscillations
+of the piezoelectric stage, the amplitude, frequency, and direction of the background flows are
+all controlled, enabling selective loading.
+To quantitatively estimate the selectivity of the flows along different angles (θ), we compute
+the flagellar loads under the flows along θ = −45◦, 0◦, and 45◦, see Fig. 1A. Computations
+based on boundary element methods (BEM) and slender-body theory (SBT) give the real-time
+drag force F on each flagellum and the power P exerted by the viscous forces on each flagellum.
+For given realistic flagellar shapes, we compare computed loads with and without external flows.
+From these we isolate the loads from the induced flows FFlow and PFlow (Methods).
+Loads on each flagellum under flows of θ = 0◦, −45◦, 45◦ are presented in Fig. 2. Upper
+panels display the magnitude of the drag force FFlow = |FFlow|; while lower panels show viscous
+power PFlow. Force magnitudes are scaled by F0 = 6πµRU0 = 9.9 pN; while the powers by
+P0 = F0U0 = 1.1 fW. F0 is the Stokes drag on a typical free-swimming cell (radius R = 5 µm,
+speed U0 = 110 µm/s, water viscosity µ = 0.95 mPa·s).
+Evidently, along θ = 0◦, flows load the flagella equally (Fig. 2A). However, at θ = −45◦,
+flows load the cis flagellum ∼ 2 times larger than the trans (Fig. 2B, F c
+Flow ≈ 2F t
+Flow); whereas
+flows at θ = 45◦ do the opposite (Fig. 2C). The selectivity also manifests in (the absolute values
+of) PFlow. We do notice that flows along θ = +45◦ are able to synchronize the flagella with
+PFlow < 0, meaning that the flagella are working against the flows, and this shall be discussed
+in later sections.
+Hereon forward, we refer to θc-flows, flows for which θ = −45◦ and the cis-flagellum is
+selectively loaded. Likewise, θt-flows denote flows on θ = +45◦ that selectively load the trans.
+θa-flows denote the axial flow along θ = 0◦. We next introduce how we quantify the flows’
+effective forcing strength (ε) on the cell.
+Phase dynamics of flagellar beating is extracted from videography (31,39,40). Recordings
+are masked and thresholded to highlight the flagella (Fig. 1B-C). Then the mean pixel values
+over time within two sampling windows (Fig. 1D) are converted to observable-invariant flagellar
+phases (41), Fig. 1E. Throughout this study, as cis and trans always beat synchronously (Fig. 1E
+inset), their phases ϕc,t are used interchangeably as the flagellar phase ϕ. The flagellar phase
+5
+
+dynamics under external periodic forcing is described by Adler equation (42–44):
+d∆ϕ
+dt
+= −2πν − 2πε sin(∆ϕ) + ζ(t).
+(1)
+∆ϕ = ϕ − 2πfft is the phase difference between the beating and the forcing, with ff the
+forcing frequency, and ε the forcing strength. The detuning ν = ff − f0 is the frequency
+mismatch between the beating (f0) and forcing. ζ(t) represents a white noise that satisfies
+⟨ζ(τ + t)ζ(τ)⟩ = 2Teffδ(t), with Teff an effective temperature and δ(t) the Dirac delta function.
+When the forcing strength outweighs the detuning (ε > |ν|), synchronization with the flow
+(d∆ϕ/dt = 0) emerges, see the plateaus marked black in Fig. 1F. We characterize synchro-
+nization with τ = tsync/ttot, where tsync is the total time of flow synchronization and ttot the
+flow duration. Fig. 1F presents the phase dynamics which are representative and range from:
+no synchronization (τ=0, i), unstable synchronization (0 < τ < 1, ii-iii), and stable synchro-
+nization (τ=1, iv). In this study, the frequency range in ν for which τ ≥ 0.5 is used to measure
+ε (see Fig. 1F inset). This method is equivalent to previous fitting-based methods (28, 31), see
+SM. Sec.S1.
+Asymmetric susceptibility to flow synchronization
+Now we examine cell responses to flows of various amplitudes and along different directions.
+First we explore flow synchronization over a broad range of amplitudes and frequencies. θa-
+flows with frequencies ff ∈ [40, 75] Hz and amplitudes U ∈ [390, 2340] µm/s are imposed. The
+scanned range covers reported intrinsic frequencies of both the cis and trans flagellum (22,24,
+26, 27); while the amplitude reaches the maximum instantaneous speed of a beating flagellum
+(∼ 2000 µm/s). Fig. 3A displays the resultant flow-synchronized time fractions τ. Up until the
+strongest flow amplitude, the large forces cannot disrupt the synchronized flagellar beating. In
+addition, synchronization is never established around frequencies other than f0. This shows that
+the inter-flagella coupling is much stronger than the maximum amplitude of forcing.
+Next we examine the synchronization with the θc-flows and θt-flows. Flows of a fixed
+amplitude (∼ 7U0) but varying frequencies around f0 are applied to each captured cell (see
+Methods). With these, the flow-synchronized time fraction τ as a function of the detuning (ν)
+and flow direction (θc,a,t) is recorded and helps quantify the flows’ effective forcing ε(θ).
+Comparing τ(ν; θc) to τ(ν; θt), with τ(ν; θa) as reference, we find that θc-flows are the most
+effective in synchronizing the beating (Fig. 3B). We illustrate this point with the profiles of an
+6
+
+exemplary cell (Fig. 3B inset). First, although both the θc-flow (red) and the θt-flow (blue)
+can synchronize the cell at small detunings (|ν| <0.5Hz), the θc-flow maintains the synchro-
+nization for the whole time ( τ(θc) =1), while the θt-flow for a slightly smaller time fraction
+( τ(θc) ≈0.85). This is due to phase-slips (step-like changes in ∆ϕ(t) in Fig. 1F) between flag-
+ella and the flow, and means that the θt-flow synchronization is less stable. Additionally, for
+intermediate detuning (0.5 Hz< |ν| <4 Hz), τ(θc) is always larger than τ(θt) . In some cases,
+the θc-flow synchronizes the cell fully whereas the θt-flow fails completely (e.g., at ν = −2
+Hz). Together, these results imply that a flow of given amplitude synchronizes flagellar beating
+more effectively if it selectively loads the cis.
+We repeat the experiments with cells from multiple cultures, captured on different pipettes,
+and with different eyespot orientations (∼50% heading rightward in the imaging plane) to rule
+out possible influence from the setup. τ(ν; θ) of N=11 wt cells tested in the TRIS-minimal
+medium (pH=7.0) are displayed in Fig. 3B (labeled as ”TRIS”). On average, ε(θc) = 2.9 Hz
+and is 70% larger than ε(θt) = 1.7 Hz. It bears emphasis that ε(θc) > ε(θt) holds true for every
+single cell tested (11/11). In Fig. 3C, we show this by representing each cell as a point in the
+ε(θc) - ε(θt) plane. A point being below the first bisector line ( ε(θc) = ε(θt) ) indicates that
+ε(θc) > ε(θt) for this cell. All cells cluster clearly below the line. This asymmetry manifest
+equivalently through τ. In Fig. 3D, each point represents the time fractions of the same cell
+synchronized by the θc-flow and the θt-flow at the same frequency. Most points (>90%) are
+below the first bisector line, meaning that τ(θc) > τ(θt) . Altogether, all results show that
+selectively loading the cis flagellum establishes synchronization with the flow more effectively,
+pointing to cis and trans playing differential roles in the coordinated beating.
+We next study whether this newly observed cis-trans asymmetry is affected by calcium
+depletion. Calcium is a critical second messenger for modulating flagellates motility and is
+deeply involved in phototaxis (45). The depletion of the free environmental calcium is known
+to degrade flagellar synchronization and exacerbate flagellar asymmetry (22). Here we focus
+on whether calcium depletion affects the asymmetry ε(θc) > ε(θt) . We deplete environmental
+calcium by EGTA-chelation, following the protocol in Ref. (46). Similar to previous reports (22,
+47), the number of freely swimming cells drops significantly in EGTA-containing medium.
+However, the remaining cells beat synchronously for hours after capture. For these beating cells,
+calcium depletion is first confirmed by characterizing their deflagellation behavior. Indeed,
+calcium depletion is reported to inhibit deflagellation (28, 48). In experiments with standard
+calcium concentration, all cells deflagellated under pipette suction (20/20). For experiments
+7
+
+conducted in calcium depleting EGTA-containing medium, we observe deflagellation to occur
+in none but one cell (1/19).
+After confirming the calcium depletion, we perform the same sets of flow synchronization
+experiments. The dashed lines in Fig. 3B show the median synchronization profiles τ(ν; θ)
+(N=6 cells, labeled as ”EGTA”). The flagellar asymmetry is unaffected, see also Fig. 3E. Note
+that ε(θc) > ε(θt) again applies for every single cell tested. The mean values of ε drop slightly.
+However, the different effectiveness between θc-flows and θt-flows, ε(θc) − ε(θt) , is not af-
+fected, see Fig. 3E inset.
+Finally, we determine how the forcing strength of the flow depends on the hydrodynamic
+forces exerted by the flow on the flagella. We compute the hydrodynamic beat-averaged loads,
+F Flow =
+� 2π
+0
+FFlowdϕ/2π, P Flow =
+� 2π
+0
+PFlowdϕ/2π, induced by the flow on the trans and on
+the cis flagella, see the horizontal lines in Fig. 2. These loads are computed for the θc-flow,
+θt-flow, θa-flow and we also include experiments and computations performed with flows along
+θ = 90◦ (circles), see SM. Sec.S2. Fig. 3F and G represent ε as a function of the loads on the
+cis and trans flagellum respectively, with each symbol representing one of the four different
+flow directions, see the drawings. We find that the effective forcing strength scales with the
+time-averaged drag on the cis, ε ∼ F
+c
+Flow, while we find no such correlation between ε and
+F
+t
+Flow. The linear relation between ε and F
+c
+Flow has an intercept near zero (ε|F c
+Flow=0 ≈ 0).
+Given the total forces on both flagella (F
+c
+Flow + F
+t
+Flow) for these flows remains almost constant
+(0.74-0.79F0), the zero-intercept implies that for a hypothetical flow that exerts no load on the
+cis but solely forces the trans, it will not be able to synchronize the cell at all. This suggests a
+negligible contribution of the forcing on the trans in establishing synchronization with flows.
+The asymmetry is lost in ptx1 mutants
+Furthermore, we examine the flagellar dominance mutant ptx1. In this mutant, both flagella re-
+spond similarly to changes of calcium concentrations (38) and have similar beating frequencies
+when demembranated and reactivated (23).
+Ptx1 mutants have two modes of coordinated beating, namely, the in-phase (IP) synchro-
+nization and the anti-phase (AP) synchronization (29, 49). First, we apply θa-flow in the same
+frequency and amplitude ranges as for wt cells. We find that the IP mode around f0 ≈ 50 Hz
+is the only mode that can be synchronized by external flows. We focus on this mode and report
+τ as τ = tsync/tIP for this mutant, where tIP is the total time of IP-beating under the applied
+8
+
+flows, see Fig. 4A. Synchronization profiles τ(ν; θ) of ptx1 are shown in Fig. 4B. The median
+profiles are of similar width and height, indistinguishable from each other, and hence indicate
+a loss of asymmetric susceptibility to flow synchronization. The loss is further confirmed by
+the extracted ε(θ) (31) and τ(θ) (Fig. 4C-D). Cells and synchronization attempts are distributed
+evenly across the first bisector lines (7/14 cells are below ε(θc) = ε(θt) in Fig. 4C, and ∼50%
+points are below τ(θc) = τ(θt) in Fig. 4D). Altogether, all results show consistently that the
+asymmetry is lost in ptx1.
+Modeling
+Framework
+To investigate the implications of our experimental results on the coupling between flagella
+and their dynamics, we develop a model for the system (SM. Sec.S3), representing flagella and
+external flows as oscillators with directional couplings:
+�
+�
+�
+�
+�
+˙ϕf = 2πff
+˙ϕc = 2π[fc − λt sin(ϕc-ϕt) − εc sin(ϕc-ϕf)] + ζc(t)
+˙ϕt = 2π[ft − λc sin(ϕt-ϕc) − εt sin(ϕt-ϕf)] + ζt(t).
+(2)
+ϕf,c,t(t) respectively represent the phase of the flow, the cis, and the trans flagellum. ff,c,t
+represents the inherent frequency of the forcing (flow), the cis, and the trans respectively. The
+phase dynamics of each flagellum is modulated by its interactions with the other flagellum as
+well as the background flow. Take the cis ( ˙ϕc) for example, the effect of the trans and the forcing
+on the cis are respectively accounted for by the λt-term and the εc-term, see Eq. (2). In other
+words, λt and εc measure the sensitivity of the actual cis-frequency to the phase differences
+between oscillators (ϕc − ϕt,f), see the arrows in Fig. 5A. Lastly, ζc,t represent the white noise
+of the cis and trans flagellum respectively. In the following parts, without loss of generality, the
+noise are assumed equally strong and uncorrelated (⟨ζ2
+c ⟩ = ⟨ζ2
+t ⟩, or T c
+eff = T t
+eff). Nuanced phase
+dynamics under differential noise levels can be found in SM. Sec.S4.
+Eq. (2) can be readily reduced to Eq. (1), which allows us to write the experimentally mea-
+sured values (f0, ε(θ), Teff) analytically with εc,t, λc,t, and ζc,t. The asymptotic behavior of the
+9
+
+model under the condition ϕc ≈ ϕt ≈ ϕf are (SM. Sec.S3):
+�
+�
+�
+�
+�
+f0
+= αfc + (1 − α)ft,
+Teff
+= α2T c
+eff + (1 − α)2T t
+eff,
+ε(θ)
+= αεc(θ) + (1 − α)εt(θ),
+(3)
+with α = λc/(λc + λt) representing the dominance of cis. It is then clear that when α ≈ 1, the
+coordinated beating will display dynamic properties of the cis flagellum.
+Fig. 5A illustrates an exemplary modeling scheme describing flagellar beating subjected
+to θc-flows. The direction and thickness of arrows represent coupling direction and strength
+respectively. The selective loading on the cis is represented by εc > εt; while λc > λt reflects
+that the cis has a more dominant role in the coordinated beating. We run Monte-Carlo simulation
+with Eq. (2) using customized MATLAB scripts.
+Coordinated beating under symmetric forcing
+We first model the flow synchronization induced by θa-flow (symmetric flagellar loads). In this
+case, ε(θ) = αεc(θ) + (1 − α)εt(θ) (Eq. (3)) reduces to ε = εc,t and is independent of α. We
+set εc,t as 2.4 Hz to match the measured ε(θa) =2.4 Hz (Fig. 3B).
+At similar detunings as in the experimental results in Fig. 1F, our Monte-Carlo simulations
+reproduces the phase dynamics with: (i) no flow synchronization, (ii-iii) unstable synchroniza-
+tion, and (iv) stable synchronization (Fig. 5B). Repeating the simulations for varying forcing
+strength ε (= εc,t) and frequency ff yields Arnold tongue diagrams in agreement with those
+reported from our experiments. The Arnold Tongue for wt in Fig. 3A and ptx1 in Fig. 4A are
+reproduced with simulations shown in Fig. 5C and D respectively. The only parameter value
+changed between Fig. 5C and D is the level of noise (T c,t
+eff ), which is increased by an order of
+magnitude. The differences in phase dynamics between wt and ptx1, when subjected to sym-
+metric external loading, are therefore accounted by solely varying the noise.
+Coordinated beating under selective loading
+We next model flow synchronization by the θc-flows and the θt-flows. The selective forcing
+(εc ̸= εt) allows the effect of flagellar dominance (λc ̸= λt) to manifest in the effective forcing
+strength ε(θ) and hence in the synchronization profiles τ(ν; θ), Fig. 5E. Similar to our exper-
+imental observations, θc-flow synchronizes the coordinated beating over the broadest range of
+ν (i.e. largest ε). This is directly attributed to the dominance λc > λt: by setting λc = λt,
+10
+
+the differences between τ(θc) and τ(θt) disappear even under selective loading (Fig. 5E inset).
+Fig. 5F details how the asymmetry of inter-flagellar coupling (λc/λt) affects the asymmetry
+between τ(θc) and τ(θt) . The open symbols represent ε(θ) measured from modeled τ(ν; θ)
+and the lines represent Eq. (3). The difference between ε(θc) and ε(θt) increases with λc/λt,
+and they each saturates to reflect only the forcing on the cis (εc, the grey dashed lines). With
+fc = 45 Hz, ft = 65 Hz (23, 26), and f0 ≈ 50 Hz, we deduce from Eq. (3) that λc = 4λt for
+wt cells. For wt cells under calcium depletion, experimental results are reproduced with a lower
+total forcing strength (Fig. 5G). εc + εt is set to 4.08 Hz (15% lower) to reflect the 7% − 20%
+decrease in ε(θ) induced by calcium depletion.
+The ptx1 results are reproduced with a stronger noise (T c,t
+eff = 9.42 rad2/s) and a symmetric
+inter-flagellar coupling λc/λt = 1, see Fig. 5H and Table. 1. Both changes are necessary for
+reproducing the synchronization profiles of ptx1 in Fig. 5H: while the stronger noise lowers
+the maximal values of τ(θ, ν), setting λc/λt = 4 would still result in τ(θc) > τ(θt) in the
+central range (|ν| ≲ 2.4 Hz). Finally, it is noteworthy that the noise in ptx1 increases not only
+because a higher noise value for individual flagella, but also because the cis-trans coupling has
+become symmetric. As shown by Eq. (3), the unilateral coupling promotes not only the cis-
+frequency in the synchrony but also the cis-noise. Given T c
+eff ≪ T t
+eff and λc = 4λt, we confirm
+with simulations that the cis stabilizes the beating frequency of the trans and decreases its
+beating noise. The simulations are in good agreement with experimental noise measurements,
+see SM. Sec.S4 for details.
+Discussion
+The two flagella of C. reinhardtii have long been known to have inherently different dynamic
+properties such as frequency, waveform, level of active noise, and responses to second messen-
+gers (23,25,26,29,30). Intriguingly, when connected by basal fibers and beating synchronously,
+they both adopt the kinematics of the cis-(eyespot) flagellum, which led to the assumption that
+the flagella may have differential roles in coordination. In this work, we test this hypothesis by
+employing oscillatory flows applied from an angle with respect to the cells’ symmetry axis and
+thus exert biased loads on one flagellum.
+Without an exception, in wt cells, θc-flows, the ones that selectively load the cis flagellum,
+are always more effective in synchronizing the flagellar beating than the θt-flows. This is shown
+by the larger effective forcing strengths ( ε(θc) > ε(θt) , Fig. 3B-C) and larger synchronized time
+11
+
+fractions ( τ(θc) > τ(θt) , Fig. 3D). Mapping the measured forcing strength ε(θ) as a function
+of the loads, we find empirically that ε ∝ F
+c
+Flow (Fig. 3F) and that trans-loads appear to mat-
+ter negligibly. These observations all indicate that the cis-loads determine whether an external
+forcing can synchronize the cell. Moreover, this point is further highlighted by an unexpected
+finding: when θt-flows are applied, the trans flagellum always beats against the external flow
+(P t
+Flow < 0) and the only stabilizing factor for flow synchronization is the cis flagellum working
+along with the flow during the recovery stroke (Fig. 2C lower panel). These observations defini-
+tively prove that the two flagella have differential roles in the coordination and interestingly
+imply that flagella are coupled to external flow only through the cis.
+To have a mechanistic understanding of this finding, we model the system with Eq. (2). In
+the model, selective hydrodynamic loading and flagellar dominance in the coordinated beating
+are respectively represented by εc ̸= εt and λc ̸= λt. Setting out from the model, we obtain
+closed-form expressions for observables such as f0 and ε (Eq. (3)), which illustrate how flag-
+ellar dominance and selective loading affect the coordinated flagellar beating. Moreover, with
+Monte-Carlo simulation, we clarified the interplay between flows and flagella (SM. Sec.S3),
+and reproduces all experimental observations.
+With the model, we show that a ”dominance” of the cis (λc > λt) is sufficient to explain
+why the coordinated flagellar beating bears the frequency and the noise level of the cis flag-
+ellum. In the model, such dominance means that the cis-phase is much less sensitive to the
+trans-phase than the other way around. We then reproduce the phase dynamics of flow synchro-
+nization at varying detunings (Fig. 5B), amplitudes (Fig. 5C), and noise (Fig. 5D). Exploiting
+the observation that the coordination between flagella cannot be broken by external flows up
+to the strongest ones tested (εmax ∼ 10 Hz, Fig. 3A), we quantify the lower limit of the total
+basal coupling, λc + λt, to be approximately 40 Hz (deduced in SM. Sec.S3), which is an order
+magnitude larger than the hydrodynamic inter-flagellar coupling (31,50–52).
+The modulation of flagellar dominance mediates tactic behaviors (22, 23, 38, 47). Calcium
+is hypothesized to be underlying the modulation of dominance, as it causes the connecting
+fiber between flagella to contract (53), modulates the cis- and trans activity (e.g. beating am-
+plitude) differentially (22), and calcium influx comprises the initial step of CR’s photo- (54)
+and mechanoresponses (45). We therefore investigate flagellar coupling in the context of tactic
+steering by depleting the environmental free calcium and hence inhibiting signals of calcium
+influxes. Cells are first acclimated to calcium depletion, and then tested with the directional
+flows. Our results show that the cis dominance does not require the involvement of free envi-
+12
+
+ronmental calcium. Calcium depletion merely induces an overall drop in the forcing strength
+perceived by the cell ε(θ) (7% − 20%), which is captured by reducing εc + εt for 15% (mean
+drop) in the model (Fig. 5G). Together, our results indicate that the leading role of cis, is an
+inherent property, that does not require active influx of external calcium, and possibly reflects
+an intrinsic mechanical asymmetry of the cellular mesh that anchors the two flagella into the
+cell body.
+In ptx1 cells, a lack of flagellar dominance (λc = λt) and a stronger noise level help repro-
+duce our experimental observations. Previous studies suggested that both flagella of ptx1 are
+similar to the wildtype trans (23), and that the noise levels of this mutant’s synchronous beating
+are much greater than those of wt (29) (see also SM. Sec.S4). If both flagella and their anchoring
+roots indeed have the composition of the wildtype trans, such symmetry would predict λc = λt.
+This symmetric coupling renders the noise of ptx1 Teff = T t
+eff (Eq. (3)), which is about an order
+of magnitude larger than the noise of wt Teff ≈ T c
+eff.
+The comparison between ptx1 and wt highlights an intriguing advantage of the observed
+unilateral coupling (λc ≫ λt); that is, it strongly suppresses the high noise of the trans. Consid-
+ering that the trans is richer in CAH6 protein and this protein’s possible role in inorganic carbon
+sensing (14,20), the potential sensing role of the trans is worth noticing. Assuming the strong
+noise present in the trans originates from the biochemical processes related to sensing, then
+the unilateral coupling effectively prevents such noise from perturbing the cell’s synchronous
+beating and effective swimming. In this way, the asymmetric coupling may combine the benefit
+of having a stable cis as the driver while equipping a noisy trans as a sensor.
+Material and methods
+Cell culture
+CR wildtype (wt) strain cc125 (mt+) and flagellar dominance mutant ptx1 cc2894 (mt+) are
+cultured in TRIS-minimal medium (pH=7.0) with sterile air bubbling, in a 14h/10h day-night
+cycle. Experiments are performed on the 4th day after inoculating the liquid culture, when the
+culture is still in the exponential growth phase and has a concentration of ∼ 2 × 105 cells/ml.
+Before experiments, cells are collected and resuspended in fresh TRIS-minimal (pH=7.0).
+13
+
+Calcium depletion
+In calcium depletion assays, cells are cultured in the same fashion as mentioned above but
+washed and resuspended in fresh TRIS-minimal medium + 0.5 mM EGTA (pH=7.0). Free
+calcium concentration is estimated to drop from 0.33 mM in the TRIS-minimal medium, to
+0.01 µM in the altered medium (46). Experiments start at least one hour after the resuspension
+in order to acclimate the cells.
+Experimental setup
+Single cells of CR are studied following a protocol similar to the one described in (31). Cell
+suspensions are filled into a customized flow chamber with an opening on one side. The air-
+water interface on that side is pinned on all edges and is sealed with silicone oil. A micropipette
+held by micromanipulator (SYS-HS6, WPI) enters the chamber and captures single cells by as-
+piration. The manipulator and the captured cell remain stationary in the lab frame of reference,
+while the flow chamber and the fluid therein are oscillated by a piezoelectric stage (Nano-Drive,
+Mad City Labs), such that external flows are applied to the cell. Frequencies and amplitudes of
+the oscillations are individually calibrated by tracking micro-beads in the chamber. Bright field
+microscopy is performed on an inverted microscope (Nikon Eclipse Ti-U, 60× water immersion
+objective). Videos are recorded with a sCMOS camera (LaVision PCO.edge) at 600-1000 Hz.
+Measurement scheme
+The flagellar beating of each tested cell is recorded before, during, and after the application
+of the flows. We measure the cell’s average beating frequency f0 over 2 s (∼100 beats). For
+ptx1 cells, f0 is reported for the in-phase (IP) synchronous beating. Unless otherwise stated,
+directional flows (θ = 0, ±45◦) are of the same amplitude (780±50 µm/s, mean±std), similar
+to those used in Ref. (31). Flow frequencies ff are scanned over [f0 − 7, f0 + 7] Hz for each
+group of directional flows.
+Computation of the flagellar loads
+To quantify the hydrodynamic forces on the flagella, we first track realistic flagellar deforma-
+tion from videos wherein background flows are applied. Then we employ a hybrid method
+combining boundary element method (BEM) and slender-body theory (40, 55) to compute the
+14
+
+drag forces exerted on each flagellum and the forces’ rates of work. In this approach, each flag-
+ellum is represented as a slender-body (55) with 26 discrete points along its centerline and the
+time-dependent velocity of each of the 26 points is calculated by its displacement across frames.
+The cell body and the pipette used to capture the cell are represented as one entity with a com-
+pleted double layer boundary integral equation (56). Stresslet are distributed on cell-pipette’s
+surface; while stokeslet and rotlet of the completion flow are distributed along cell-pipette’s
+centerline (57). The no-slip boundary condition on the cell-pipette surface is satisfied at col-
+location points. Lastly, stokeslets are distributed along the centerlines of the flagella, so that
+no-slip boundary conditions are met on their surfaces. Integrating the distribution of stokeslets
+f(s) over a flagellar shape, one obtains the total drag force F =
+�
+f(s)ds is obtained. Similarly,
+the force’s rate of work is computed as P =
+�
+f(s) · U(s)ds, where U(s) is the velocity of the
+flagellum at the position s along the centerline.
+The computations shown in this study are based on videos of a representative cell which
+originally beats at ∼50 Hz. The cell is fully synchronized by flows along different directions
+(θ = 0◦, ±45◦ and 90◦) at 49.2 Hz. In the computations, the applied flows are set to have an
+amplitude of 780 µm/s to reflect the experiments. Computations begin with the onset of the
+background flows (notified experimentally by a flashlight event), and last for ∼30 beats (500
+frames sampled at 801 fps). Additionally, we confirm the results of θt-flow-synchronization,
+that both flagella spend large fractions of time beating against the flows, with other cells and
+with θt-flows at other frequencies.
+Isolate loads of external flows
+The total loads (F and P) computed consist of two parts, one from the flow created by the two
+flagella themselves and the other from the applied flow. In the low Reynolds number regime,
+the loads of the two parts add up directly (linearity): F = FSelf + FFlow, and P = PSelf + PFlow.
+To isolate FFlow and PFlow, we compute F′ = FSelf and P ′ = PSelf by running the computation
+again but without the external flows, and obtain FFlow = F − F′ and PFlow = P − P ′.
+Modeling parameters
+We assume the flagellar intrinsic frequencies fc and ft to be 45 Hz and 65 Hz respectively
+(23, 26, 28). On this basis, λc : λt is assumed to be 4:1 to account for the observed f0 (∼ 50
+Hz). εc : εt is set as 2:1, 1:1, and 1:2 for the θc-flows, the θa-flows, and the θt-flows respectively,
+15
+
+see Fig. 2A-C. Additionally, εc + εt is assumed to be constant to reflect the fact that F
+c
+Flow +
+F
+t
+Flow approximately does not vary with flow directions. We take a typical value of T c,t
+eff =
+1.57 rad2/s (31). The sum of inter-flagellar coupling λtot = λc + λt is set to be large enough,
+i.e., λtot = 3νct with νct = |ft − fc|, to account for the fact that: 1) the coordinated beating
+is approximated in-phase, and 2) up until the strongest flow applied, the coordinated beating
+cannot be broken (quantitative evaluation is detailed in SM. Sec.S3). To model wt cells under
+calcium depletion, we decrease εc + εt by 15% - which is the mean decrease in the observed
+ε(θc) , ε(θa) , and ε(θt) (Fig. 3E). For ptx1 cells, we assume a symmetric inter-flagellar coupling
+(λc = λt) and a stronger noise level (SM. Sec.S4). The parameters are summarized in Table. 1.
+Table 1: Modeling parameters
+variable
+symbol (unit)
+TRIS
+EGTA
+ptx1
+Intrinsic freq. (23,26)
+fc, ft (Hz)
+45,65
+45,65
+45,65
+Basal coupling∗
+λc + λt (Hz)
+60
+60
+60
+cis dominance (23,38)
+λc : λt (-)
+4:1
+4:1
+1:1
+Flow detuning
+ν (Hz)
+[-10,10]
+[-10,10]
+[-10,10]
+Total forcing (51)
+εc + εt (Hz)
+4.8
+4.08
+4.8
+Noise∗ (31)
+T c,t
+eff (rad2/s)
+1.57
+1.57
+9.42
+∗ detailed in SM. Sec.S3
+16
+
+References
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+19
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+coupling. Journal of The Royal Society Interface 15 (2018).
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+synchrony. Journal of The Royal Society Interface 18, 20200660 (2021).
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+47. Pazour, G. J., Agrin, N., Leszyk, J. & Witman, G. B. Proteomic analysis of a eukaryotic
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+can trigger deflagellation in C. reinhardtii.
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+(1994).
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+Flagellar coordination in Chlamydomonas cells held on mi-
+cropipettes. Cell Motility and the Cytoskeleton 41, 297–307 (1998).
+50. Brumley, D. R., Wan, K. Y., Polin, M. & Goldstein, R. E. Flagellar synchronization through
+direct hydrodynamic interactions. eLife 3, e02750 (2014).
+51. Klindt, G. S., Ruloff, C., Wagner, C. & Friedrich, B. M. Load response of the flagellar beat.
+Physical Review Letters 258101, 1–5 (2016).
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+namic forces. Proceedings of the National Academy of Sciences 117, 8315–8325 (2020).
+53. Hayashi, M., Yagi, T., Yoshimura, K. & Kamiya, R. Real-time observation of Ca2+-induced
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+(1998).
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+ture 351, 489–491 (1991).
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+Mechanics 75, 705–714 (1976).
+56. Power, H. & Miranda, G. Second kind integral equation formulation of stokes’ flows past
+a particle of arbitrary shape. SIAM Journal on Applied Mathematics 47, 689–698 (1987).
+57. Keaveny, E. E. & Shelley, M. J. Applying a second-kind boundary integral equation for
+surface tractions in stokes flow. Journal of Computational Physics 230, 2141 – 2159 (2011).
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+Methods 22, 383–387 (2000).
+21
+
+Acknowledgments
+The authors thank Roland Kieffer for technical support. D.W. thanks Ritsu Kamiya for helpful
+discussions. The authors acknowledge support by the European Research Council (ERC starting
+grants no. 716712 and no. 101042612).
+Author Contributions
+D.W. performed experiments, computations, designed the model, and drafted the manuscript.
+G.Q. performed early experiments and obtained preliminary results. M.A. and D.T. conceived
+the study, supervised the project and critically revised the manuscript.
+Competing interests
+Authors declare that they have no competing interests.
+Supplementary materials
+Supplementary Text
+Figs. S1 to S5
+References (23,26,28,29,31,38,43,58)
+22
+
+Figure 1: Experimental workflow. (A) Captured CR cells are subjected to sinusoidal flows of
+frequency ff along given angles (θ) in the xy-plane. Flows along θ = −45◦, 0◦, 45◦ of same
+amplitude (780±50 µm/s, mean±std.) are used and termed as shown. (B-E) Extracting flagellar
+phase ϕc and ϕt by image processing. Raw images (B) are thresholded and contrast-adjusted to
+highlight the flagella (C). Mean pixel values within the user-defined interrogation windows (red
+and blue circles) capture the raw phases of beating (D), which are then converted to observable-
+independent phases (E). Inset: phase difference ϕc − ϕt. (F) Flagella-flow phase dynamics at
+decreasing detuning ν = ff − f0 with f0 the cell’s beating frequency without external flow.
+Traces i to iv are taken at detunings marked in the inset. Plateaus marked black represent
+flow synchronization, whose time fractions τ = tsync/ttot are noted. ttot is the total time of
+recording. Inset: the flow synchronization profile, τ(ν), reports the effective forcing strength
+2ε by its width.
+23
+
+A
+Oa- flow,0
+B
+eyespot
+Ot- flow, 45
+c -flow,-45°
+A
+(s/ur)
+-780μm/s
+D
+D
+t ()
+E
+c
+pt
+(2元)
+0.5
+0.5S
+0
+4444442
+0.5
+0
+4
+8
+12
+16
+t (beat)
+F 12
+IV
+ii
+()  - )
+i, T=0
+T
+2
+8
+ii
+ii, T = 0.18
+1
+0
+-6
+0
+ii, T = 0.80
+v (Hz)
+4
+iv, T = 1.00
+0
+0
+2
+4
+6
+8
+10
+t (s)Figure 2: External flagellar loads when beating is synchronized. Force magnitude (upper pan-
+els) and power (lower panels) exerted by external flows of θ = 0◦ (A, θa-flow), −45◦ (B,
+θc-flow), and +45◦ (C, θt-flow). The medians (solid lines) and interquartile ranges (shadings)
+are computed over ∼20 synchronized beats. Dashed horizontal lines: loads averaged over a
+synchronized beat. Force magnitudes and powers are scaled by F0=9.9 pN and P0=1.1 fW
+respectively. Flagellar phase corresponds to the displayed shapes in the middle x-axis.
+24
+
+I cis loads
+ trans loads
+= Median
+Interquartile
+:.: Beat-averaged
+A
+B
+FFlow/Fo
+0.5
+0
+H
+/Po
+10
+0
+-10
+0元/2元3元/22元
+0元/2元3元/2 2元
+0元/2元3元/2 2元
+Flagellar phase (rad)
+Flagellar phase (rad)
+Flagellar phase (rad)Figure 3: Flow synchronization of wt cells. (A) Arnold tongue of a representative cell tested
+with θa-flow. The contour is interpolated from N=132 measurements (6 equidistant amplitudes
+× 22 equidistant frequencies), and color-coded by the entrained time fraction τ. (B) The syn-
+chronization profiles τ(ν; θ) of a representative wt cell (inset), the median profile of the TRIS
+group wt cells (N=11, solid lines) and the EGTA group (N=6, dashed lines), with either θc-
+flows (red), θa-flows (yellow) or θt-flows (blue). Shaded areas are the interquartile ranges for
+the TRIS group. (C) Tested wt cells represented on the ε(θc) − ε(θt) plane (TRIS group). Solid
+line: the first bisector line (y = x). (D) Comparing τ(ν; θc) and τ(ν; θt) for each cell at each ap-
+plied frequency. N=132 pairs of experiments are represented on the τ(θc) − τ(θt) plane. More
+than 90% of them are below the first bisector line. (E) The coupling strengths ε(θ) of the TRIS
+group (black) and the EGTA group (gray). Bars and error bars: mean and 1 std., respectively.
+Inset: δε = ε(θc) − ε(θt) . NS: not significant, p>0.05, Kruskal-Wallis test, One-Way ANOVA.
+Relations between the forcing strength ε and the loads on the cis (F) and the trans flagellum
+(G). Markers represent different flow angles, see the drawings.
+25
+
+20
+C
+4
+0
+(zH) 
+(10)3
+2
+8
+0
+0
+-10-5
+0
+510
+15
+2025
+0
+2
+4
+6
+v (Hz)
+ε(0c) (Hz)
+B
+D
+A single celi
+4
+0
+4
+Median over population
+T=1
+TRIS
+0
+EGTA :
+2 = 6.0 Hz
+T
+0
+t(Gc) (-)
+1
+E
+6
+(zH)
+3NS
+5.3 Hz
+E
+3
+3.8 Hz
+m
+0
+-8
+-4
+0
+4
+8
+v (Hz)
+TRIS EGTA
+ cis loads
+ trans loads
+F
+4
+G 4
+2
+2
+口
+m
+3
+0
+0
+0
+0.5
+0
+5
+0
+0.5
+0
+5
+FFlow/Fo
+Plow/Po
+FFlow/Fo
+Pflow/PoFigure 4: The asymmetric susceptibility to flow synchronization is lost in the flagellar domi-
+nance mutant ptx1. (A) Arnold tongue of a representative ptx1 cell tested with θa-flow. The
+contour is interpolated from N=132 measurements (6 equidistant amplitudes × 22 equidistant
+frequencies). Color bar: the entrained time fraction τ = tsync/tIP. (B) Flow synchronization
+profiles τ(ν; θ) of N=14 ptx1 cells, tested with θc-flows (red), θa-flows (yellow) and θt-flows
+(blue). (C) ε(θc) and ε(θt) of the tested cells. The first bisector line (solid): y = x. (D)
+τ(ν; θc,t) for each cell at each applied frequency. N=154 points are present.
+26
+
+A
+20
+6
+() n/n
+口
+10
+(zH)
+4
+口
+口
+口
+8
+口
+@2
+口
+口
+-10
+-5
+0
+5
+10
+15
+20 25
+口
+3
+v (Hz)
+0
+B
+Median
+Interquartile
+0
+2
+4
+6
+(0c) (Hz)
+0
+(-) (0)1
+T
+0
+1
+0
+0
+-8
+-4
+0
+4
+8
+0
+1
+v (Hz)
+T(c) (-)Figure 5: Modeling the asymmetric flow synchronization. (A) Modeling scheme describing a
+cell beating under directional flow (θc-flow as an example). Arrows represent the directional
+coupling coefficients with line thickness representing the relative strength. For example, λc
+points from cis to trans, representing how the latter (ϕc) is sensitive to the former (ϕt); mean-
+while, the arrow of λc being thicker than λt means that ϕt is much more sensitive to ϕc than
+the other way around. (B) Modeled phase dynamics of flow synchronization under θa-flows,
+analogous to Fig. 1F. Reproducing the Arnold tongue diagrams at the noise level of wt (C)
+and ptx1 (D), analogous to Fig. 3A and Fig. 4A respectively. (E) Flow synchronization profiles
+τ(ν; θ) obtained experimentally (upper panel) and by modeling (lower panel). Inset: the mod-
+eling results with symmetric inter-flagellar coupling. (F) Effective forcing strength ε(θ) as a
+function of the inter-flagellar coupling asymmetry λc/λt. Points: measured from simulation;
+lines: analytical approximation (Eq. (3)); dashed lines: εc respectively for the θc-flow, θa-flow,
+and θt-flow (from top to bottom). (G) Reproducing the flow synchronization of wt cells under
+calcium depletion (H) Reproducing results of ptx1. See Table. 1 for the modeling parameters.
+27
+
+A
+B
+trans
+cis
+(fe, pc)
+12
+(ft, Pt)
+111
+1
+ii
+1
+()
+fi, -0
+6
+0
+6
+Idh.
+v (Hz)
+Λt
+ji, {-0.27
+i, t=0.85
+Et
+Ec
+iv, T=1.00
+(fr, Pf)
+0
+0
+2
+4
+6
+8
+Induced flow (0=-45°)
+t (s)
+C
+D
+10
+10
+Ec,t (Hz)
+(zH)
+5
+Low noise level
+Ec,t
+High noise level
+0
+0
+-10-5
+¥05101520
+25
+-10-5
+051015 20 25
+v (Hz)
+v (Hz)
+E
+F
+Exp, wt
+3
+0
+(zH)
+2
+0
+0
+Model,wt
+m
+Analytical
+2e/t=4
+1
+6
+Monte-Carlo
+Ac=入t
+000
+Ec
+0
+0
+-6
+6
+0
+0
+5
+10
+15
+20
+8
+入/M
+v (Hz)
+G
+H
+Exp,wt
+Exp, ptx1
+EGTA
+0
+0
+T
+1
+L
+[Model,wt
+Model, ptx1
+EGTA
+0
+0
+6
+0
+6
+-6
+0
+6
+v (Hz)
+v (Hz)Supplementary materials for
+The younger flagellum coordinates the beating in
+C. reinhardtii
+Da Wei1,3,Greta Quaranta2, Marie-Eve Aubin-Tam1†, Daniel S.W. Tam2∗
+1Department of Bionanoscience, Delft University of Technology,
+2628CJ Delft, Netherlands.
+2Laboratory for Aero and Hydrodynamics, Delft University of Technology,
+2628CD Delft, Netherlands.
+3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,
+Chinese Academy of Sciences; Beijing 100190, China.
+†Corresponding author. Email: [email protected];
+∗Corresponding author. Email: [email protected].
+1
+arXiv:2301.13278v1  [physics.bio-ph]  30 Jan 2023
+
+S1
+Extracting coupling strength by fitting phase dynamics
+In the work described in the manuscript, the flagellum-flow coupling strength ε in wt cells is
+mainly extracted by the synchronization profile τ(ν) ≥50%. Meanwhile, in previous works [1,
+2], fitting the distribution of phase dynamics is employed to extract ε. In the latter approach, the
+idea is that the phase locking during synchronization leads to a peaked probability distribution
+of ∆ϕ, whose width is affected by the effective noise Teff. The distribution, P(∆ϕ), can be
+derived from the Adler equation Eq. (1) as:
+P(∆ϕ) =
+� ∆ϕ+2π
+δct
+exp(V (∆ϕ′) − V (∆ϕ)
+Teff
+)d∆ϕ′.
+(S1)
+Here V (∆ϕ) = ν∆ϕ + ε cos(∆ϕ) is a wash-board potential, Teff is the noise, and ∆ϕ is the
+difference between the flagellar phase and the flow’s phase.
+Here, we demonstrate that these two approaches are equivalent in extracting ε. For all
+wt cells tested in the TRIS-minimal medium (N=11), their ε(θ) measured by the τ(ν) width
+and extracted from fitting are plotted against each other, Fig. S1. All points center around the
+identity line, showing the equivalence in obtaining ε by the two methods. For the ptx1 dataset,
+ε are extracted from fitting the phase dynamics.
+2
+
+��������������
+�����������������
+����������
+���������
+�������������
+Figure S1: Equivalence of extracting coupling strength ε by different methods. Each point
+represents one cell under either the θa-flow (green square), the θc-flow (red circle), or the θt-
+flow (blue triangle). The x coordinate is the coupling strength ε measured by the half width of
+synchronization profile τ(ν) ≥ 50%; and the y coordinate is obtained by fitting the flagellar
+phase dynamics.
+3
+
+S2
+Hydrodynamic computation for flow along 90 degree
+Similar to Fig. 2 in the main text, we present the computed drag force and power for the
+flow along 90◦. The solid lines and the shadings represent the median and the interquartile
+range of FFlow and PFlow over the flow-synchronized beats, respectively. Force magnitudes
+are scaled by F0 = 6πµRU0 = 9.9 pN, which is the Stokes drag on a typical free-swimming
+cell (radius R = 5 µm, swim velocity U0 = 110 µm/s); while the viscous powers are scaled by
+P0 = F0U0 = 6πµRU 2
+0 = 1.1 fW. Here µ = 0.95 mPa·s is the dynamic viscosity of water at
+22 oC. Quantitatively, the mean force is 0.37F0 and 0.34F0 (Fig. S2 top panel) while the mean
+power is -0.2P0 and -0.4P0 (Fig. S2 bottom panel), for the cis and the trans respectively.
+Figure S2: Computed hydrodynamic loads on the flagella. Computation results of the drag
+force (upper panel) and the force’s rate of work (lower panel) on the cis (red) and the trans
+(blue) flagellum during synchronized cycles, when the cell is subjected to the flow with θ = 90◦.
+Scaling factors F0=9.9 pN and P0=1.1 fW.
+4
+
+FFlow/Fo
+1
+0.5
+0
+ cis loads
+10
+PFlow/Po
+- trans loads
+0
+-10
+0
+元/2
+2元3元/22元
+Flagellar phase (rad)S3
+The model
+The external flow and the two flagella are described by three coupled ordinary differential equa-
+tions (ODEs). Phase dynamics of these equations are examined by Monte-Carlo simulation.
+The temporal resolution of simulation (dt) is 1 ms, which corresponds to the experimental
+frame rates (801 Hz).
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+dϕf
+dt = 2πff
+(S2a)
+dϕc
+dt = 2πfc − 2πλt sin(��c − ϕt) − 2πεc sin(ϕc − ϕf) + ζc(t)
+(S2b)
+dϕt
+dt = 2πft − 2πλc sin(ϕt − ϕc) − 2πεt sin(ϕt − ϕf) + ζt(t).
+(S2c)
+The cis, the trans, and the external flow are described as oscillators, whose intrinsic fre-
+quencies are fc,t,f and phases ϕc,t,f, respectively. The flow is assumed to be noise free and the
+two flagella are assumed to have the same level of noise (ζc = ζt). The noises are assumed to
+be Gaussian, ⟨ζc,t(τ + t)ζc,t(τ)⟩ = 2T c,t
+eff δ(t).
+S3.1
+Flagellar synchronization
+Setting εc and εt to 0, the interaction between the two flagella in the absence of the flow is
+modeled by:
+�
+�
+�
+�
+�
+dϕt
+dt = 2πfc − 2πλt sin(ϕc − ϕt) + ζc(t)
+(S3a)
+dϕc
+dt = 2πft − 2πλc sin(ϕt − ϕc) + ζt(t).
+(S3b)
+When the two flagella are able to beat synchronously, dϕc
+dt = dϕt
+dt = f0, we can obtain the
+analytical expression of f0 by adding up λc×Eq. (S3a) and λt×Eq. (S3b):
+f0 = λtft + λcfc
+λc + λt
+.
+(S4)
+5
+
+Meanwhile, the steady-state phase difference δct = ϕc −ϕt is obtained by subtracting Eq. (S3a)
+from Eq. (S3b):
+sin(δct) = fc − ft
+λc + λt
+= νct
+λtot
+.
+(S5)
+It is therefore obvious that the two flagella can only beat at the same frequency (dϕc/dt =
+dϕt/dt = f0) if |νct/λtot| ≤ 1.
+S3.2
+Interaction between three oscillators
+Now we put the flow back into the picture. According to experimental observations, the two
+flagella mostly beat synchronously, we therefore focus on this case and first simplify the equa-
+tions. By adding up λc×Eq. (S2b) and λt×Eq. (S2c), and substituting ϕc,t as ϕ0 = ϕc −δct/2 =
+ϕt + δct/2, we obtain:
+dϕ0
+dt = 2πf0−2π λcεc
+λc + λt
+sin
+�
+ϕ0 − ϕf − δct
+2
+�
+−2π
+λtεt
+λc + λt
+sin
+�
+ϕ0 − ϕf + δct
+2
+�
++λtζt + λcζc
+λc + λt
+.
+(S6)
+Given different choices of coupling constants (λc,t, εc,t), this equation would generate com-
+plex phase dynamics - as we shall see in the following sections. We first limit the discussion to
+small δct - as it is observed in our experiment as well as in [3]. The model’s asymptotic behavior
+at δct ≈ 0 is:
+dϕ0
+dt = 2πf0 − 2πε sin(ϕ0 − ϕf) + ζ0(t),
+(S7)
+where
+f0 = λtft + λcfc
+εtc + λt
+, ε = λtεt + λcεc
+λc + λt
+, ζ0 = λtζt + λcζc
+λc + λt
+.
+(S8)
+In this strong-coupling limit (δct ≈ 0, or equivalently, λtot ≫ νct), the coupled flagella
+behaves as a single oscillator whose beating frequency f0 will not be interfered by the external
+flow. The analytical form well captures the system’s behavior, as shown by Fig. 5F. Next we
+explore the model’s behaviors when λtot − νct is comparable with ε.
+6
+
+���
+���
+���
+���
+���
+���
+������������
+f�
+f�
+��������cis���������
+��������trans���������
+��������trans ����cis
+���
+���
+���
+������
+������
+������
+Figure S3: Determine the lower limit of λtot. The time fractions of the cis (a) and the trans
+flagellum (b) synchronized by the flow. (c) The time fraction of where cis and trans are syn-
+chronized. Arrows points towards increasing (λtot − ν)/ε.
+S3.3
+Lower limit of inter-flagellar coupling
+The value (λtot − νct)/ε determines if the flow can disrupt the synchronization between cis
+and trans. We assume νct = 20 Hz[4, 5, 6, 3] and focus on synchronization of the θa-flow.
+We plot the synchronization time fractions with increasing λtot in Fig. S3. When it satisfies
+(λtot − νct)/ε ≥ 2, external flows cease to affect the flagellar synchronization observably. As
+the strongest flow (21U0) applied experimentally corresponds to ε ≈ 10 Hz, altogether, we
+conclude that λtot ≳ νct + 2εmax = 40 Hz. In the main text, we set λtot = 60 = 3νct Hz, which
+satisfies this relation and matches the observation that the phase lag between the flagella (δct) is
+small.
+7
+
+S4
+Flagellar noise of the ptx1 mutant
+Here we show an as-yet uncharacterized strong noise present in the synchronous beating of the
+mutant ptx1. The in-phase (IP) mode of ptx1 cells and the breaststroke beating of the wt cells
+are similar in waveform and frequency [7, 8]. However, the former has a much stronger noise.
+Figure S4: Stronger frequency fluctuation of the IP mode of ptx1 cells. (a-e) Representative
+probability distributions of the beating frequency of a wt (a) and four ptx1 cells (b-e) over 30
+seconds. Probability distributions of the IP (purple) and AP mode (yellow) are respectively nor-
+malized for better visualization. The time fractions of the AP mode are noted in each panel. (f)
+The wt and ptx1 cells represented by its mean beating frequency ⟨f0⟩ and the standard deviation
+of the beating frequencies over time σ(f0).
+The strong noises show obviously in fluctuations of IP beating frequencies [8].
+In Fig. S4, we display the distribution of beating frequency of a representative wt cell (panel
+a) and four representative ptx1 cells (panels b-e). The broad peaks of the IP (purple) and AP
+(yellow) beating of ptx1 sharply contrast the narrow peak of wt. We quantify the frequency
+fluctuations of all the cells in the main text (N=11 for wt and N=14 for ptx1), Fig. S4f. The
+cells are represented by its mean beating frequency over time ⟨f0⟩ and the frequency’s standard
+deviation σ(f0). Clearly, the breaststroke beating of wt, the IP, and the AP mode of ptx1 each
+forms a cluster. The wt cluster is at (⟨f0⟩, σ(f0)) = (50.5 ± 2.6, 0.8 ± 0.3) Hz (mean± 1 std.
+the over cell population); and it is evidently less dispersed than both the IP and the AP mode
+8
+
+0.1
+(a)
+(f)
+wt
+wildtype
+ptxl, IP mode
+0
+40 
+50
+60
+70
+80
+ptxl, AP mode
+0.06
+4
+AP: 6.9%
+(b)
+ptx1
+0
+(zH) (°J)o
+0.04
+AP: 10.2%.
+(c)
+PDF
+-
+0
+2
+0.06
+AP: 39.7% 
+(d)
+口
+0
+口
+0.04
+AP: 20.4% 
+口
+(e)
+0
+0
+40
+50
+60
+70
+80
+40
+50
+60
+70
+80
+Frequency (Hz)
+(fo) (Hz)of ptx1, which are at (47.4 ± 3.1, 3.4 ± 0.9) Hz and (67.6 ± 2.1, 1.9 ± 0.7) Hz, respectively.
+Under the assumption of a white (Gaussian) noise, σ(f0) is proportional to the noise level ζ,
+and thus scales with √Teff. Consider that σ(f0) for ptx1 is 3-5 folds larger than that of wt,
+we therefore conclude that the noise level in ptx1 is an order of magnitude larger than wt,
+T ptx1
+eff
+/T wt
+eff ∼ O(10).
+Figure S5: Effect of a low-noise cis in stabilizing the beating of the trans (a) Fluctuations in
+beating frequency (σ(f0)) under different coupling schemes and flagellar noises. Other model
+parameters are the same as used in the main text. The red and blue shaded area represent the
+experimentally observed range for ptx1 and wt cells, respectively, with short bars marking the
+mean values. (b) the rate of slip under the conditions. Error bars correspond to 1 std. over N=9
+repetitions.
+The stronger noise in ptx1 can be attributed to two sources, namely, the loss of a stable
+cis and the loss of the unilateral coupling, Fig. S5. We perform Monte-Carlo simulations of
+the coupled beating of cis and trans under three conditions: (1) a stable cis (T c
+eff = T 0
+eff =
+1.57 rad/s2) coupled with the trans unilaterally (λc = 4λt), (2) a stable cis coupled with the
+trans bilaterally (λc = λt), and (3) an equally noisy cis (T c
+eff = T t
+eff) bilaterally coupled with
+trans, see the blue, yellow, and red data in Fig. S5 respectively. It is obvious that, when the trans
+is coupled to a stable cis, varying its noise over an order of magnitude only leads to a ∼ 20%
+stronger frequency fluctuation (the blue line in Fig. S5(a)). On the contrary, lacking either
+the unilateral coupling or the low-noised cis would increase the fluctuation for 200% (yellow
+9
+
+= =
+=
+=
+0.1
+(a)
+(b)
+3
+ptx1
+0.08
+Slip rate (Hz)
+o(fo) (Hz)
+0.06
+2
+0.04
+1
+0.02
+im
+0
+0
+100
+101
+10°
+101
+Teff / Teff
+Teff / Teffline) or 300% (red line). Qualitatively, simulation results are in agreement with experimental
+measurements assuming that T t
+eff/T c
+eff ∼ O(10), see the red and blue shaded areas in Fig. S5(a).
+Moreover, a low-noise cis is already sufficient to prevent slips from interrupting the synchrony
+between cis and trans, even for bilateral coupling. In Fig. S5(b), as long as the cis-noise remains
+low, slips will be sparse (< 0.01 Hz). Together, these simulation results highlight the stabilizing
+effect of a low-noise cis flagellum, and illustrates the contribution of unilateral coupling in
+further enhancing the stabilization.
+References
+[1] Polin, M., Tuval, I., Drescher, K., Gollub, J. P. & Goldstein, R. E. Chlamydomonas swims
+with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325, 487–
+490 (2009).
+[2] Quaranta, G., Aubin-Tam, M.-E. & Tam, D. Hydrodynamics versus intracellular coupling
+in the synchronization of eukaryotic flagella. Physical Review Letters 115, 238101 (2015).
+[3] Wan, K. Y., Leptos, K. C. & Goldstein, R. E. Lag, lock, sync, slip: the many phases of
+coupled flagella. Journal of The Royal Society Interface 11, 20131160 (2014).
+[4] Kamiya, R. & Hasegawa, E. Intrinsic difference in beat frequency between the two flagella
+of Chlamydomonas reinhardtii. Experimental Cell Research 173, 299–304 (1987).
+[5] Kamiya, R. Analysis of cell vibration for assessing axonemal motility in Chlamydomonas.
+Methods 22, 383–387 (2000).
+[6] Okita, N., Isogai, N., Hirono, M., Kamiya, R. & Yoshimura, K. Phototactic activity in
+Chlamydomonas ’non-phototactic’ mutants deficient in Ca2+-dependent control of flagellar
+dominance or in inner-arm dynein. Journal of Cell Science 118, 529–537 (2005).
+10
+
+[7] Horst, J. & Witman, G. B. Ptx1, a nonphototactic mutant of Chlamydomonas, lacks control
+of flagellar dominance. The Journal of Cell Biology 120, 733–741 (1993).
+[8] Leptos, K. C. et al. Antiphase synchronization in a flagellar-dominance mutant of Chlamy-
+domonas. Physical Review Letters 111, 1–5 (2013).
+11
+

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+Lindbladian-Induced Alignment in Quantum
+Measurements
+R. Englman and A. Yahalom
+Ariel University, Ariel 40700,Israel
+January 10, 2023
+Keywords: Quantum measurement theory, Density matrix evolution, Quan-
+tum state resolution, Lindblad operators, Quantum speed limit.
+Abstract
+An expression of the Lindbladian form is proposed that ensures an un-
+ambiguous time-continuous reduction of the initial system-pointer wave-
+packet to one in which the readings and the observable’s values are aligned,
+formalized as the transition from an outer product to an inner product of
+the system’s and apparatus’ density matrices. The jump operators are in
+the basis of the observables, with uniquely determined parameters derived
+from the measurement set-up (thereby differing from S. Weinberg’s Lind-
+bladian resolution of wave-packet formalism) and conforming to Born’s
+probability rules. The novelty lies in formalising the adaptability of the
+surroundings (including the measuring device) to the mode of observa-
+tion. Accordingly, the transition is of finite duration (in contrast to its
+instantaneousness in the von Neumann’s formulation). This duration is
+estimated for a simple half-spin-like model.
+1
+Introduction
+In the century-run of quantum physics (plus 4 years, if one marks its beginning
+with the award of a Nobel Prize in 1918 to Max Planck for ”his discovery of
+quanta”) a single shadow of non-sequitur has darkened its glorious achievements,
+one that goes variously under the names of wave-function collapse, reduction of
+the wave-packet, quantum measurement, einselection, etc. Aspects of the prob-
+lem (or its articulations) were manifold, such as the breakdown of the predicted
+time-development in accordance with the Schr¨odinger equation, the abruptness
+of change in a measurement (”natura non facit saltum”, where art thou?), the
+apparent non-applicability of quantum rules to macroscopic systems, imputed
+arbitrariness of Born’s probability rules, the requirement of ”infinite regress”
+1
+arXiv:2301.02664v1  [quant-ph]  6 Jan 2023
+
+for the measuring apparatus and others. Numerous papers enlarged on these is-
+sues [1, 2] and various proposals for resolution of the problem were put forward.
+These include the observer’s cognition [3], stochastic effects [4], in particular
+spontaneous localization [2, 5, 6], a many world scenario [7], non-linearity addi-
+tion to the Schr¨odinger equation [8], Poincar´e recurrent state [9], gravitationally
+induced collapse [10, 11, 12], etc.
+Common to these works, and with the specific purpose of providing a
+blue-print for measurements compatible with the Copenhagen formulation of
+quantum theory, was the need to give expression to the coupling of the micro-
+scopic system with its macroscopic environment. Standing apart from these and
+belonging to the field of non-equilibrium thermodynamics and to the establish-
+ment of equilibrium, a general form for this interaction was given by Lindblad
+[13] and by Gorin , Kossakowski and Sudarshan [14], satisfying some necessary
+conditions. Constructing a merger between the two separately oriented fields,
+S. Weinberg recently proposed a Lindblad-operator mechanism for the collapse
+of the density matrix (DM) in the course of a complete measurement [15]. No-
+tably, the mechanism was linear in the state’s DM. The collapsed state (Eq. (1)
+in [15]) comprises the set of projection operators of the measurable item; the
+system’s Hamiltonian is described by a spectral decomposition onto the same
+operators (Eq.
+(16) in [15]) (although in the verbal discussion a more gen-
+eral situation is considered): collapse is achieved ”independent[ly] of the details
+of these [Lindblad] operators”. Decay between energy eigenstates had earlier
+been treated by the Lindblad formalism (for a pedagogical presentation the
+volume [16], Chapter 8 may be consulted) employing the interaction represen-
+tation. However, this is not convenient for treating measurements of observables
+that do not commute with the Hamiltonian. Detailed theories relate to the out-
+come (”mapping”) of quantum operations, including measurements; the present
+work describes the process of these happening.(For a pedagogical introduction
+to stochasticity-induced wave-packet- reduction, obviating pointer reading, one
+may refer to [17].)
+2
+Overview of the Method and Terms
+2.1
+The leading idea, also in review
+While the concept of unity of observer and observation had already featured
+in Bohr’s view: ”The answer that we get is built up from the combined interac-
+tion of [the observer’s] state and the object of interrogation.” [18], this was not
+given a formal expression in the Copenhagen interpretation. It was more em-
+phatically asserted both by J. Bell: ”I meant that the ’apparatus’ should not be
+severed from the rest of the world in boxes ...[19]” and A. Peres: ” A measure-
+ment both creates and records a property of the system [20]”. This change in
+2
+
+the course of a measurement a���ects also the environment outside the observed
+system ; in the words of A. Leggett ”...under these conditions the macroscopic
+apparatus, and more generally any part of the macro-world which has suffered
+changes in the course of the measurement process, does not end up in a state
+with definite macroscopic properties at all,... [1]”.
+The same line of thought appears to motivate S. Weinberg, who wrote in his
+preamble to a 2016 Lindbladian formulation of the masurement process[15], that
+”We will instead [of the original formulation of the Copenhagen interpretation,
+(which we will not dwell on here)] adopt the popular modern
+view that the
+Copenhagen interpretation refers to open systems in which the transition is
+driven by the ineraction of the microscopic system under study (which may
+include an observer) chosen to bring the transition about.” (Our italics.)
+These developments indicate the justification for a formulation in which the
+effect of the apparatus is incorporated in the equation defining the evolution of
+the system, rather than one in which the two entities are separate, barring an
+interaction between them.
+2.1.1
+”Alignment”
+The process whereby the pointer readings become in correspondence with the
+possible values of the observable. Formally, for I possible values, the combined
+density matrix reduces from comprising I2 terms to one having I terms. (E.g.,
+equation (2.5) in [1].)
+2.1.2
+”Dissipator”
+Added term (in the form of sums of appropriately weighted jump-operators)
+to the standard time dependent Schr¨odinger equation, inducing non-unitary
+evolution in the system, accompanied by changes of its information entropy.
+2.2
+Motivation for the choice of formalism
+Thermalization of open systems can be described by a Lindbladian formal-
+ism in which Gibbsian probabilities are so inserted as parameters, that the
+”Dissipator” vanishes at these values of the density matrix. Replacement of
+the Gibbsian probabilities by Born probabilities achieves alignment in a state
+reduction and does so continuously.
+Limitations: Born’s probability rules are assumed, not derived; the interac-
+tion term is not traced to a microscopic mechanism.
+The source of this interaction term, shown in Eqn. 6 below, incorporating
+the coupling between the observed system and its surroundings (including the
+3
+
+measuring device) is an open question (also raised by a referee). In its appli-
+cation to a thermalization process, the Lindbladian jump operators have been
+derived, though with the aids of several approximations (e.g.,[21]), as well as,
+more recently, for the dissipation in a Dicke system with a bosonic background
+[22]. We are not in the position to provide such first principle derivation for
+the Lindbladian jump-operators bringing about a transition and incorporat-
+ing the Born rules. It seems to be specific to the type of measurement under
+consideration and it is clear that just any jump operator, as in Weinberg’s
+Lindbladian formulation will not do the job . Likely, one would need to in-
+clude non-Markovian dynamics, so that the coupling to the device and eventual
+pointer reading are two separate consecutive events. Inclusion of such dynamics
+is outside the scope of the present work.
+3
+Assumptions
+We explore the time (t)-development of the combined density matrix ρ(t) of
+the measured system (S) and of the reading (pointer, dial, etc.) on the mea-
+suring apparatus (A) for a complete and discrete measurement , expressing the
+underlying assumptions by three propositions.
+Proposition 1. In accord with the long-time historical approach, the mea-
+sured object S and the pointer of the measuring set-up A are treated on equal
+footings as subject to microscopic quantum laws, and formally describable by
+their respective Hamiltonians. Aware of the difficulties connected with an ”in-
+finite regress”, the effects of the rest of the Universe on S+A are not included
+in the formalism; instead, for a phenomenological, approximative description, a
+Lindbladian term appears in the master equation.
+Proposition 2. Prior to the measurement with A and S decoupled, and being
+free of external influence for a long time, both are in energy quantum states,
+pure or mixed. After the measurement, the state is not an energy eigenstate
+and subsequently it will spread over to a superposition of energy eigenstates.
+The fast decoherence case treated below in section 5 is akin to the Zeno effect
+[23].
+Proposition 3. Only those states of the reading apparatus (e.g., the right
+or left positions of a pointer) that may be in direct correspondence with the
+measured states of the system (e.g., spin up or down) are given expression in
+the formalism. (At a beginning, the case treated is one in which there is a one-
+to-one correspondence between the states of the system and the readings of the
+apparatus; a generalization is given subsequently.) A discussion in section
+8
+touches on the epistemological status of the Lindbladian terms in a measurement
+process.
+4
+
+4
+Analysis
+Considering (for simplicity) a pure state for the system, its initial state-vector
+written in the basis of the observed property |S, i > takes the form
+ψS(t = 0) =
+�
+i=1,..,I
+cS
+i |S, i >
+(1)
+Born’s rule for the probability of observing the i-component is |cS
+i ]2 ≡ pi, sum-
+ming to unity. Likewise, for the apparatus readings j, numbering J, one has
+the superposition with (complex and normalized) coefficients cA
+j
+ψA(t = 0) =
+�
+j=1,..,J
+cA
+j |A, j >
+(2)
+We start with the one-to-one correspondence situation, for which I = J, and the
+reading j on A establishes uniquely the value i = j for the system’s measured
+property.
+For the combined state-vector the density operator has the outer-product
+form (where the stars denote complex conjugates):
+�
+i,j,i′,j′
+|S, i > |A, j > cS∗
+i cA∗
+j cA
+j′cS
+i′ < A, j′| < S, i′| ≡
+�
+i,j,i′,j′
+|i, j > Ciji′j′ < i′, j′|
+(3)
+the right hand side written in an obvious shortened notation, in which Ciji′j′ =
+cS∗
+i cA∗
+j cA
+j′cS
+i′. After collapse, the density operator takes the aligned, single-sum
+form
+�
+i
+|S, i > |A, i > |cS
+i |2 < S, i| < A, i|
+(4)
+It will be now shown that this is the time-asymptotic solution of the Lind-
+bladian master equation properly parametrized.
+We recall Lindblad’s equation for the time varying density of states operator
+ρ ≡ ρ(t), as being of the following general form:
+∂ρ
+∂t = − i
+¯h[H, ρ] +
+�
+n
+γn⟨LnρL†
+n − 1
+2(L†
+nLnρ + ρL†
+nLn)⟩
+(5)
+The second term, here named the ”Lindblad term” [13, 14] though in different
+contexts also referred to as the Dissipator [24], contains Ln’s that are Lindblad
+jump-operators. We shall consistently work in the observable + pointer’s basis
+(i.e., not in an energy basis). In this basis, neither the density operator ρ = ρ(t),
+nor the A+S Hamiltonian H is diagonal at the beginning or in the course of
+the development. But, as will be demonstrated, the Lindbladian formalism, by
+a proper choice of its form, drives A+S to the desired diagonal form for the
+combined observable +pointer basis. We postulate just one single term in the
+previous n-sum, as well as off-diagonal forms, namely |i, j >< i′, j′|, (i, j ̸=
+5
+
+i′, j′), for the jump-operators in the observable basis, leading to the following
+parametrized form of the Lindblad term
+Lρ
+≡
+ΓΩ
+�
+i′,j′̸=i,j
+r(i, j)
+r(i′, j′)⟨|i, j >< i′, j′|ρ|i′, j′ >< i, j|
+−
+1
+2(|i′, j′ >< i, j|i, j >< i′, j′|ρ + ρ|i′, j′ >< i, j|i, j >< i′, j′|⟩
+(6)
+Here a circular frequency Ω is inserted, so as to make Γ , that quantifies the
+strength of the system-environment coupling, dimensionless. One notes that in
+the pre-factor appear the parameters r(i, j), r(i′, j′)(i, j, i′, j′ = 1, ..., I) whose
+significance will be clear by deriving the matrix elements of the above operator.
+These are
+Lρi,j,i′,j′
+=
+δi,i′δj,j′r(i, j)
+�
+k,l
+r−1(k, l)ρk,l,k,l
+−
+1
+2[r−1(i, j) + r−1(i′, j′)]ρi,j,i′,j′
+�
+k,l
+r(k, l)
+(7)
+It can be seen that the trace of the above vanishes and that each matrix element
+vanishes upon the substitution
+ρi,j,i′,j′ = δi,i′δj,j′r2(i, j)
+(8)
+While these properties hold for any arbitrary r(ij), the observable-pointer align-
+ment is achieved by identifying the r parameters with the system’s superposition
+coefficient: r(ij) = |cS
+i |δi,j, or
+r(i, j)2 = |cS
+i |2 ≡ piδi,j
+(9)
+the last being the Born probabilities appearing in the collapsed state. As already
+noted, this identification of probabilities relates to the well known procedure for
+the Lindblad-induced thermalization of open systems, for which detailed balance
+imposes the relation between the pre-factors γ(δE)/γ(−δE) = e−βδE/Z, the
+latter being the canonical probabilities (with β = 1/kBT, kB the Boltzmann
+constant, T the ambient temperature and Z the partition function [24, 25, 21])
+.
+[It also seems fair to point out that also in the standard (Copenhagen, or von
+Neumannian) description of the alignment stage, as appears in e.g. Eq.(2.5) of
+[1], this development is summarily stated, without specification of the underlying
+mechanism.]
+5
+Fast Decoherence Limit
+We now consider the case that the time development in the state is predom-
+inantly due to the coupling to the environment, rather than to the unitary
+6
+
+change induced by the Hamiltonian, meaning that the second term on the right
+hand side in Eq. 5 dominates the first. Quantitatively: Γ >> ||H||/¯hΩ. Ne-
+glecting the commutator we now form matrix elements of the Lindblad term in
+Eq. 5 in the observable+pointer basis. Because of the approximation made, the
+off-diagonal matrix elements are decoupled from the diagonal ones. The master
+equation of the off-diagonal terms reads (with a notation simplified by writing for
+the index pairs i, j → r, i′, j′ → s and consequently for ρi,j,i′,j′ → ρrs ≡ ρrs(t)
+dρrs
+dt
+= −ΓΩ
+��√pr + √ps
+2
+ρrs
+�
+m
+√pm
+�
+, r ̸= s
+(10)
+This shows that off-diagonal matrix elements decay exponentially in time (de-
+cohere), maintaining their real character that they had initially. Had we kept
+the (imaginary) commutator term, we would have found that the decay is mod-
+ulated by the eigen-energies of the Hamiltonian.
+For the diagonal matrix elements we find,
+dρrr
+dt
+= ΓΩ
+�
+√pr
+�
+m
+ρmm
+√pm
+− ρrr
+√pr
+�
+m
+√pm
+�
+(11)
+Again, it can be seen that the trace of the last expression vanishes, and so
+does the right-hand side under the substitution ρrr → pr. With these taking the
+values as in Eq. 9, one arrives at the aligned form (written out in the original,
+system-pointer indexes)
+ρ(t → ∞) =
+�
+i
+|ψA
+i > |ψS
+i > |cS
+i |2 < ψS
+i | < ψA
+i |
+(12)
+5.1
+Illustrative example for a two-way experiment
+Exemplifying the foregoing for a two-valued system (such as a 1
+2-spin electron),
+prepared as an eigenstate of a Zeeman-field with the magnetic field inclined at an
+angle 2αS to the vertical, in conjunction with an apparatus pointer, represented
+as being likewise in an eigenstate of a quasi-Zeeman field inclined at an angle 2αA
+to the vertical. The eigenstates are linear superpositions of their z- spins; these
+are the observables that are to be determined by the measurement. Initially, the
+system and the pointer are in the superposition states as shown above in Eqs.
+1 and 2 and whose superposition coefficients cS
+i and cA
+j now have the values,
+sin / cos(αS) and sin / cos(αA), respectively. The DM in the observable basis
+is now a 4x4 matrix, in which appear all the combinations of the products of
+the above circular functions. As the outcome of the application of the Lindblad
+operator in the rate equation, at long times the matrix becomes reduced to the
+diagonal form discussed earlier. In these, cos2(αS) = p1 and sin2(αS) = p4
+belonging to the aligned observable lie on the diagonal and are non-zero; the
+other two diagonal entries for the anti-aligned situations are zero.
+Plotted in Figure 1 are computed DM eigenvalues as functions of time
+(in red and blue), normalized to their respective Born probabilities, showing
+7
+
+-4
+-3
+-2
+-1
+0
+1
+2
+Log10time[invfrequn]
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+P1,P2,Decoh
+Figure 1: Density matrix eigenvalues normalized to their asymptotic (pointer-
+aligned) values for the two aligned terms in the illustrative example (in red and
+blue), plotted against time in inverse circular frequency unit. In green is shown
+a decohering off-diagonal matrix element. Lindblad coupling strength Γ = 5, α
+angles .37 π and .65 π.
+their asymptotic convergence.
+In green, the typical decohering tendency of
+an off-diagonal element is demonstrated. Figure 2 depicts the entropy S(t) =
+− �
+r Pr(t) log Pr(t) of the system and apparatus-pointer, (in which Pr(t) are
+computed eigenvalues of the DM.) The non-monotonic behavior is characteristic
+of of the Lindblad formalism, in which the environment’s entropy change is not
+taken into account.
+[In numerical work, based on forward integration, putting zeros for some
+of the pi’s introduces singularities, eventually algebraically cancelling out, but
+preventing flow of computation. Therefore, instead, one puts arbitrarily small
+values for these and obtains for the aligned DM one that is arbitrarily close to,
+but not exactly equal to the true one.]
+6
+Eigenvalue analysis
+An alternative to the numerical solution of the differential rate equation is eigen-
+value analysis, already treated in [15], based on the Landbladian term being a
+linear function of the diagonals in the density matrix. Thereby, the resulting
+rate equations have solution of the form
+ρnn(t) =
+�
+k
+vn,keλkt
+(13)
+in which λk and vn,k are the diagonalized eigenvalues and eigenvectors of the
+Lindbladian matrix diagonals in Eq. 11. Calculation shows that for the 4 x
+8
+
+-4
+-3
+-2
+-1
+0
+Log10time[invfrequn]
+0.05
+0.10
+0.15
+0.20
+S
+Figure 2: Entropy of the combined system plus apparatus. Noteworthy is the
+initial peak common to the Lindblad formalism .
+4 matrix considered above there are three negative eigenvalues and one zero
+eigenvalue, which alone is of interest at the long term behavior. Belonging to
+this eigenvalue, the (transposed) eigenvector is found to be {p1, p2, p3, p4} ≈
+{cS
+1 , 0, 0, cS
+4 }, as required for the alignment between the quantum states and the
+reading in the measuring apparatus.
+6.1
+Measurement speed
+Figure 1 shows that alignment is achieved for the model with the chosen strength
+parameter (Γ = 5) by a time of cca. 0.1/Ω. By varying the strength in the
+computed model, we find a shortening of this time that is inversely proportional
+to the strength. This is expected from the quantum speed limit (QSL) results
+that border quantum transition times τ from below.
+Essentially, QSL is the ratio of two norms [26, 27], that of the ”quantum
+distance” [28] and of the speed of the state evolution. Formally
+τ > ||ρ(t → ∞) − ρ(t = 0)||
+|| dρ(t)
+dt ||
+= ||ρ(t → ∞) − ρ(t = 0)||
+||[Lρ(t)]||
+(14)
+Ways of calculating the norms vary, e.g., [29, 30]. Recently, for a system de-
+veloping due to a Lindbalian operator, three contributions to the speed were
+discerned [24]. To estimate ||ρ(t → ∞) − ρ(t = 0)|, we have used the ”Trace
+Distance ”defined as
+T(ρ, σ) = 1
+2Tr[
+�
+(ρ − σ)] = 1
+2
+�
+i
+|µi|
+(15)
+[31], where µi are the eigenvalues of the matrix differences. The DM velocity,
+as defined above , changes (decreases) with time, ultimately vanishing at the
+9
+
+fulfilment of alignment; we have taken the root-mean-square sum of the rate of
+the diagonal matrix elements at initial times. These yield a very low limit of
+τ > 1
+2
+1.41
+5 ∗ 69.03 = .0045/Ω
+(16)
+to be compared with the actually computed value, about 20 times longer. Better
+(higher) limits of transition times may be generated by different ways of forming
+the norm for the DM velocity (e.g. not at the beginning).
+6.2
+Multiple Reading-System correspondence
+A simple generalization of the foregoing applies when each (eigen-)value of the
+observable is in correspondence with not just one reading of the pointer, but
+with several (say, R) readings, all of the same significance for the outcome. Then
+one simply inserts pi/R into the corresponding Lindblad term, in place of just
+pi. In the more complex case, that not all readings have the same likelihood,
+pi would have to be weighted by a probability factot, rather than by a constant
+denominator.
+7
+The Lindbladian, ”Who ordered this?”
+Historically, Lindblad terms were introduced as the most general forms that
+maintain complete positivity of the DM’s and preserve their trace [13, 14]. The
+various derivations that have been presented (and among these a recent one by
+[32]), involve several approximations for the coupling between the system and its
+environment. Insomuch that the derivation involves also tracing over the degrees
+of freedom of the environment, much detail of the latter is lost and of course
+it is impossible to work backwards from the Lindbladian to the environment.
+What is remarkable is that for special purposes the appropriate Lindbladian
+operators take a very special, practically unique form. Such is the case for the
+accepted description of thermalization [21, 24] by a Lindblad formalism. The
+parametrization of the Lindblad term employed in the present work, though it
+may appear arbitrary and particular for each case, is in fact identical to the
+one used for thermalization subject to the relabelling of the Gibbsian thermal
+distribution function as (the Born) probabilities (p1, p2, ...),with the proviso of
+working in the observable, rather than in the energy basis.
+(This contrasts
+with the different approach in [15], which claims attainment of collapse for any
+Lindbladian operator.) At the same time, it needs to be noted that the analog
+of detailed balance is missing in wave-function collapse.
+How come to have
+such a specific Lindbladian, whose source may be any measurement device and
+procedure? One is left to wonder about the possibility of a special meta-physical
+status of the Lindblad terms, or query with Wheeler ”Who ordered this?”
+10
+
+8
+Conclusion
+The well-known Lindbladian extension to the quantum theory of motion to
+environmental effects is here adapted to establish the resolution of a wave-packet
+in a measurement as a smooth process. This is enabled by an unambiguous
+parametrization of the jump-operators describing the interaction of the broad
+environment with the observed system, both regarded as quantal entities.
+Above, in section 2.1, a brief historically oriented preview has been provided
+for the distinct approach in this work, namely, one based on the wholeness of
+the entities (observed system and observing device), through the (Lindbladian)
+equation yielding the evolution of the system.
+A main result emerging from the formalism, and capable of experimental
+verification, is the finitely temporal variation of the system, and this in a de-
+terministic way rather than just statistically, on the average, contrasting also
+with the instantaneous collapse description by (e.g.) von Neumann. Such tem-
+poral variation in continuous-thermalization processes has been proposed quite
+recently [33, 34], also by employment of a Lindbladian formalism and within a
+Markovian framework.
+Experimentally, verification of the time dependence of the transition in any
+particular measurement, implicit in our formulae, could be observed by re-
+peated observation performed on the system subject to non-demolition tran-
+sitions. These observations would be akin to the Zeno-effect measurement, such
+as has been achieved in the form of quasi-periodic oscillation of the result for
+a Superconducting flux cubit[35]. Further work is needed for quantifying the
+information-entropy change in the environment [31, 36].
+Acknowledgement
+The authors thank the referee for meticulous reading and insightful ques-
+tioning of a previous version of this paper.
+References
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+13
+

EdE0T4oBgHgl3EQfywKq/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf,len=470
+page_content='Lindbladian-Induced Alignment in Quantum  Measurements  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Englman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Yahalom  Ariel University, Ariel 40700,Israel  January 10, 2023  Keywords: Quantum measurement theory, Density matrix evolution, Quan-  tum state resolution, Lindblad operators, Quantum speed limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Abstract  An expression of the Lindbladian form is proposed that ensures an un-  ambiguous time-continuous reduction of the initial system-pointer wave-  packet to one in which the readings and the observable’s values are aligned,  formalized as the transition from an outer product to an inner product of  the system’s and apparatus’ density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The jump operators are in  the basis of the observables, with uniquely determined parameters derived  from the measurement set-up (thereby differing from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Weinberg’s Lind-  bladian resolution of wave-packet formalism) and conforming to Born’s  probability rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The novelty lies in formalising the adaptability of the  surroundings (including the measuring device) to the mode of observa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Accordingly, the transition is of finite duration (in contrast to its  instantaneousness in the von Neumann’s formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' This duration is  estimated for a simple half-spin-like model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  1  Introduction  In the century-run of quantum physics (plus 4 years, if one marks its beginning  with the award of a Nobel Prize in 1918 to Max Planck for ”his discovery of  quanta”) a single shadow of non-sequitur has darkened its glorious achievements,  one that goes variously under the names of wave-function collapse, reduction of  the wave-packet, quantum measurement, einselection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Aspects of the prob-  lem (or its articulations) were manifold, such as the breakdown of the predicted  time-development in accordance with the Schr¨odinger equation, the abruptness  of change in a measurement (”natura non facit saltum”, where art thou?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' ), the  apparent non-applicability of quantum rules to macroscopic systems, imputed  arbitrariness of Born’s probability rules, the requirement of ”infinite regress”  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='02664v1  [quant-ph]  6 Jan 2023  for the measuring apparatus and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Numerous papers enlarged on these is-  sues [1, 2] and various proposals for resolution of the problem were put forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  These include the observer’s cognition [3], stochastic effects [4], in particular  spontaneous localization [2, 5, 6], a many world scenario [7], non-linearity addi-  tion to the Schr¨odinger equation [8], Poincar´e recurrent state [9], gravitationally  induced collapse [10, 11, 12], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Common to these works, and with the specific purpose of providing a  blue-print for measurements compatible with the Copenhagen formulation of  quantum theory, was the need to give expression to the coupling of the micro-  scopic system with its macroscopic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Standing apart from these and  belonging to the field of non-equilibrium thermodynamics and to the establish-  ment of equilibrium, a general form for this interaction was given by Lindblad  [13] and by Gorin , Kossakowski and Sudarshan [14], satisfying some necessary  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Constructing a merger between the two separately oriented fields,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Weinberg recently proposed a Lindblad-operator mechanism for the collapse  of the density matrix (DM) in the course of a complete measurement [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' No-  tably, the mechanism was linear in the state’s DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The collapsed state (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (1)  in [15]) comprises the set of projection operators of the measurable item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' the  system’s Hamiltonian is described by a spectral decomposition onto the same  operators (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  (16) in [15]) (although in the verbal discussion a more gen-  eral situation is considered): collapse is achieved ”independent[ly] of the details  of these [Lindblad] operators”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Decay between energy eigenstates had earlier  been treated by the Lindblad formalism (for a pedagogical presentation the  volume [16], Chapter 8 may be consulted) employing the interaction represen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' However, this is not convenient for treating measurements of observables  that do not commute with the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Detailed theories relate to the out-  come (”mapping”) of quantum operations, including measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' the present  work describes the process of these happening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (For a pedagogical introduction  to stochasticity-induced wave-packet- reduction, obviating pointer reading, one  may refer to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=')  2  Overview of the Method and Terms  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1  The leading idea, also in review  While the concept of unity of observer and observation had already featured  in Bohr’s view: ”The answer that we get is built up from the combined interac-  tion of [the observer’s] state and the object of interrogation.” [18], this was not  given a formal expression in the Copenhagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' It was more em-  phatically asserted both by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Bell: ”I meant that the ’apparatus’ should not be  severed from the rest of the world in boxes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='[19]” and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Peres: ” A measure-  ment both creates and records a property of the system [20]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' This change in  2  the course of a measurement affects also the environment outside the observed  system ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' in the words of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Leggett ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='under these conditions the macroscopic  apparatus, and more generally any part of the macro-world which has suffered  changes in the course of the measurement process, does not end up in a state  with definite macroscopic properties at all,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' [1]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  The same line of thought appears to motivate S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Weinberg, who wrote in his  preamble to a 2016 Lindbladian formulation of the masurement process[15], that  ”We will instead [of the original formulation of the Copenhagen interpretation,  (which we will not dwell on here)] adopt the popular modern  view that the  Copenhagen interpretation refers to open systems in which the transition is  driven by the ineraction of the microscopic system under study (which may  include an observer) chosen to bring the transition about.” (Our italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=')  These developments indicate the justification for a formulation in which the  effect of the apparatus is incorporated in the equation defining the evolution of  the system, rather than one in which the two entities are separate, barring an  interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1  ”Alignment”  The process whereby the pointer readings become in correspondence with the  possible values of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Formally, for I possible values, the combined  density matrix reduces from comprising I2 terms to one having I terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=',  equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='5) in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='2  ”Dissipator”  Added term (in the form of sums of appropriately weighted jump-operators)  to the standard time dependent Schr¨odinger equation, inducing non-unitary  evolution in the system, accompanied by changes of its information entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='2  Motivation for the choice of formalism  Thermalization of open systems can be described by a Lindbladian formal-  ism in which Gibbsian probabilities are so inserted as parameters, that the  ”Dissipator” vanishes at these values of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Replacement of  the Gibbsian probabilities by Born probabilities achieves alignment in a state  reduction and does so continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Limitations: Born’s probability rules are assumed, not derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' the interac-  tion term is not traced to a microscopic mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  The source of this interaction term, shown in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 6 below, incorporating  the coupling between the observed system and its surroundings (including the  3  measuring device) is an open question (also raised by a referee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In its appli-  cation to a thermalization process, the Lindbladian jump operators have been  derived, though with the aids of several approximations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=',[21]), as well as,  more recently, for the dissipation in a Dicke system with a bosonic background  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' We are not in the position to provide such first principle derivation for  the Lindbladian jump-operators bringing about a transition and incorporat-  ing the Born rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' It seems to be specific to the type of measurement under  consideration and it is clear that just any jump operator, as in Weinberg’s  Lindbladian formulation will not do the job .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Likely, one would need to in-  clude non-Markovian dynamics, so that the coupling to the device and eventual  pointer reading are two separate consecutive events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Inclusion of such dynamics  is outside the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  3  Assumptions  We explore the time (t)-development of the combined density matrix ρ(t) of  the measured system (S) and of the reading (pointer, dial, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=') on the mea-  suring apparatus (A) for a complete and discrete measurement , expressing the  underlying assumptions by three propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In accord with the long-time historical approach, the mea-  sured object S and the pointer of the measuring set-up A are treated on equal  footings as subject to microscopic quantum laws, and formally describable by  their respective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Aware of the difficulties connected with an ”in-  finite regress”, the effects of the rest of the Universe on S+A are not included  in the formalism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' instead, for a phenomenological, approximative description, a  Lindbladian term appears in the master equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Prior to the measurement with A and S decoupled, and being  free of external influence for a long time, both are in energy quantum states,  pure or mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' After the measurement, the state is not an energy eigenstate  and subsequently it will spread over to a superposition of energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  The fast decoherence case treated below in section 5 is akin to the Zeno effect  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Only those states of the reading apparatus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=', the right  or left positions of a pointer) that may be in direct correspondence with the  measured states of the system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=', spin up or down) are given expression in  the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (At a beginning, the case treated is one in which there is a one-  to-one correspondence between the states of the system and the readings of the  apparatus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' a generalization is given subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=') A discussion in section  8  touches on the epistemological status of the Lindbladian terms in a measurement  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  4  4  Analysis  Considering (for simplicity) a pure state for the system, its initial state-vector  written in the basis of the observed property |S, i > takes the form  ψS(t = 0) =  �  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='.,I  cS  i |S, i >  (1)  Born’s rule for the probability of observing the i-component is |cS  i ]2 ≡ pi, sum-  ming to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Likewise, for the apparatus readings j, numbering J, one has  the superposition with (complex and normalized) coefficients cA  j  ψA(t = 0) =  �  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='.,J  cA  j |A, j >  (2)  We start with the one-to-one correspondence situation, for which I = J, and the  reading j on A establishes uniquely the value i = j for the system’s measured  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  For the combined state-vector the density operator has the outer-product  form (where the stars denote complex conjugates):  �  i,j,i′,j′  |S, i > |A, j > cS∗  i cA∗  j cA  j′cS  i′ < A, j′| < S, i′| ≡  �  i,j,i′,j′  |i, j > Ciji′j′ < i′, j′|  (3)  the right hand side written in an obvious shortened notation, in which Ciji′j′ =  cS∗  i cA∗  j cA  j′cS  i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' After collapse, the density operator takes the aligned, single-sum  form  �  i  |S, i > |A, i > |cS  i |2 < S, i| < A, i|  (4)  It will be now shown that this is the time-asymptotic solution of the Lind-  bladian master equation properly parametrized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  We recall Lindblad’s equation for the time varying density of states operator  ρ ≡ ρ(t), as being of the following general form:  ∂ρ  ∂t = − i  ¯h[H, ρ] +  �  n  γn⟨LnρL†  n − 1  2(L†  nLnρ + ρL†  nLn)⟩  (5)  The second term, here named the ”Lindblad term” [13, 14] though in different  contexts also referred to as the Dissipator [24], contains Ln’s that are Lindblad  jump-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' We shall consistently work in the observable + pointer’s basis  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=', not in an energy basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In this basis, neither the density operator ρ = ρ(t),  nor the A+S Hamiltonian H is diagonal at the beginning or in the course of  the development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' But, as will be demonstrated, the Lindbladian formalism, by  a proper choice of its form, drives A+S to the desired diagonal form for the  combined observable +pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' We postulate just one single term in the  previous n-sum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' as well as off-diagonal forms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' namely |i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j ̸=  5  i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' for the jump-operators in the observable basis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' leading to the following  parametrized form of the Lindblad term  Lρ  ≡  ΓΩ  �  i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′̸=i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j  r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j)  r(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′)⟨|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′|ρ|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j|  −  1  2(|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′|ρ + ρ|i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′ >< i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j|i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j >< i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′|⟩  (6)  Here a circular frequency Ω is inserted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' so as to make Γ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' that quantifies the  strength of the system-environment coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' One notes that in  the pre-factor appear the parameters r(i, j), r(i′, j′)(i, j, i′, j′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=', I) whose  significance will be clear by deriving the matrix elements of the above operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  These are  Lρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′  =  δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='i′δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j)  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='l  r−1(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' l)ρk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='l  −  1  2[r−1(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j) + r−1(i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j′)]ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′  �  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='l  r(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' l)  (7)  It can be seen that the trace of the above vanishes and that each matrix element  vanishes upon the substitution  ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='i′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′ = δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='i′δj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j′r2(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j)  (8)  While these properties hold for any arbitrary r(ij),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' the observable-pointer align-  ment is achieved by identifying the r parameters with the system’s superposition  coefficient: r(ij) = |cS  i |δi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' or  r(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' j)2 = |cS  i |2 ≡ piδi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='j  (9)  the last being the Born probabilities appearing in the collapsed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' As already  noted, this identification of probabilities relates to the well known procedure for  the Lindblad-induced thermalization of open systems, for which detailed balance  imposes the relation between the pre-factors γ(δE)/γ(−δE) = e−βδE/Z, the  latter being the canonical probabilities (with β = 1/kBT, kB the Boltzmann  constant, T the ambient temperature and Z the partition function [24, 25, 21])  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  [It also seems fair to point out that also in the standard (Copenhagen, or von  Neumannian) description of the alignment stage, as appears in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='5) of  [1], this development is summarily stated, without specification of the underlying  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=']  5  Fast Decoherence Limit  We now consider the case that the time development in the state is predom-  inantly due to the coupling to the environment, rather than to the unitary  6  change induced by the Hamiltonian, meaning that the second term on the right  hand side in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 5 dominates the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Quantitatively: Γ >> ||H||/¯hΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Ne-  glecting the commutator we now form matrix elements of the Lindblad term in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 5 in the observable+pointer basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Because of the approximation made, the  off-diagonal matrix elements are decoupled from the diagonal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The master  equation of the off-diagonal terms reads (with a notation simplified by writing for  the index pairs i, j → r, i′, j′ → s and consequently for ρi,j,i′,j′ → ρrs ≡ ρrs(t)  dρrs  dt  = −ΓΩ  �√pr + √ps  2  ρrs  �  m  √pm  �  , r ̸= s  (10)  This shows that off-diagonal matrix elements decay exponentially in time (de-  cohere), maintaining their real character that they had initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Had we kept  the (imaginary) commutator term, we would have found that the decay is mod-  ulated by the eigen-energies of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  For the diagonal matrix elements we find,  dρrr  dt  = ΓΩ  �  √pr  �  m  ρmm  √pm  − ρrr  √pr  �  m  √pm  �  (11)  Again, it can be seen that the trace of the last expression vanishes, and so  does the right-hand side under the substitution ρrr → pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' With these taking the  values as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 9, one arrives at the aligned form (written out in the original,  system-pointer indexes)  ρ(t → ∞) =  �  i  |ψA  i > |ψS  i > |cS  i |2 < ψS  i | < ψA  i |  (12)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1  Illustrative example for a two-way experiment  Exemplifying the foregoing for a two-valued system (such as a 1  2-spin electron),  prepared as an eigenstate of a Zeeman-field with the magnetic field inclined at an  angle 2αS to the vertical, in conjunction with an apparatus pointer, represented  as being likewise in an eigenstate of a quasi-Zeeman field inclined at an angle 2αA  to the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The eigenstates are linear superpositions of their z- spins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' these  are the observables that are to be determined by the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Initially, the  system and the pointer are in the superposition states as shown above in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  1 and 2 and whose superposition coefficients cS  i and cA  j now have the values,  sin / cos(αS) and sin / cos(αA), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The DM in the observable basis  is now a 4x4 matrix, in which appear all the combinations of the products of  the above circular functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' As the outcome of the application of the Lindblad  operator in the rate equation, at long times the matrix becomes reduced to the  diagonal form discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In these, cos2(αS) = p1 and sin2(αS) = p4  belonging to the aligned observable lie on the diagonal and are non-zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' the  other two diagonal entries for the anti-aligned situations are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Plotted in Figure 1 are computed DM eigenvalues as functions of time  (in red and blue), normalized to their respective Born probabilities, showing  7  4  3  2  1  0  1  2  Log10time[invfrequn]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='4  P1,P2,Decoh  Figure 1: Density matrix eigenvalues normalized to their asymptotic (pointer-  aligned) values for the two aligned terms in the illustrative example (in red and  blue), plotted against time in inverse circular frequency unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In green is shown  a decohering off-diagonal matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Lindblad coupling strength Γ = 5, α  angles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='37 π and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='65 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  their asymptotic convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  In green, the typical decohering tendency of  an off-diagonal element is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Figure 2 depicts the entropy S(t) =  − �  r Pr(t) log Pr(t) of the system and apparatus-pointer, (in which Pr(t) are  computed eigenvalues of the DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=') The non-monotonic behavior is characteristic  of of the Lindblad formalism, in which the environment’s entropy change is not  taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  [In numerical work, based on forward integration, putting zeros for some  of the pi’s introduces singularities, eventually algebraically cancelling out, but  preventing flow of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Therefore, instead, one puts arbitrarily small  values for these and obtains for the aligned DM one that is arbitrarily close to,  but not exactly equal to the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=']  6  Eigenvalue analysis  An alternative to the numerical solution of the differential rate equation is eigen-  value analysis, already treated in [15], based on the Landbladian term being a  linear function of the diagonals in the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Thereby, the resulting  rate equations have solution of the form  ρnn(t) =  �  k  vn,keλkt  (13)  in which λk and vn,k are the diagonalized eigenvalues and eigenvectors of the  Lindbladian matrix diagonals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Calculation shows that for the 4 x  8  4  3  2  1  0  Log10time[invfrequn]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='20  S  Figure 2: Entropy of the combined system plus apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Noteworthy is the  initial peak common to the Lindblad formalism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  4 matrix considered above there are three negative eigenvalues and one zero  eigenvalue, which alone is of interest at the long term behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Belonging to  this eigenvalue, the (transposed) eigenvector is found to be {p1, p2, p3, p4} ≈  {cS  1 , 0, 0, cS  4 }, as required for the alignment between the quantum states and the  reading in the measuring apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1  Measurement speed  Figure 1 shows that alignment is achieved for the model with the chosen strength  parameter (Γ = 5) by a time of cca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' By varying the strength in the  computed model, we find a shortening of this time that is inversely proportional  to the strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' This is expected from the quantum speed limit (QSL) results  that border quantum transition times τ from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Essentially, QSL is the ratio of two norms [26, 27], that of the ”quantum  distance” [28] and of the speed of the state evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Formally  τ > ||ρ(t → ∞) − ρ(t = 0)||  || dρ(t)  dt ||  = ||ρ(t → ∞) − ρ(t = 0)||  ||[Lρ(t)]||  (14)  Ways of calculating the norms vary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=', [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Recently, for a system de-  veloping due to a Lindbalian operator, three contributions to the speed were  discerned [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' To estimate ||ρ(t → ∞) − ρ(t = 0)|, we have used the ”Trace  Distance ”defined as  T(ρ, σ) = 1  2Tr[  �  (ρ − σ)] = 1  2  �  i  |µi|  (15)  [31], where µi are the eigenvalues of the matrix differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The DM velocity,  as defined above , changes (decreases) with time, ultimately vanishing at the  9  fulfilment of alignment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' we have taken the root-mean-square sum of the rate of  the diagonal matrix elements at initial times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' These yield a very low limit of  τ > 1  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='41  5 ∗ 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='03 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='0045/Ω  (16)  to be compared with the actually computed value, about 20 times longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Better  (higher) limits of transition times may be generated by different ways of forming  the norm for the DM velocity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' not at the beginning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='2  Multiple Reading-System correspondence  A simple generalization of the foregoing applies when each (eigen-)value of the  observable is in correspondence with not just one reading of the pointer, but  with several (say, R) readings, all of the same significance for the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Then  one simply inserts pi/R into the corresponding Lindblad term, in place of just  pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' In the more complex case, that not all readings have the same likelihood,  pi would have to be weighted by a probability factot, rather than by a constant  denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  7  The Lindbladian, ”Who ordered this?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Historically, Lindblad terms were introduced as the most general forms that  maintain complete positivity of the DM’s and preserve their trace [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The  various derivations that have been presented (and among these a recent one by  [32]), involve several approximations for the coupling between the system and its  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Insomuch that the derivation involves also tracing over the degrees  of freedom of the environment, much detail of the latter is lost and of course  it is impossible to work backwards from the Lindbladian to the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  What is remarkable is that for special purposes the appropriate Lindbladian  operators take a very special, practically unique form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Such is the case for the  accepted description of thermalization [21, 24] by a Lindblad formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' The  parametrization of the Lindblad term employed in the present work, though it  may appear arbitrary and particular for each case, is in fact identical to the  one used for thermalization subject to the relabelling of the Gibbsian thermal  distribution function as (the Born) probabilities (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='),with the proviso of  working in the observable, rather than in the energy basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  (This contrasts  with the different approach in [15], which claims attainment of collapse for any  Lindbladian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=') At the same time, it needs to be noted that the analog  of detailed balance is missing in wave-function collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  How come to have  such a specific Lindbladian, whose source may be any measurement device and  procedure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' One is left to wonder about the possibility of a special meta-physical  status of the Lindblad terms, or query with Wheeler ”Who ordered this?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  10  8  Conclusion  The well-known Lindbladian extension to the quantum theory of motion to  environmental effects is here adapted to establish the resolution of a wave-packet  in a measurement as a smooth process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' This is enabled by an unambiguous  parametrization of the jump-operators describing the interaction of the broad  environment with the observed system, both regarded as quantal entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Above, in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1, a brief historically oriented preview has been provided  for the distinct approach in this work, namely, one based on the wholeness of  the entities (observed system and observing device), through the (Lindbladian)  equation yielding the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  A main result emerging from the formalism, and capable of experimental  verification, is the finitely temporal variation of the system, and this in a de-  terministic way rather than just statistically, on the average, contrasting also  with the instantaneous collapse description by (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=') von Neumann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Such tem-  poral variation in continuous-thermalization processes has been proposed quite  recently [33, 34], also by employment of a Lindbladian formalism and within a  Markovian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Experimentally, verification of the time dependence of the transition in any  particular measurement, implicit in our formulae, could be observed by re-  peated observation performed on the system subject to non-demolition tran-  sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' These observations would be akin to the Zeno-effect measurement, such  as has been achieved in the form of quasi-periodic oscillation of the result for  a Superconducting flux cubit[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content=' Further work is needed for quantifying the  information-entropy change in the environment [31, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='  Acknowledgement  The authors thank the referee for meticulous reading and insightful ques-  tioning of a previous version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
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+page_content=' doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
+page_content='5115323  [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
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+page_content='Lostaglio and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE0T4oBgHgl3EQfywKq/content/2301.02664v1.pdf'}
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+Quasi-equilibrium configurations of binary systems of dark matter admixed neutron
+stars
+Hannes R. R¨uter
+,1 Violetta Sagun
+,1 Wolfgang Tichy
+,2 and Tim Dietrich
+3, 4
+1CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
+2Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA
+3Institut f¨ur Physik und Astronomie, Universit¨at Potsdam, Haus 28, Karl-Liebknecht-Str. 24/25, Potsdam, Germany
+4Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, Potsdam 14476, Germany
+(Dated: January 10, 2023)
+Using an adapted version of the SGRID code, we construct for the first time consistent quasi-
+equilibrium configurations for a binary system consisting of two neutron stars in which each is
+admixed with dark matter. The stars are modelled as a system of two non-interacting fluids min-
+imally coupled to gravity. For the fluid representing baryonic matter the SLy equation of state is
+used, whereas the second fluid, which corresponds to dark matter, is described using the equation
+of state of a degenerate Fermi gas. We consider two different scenarios for the distribution of the
+dark matter. In the first scenario the dark matter is confined to the core of the star, whereas in the
+second scenario the dark matter extends beyond the surface of the baryonic matter, forming a halo
+around the baryonic star. The presence of dark matter alters the star’s reaction to the companion’s
+tidal forces, which we investigate in terms of the coordinate deformation and mass shedding pa-
+rameters. The constructed quasi-equilibrium configurations mark the first step towards consistent
+numerical-relativity simulations of dark matter admixed neutron star binaries.
+I.
+INTRODUCTION
+In the present era of gravitational wave (GW) astron-
+omy, the internal properties of compact stars can be
+probed during their mergers. Using numerical-relativity
+(NR) simulations of the last stages of a binary coales-
+cence, it is possible to relate observational GW data to
+properties of the source. While these simulations have
+undergone significant improvements in the past, the im-
+pact of dark matter (DM) on the binary neutron star
+(NS) dynamics has not yet been investigated in detail
+and is not taken into account in standard GW analyses.
+In fact, considering a coalescence of compact objects to
+occur in pure vacuum, could be an oversimplification that
+may lead to incorrect conclusions.
+Due to their high compactness, NSs can trap and ac-
+cumulate DM in their interior throughout the star’s evo-
+lution. DM alters the compact star’s properties, e. g., its
+mass, its radius, its tidal deformability, its energy density
+and speed of sound profiles [1–15]. Its effect depends on
+the relative fraction of DM and on the exact equation of
+state (EoS) for the DM and baryonic matter (BM). For an
+extended discussion of the impact of DM on compact star
+properties and its smoking gun signals, see Refs. [16–18].
+While the effect of DM on isolated NSs can be probed
+through electromagnetic observations, GW observations
+of binary systems of DM admixed compact stars open up
+a new observational window and the possibility to probe
+a density and temperature range larger that of isolated
+stars. To push forward our understanding of the imprint
+of DM, we construct quasi-equilibrium configurations of
+DM admixed NS binary system and study the impact of
+DM focusing on quantities pertaining to binary system,
+such as the orbital binding energy and the tidal deforma-
+tions.
+It is worth noting that not only NSs, but also black
+holes could be embedded into DM. A step towards un-
+derstanding the impact of DM on black hole mergers was
+made in [19], where the behaviour of wave DM around
+equal mass black hole binaries was studied in numerical
+simulations. Furthermore, GW signals from binary coa-
+lescences contain information of the binaries surrounding
+medium [20].
+The effect of DM on the inspiral and post-merger
+phases of DM admixed NSs has been studied by a few
+groups. A first study by Ellis et al. [21] used a simple
+mechanical model, and showed that a DM core can lead
+to the appearance of additional peaks in the post-merger
+GW spectrum. In [22] NR simulations of equal-mass bi-
+naries consisting of BM admixed with a bosonic Klein-
+Gordon field were performed. For a DM mass fraction of
+10%, a redistribution of fermionic matter by the bosonic
+cores was found, followed by the formation of a one-arm
+spiral instability. Another approach approximating com-
+pact dark component as test particles was studied in [23].
+The simulations show the DM component to remain grav-
+itationally bound after the merger of BM and orbit the
+center of the remnant with an orbital separation of a few
+km. The DM core and a host star are likely to spin at
+different rotational frequencies just after the merger due
+to the absence of non-gravitational interaction. Further
+on, they may synchronise via the gravitational angular
+momentum transfer, including tidal effects [24].
+Up to our knowledge, the first two-fluid NR simulations
+describing binaries of DM admixed NSs were performed
+by Emma et al. [25] for a mixture of BM and mirror DM
+only interacting via the gravitational field. The results
+demonstrate that these systems tend to have a longer in-
+spiral phase with increasing amount of DM, which could
+be associated to the lower deformability of DM admixed
+NSs. These simulations however, did not start from ini-
+tial data satisfying the Hamiltonian and momentum con-
+arXiv:2301.03568v1  [gr-qc]  9 Jan 2023
+
+2
+straints [26–28] and the fluids did not start in an equilib-
+rium configuration. Instead the initial data was approx-
+imated by superimposing TOV-like solutions of isolated
+DM admixed NSs. In this work we construct consistent,
+constraint-solved, quasi-equilibrium conditions for a two-
+fluid system of BM and DM.
+One possible scenario for the formation of DM admixed
+NSs is the capture of DM particles during the lifetime of
+the star, from a progenitor to the equilibrated NS stages.
+The core of a NS is very dense and hence the chance of
+a DM particle experiencing scattering is relatively high.
+In this scattering process the particle transfers its kinetic
+energy to the star, becoming gravitationally bound [29–
+31].
+This process is more efficient towards the Galac-
+tic center, where the density of DM is many orders of
+magnitude greater than in the galaxy’s arms [32, 33]. A
+conservative estimate of DM capture in the most cen-
+tral part of the Galaxy shows that stars accumulate up
+to 0.01% of DM during the main sequence and equili-
+brated NS stages combined [8]. However, also higher DM
+factions inside compact stars can be achieved through
+other scenarios, e.g., DM production during a supernova
+explosion, accretion of DM clumps formed at the early
+stage of the Universe, or initial star formation on a pre-
+existing DM seed or local DM rich environments [34, 35].
+If DM is symmetric, it cannot reach a high fraction due
+to self-annihilation, producing an electromagnetic or neu-
+trino signal [36]. The latter scenario could lead to addi-
+tional heating of isolated NSs as well as post-merger rem-
+nants [37, 38], modification of kinematic properties [39].
+Moreover, production of light DM particles, e.g., axions,
+in nucleon bremsstrahlung or in Cooper pair breaking
+and formation processes in the NS interior [40–43], could
+speed up the thermal evolution of a star by contributing
+an additional cooling channel.
+We consider DM to be either concentrated in a core or
+extending beyond the surface of BM, forming a DM halo
+around it. As a first step, we consider non-interacting,
+fermonic DM with spin 1
+2. The single star properties of
+this DM candidate have been discussed in Ref. [8]. The
+baryonic component is modelled through a piecewiese-
+polytropic fit [44] of the SLy EoS [45] that reproduces
+nuclear matter ground state properties, fulfils heaviest
+pulsars measurements [46, 47], X-ray observations by
+NICER [48–52], and tidal deformability constraints from
+GW170817 [53] and GW190425 [54] binary NS mergers.
+The two components interact only through gravity, and
+therefore do not repel each other, but overlap due to the
+absence of non-gravitational interaction. This assump-
+tion is in very good agreement with the observations of
+the Bullet Cluster [55, 56] and direct DM searches [57],
+which show that the DM-BM cross section to be many
+orders of magnitude lower than the typical nuclear one,
+σDM−BM ≈ 10−45 cm2 ≪ σBM ∼ 10−24 cm2.
+By varying the particle mass and relative fraction of
+DM, we obtain either a core configuration with a ra-
+dius of the DM component less or equal to the baryonic
+one, RD ≤ RB, or a halo with RD > RB [58].
+For
+both scenarios, we construct initial configurations em-
+ploying SGRID [59, 60]. Many other codes exist for the
+construction of quasi-equilibrium NS binary systems, no-
+tably the spectral codes LORENE [61, 62], Spells [63],
+FUKA [64, 65], Elliptica [66], and the finite difference
+based code COCAL [67, 68]. Up to our knowledge, these
+codes are only able to solve systems consisting of a sin-
+gle fluid.
+Here we construct for the first time quasi-
+equilibrium binary configurations with two fluids.
+The formalism and results are presented in geometric
+units in which the gravitational constant G = 1 and the
+speed of light c = 1. In these units, lengths are given
+as multiples of the solar mass, M⊙. For the conversion
+to SI units a spatial length must be multiplied by L0 =
+1476.6250 m/M⊙ and a time by T0 = 4.9254909 × 10−6
+s/M⊙. Where appropriate we also use MeV to specify en-
+ergy and mass of particles, as well as SI units. Through-
+out the paper, Greek letter indices denote four dimen-
+sional, spacetime indices, whereas Latin indices denote
+three-dimensional, spatial indices.
+The paper is organized as follows.
+In Section II we
+summarize the two-fluid formalism and DM distribu-
+tion regimes. Its implementation to the SGRID code is
+described in Section III. In Section IV we analyse the
+convergence properties of the constructed configurations,
+quantify the difference in the velocities of the two flu-
+ids and investigate some physical properties of the quasi-
+equilibrium configuration over a sequence of separations.
+Section V summarizes the results and discusses future
+perspectives.
+II.
+FORMALISM
+We describe the matter as a system of two non-
+interacting perfect fluids only indirectly coupled through
+the gravitational field. This model is well justified, be-
+cause the interaction between standard model BM and
+DM is weak. Utilisation of the perfect fluid model for DM
+is also justified, as the mean free path and the scattering
+time scale of DM particles can be small compared to the
+characteristic time scales of the binary. In the following,
+we estimate the mean free path and scattering time in
+a semi-classical approach for a degenerate Fermi gas of
+particles with the mass of 170 MeV (≈ 3 × 10−28 kg).
+The Fermi gas consists of non-interacting fermions, for
+which a self-scattering cross section σDM formally does
+not exist. Instead, we use the value of the upper limit
+obtained from observations of merging galaxies, which
+yield σDM/m(DM)
+p
+< 1.25 cm2/g, with m(DM)
+p
+the mass
+of the DM particles [56, 69]. In this work we construct
+configurations with a particle density n(DM) of 0.7 fm−3
+in the center of the star. Together with the upper limit
+for σDM this yields a mean free path λ = 1/(n(DM)σDM)
+of 3.7 × 10−17 m, much smaller than the typical length
+scale of a NS, which is on the order of 104 m. The scatter-
+ing time scale can be estimated using the Fermi velocity,
+which reaches values up to 0.8 c in the centre of the star.
+
+3
+Finally, using the value of the mean free path, this yields
+a scattering time of tc = λ/vDM = 1.5 × 10−25 s, much
+smaller than for example the orbital period of the binary,
+which in our configurations is a small as 3 × 10−4 s. At
+the surface of the stars DM reaches the free streaming
+limit and the perfect fluid limit breaks down, but there
+the density is so small, that the impact on the gravita-
+tional field is low and hence the matter in this region can
+be neglected.
+For non-interacting fluids, the energy-momentum ten-
+sor can be split into the two individual fluid components
+given by:
+T (s)
+µν = (e(s) + p(s))u(s)
+µ u(s)
+ν
++ p(s)gµν ,
+(1)
+where e is the proper energy density, p is the pressure, uµ
+is the four velocity of the fluid and the label (s) denotes
+the particles species, which is either BM or DM. The
+Einstein field equations are then given by
+Rµν + 1
+2gµνR = 8π(T (BM)
+µν
++ T (DM)
+µν
+)
+(2)
+and, because the two particle species do not interact,
+each fluid satisfies the equations of motion of a single
+fluid. Consequently, each fluid satisfies energy momen-
+tum conservation separately: ∇µT (s)
+µν = 0.
+For each fluid, we also define the rest mass density ρ(s)
+0 ,
+which is computed from the number density n(s) by
+ρ(s)
+0
+= m(s)
+p n(s) ,
+(3)
+with m(s)
+p
+being the mass of the particles. Furthermore,
+the specific enthalpy is then given by
+h(s) = e(s) + p(s)
+ρ(s)
+0
+.
+(4)
+To make the equations tractable, the spacetime metric
+gµν is decomposed into a temporal and a spatial part by
+introducing the spatial metric γij, the lapse α, and the
+shift βi [27, 70, 71]. The line element in this 3+1 split
+reads
+ds2 = −α dt2 + γij (βidt + dxi)(βjdt + dxj) .
+(5)
+The extrinsic curvature Kij is related to the time deriva-
+tive of γij, by the formula
+Kij = − 1
+2α(∂tγij − Diβj − Djβi) ,
+(6)
+where Di denotes the covariant derivative compatible
+with the spatial metric γij.
+We construct the partial differential equations govern-
+ing quasi-equilibrium by following the derivation in [72],
+which is trivially applied to a system of non-interacting
+fluids. To generate quasi-equilibrium configurations, we
+solve equations for velocity potentials φ(s), which are de-
+fined through the following split of the four-velocity
+γi
+µu(s)µ =
+1
+h(s) (Diφ(s) + w(s)i) ,
+(7)
+where w(s)i is a divergence free vector, i.e., Diw(s)i = 0,
+describing the rotational part of the fluid. Following the
+derivation of [72], we fix the time derivatives of the fields
+by assuming the existence of an approximate Killing vec-
+tor ξ and a set of quasi-equilibrium conditions for the two
+fluids
+Lξe(s) ≈ 0 ,
+(8)
+Lξp(s) ≈ 0 ,
+(9)
+γi
+µLξ(∇µφ(s)) ≈ 0 ,
+(10)
+γi
+µL
+∇φ(s)
+h(s)u(s)0 w(s)
+µ
+≈ 0 .
+(11)
+We omit further details of the derivation, since for non-
+interacting fluids everything can be directly carried over
+to the individual fluid components, and we state only
+the resulting partial differential equation for the velocity
+potentials φ(s):
+Di
+�
+ρ(s)
+0 α
+h(s) (Diφ(s) + w(s)i) − ρ(s)
+0 αu(s)0(βi + ξi)
+�
+= 0 ,
+(12)
+where the boost factor u(s)0 is given by
+u(s)0 =
+�
+h(s)2 + (Diφ(s) + w(s)
+i )(Diφ(s) + w(s)i)
+αh(s)
+,
+(13)
+and the specific enthalpy is given by the expression
+h(s) =
+�
+L(s)2 − (Diφ(s) + w(s)
+i )(Diφ(s) + w(s)i) , (14)
+with
+L(s)2 =
+b(s) +
+�
+b(s)2 − 4α4((Diφ(s) + w(s)
+i )w(s)i)2
+2α2
+(15)
+and
+b(s) = ((ξi+βi)Diφ(s)−C(s))2+2α2(Diφ(s)+w(s)
+i )w(s)i .
+(16)
+The variable C(s) is a constant, which can vary for each
+star and controls the mass of the fluid component.
+For the approximate Killing vector ξi we make the fol-
+lowing ansatz:
+ξi = Ω(−y, x − xCM, 0) + vr
+D (ri − ri
+CM) ,
+(17)
+where Ω is the instantaneous orbital frequency, D is the
+separation between the star centres, vr is the radial ve-
+locity, and xCM is the x-coordinate of the centre of mass.
+
+4
+At apsis the orbital frequency together with the sepa-
+ration of the stars control the orbital parameters like ec-
+centricity and length of the semi-major axis. Away from
+apsis there is a non-vanishing radial component of the
+velocity to be taken into account. In cases like the “circu-
+lar” inspiral there is no apsis, but there is an always non-
+vanishing radially inward directed velocity component.
+The configurations presented in this work are constructed
+within the quasi-circular approximation for which the ra-
+dial component is neglected, vr = 0.
+We set the value of Ω to its value at second
+Post-Newtonian order in Arnowitt-Deser-Misner (ADM)
+gauge [73–75] using the sum of the rest masses of the
+two fluids as the mass estimates of the stars, which are
+computed by
+m(s)
+0i =
+�
+Vi
+ρ(s)
+i u(s)0α
+�
+det(γjk)d3x ,
+(18)
+where Vi is the spatial volume over which the i-th star
+extends. The value of xCM is then given by
+xCM = (m(BM)
+01
++ m(DM)
+01
+)xc1 + (m(BM)
+02
++ m(DM)
+02
+)xc2
+m(BM)
+01
++ m(DM)
+01
++ m(BM)
+02
++ m(DM)
+02
+,
+(19)
+where xc1/2 are the x-coordinates of the centres of the
+stars.
+In this work, we present results for equal-mass
+configurations only, i. e., xCM = 0.
+Besides the continuity equation (Eq. (12)) governing
+the fluid velocity potentials φ(s), the metric must be fixed
+in a way satisfying the ADM constraints. To this end we
+choose a conformally flat ansatz for the spatial metric,
+i. e., γij = ψ4¯γij, with γij = δij and ∂tγij = 0, and con-
+struct the data on maximally sliced hypersurfaces, i. e.,
+the trace of the extrinsic curvature vanishes: K = 0 and
+∂tK = 0.
+The free metric components are the lapse,
+shift, and conformal factor ψ and their governing equa-
+tions are formulated in terms of the extended conformal
+thin sandwich equations (XCTS) [27, 28]. Together with
+Eq. (12), the data is constrained by a set of seven coupled
+partial differential equations, which are solved iteratively
+one-by-one in a self-consistent manner.
+III.
+SGRID
+We have adapted the pseudo-spectral SGRID code [59,
+60] to generate quasi-equilibrium configurations for two
+fluid systems.
+We use the same iteration scheme that
+is used in [60] for single-fluid NSs. We sketch the iter-
+ation scheme in the following with an emphasis on the
+adaptions and changes made.
+1. To ensure the convergence of the solver, it is nec-
+essary to provide an initial guess sufficiently close
+to the true solution. This initial guess is chosen as
+a superposition of two boosted TOV-like two fluid
+stars of a given mass. To generate solutions with
+particular rest masses for the fluid components,
+one has to find the central pressures for which the
+masses are realized. Since we are dealing with two
+fluids, this is a two-dimensional root finding prob-
+lem. In our tests, we found that using the Newton-
+Raphson method is not always reliable, because the
+masses are not a monotonous function of the central
+pressures, hence, a Newton-Raphson solver easily
+gets caught in a local extremum of the mass func-
+tion. Instead, we employ a series of bisections on
+the central pressure of one fluid component while
+keeping the central pressure of the other fluid fixed.
+The series of bisections iterates between the two
+fluid components in a self-consistent manner until
+the fluid masses are sufficiently close to the target
+parameters.
+2. If the residuals of Eq. (12) are larger than 10%
+of the combined residuals of the XCTS equations,
+we solve Eq. (12) and set the new φ(s) to be the
+average of the old solution φ(s)
+old and the just ob-
+tained solution φ(s)
+ell , using the following weights
+φ(s) = 0.8φ(s)
+old + 0.2φ(s)
+ell .
+3. We proceed by solving the XCTS equations and
+update α, β, and ψ in the same way, averaging the
+old and new solution.
+4. We do not adjust the values of Ω and xCM as in [60].
+The value of Ω would be fixed within an eccentric-
+ity reduction scheme. xCM is left at its Newtonian
+value, Eq. (19).
+5. We adjust the constants C(s), such that the rest
+masses of each component and in each star match
+our desired target masses. We then update the val-
+ues of h(s) keeping it fixed until the end of the next
+iteration.
+6. If the sum of the residuals is below a certain toler-
+ance or a prescribed maximum number of iterations
+is reached, the iteration ends here and is concluded
+with a final solving of the XCTS equations.
+7. The system of partial differential equations does
+not fix the position of the stars and, hence, they will
+slowly drift if not kept under control. To keep the
+stars in place, the center of the stars are driven back
+to the desired position. For single fluids, the center
+is usually defined in an unambiguous way as the
+point of maximum density. For two fluids the defi-
+nition is ambiguous, because the tidal deformations
+due to the companion star are different for each
+fluid component and, consequently, the maximum
+densities are at different points. In most cases, how-
+ever, the two maximum points will still be close.
+The results shown in this work are obtained by
+choosing the point with the maximum of the to-
+tal proper energy density, e(tot) = e(BM) + e(DM),
+as the center of the stars. We have chosen e(tot),
+
+5
+1
+1.05 1.1 1.15 1.2 1.25 1.3
+1
+1.05
+1.1
+1.15
+1.2
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+X
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+Y
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+X
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+Y
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+X
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+Y
+-25
+-20
+-15
+-10
+-5
+0
+5
+10
+15
+20
+25
+X
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+10
+Y
+FIG. 1.
+Specific enthalpy in the z = 0 plane for a config-
+uration with DM halo. In the upper halves only the specific
+enthalpy of DM is shown, whereas in the lower halves the
+BM component lies on top of it. The black lines indicate the
+boundaries of the spectral elements. Each NS is comprised of
+a central cubical element and six cubed sphere elements (of
+which only four intersect the z = 0 plane). The separation
+between the NS centres amounts to 32 M⊙ (47.3 km).
+in particular, because it is a covariant scalar and it
+is the major quantity determining the gravitational
+potential, hence giving an estimate for the center of
+mass of the star. To drive the center of mass back,
+the values of h(s) are transformed by
+h(s),new = h(s) + ∆ri∂ih(s) ,
+(20)
+where ∆ri = ri
+current − ri
+desired.
+8. Continue with step 2.
+The SGRID code uses surface-fitted coordinates to re-
+duce the Runge phenomenon at the surface of the star.
+Each time we update the specific enthalpy h(s) (step 5
+in the iteration), we adapt the grid such that the bound-
+aries of spectral elements coincide with the new surface
+of the outer fluid. That means we only construct con-
+figurations in which the surfaces of the two fluids do not
+intersect, which would in principle be possible given the
+different deformabilities of the fluids. Furthermore, we do
+not construct domains that are adapted to the surface of
+the inner fluid.
+Therefore, at the surface of the inner
+fluid one can expect to observe the Runge phenomenon
+and a slight degradation of the convergence in the trun-
+cation error. Fig. 1 shows a visualisation of the deformed
+spectral elements inside the NS and the distribution of
+matter in terms of the specific enthalpy.
+To close the system, the EoS is required to relate e(s),
+p(s), ρ(s)
+0 , and h(s). For the EoS, SGRID reads in either
+parameters of piecewise polytropes or EoS tables. EoS
+tables are interpolated in a thermodynamically consis-
+tent manner [76] using a cubic Hermite interpolation. To
+find the thermodynamic quantities for a given specific
+enthalpy a Newton-Raphson root finder is used. At low
+densities we use a polytrope that is matched at the lowest
+density of the table.
+IV.
+RESULTS
+A.
+Parameters of Constructed Configurations
+We consider two different two-fluid configurations, one
+in which DM is confined to the core of the NS, the dark
+core configuration, and one in which DM extends beyond
+the surface of the BM, so that the NS has a DM halo,
+which we will refer to as the dark halo configuration.
+Furthermore we compare to configurations consisting of
+BM only: the single fluid configuration.
+We describe BM by a piecewise-polytropic fit [44] to
+the SLy EoS [45]. As a model of DM, we investigate the
+degenerate, relativistic Fermi gas of spin- 1
+2 particles at
+zero temperature, for which the EoS is read in as tabu-
+lated data. EoSs at zero temperature are sufficient for
+our calculations, because the Fermi energy of the sys-
+tem is much higher than its temperature. The typical
+temperature T0 of NS cores is of the order of 106 − 108
+K [77, 78]. We assume that DM has the same tempera-
+ture as the BM, because the captured DM particles keep
+scattering with baryons, rarely but often enough to ther-
+malise with the BM component. A core temperature of
+approximately 108 K is much lower than the Fermi en-
+ergy of BM. This is also true for the Fermi gas EoS we
+consider, e. g. in the dark halo case the Fermi energy of
+DM reaches 403 MeV in the center of the star, an energy
+smaller than that of the BM, but still much larger than
+the temperature of the star, kBT0 �� 0.009 MeV.
+For the dark core configuration the DM particles have a
+mass of 1000 MeV and DM provides 5% of the NSs’ total
+rest mass.
+Fermionic DM particles with mass of 1000
+MeV present an interesting case, because they resemble
+nucleons.
+In the dark halo case we model DM by particles with
+a mass of 170 MeV, for which the fluid is less dense and
+hence easily forms a halo. Furthermore in the dark halo
+configuration DM only contributes 0.5% of the total rest
+mass. The choice of these values for the particle masses
+is motivated by the results of [8], where it was shown that
+for the DM particle masses below 174 MeV DM admixed
+NSs are in agreement with astrophysical observations of
+the heaviest NSs for arbitrary relative fraction of DM.
+Moreover, the chosen mass of 170 MeV and the fraction
+of 0.5% leads to a relatively small halo of approximately
+twice the radius of the BM component, which is easy to
+model. When the size of the halos is big enough so that
+they touch each other, it is no longer possible to fit the
+element surfaces to the outer fluid of a star. Hence, we
+are discarding configurations with separations at which
+the two halos merge. In all configurations the individual
+NSs have the same total rest mass, i. e., the combined
+rest mass of BM and DM is 1.4M⊙. In all setups, the
+NSs have equal masses and are irrotational, i. e., they
+have zero spins.
+
+6
+B.
+Convergence
+To validate the code, we check the convergence of the
+Hamiltonian constraint for a dark halo configuration of
+NSs with a separation of 44 M⊙ (65.0 km) on a quasi-
+circular orbit.
+Fig. 2 shows the magnitude of the Hamiltonian con-
+straint H on the z = 0 plane. The constraint violations
+are largest in the interior of the star, where they reach
+values up to 4×10−5, whereas in the vacuum regions the
+error drops to values below 10−9, but with some spikes
+on the order of 10−7 at the element boundaries. A be-
+haviour commonly seen for spectral codes. The Hamil-
+tonian constraint is largest in the region where the inner
+fluid is non-vanishing. In Fig. 2 one can observe a clear
+transition on the surface of the baryonic fluid to lower
+constraint violations in the DM halo.
+Fig. 3 demonstrates the development of the volume-
+normalised L2-norm of the Hamiltonian constraint for the
+inner cube of one of the stars during the iterative solv-
+ing process. The figure shows the behaviour for different
+number of points n in each dimension, which is the same
+for each spectral element. All curves show a saturation
+in the norm of the Hamiltonian constraint towards the
+end of the iteration process, which for all configurations
+is stopped after 40 iterations. Furthermore, it is visible
+that higher resolution leads to smaller violations of the
+Hamiltonian constraint in the final solution. For compar-
+ison Fig. 3 also shows the sequence for a corresponding
+single fluid configuration with the same mass and sepa-
+ration. After 40 iterations its Hamiltonian constraint is a
+factor 10 smaller than the dark halo configurations and it
+does not show any signs of saturation, i. e. it would prob-
+ably reach even smaller constraint violations if iterated
+further.
+The convergence in the final solution is further inves-
+tigated in Fig. 4, which shows its L2-norm of the Hamil-
+tonian constraint with respect to the number of colloca-
+tion points in the spectral elements. The figure shows
+the constraint violation for the inner cube element and
+for the cubed sphere facing towards the companion star,
+which is also representative for all other cubed sphere el-
+ements inside the NSs. The curves are almost straight
+lines on the log-log-plot of Fig. 4, which is compatible
+with a polynomial convergence of the constraints, i. e.,
+|H|L2 ∼ n−p, with p the order of convergence.
+This
+is the expected convergence behaviour for non-smooth
+data, which we have due to the surface of the inner fluid.
+Using the highest and lowest resolution we can estimate
+the order of convergence in the inner cube element to be
+p ≈ log22/10(|H|L2,n=10/|H|L2,n=22) ≈ 2.7.
+To investigate the convergence of the actual solution
+variables we interpolate the data from different resolu-
+tions on a common set of points and compute norms
+of the estimated errors on these points.
+We interpo-
+late the solution onto a 10 × 10 × 10-grid equidistant
+in each direction, with coordinate components given by
+ri ∈ {20m/9, m ∈ [0..9]}. This grid includes some points
+1e-13
+1e-12
+1e-11
+1e-10
+1e-9
+1e-8
+1e-7
+1e-6
+1e-5
+1.00  
+1.33  
+1.05 1.1 1.15 1.2 1.25
+1.00  
+1.22  
+1.05
+1.1
+1.15
+FIG. 2.
+Hamiltonian constraint in a dark halo configuration
+in the z = 0 plane. In the lower half the specific enthalpy of
+the two fluids is overlaid.
+with pure vacuum, points with only one fluid present
+and points with both fluids present.
+The error in the
+solution is estimated by taking the difference to the so-
+lution with the highest resolution, i. e., the solution that
+has 22 points in each dimension of the spectral elements.
+In Fig. 5 we show the convergence of the 1-norm and
+the maximum norm over the set of interpolated points
+for the gxx component of the metric and the lapse α.
+Both quantities do not show a monotonic decay of the er-
+ror, but there is an overall trend of decaying error. This
+somewhat broken convergence behaviour can again be
+attributed to the presence of non-smooth fields on the
+surface of the inner fluid. Fig. 6 shows the convergence
+of the error in the specific enthalpy.
+The DM in this
+configuration is fitted to the element boundaries and its
+specific enthalpy displays a relatively clear convergence
+behaviour.
+The BM fluid on the other hand shows a
+very broken convergence and only very little improve-
+ment from the lowest to the highest number of points.
+The maximum norm of the error is actually growing for
+the two largest number of points, whereas the 1-norm
+of the error is also slightly broken, but with an overall
+behaviour similar to that of gxx and α.
+It should be noted, that it is not clear whether the
+formalism of Sec. II actually possesses a unique solution.
+The partial differential equation (12) is not strictly ellip-
+tic on the fluid surface and hence the standard theorems
+for the uniqueness of the solution can not be applied. In-
+stead our algorithm might find a solution of many possi-
+ble, which is another possible explanation for the slightly
+broken convergence behaviour.
+C.
+Difference in the Fluid Velocities
+It is worth pointing out that even if the BM and
+DM fluid components are both irrotational, i. e., non-
+spinning, the exact velocity profiles are not the same.
+The reason for this does not lie in the notion of an irro-
+tational fluid, but is caused by differences in the fluids’
+equations of motion. An irrotational fluid is defined by
+
+7
+0
+5
+10
+15
+20
+25
+30
+35
+40
+iteration
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+|H|
+L
+2
+/V
+element
+n
+=
+22, single fluid
+n
+=
+12, dark halo
+n
+=
+14, dark halo
+n
+=
+18, dark halo
+n
+=
+22, dark halo
+FIG. 3.
+L2-norm over the inner cube in one of the stars,
+normalised by the volume of the inner cube. The different
+lines show configurations with different number of points n in
+each dimension.
+10
+12
+14
+16
+18
+20
+22
+numer of points in each dimension n
+10
+-4
+|H|
+L
+2
+/V
+element
+inner cube
+left cubed sphere
+FIG. 4.
+Normalised L2-norm of the Hamiltonian constraint
+in a dark halo configuration for a different number of points
+per dimension. The norm is normalised by the volume of the
+spectral element. Note that the x-axis and y-axis are scaled
+logarithmically.
+10
+12
+14
+16
+18
+20
+numer of points in each dimension n
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+10
+1
+error norm of variable Y, ||Y
+n
+−
+Y
+22
+||
+g
+xx
+, 1-norm
+g
+xx
+, maximum norm
+α, 1-norm
+α, maximum norm
+FIG. 5.
+Self-convergence of metric variables in dark halo
+configurations. Black: error norm of the gxx component of
+the metric. Blue: error norm of the lapse, α. We not that the
+1-norm is not normalised by the number of points.
+10
+12
+14
+16
+18
+20
+numer of points in each dimension n
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+||h
+(s)
+n
+−
+h
+(s)
+22
+||
+h
+BM
+, 1-norm
+h
+BM
+, maximum norm
+h
+DM
+, 1-norm
+h
+DM
+, maximum norm
+FIG. 6.
+Self-convergence of the specific enthalpy in dark halo
+configurations.
+Black: error norm of the baryonic specific
+enthalpy h(BM), which is the inner fluid. Blue: error norm of
+the specific enthalpy of DM, h(DM). We not that the 1-norm
+is not normalised by the number of points.
+the vanishing of its kinematic vorticity tensor [79]
+ωαβ := P µ
+α P ν
+β ∇[µuν] = 0 ,
+(21)
+with P µ
+α = δµ
+α + uµuα.
+This notion does not depend
+on the thermodynamic properties of the fluid and hence
+differences in the velocities can only be the result of the of
+the equations of motion, that are used in the derivation of
+the formlalism in Sec. II, i. e. the Euler equations [72, 80]
+u(s)µ∇µ(h(s)u(s)
+ν
++ ∇νh(s)) = 0 ,
+(22)
+which follow from ∇µT (s)
+µν = 0, and the continuity equa-
+tion
+∇µ(ρ(s)
+0 u(s)µ) = 0 .
+(23)
+If for example the DM would have the same four-velocity
+as the BM, it would still be irrotational, but might be
+incompatible with the laws of energy-momentum or par-
+ticle number conservation.
+In nature the disparity in the fluid velocities is affected
+by two counter-acting effects, particle scattering between
+BM and DM on the one hand and physics determining
+spin-down on the other hand.
+In our formulation the
+two fluids are modelled as non-interacting, but the BM-
+DM scattering cross-section might be non-zero in nature,
+which would drive the two fluids towards a common ve-
+locity. This process is counter-acted by effects driving the
+fluid into an irrotational state, as for example magnetic
+braking for BM [81–83]. It is unclear whether a similar
+effect exists for DM and whether it is dominant over the
+effect of BM-DM scattering. By assuming vanishing of
+the kinematic vorticity for the DM component, we as-
+sume that such an effect exists and it is also dominating
+over the scattering with BM.
+We find that both fluids move with basically the same
+velocity, with coinciding velocities in the star center,
+
+8
+12
+14
+16
+18
+20
+x
+−0.15
+−0.10
+−0.05
+0.00
+0.05
+0.10
+relative difference, V
+(s)x
+1
+−
+V
+(DM)x
+/V
+(BM)x
+, dark halo
+1
+−
+V
+(DM)x
+/V
+(BM)x
+, dark core
+V
+(BM)x
+, dark halo
+V
+(DM)x
+, dark core
+FIG. 7.
+Relative difference in the velocities for configurations
+with a separation of 32 M⊙. The difference is shown along a
+diagonal with the parametrization ri(s) = s(1, 1, 0)+ri
+c, going
+through the center of the star located at ri
+c = (16M⊙, 0, 0).
+V (BM)x (black, dash-dotted line) and V (DM)x (grey, dotted
+line) show the x-component of the velocity of the respective
+inner fluid.
+but increasing difference towards the surface of the in-
+ner fluid. We quantify this effect in terms of the residual
+three-velocity V (s)i, in which the orbital movement given
+by the Killing vector ξµ is split off,
+V (s)i = u(s)i/u(s)0 − ξi .
+(24)
+Fig. 7 shows the x-component of V (s)i and the relative
+difference of the fluid velocities for the region in which
+both fluids are present. We present results for configu-
+rations at a separation of 32 M⊙, a separation at which
+the DM halos in the dark halo configurations are already
+relatively close and deformed (Fig. 1). We find that dif-
+ferences in the two fluids are smaller for larger separation,
+which is intuitively understandable, because for large sep-
+arations the system goes to the limit of isolated NSs in
+which the fluid velocities coincide.
+The data in Fig. 7 is shown along a diagonal through
+the star parametrized in the following way:
+ri(s) =
+s(1, 1, 0)+ri
+c, where ri
+c is the center of the star. We choose
+to present the data along this diagonal because the differ-
+ence V (BM)i − V (DM)i has a quadrupolar structure with
+nodes going through ri
+c and being approximately parallel
+to the x and y axes. Hence the difference is basically zero
+on the x and y-axis, but very prominent along the spec-
+ified diagonal. The relative difference between the resid-
+ual velocities is below 0.2% near the center of the star
+and reaches up to 10% on the surfaces of the inner fluids.
+The difference between the velocities of the dark halo and
+dark core configurations is relatively small, which can be
+seen from the fact the curves of the velocities of the inner
+fluids lie on top of each other.
+D.
+Binding Energy
+NSs with a DM component are more tightly bound,
+because the DM component adds gravitating mass, but
+provides no additional repulsion to balance the gravita-
+tional pressure [8]. The gravitational binding energy of
+the particles is the difference of the ADM mass [27, 84, 85]
+and the sum of the rest masses m(s)
+0i of the components.
+If all fluid particles would fall in from infinity, the true
+ADM mass would equal the total rest mass. However, the
+configurations that we construct do not contain GWs and
+therefore they do not model the energy lost in gravita-
+tional radiation. The difference in our ADM mass esti-
+mate and the total rest mass is, therefore, a measure of
+the particle binding energy:
+Ebind,p = MADM − m(BM)
+01
+− m(DM)
+01
+− m(BM)
+02
+− m(DM)
+02
+.
+(25)
+Fig. 8 shows the particle binding energy as a function of
+our estimate for the ADM angular momentum JADM. It
+can be seen that dark core configurations are more tightly
+bound than single fluid configurations. The dark halo
+configurations seemingly coincide with the single fluid
+case. This can be attributed to the relatively low DM
+fraction of only 0.5% in these configurations. All config-
+urations are more tightly bound for smaller JADM cor-
+responding to smaller stellar separations. This is due to
+the stronger orbital binding between the two stars.
+Most of the binding energy is contained in the indi-
+vidual stars and the contribution of the orbital binding
+energy is universal in all configurations. The orbital bind-
+ing energy Ebind,orb is the energy necessary for the two
+NSs to escape to infinity. It can be computed using the
+gravitational mass m(s)
+i
+of the components, by
+Ebind,orb = MADM −m(BM)
+1
+−m(DM)
+1
+−m(BM)
+2
+−m(DM)
+2
+.
+(26)
+The gravitational masses m(s)
+i
+are obtained by solving a
+TOV-like equation for isolated stars that have the same
+rest masses. The gravitational mass m(s)
+i
+is smaller than
+the rest mass m(s)
+i0 , because it accounts for the binding
+energy. Hence, Ebind,orb contains only contributions of
+the binding energy that are due to the mutual binding
+between the stars. Fig. 9 shows that the orbital binding
+energy is mostly independent of the DM configuration.
+The biggest effect is seen for the dark core configurations
+for which the magnitude of Ebind,orb is about 2% smaller
+than that of the other configurations.
+E.
+Deformation
+To quantify the deformation of the stars we compute
+the ratio of the diameters along the orbital radius and
+along the polar axes. The diameter along the orbital ra-
+dius is taken as ∆x, largest difference in the x-coordinates
+of two points on the fluid surface. The polar diameter
+
+9
+6.00
+6.25
+6.50
+6.75
+7.00
+7.25
+7.50
+7.75
+J
+ADM
+/M
+2
+⊙
+−0.300
+−0.295
+−0.290
+−0.285
+−0.280
+−0.275
+−0.270
+particle binding energy E
+bind,
+p
+/M
+⊙
+single fluid
+dark halo
+dark core
+FIG. 8.
+Particle binding energy Ebind,p as a function of the
+ADM angular momentum.
+6.00
+6.25
+6.50
+6.75
+7.00
+7.25
+7.50
+7.75
+J
+ADM
+/M
+2
+⊙
+−0.0300
+−0.0275
+−0.0250
+−0.0225
+−0.0200
+−0.0175
+−0.0150
+orbital binding energy E
+bind,
+orb
+/M
+⊙
+single fluid
+dark halo
+dark core
+FIG. 9.
+Orbital binding energy Ebind,orb as a function of the
+ADM angular momentum.
+∆z, is the largest difference in the z-coordinate of two
+points on the fluid surface. The tidal force of the com-
+panion stretches the star in x-direction, whereas the poles
+are slightly flattened. This measure of deformation is of
+course coordinate-dependent, but it still provides some
+physical insights. Fig. 10 shows the deformation ∆x/∆z
+for each fluid surface. When the NSs are closer, the tidal
+forces on the companion are stronger and hence the de-
+formation is stronger. It can be observed that NSs with
+a DM core are systematically less deformed than their
+one-fluid counterparts.
+The strong deformation in the dark halo case can also
+be seen in Fig. 1, which shows a cut through the z = 0
+plane.
+For a separation of 32 M⊙ (47.3 km) the de-
+formation is clearly visible by eye. At a separation of
+28 M⊙ (41.3 km) the deformation becomes already so
+strong that the surfaces of the NSs touch and mass shed-
+ding occurs.
+The closeness to mass shedding can be quantified in
+terms of the mass-shedding parameter χ, which was first
+introduced in [86] and which we define as
+χ(s) =
+∂xh(s)|eq
+∂zh(s)|pole,avg
+,
+(27)
+25
+30
+35
+40
+45
+50
+55
+60
+65
+separation D
+[km]
+20
+25
+30
+35
+40
+45
+separation D/M
+⊙
+1.000
+1.025
+1.050
+1.075
+1.100
+1.125
+1.150
+deformation ratio ∆x/∆z
+BM, single fluid
+BM, dark halo
+BM, dark core
+DM, dark halo
+DM, dark core
+FIG. 10.
+Deformation ∆x/∆z of the fluid surfaces as func-
+tion of the NS centres. The deformation is computed as the
+ratio of the largest extents in x and z direction. Curves la-
+beled BM show the deformation of the surface of the baryonic
+fluid, whereas curves labeled DM show the deformation of the
+DM surface.
+where the label ”eq” denotes the point on the surface,
+which is facing towards the other companion and for
+which the x-coordinate is extremal.
+The label ”pole”
+denotes the surface points at which the z-coordinate is
+extremal and where in Eq. (27) the label ”avg” indicates
+that we have averaged over the values at the ”north and
+south pole”. Note that for non-spinning stars the ”north”
+and ”south pole” values only differ slightly due to round-
+off error. In the mass shedding limit χ(s) will tend to
+0. We evaluate the χ(s) for each fluid component indi-
+vidually on the respective fluid surfaces. We show the
+resulting χ(s) as a function of the distance of the centres
+of the stars in Fig. 11. The DM fluid in the dark halo
+scenario is easily deformable, which leads to a relatively
+small mass shedding parameter of 0.9 already at a sep-
+aration of 44 M⊙. We find that a separation of 28 M⊙
+leads to a configuration with touching star surfaces, from
+which we conclude that mass shedding occurs somewhere
+at a separation between 28 and 29 M⊙, which means the
+system will transition relatively slowly to the mass shed-
+ding regime over a time where the two NSs decrease their
+separation by 16 M⊙. For the dark core configurations,
+on the other hand, the transition to mass shedding is
+rather sudden with χ reaching a value of 0.9 at sepa-
+ration of approximately 23 M⊙ and the mass shedding
+occurring for the baryonic fluid at a separation of 16 M⊙.
+V.
+CONCLUSION
+We have extended the SGRID code to construct
+constraint-solved, quasi-equlibrium configurations of bi-
+naries of NSs consisting of two non-interacting fluids.
+The second fluid represents DM that can comprise some
+part of the matter of NS. In this study we have used the
+
+10
+25
+30
+35
+40
+45
+50
+55
+60
+65
+separation D
+[km]
+20
+25
+30
+35
+40
+45
+separation D/M
+⊙
+0.4
+0.6
+0.8
+1.0
+mass shedding parameter χ
+BM, single fluid
+BM, dark halo
+BM, dark core
+DM, dark halo
+DM, dark core
+FIG. 11.
+Mass shedding parameter χ as a function of the sep-
+aration of the NS. Curves labeled BM show the deformation
+of the surface of the baryonic fluid, whereas curves labeled
+DM show the deformation of the DM surface.
+EoS of a degenerate, relativistic Fermi gas with different
+particle masses to model the DM fluid.
+These quasi-
+equlibrium configurations can be used as initial data for
+NR inspiral simulations of DM admixed NS binaries. The
+BAM code can already evolve mirror DM [25] and could
+be easily extended to allow for general EoS for the DM
+fluid.
+Another possible application of the two fluid approach
+are superfluid NS cores. At sufficiently high density BM
+forms a state made of superfluid neutrons and supercon-
+ducting protons, which can be described in a two fluid
+approach.
+However, the two fluids still interact with
+each other due to the entrainment effect and the con-
+dition of beta-equilibrium [87]. Solutions of isolated NS
+with superfluid cores are constructed in [88, 89] taking
+into account the interaction of the fluids. For a study of
+superfluid and superconducting cores in binary NS the
+formalism in this work could be extended using a similar
+model for the interactions. In binary NS collisions the
+temperature will rise above the critical temperature for
+superfluidity and superconductivity, so that it becomes
+necessary to include even a third fluid representing the
+non-superfluid component.
+We have tested the convergence of the constructed con-
+figurations with respect to resolution. The Hamiltonian
+constraint converges polynomially with an order of ≈ 2.7.
+The lack of exponential convergence can be attributed to
+the presence of the non-smooth transition of the density
+at the surface of the inner fluid, which is not fitted to
+the boundaries of the spectral elements. Self-convergence
+tests for metric components and the specific enthalpies
+show that the solution improves with increasing reso-
+lution, but with a slightly broken convergence towards
+higher resolution, which we again attribute to the sur-
+face of the inner fluid. For future improvements to the
+code it is a worthwhile consideration to implement a new
+grid layout that allows fitting to the surface of a second
+fluid
+We have shown that the two fluids do not have the ex-
+act same velocities, but that the difference in the resid-
+ual velocities reaches up to 10% on the surface of the
+inner fluids. The difference in the velocity profiles will
+be even stronger if one assumes independent rotational
+states for the components. In this work we only inves-
+tigated only purely irrotational configurations, but our
+formalism, in principle, allows for to construct configu-
+rations with arbitrary spin for the individual stars and
+fluid components. This is relevant in particular for the
+DM component, which might only have insufficient mech-
+anisms to lose angular momentum and hence could be in
+a state of rapid rotation.
+The presence of DM affects the compactness and de-
+formability of NSs, which will change the merger dynam-
+ics. We have shown that the presence of DM can delay
+the point of mass-shedding to a later stage of the inspi-
+ral, i. e., towards closer separations. This is in accordance
+with the findings in numerical evolutions of two-fluid bi-
+nary mergers [25]. In the case of a DM halo, mass shed-
+ding could occur much earlier than for the baryonic com-
+ponent. However the matter contained in the DM halo
+is rather low and hence the impact of DM mass shedding
+on the dynamics of the BM is potentially small, never-
+theless, dynamical simulations are needed to verify this
+assumption.
+ACKNOWLEDGMENTS
+This work was supported by funding from the FCT
+– Funda¸c˜ao para a Ciˆencia e a Tecnologia, I.P., within
+the Project No.
+EXPL/FIS-AST/0735/2021.
+H.R.R.
+and V.S. also acknowledge the support from the project
+No. UIDB/04564/2020, and UIDP/04564/2020. W.T.
+acknowledges funding from the National Science Foun-
+dation under grant PHY-2136036.
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+[85] E. Gourgoulhon, 3+1 formalism and bases of numerical
+relativity (2007), arXiv:gr-qc/0703035.
+[86] E. Gourgoulhon, P. Grandcl´ement, K. Taniguchi, J.-A.
+Marck, and S. Bonazzola, Quasiequilibrium sequences
+of synchronized and irrotational binary neutron stars in
+general relativity: Method and tests, Phys. Rev. D 63,
+064029 (2001), gr-qc/0007028.
+[87] G. L. Comer and R. Joynt, Relativistic mean field model
+for entrainment in general relativistic superfluid neutron
+stars, Phys. Rev. D 68, 023002 (2003), gr-qc/0212083.
+[88] R. Prix, J. Novak, and G. L. Comer, Relativistic numeri-
+cal models for stationary superfluid neutron stars, Phys.
+Rev. D 71, 043005 (2005), gr-qc/0410023.
+[89] N. Chamel, Two-fluid models of superfluid neutron star
+cores, Monthly Notices of the Royal Astronomical Society
+388, 737 (2008), arXiv:0805.1007 [astro-ph].
+

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+EFFICIENT INTRA-RACK RESOURCE DISAGGREGATION FOR
+HPC USING CO-PACKAGED DWDM PHOTONICS
+A PREPRINT
+George Michelogiannakis
+Lawrence Berkeley National Laboratory
+[email protected]
+Yehia Arafa
+Qualcomm Technologies, Inc
+[email protected]
+Brandon Cook
+Lawrence Berkeley National Laboratory
+[email protected]
+Liang Yuan Dai
+Columbia University
+[email protected]
+Abdel-Hameed Badawy
+New Mexico State University
+[email protected]
+Madeleine Glick
+Columbia University
+[email protected]
+Yuyang Wang
+Columbia University
+[email protected]
+Keren Bergman
+Columbia University
+[email protected]
+John Shalf
+Lawrence Berkeley National Laboratory
+[email protected]
+January 11, 2023
+ABSTRACT
+The diversity of workload requirements and increasing hardware heterogeneity in emerging high
+performance computing (HPC) systems motivate resource disaggregation. Disaggregation separates
+servers into their constituent compute and memory resources so that they can be allocated as required
+to each workload. Previous work has shown the potential of intra-rack resource disaggregation,
+but it is not clear how to realize these gains and cost-effectively meet the stringent bandwidth
+and latency requirements of HPC applications. To that end, we describe how modern photonic
+components can be co-designed with modern HPC racks to implement flexible intra-rack resource
+disaggregation and fully meet the high escape bandwidth of contemporary multi chip module (MCM)
+packages and all chip types in modern HPC racks with negligible power overhead. We show how to
+use distributed indirect routing to meet these demands without the need for significant complexity
+for reconfiguration that spatial optical switches require. We then show that intra-rack resource
+disaggregation implemented using emerging photonics and parallel optical wavelength-selective
+switches satisfies bit error rate (BER) and bandwidth constraints and provides an average application
+speedup of 23.9% for 31 out-of-order CPU and 61.5% for 27 GPU benchmarks compared to a similar
+system that instead uses modern electronic switches for disaggregation, due to their higher latency.
+Using observed resource usage from a production system, we estimate that an iso-performance
+intra-rack disaggregated HPC system using photonics would require 4× fewer memory modules and
+2× fewer NICs than a non-disaggregated baseline.
+Keywords Photonics · disaggregation · AWGR · spatial
+arXiv:2301.03592v1  [cs.DC]  9 Jan 2023
+
+arXiv Template
+A PREPRINT
+1
+Introduction
+Leading high performance computing (HPC) systems are steadily embracing heterogeneity of compute and memory
+resources as a means to preserve performance scaling and reduce system power Liu et al. [2012], Top [2018], Ujaldón
+[2016]. This trend is already apparent with the integration of GPUs Mittal and Vetter [2015], Tiwari et al. [2015],
+Gao and Zhang [2016] and is expected to continue with fixed-function or reconfigurable accelerators such as field
+programmable gate arrays (FPGAs), Milojicic [2020], Asaadi and Chapman [2017], Segal et al. [2014], Hogervorst
+et al. [2021], Lant et al. [2020], Dimond et al. [2011], Ramirez-Gargallo et al. [2019], emerging customized accelerators,
+and heterogeneous memory Venkata et al. [2017]. In addition, key HPC workloads show considerable diversity in
+computational and memory access patterns Michelogiannakis et al. [2022], Rodrigo et al. [2016].
+This expectation of resource heterogeneity, workload diversity, and today’s method of allocating resources to applications
+in units of statically-configured nodes where every node is identical and unused resources are left to idle, raises the
+concern of resource underutilization (referred to as “marooned resources”). These marooned resources increase both
+capital and operational costs without improving performance. This has led to the emergence of resource disaggregation.
+Disaggregation refers to decomposing servers into their constituent compute and memory resources so that these can be
+allocated as required according to the needs of each workload. Hyperscale datacenters have readily embraced resource
+disaggregation and have demonstrated that it significantly improves utilization of GPUs and memory Guleria et al.
+[2019a], Peng et al. [2020], Taylor [2015], Li et al. [2022], Koh et al. [2019], Papaioannou et al. [2016], Gonzalez et al.
+[2020], Cheng et al. [2019a], Guleria et al. [2019b].
+Although file storage is routinely disaggregated in modern systems Per, Michelogiannakis et al. [2022], Petersen and
+Bent [2017], HPC has been slow to embrace disaggregation of compute and memory resources Glick et al. [2020], Guo
+et al. [2021] due to the sensitivity of HPC workloads to bandwidth and latency that cannot be met by current PCIe/CXL
+or Ethernet link technologies used in contemporary disaggregated architectures. Studies showed that disaggregation only
+among resources in the same rack (i.e., intra-rack resource disaggregation) in HPC could reduce resources by 5.36% to
+69.01% while avoiding the overhead of full-system disaggregation Michelogiannakis et al. [2022], but the impact of
+increased memory latency and specific architectural trade offs have not been explored. Thus, although disaggregation
+using electronic networks have been demonstrated in hyperscale datacenters Lin et al. [2020], Papaioannou et al. [2016],
+Call et al. [2020], minimizing adverse effects to and addressing the stringent bandwidth density and latency demands of
+HPC workloads requires a thorough investigation.
+In this work, our contributions are as follows. Firstly, we describe how to use emerging photonic links and switches to
+design modern and practical resource-disaggregated HPC racks based on an existing GPU-accelerated HPE/Cray EX
+supercomputer Per. Secondly, we show how state of the art commercially available photonics and advanced packaging
+multi chip modules (MCMs) meet bit error rate (BER) requirements, impose a negligible power overhead, and deliver
+sufficient bandwidth to satisfy the escape bandwidth of all chips in modern HPC racks. Thirdly, we show how to use
+distributed indirect routing and arrayed waveguide grating routers (AWGRs) Liu et al. [2020], Zhang et al. [2019] to
+satisfy all bandwidth requirements without the overhead and latency for reconfiguration that spatial Seok et al. [2019a],
+Ding et al. [2016] and wave-selective Huang et al. [2020] switches require. Moreover, having demonstrated negligible
+adverse impact to all other metrics, we show that intra-rack disaggregation using emerging photonics provides an
+average application speedup of 23.9% for 31 out-of-order (OOO) CPU and 61.5% for 27 GPU benchmarks compared to
+a similar system that instead uses state of the art electronic switches, which also increase power overhead by three orders
+of magnitude. Finally, based on observed resource usage, we estimate that a system based on racks using photonics
+for resource disaggregation can have 43% fewer overall chips compared to a non-disaggregated system with the same
+computational throughput.
+2
+Related Work
+Hyperscale datacenters predominantly focus on full-system resource disaggregation where applications can allocate
+fine-grain resources of different types, today typically graphical processing units (GPUs) Guleria et al. [2019a] and
+memory Peng et al. [2020], Lim et al. [2009], Koh et al. [2019], Gonzalez et al. [2020], which are located anywhere in
+the system or within a group of racks Lin et al. [2020], Papaioannou et al. [2016], Call et al. [2020]. In such a system,
+resources of the same type are typically placed in the same rack.
+However, full-system, flexible, and fine-grain disaggregation introduces significant overhead because of the higher
+latency and lower bandwidth density of contemporary hardware that is used to implement resource disaggregation –
+typically PCIe, 100Gig Ethernet, and eventually compute express link (CXL) Van Doren [2019] over electronic links.
+This overhead does not simply increase power and procurement cost, but rather adds potentially substantial latency
+between key resources such as central processing units (CPUs) and memory that traditionally exhibit latency-sensitive
+2
+
+arXiv Template
+A PREPRINT
+communication. The aforementioned studies quote a several orders of magnitude increase in network and memory
+latency due to full-system resource disaggregation to improve resource utilization by 35% at most Zervas et al. [2018].
+Another study found that application performance degradation depends on both network bandwidth and latency, but
+can still reach up to 40% even with high bandwidth, low-latency networks Gao et al. [2016]. Work on SPEC and
+commercial benchmarks also found an up to 27% application slowdown due to the additional memory latency Abali
+et al. [2015]. A study on Microsoft’s Azure found a range of performance slowdowns up to 30% from an extra 65 ns to
+access main memory Li et al. [2022]. Software defined networks (SDNs) based on electrical networks fare no better in
+terms of overhead Gao et al. [2016], Han et al. [2013], Call et al. [2020].
+Hybrid full-system photonic–electronic approaches have also been proposed that rely on circuit switching Zervas et al.
+[2018] for reconfiguration. As a result, a few studies call for intra-rack disaggregation in datacenters Taylor [2015],
+Lim et al. [2009], Guleria et al. [2019b]. Even the low latency and high bandwidth density of modern photonics cannot
+fully satisfy the bandwidth, energy, and latency requirements of full system disaggregation. This makes system-wide
+disaggregation impractical in many cases Lin et al. [2020], Zervas et al. [2018], Cheng et al. [2019a], Cheng et al.
+[2018].
+Recent full system approaches in high performance computing (HPC) rely on optics to connect CPUs and memory, and
+electronic switches for hard disk drives (HDDs) to increase resource CPU utilization by 36.6% and memory 21.5% Guo
+et al. [2021]. In contrast, another study confirms that production HPC systems can reduce resources from 5.36%
+to 69.01% with intra-rack disaggregation and still satisfy the worst-case average rack utilization Michelogiannakis
+et al. [2022]. Similar to datacenters, intra-rack disaggregation in HPC promises the lowest overhead and impact to
+applications Glick et al. [2020], Taylor [2015], Guleria et al. [2019b].
+Related work has research other aspects necessary to make resource disaggregation practical in a system, such as
+job scheduling Fan et al. [2019], Agosta et al. [2018], Amaral et al. [2021], Domeniconi et al. [2019], how the
+operating system (OS) and runtime should adapt Maccabe [2017], Hwu et al. [2015], Shan et al. [2018], programming
+of code portability in heterogeneous systems Gioiosa et al. [2020], Agosta et al. [2018], partitioning of application
+data Khaleghzadeh et al. [2020], fault tolerance Hussain [2020], how to fairly compare the performance of different het-
+erogeneous systems Jamieson et al. [2018], and the impact of heterogeneous resources to application performance Tang
+et al. [2017], Lastovetsky [2015], Venkata et al. [2017]. These are important but out of scope topics for our study.
+2.1
+Under-utilization in Production Systems
+We use NERSC’s Cori system as an exemplar production HPC system, while recognizing workload requirements
+on other systems may differ. In NERSC’s Cori, before Perlmutter came online and thus Cori was runnign the full
+NERSC workload, three quarters of the time Haswell nodes use less than 17.4% of memory capacity (50.1% for
+KNL nodes) and less than 0.46 GB/s of memory bandwidth Michelogiannakis et al. [2022]. These observations are
+similar to observations collected on LANL clusters Peng et al. [2020] and Alibaba machines that execute batch jobs.
+Likewise, half of the time Cori nodes use no more than half of their compute cores and three quarters of the time 1.25%
+of available network interface controller (NIC) bandwidth. Similarly, in Lawrence Livermore National Laboratory
+(LANL) clusters, approximately 75% of the time, no more than 20% of memory capacity is used Peng et al. [2020].
+Alibaba’s published data Guo et al. [2019] show that memory is underutilized similarly to Cori for machines that
+execute batch jobs. Data from Google systems shows that memory and disk capacity of tasks is spread over three orders
+of magnitude and typically underutilized Han et al. [2013]. Azure reports 25% of memory under-utilization Li et al.
+[2022]. Datacenters have also reported 28% to 55% CPU idle in the case of Google trace data Patel et al. [2015] and
+20% to 50% most of the time in Alibaba Guo et al. [2019]. Early studies also suggest GPU under-utilization Li et al.
+[2015], Jeon et al. [2019], Li et al. [2011].
+3
+Photonics for Resource Disaggregation
+Here we walk through available optical link and switch technologies and argue that photonics today meet the strict
+performance and error rate requirements to efficiently implement intra-rack resource disaggregation in HPC.
+3.1
+Memory Technologies and Requirements
+IO systems in HPC are already largely disaggregated over conventional system-scale interconnects since the underlying
+technologies (disk or SSD) are relatively high latency and lower bandwidth Michelogiannakis et al. [2022], Terzenidis
+et al. [2018]. By contrast, memory technologies (particularly high bandwidth memorys (HBMs) needed by GPUs) are
+much higher bandwidth and much less tolerant of latency and require much lower bit error rates (BERs). Given that
+memory disaggregation imposes the most challenging constraints among other resources in today’s compute nodes, we
+3
+
+arXiv Template
+A PREPRINT
+Figure 1: Logical schematic of a DWDM link using ring resonator technology and a comb-laser source. Each ring is
+tuned to a different frequency of light and can be used to modulate that specific wavelength of light (a channel). Comb
+laser sources provide a comb of frequencies of light to provide those wavelengths for encoding. All of the encoded
+optical channels share the same optical fiber and are decoded using the rings on the receiving side to route channels to
+the photodetectors.
+Aggregated comb laser sources
+Active photonic interposer
+(a) Active Optical MCM
+(b) Blade
+Optical Circuit Switches
+Optical Fiber
+(c) Rack/Pod
+Figure 2: Overall physical structure of Rack/Pod scale resource disaggregation from photonically connected MCMs
+to the Rack/Pod scale pooling of disaggregated resources. The conversion from CXL-over-fiber to HBM or NVM
+electrical protocol is implemented in the active interposer for the photonics MCM.
+will use DDR and HBM memory technology to set our performance target. A typical DDR4 memory has a response
+latency of approximately 90 ns and for HBM the average response latency is 90-140 ns Wang et al. [2020]. Still, any
+added latency between the CPU and memory from resource disaggregation may penalize application performance as
+we quantify later. Server-class memories typically require BERs of less than 10−18 to achieve tolerable failures in
+time (FIT) rates with conventional single-error-correct/double-error-detect (SEC-DED) protection Meza et al. [2015],
+Sridharan et al. [2015]. Forward error correction (FEC) can reduce the BER, but with additional latency Luyi et al.
+[2012].
+3.2
+Optical Link Technologies
+We consider a range of photonic link technologies that include conventional 100 Gbps Ethernet physical interfaces that
+represent the current baseline link technology for memory disaggregation. We also introduce a range of cutting-edge
+dense wavelength division multiplexing (DWDM) link technologies that are either demonstrated as research prototypes
+or are commercially available. The photonic components all come from existing commercial technologies (100 Gbps,
+400 Gbps, Ayar TeraPhy) and some research prototypes from DARPA PIPES (the 1-2 Tb link technologies). These
+higher performance link technologies must be co-packaged in order to achieve their bandwidth density. These link
+technologies are summarized in Table 1. The technology for the optical links is depicted in Figure 1. Delivering
+multiple channels of laser light to the package has been challenging to scale cost-effectively if each "color" of light
+were to require a separate laser source. This concern was alleviated by the emergence of quantum dot and soliton comb
+laser sources shown in Figure 4 that can produce hundreds of usable light frequencies with wall-plug efficiencies of up
+to 41% Kim et al. [2019a].
+4
+
+7NNN00NNN:.MMCustomAccelGPuCPUDisaggregatedresourceRack/PodarXiv Template
+A PREPRINT
+Figure 3: Copackaged optics are required for DWDM link technologies to achieve the kind of bandwidth density
+required to operate at native memory bandwidths.
+3.3
+Active Photonic MCMs
+Many CPUs and GPUs do not have the necessary off-chip bandwidth for full utilization of their compute resources
+because operating their I/O pins at higher bandwidth incurs a power cost Chen et al. [2017], Jouppi et al. [2017].
+Using emerging high-speed optical links directly to the MCM, illustrated in Figure 3 provides to the order of 10×
+gains in escape bandwidth Glick et al. [2020], Wade [2019], Bergman et al. [2018], Maniotis et al. [2021]. This is
+a necessary property to enable efficient resource disaggregation as well as handle changing bandwidth requirements
+of key applications such as machine learning that drastically shifts bandwidth between inter-GPUs and off-chip from
+inference to training.
+MCMs with integrated photonics have been demonstrated in both 2.5D and 3D interposer platforms Glick et al. [2020],
+Minkenberg et al. [2021], Sutono et al. [1998], Abrams et al. [2020]. They can use different die-to-die link standards
+such as UCIe. Active interposer platforms combine the photonic integrated circuit (PIC) and interposer into a single
+integrated substrate. The active interposer allows photonic components to be fabricated and directly integrated with
+through silicon vias (TSVs) and additional metal redistribution layers. Electronic circuits are flip-chipped on top of
+active interposers using copper pillars Dittrich et al. [2017]. Further work has embedded photonic switch fabrics within
+MCM platforms with a crosstalk suppression and extinction ratio of >50dB and on-chip loss as low <1.8dB Glick et al.
+5
+
+2.5DIntegratedTransceiver
+ComputeNode
+Optical
+Fibers
+FCBumps
+PIC
+FC Bumps
+EIC
+Interposer
+ASIC/FPGA
+PCBEIO
+PIC
+25umpitchpad
+array to be bumpec
+25umpitchpad
+arraytobeb
+MCM
+C3
+ElectricalloonPcBto
+be wirebondedto PiC
+EIC
+PICarXiv Template
+A PREPRINT
+Figure 4: Comb laser sources provide the many discrete optical frequencies for the DWDM link with up to 41%
+experimentally measured conversion efficiency.
+[2020]. This was further scaled up to support more than 100 ports with microring resonators using a scalable switch
+fabric that combined switching in the space domain with wavelength-selectivity to define fine-grained connectivity for
+node disaggregation Huang et al. [2020], Glick et al. [2020].
+3.3.1
+Link Protocol
+We adopt CXL as our link protocol Van Doren [2019]. CXL is an overlay on the PCIe-Gen6 physical layer, it includes
+guaranteed ordering of events and is a broadly adopted industry standard with published specifications. However, our
+study does not rely on any features of any particular protocol so alternatives such as UCIe also apply.
+3.3.2
+Link Propagation and Encoding/Decoding Latency
+The target reach for an intra-rack disaggregation solution is approximately 1-4 meters. Given the speed of light c
+and light propagating through optical material that has an index of refraction that is near r1.5, the effective latency
+of propagating through an optical fiber at nominally 0.75c is approximately 5 ns per meter. Therefore, rack-scale
+disaggregation adds 5-15 ns to our latency budget, less than 20% of the typical DRAM latency. The link latency for
+SERDES and photonic ring modulation is negligible. Intra-rack fiber lengths up to 4 meters require no intervening
+Electrical Optical (OEO) conversions.
+3.3.3
+Bit Error Rates and FEC
+To achieve 10−18 BER required for memory technologies, FEC Luyi et al. [2012] will likely be required. Using the
+lightweight FEC scheme that is proposed for CXL Van Doren [2019] and PCIe Gen6 Sharma [2020] as an example,
+the all-inclusive latency for FEC can be as low as 2 ns. Therefore, for 200 Gbps, serialization delay is 10 ns and the
+FEC calculations add 2-3 ns. At 400 Gbps and above, the net latency for FEC would be 5 ns plus 2-3 ns. Of note, this
+approach to achieving these BER targets is achievable with less than a 0.1% bandwidth loss.
+6
+
+100um-6 dBm
+10
+-10 dBm
+S-band
+C-band
+L-band
+0
+Bm)
+-10
+NO
+-20
+-30
+-40
+1,520
+1,540
+1,560
+1,580
+1,600
+Wavelength (nm)arXiv Template
+A PREPRINT
+BW
+(Gbps)
+Energy
+(pJ/bit)
+Link
+Gbps ×
+Chan-
+nels
+#Links
+(2
+TB/s
+es-
+cape)
+Agg.
+Ws (2
+TB/s
+es-
+cape)
+Ref.
+100
+30
+25 × 4
+160
+480
+Fathololoumi
+et
+al.
+[2021],
+Agrell
+et
+al.
+[2016]
+400
+30
+100 × 4
+40
+197
+Wei
+et
+al.
+[2015]
+768
+< 1
+32 × 24
+21
+14.4
+Wade
+[2019]
+1,024
+0.45
+16 × 64
+16
+7.2
+Kim
+et
+al.
+[2019b]
+2,048
+0.3
+16 × 128
+8
+4.8
+Kim
+et
+al.
+[2019b]
+Table 1: A range of WDM photonic link technologies.
+In terms of impact on BER, this PCIe/CXL-like correction scheme corrects all single bursts of up to 16 bits. Double
+bursts will likely be mis-corrected, but the chance of a bad flit decreases quadratically (e.g., a flit BER of 10−6 becomes
+10−12 as you need two error bursts per flit to fail). Each flit is protected with a strong 64-flit CRC such that the flit FIT
+rate (CRC escapes) is significantly less than one part per billion. Lastly, the FEC escapes become link retransmissions
+and the ASIC-to-ASIC connection sees close to zero errors. As a result, emerging memory fabric protocols such as
+CXL, which could be run over our evaluated physical links, are capable of achieving a BER rate that meets the stringent
+memory system requirements and minimizes performance loss due to retransmission.
+3.4
+Optical Circuit Switch Technologies
+Motivated by minimizing latency, our vision for a disaggregated rack is to have photonically-enabled MCMs that are
+connected via an optical circuit switch, as shown in Figure 2. Compute and memory chips would be in the center of the
+MCM and the edge of the MCM would contain co-packaged optical silicon in-package photonics (SiPs). Switches with
+all-optical paths include spatial- and wave-selective approaches, shown in Table 2.
+3.4.1
+Spatial Optical Switches
+In recent years, the primary switching cells investigated are microelectromechanical systems (MEMS) actuated
+couplers, Mach-Zehnder interferometers (MZIs), and microring resonators (MRRs). Taking after their free-space
+counterpart, photonic MEMS-actuated switches are broadband spatial switches that have demonstrated radix scaling up
+to 240×240 Seok et al. [2019b]. However, MEMS switching cells generally require high driving voltages (greater than
+20 V), which make them less attractive for co-integration with electronic drivers, but typically offer low inter-channel
+cross-talk and low optical losses. Spatial switches can also use mirrors Cal, photonic integrated circuits Ding et al.
+[2016], or tiled planar silicon photonics Seok et al. [2019a]. MZI switches are more co-integration friendly compared
+to MEMS but have only been shown to scale up to 32×32 Ikeda et al. [2020]. This limit can be seen as a consequence
+of the higher insertion-loss scaling resulting from cascaded MZI cells, as well as the susceptibility of popular MZI
+topologies to first-order crosstalk.
+The challenge for scaling-up the spatial approach is the quantization of package and MCM escape bandwidth and
+reduced configuration options. For example, at 768 Gbps (the Ayar TeraPhy Wade [2019]), the number of fibers escaping
+the package is 21 fibers, which means the package can be connected only up to 21 different potential destinations using
+a spatial switch.
+7
+
+arXiv Template
+A PREPRINT
+Switch
+Type
+Radix
+Wave-
+lengths
+per
+port
+B/W
+per
+channel
+(wave-
+length)
+Insertion
+Loss
+Crosstalk
+Mach-
+Zehnder
+based Ikeda
+et
+al.
+[2020]
+32×32
+1
+439
+Gbps
+12.8
+dB
+-26.6
+dB
+MEMS-
+actuated Seok
+et
+al.
+[2019b]
+240×240 1
+–
+9.8 dB
+-70 dB
+Microring
+res-
+onator Khope
+et
+al.
+[2017],
+Cheng
+et
+al.
+[2019b]
+8×8
+(128×128)
+8
+(128)
+100
+Gbps
+(42
+Gbps)
+5dB
+(10dB)
+(-35
+dB)
+Casc.
+AW-
+GRs Sato
+[2018]
+370×370 370
+25
+Gbps
+15 dB
+-35 dB
+Table 2: High-radix CMOS-compatible photonic switches.
+3.4.2
+Wavelength Selective Optical Switches and AWGRs
+The inherent wavelength-selectivity of MRR switching cells allows for the straightforward implementation of
+wavelength-selective switching (WSS) topologies. This enables one to establish all-to-all networks by leveraging
+wavelength-division multiplexing (WDM). Currently, MRR-based switches with the largest radix include the 8×8
+crossbar Khope et al. [2017] and switch-and-select Nikolova et al. [2017], but have been experimentally emulated to
+include a 16×16 Clos Dai et al. [2020]. The metrics in Dai et al. [2020] can be seen to correlate very closely with the
+scaling proposed in Cheng et al. [2019b], making a practical case for the 128×128 shown in Table 2.
+All-to-all networks via WDM signals can also be achieved by arrayed waveguide grating routers (AWGRs) Liu et al.
+[2020], Zhang et al. [2019], Proietti et al. [2013], Lea [2015], Terzenidis et al. [2018]. As AWGRs are passive optical
+elements, no reconfiguration is possible within the routing fabric itself. Instead, fast wavelength-tunable lasers must
+be leveraged at the transmitter of every node if it wishes to address a different destination since AWGRs shuffle the
+light frequencies such that one lambda goes to each endpoint from each source. AWGRs enable us to implement an
+N×N all-to-all topology using just O(N) fibers (each carrying N frequencies of light) whereas an implementation
+using copper would require N2 wires. Although the cost of fast wavelength-tunable lasers is still an ongoing research
+topic Dhoore, Sören and Roelkens, Günther and Morthier, Geert [2019], AWGRs are mature, commercially available,
+and well established in literature FSp.
+In AWGRs, only a limited number of ports can be practically supported due to the walk-off of passband center
+frequencies from the carrier wavelength grid and the worse crosstalk associated with a larger number of ports (N).
+A feasible implementation of AWGR-based optical switches with large N has been demonstrated utilizing cascaded
+small-size AWGRs Sato [2018]. Specifically, N M × M AWGRs (front-AWGRs) are interconnected with M N × N
+AWGRs (rear-AWGRs) to effectively act as an MN × MN AWGR. Each output port of a front-AWGR is connected
+to an input port of a rear-AWGR, where the interconnection pattern can be optimized with knowledge of port-specific
+insertion losses to minimize the worst-case insertion loss of the aggregated AWGR. Further up-scaling of the switch
+radix can be achieved by interconnecting small K × K delivery-coupling switchs (DC-switchs) with multiple copies
+of the MN × MN AWGRs, yielding a KMN × KMN switching capability. This architecture has been verified by
+hardware prototypes of 270 × 270 and 1440 × 1440 Sato et al. [2013], Ueda et al. [2016], showing ∼15 dB insertion
+loss and below −35 dB crosstalk suppression. In order to accommodate the 350 MCMs of our rack, a reasonable
+8
+
+arXiv Template
+A PREPRINT
+1
+5
+3
+2
+8
+4
+6
+7
+Figure 5: With an AWGR, endpoint 1 has one wavelength directly connecting it to endpoint 3. If it desires more
+bandwidth, it can route through another intermediate endpoint (indirect routing) chosen in a Valiant fashion Liu et al.
+[2020], Teh et al. [2020]. Here, the link from 1 to 7 is available (green) but the link from 7 to 3 is not (red). The chosen
+path is from 1 to 6 to 3 because both links are available.
+configuration is KMN = 3 × 12 × 11 = 396. This results in 370 ports and 370 wavelengths per port (Table 2). Since
+AWGRs typically have a 25 GHz optical bandwidth if the wavelength grid is 50 GHz, with PAM4 we assume 25 Gbps
+per wavelength Dai et al. [2020], Bhoja [2017].
+Wave-selective switches Huang et al. [2020], Marom et al. [2017] can steer any subset of wavelengths to a given
+destination, not just all (spatial) or one (AWGR). Dynamic programming methods can avoid sending the same frequency
+of light from two different sources to the same destination. Since this is a relatively new technology, we constructed a
+model shown in Table 2 that projects the performance of a larger radix switch that is comprised of smaller demonstrated
+building blocks.
+3.4.3
+Reconfiguration Time
+Spatial and wave-selective switches typically require centralized scheduling Teh et al. [2020] to reach a steady globally
+optimal solution. The reconfiguration time can range from tens of nanoseconds to tens of milliseconds. In production
+HPC systems, multi-node jobs start every few seconds and last from minutes to hours Michelogiannakis et al. [2019,
+2022]. Also, job resource usage and communication becomes predictable early, does not change fast, and typically
+remains predictable throughout a job’s execution time Michelogiannakis et al. [2022, 2019], Shalf et al. [2005], Vetter
+and Mueller [2002]. Therefore, even milliseconds of reconfiguration time is ample.
+4
+Control Logic
+Here we describe how we can perform indirect routing to increase point-to-point bandwidth using only per-source logic.
+4.1
+Indirect routing in AWGRs
+AWGRs dedicate exactly one wavelength between any source–destination pair. If a source–destination pair requests
+more bandwidth than what a single wavelength can satisfy, sources can use indirect routing an example of which is
+shown in Figure 5. Sources can split traffic to N intermediate destinations in parallel in order to use the bandwidth of
+9
+
+arXiv Template
+A PREPRINT
+N wavelengths. This does not consume additional power in the photonic components assuming lasers are constantly
+powered. Sources consider indirect paths only if the direct (single-hop) bandwidth to their desired destination does
+not suffice. A source considers indirect destinations for which the direct bandwidth from the source is available and
+whose wavelengths from the intermediate hop to the desired final destination is available. Among potentially multiple
+candidates, sources choose one in a Valiant fashion Liu et al. [2020], Teh et al. [2020], Domke et al. [2019]. This
+is done on a per-flow basis in order to avoid out of order packet delivery. This routing logic can be modelled as an
+allocator problem and implemented with a low latency and area penalty Ma et al. [2014], Becker and Dally [2009].
+Indirect routing relies on sources knowing which other sources attached to the same AWGR are utilizing their local
+wavelengths in order to identify a productive intermediate destination. For instance, in Figure 5 endpoint 1 should
+know whether the wavelengths from 7 to 3 and 6 to 7 are occupied. For that, we rely on piggybacking where traffic
+between a source and a destination periodically includes the state of the sources’s wavelengths as a way to broadcast
+local state to the rest of the endpoints attached to the same AWGR Jiang et al. [2009]. In the case of a N×N AWGR,
+each source uses N bits to encode which of its N local wavelengths it is using with one-hot encoding. Even if we
+piggyback this information multiple times a second, the bandwidth impact is negligible. For instance, if we multiplex
+multiple flows into a wavelength and therefore denote 8 bits per wavelength, the status vector per source becomes
+256 × 8 = 2048bits = 256bytes. If, due to stale information, sources pick an intermediate destination whose
+wavelength direct to the final destination is not available, the intermediate destination performs indirect routing through
+a second intermediate destination, and so on. If no data is exchanged between a pair, thus presenting no opportunity for
+piggybacking, that pair can exchange a separate control message with the same information.
+4.2
+Spatial and Wave-Selective Switches
+Spatial and wave-selective switches can use indirect routing in tandem with reconfiguration. Indirect routing reduces
+the need for reconfiguration, but intermediate hops should be chosen among hops that already have a direct connection
+with the final destination; otherwise, the intermediate hop itself may trigger a reconfiguration. The synergy between
+indirect routing and switch reconfiguration was explored in Teh et al. [2020].
+5
+Disaggregated Rack Design
+For the rest of our study, we will model an HPC rack based on a GPU-accelerated HPE/Cray EX Supercomputer Per
+where a rack contains 128 GPU-accelerated nodes. Each node of our model system contains an AMD Milan CPU
+that has eight memory controllers each supporting a 3200MHz DDR4 module. Therefore, each CPU has 256 GB of
+memory with a maximum bandwidth of 204.8 GBps. A compute node also has four NVIDIA Ampere A100 GPUs.
+Each GPU supports 12 third generation NVLink links each supporting 25 GBps per direction. Each GPU also has 40
+GB of co-located HBM with a bandwidth of 1555.2 GBps. Each node also has four 31.5 GBps PCI Gen4 links to
+connect each GPU to the CPU. The CPU also connects to four Slingshot 11 NICs with 200 Gbps per direction De Sensi
+et al. [2020a]. Note that our photonic disaggregation hardware is orthogonal to and thus does not impair past work
+related to disaggregation such as runtimes, OS support, endpoint sharing management, and security.
+5.1
+MCMs and Escape Bandwidth
+We organize chips within each rack into an MCMs package. For simplicity, we restrict all MCMs to have the same
+escape bandwidth and we place chips of only the same type in MCMs. We then make conservative assumptions for next
+generation photonics that are entering the market today based on our analysis of Section 3. In particular, each MCM has
+32 optical fibers attached to it, a conservative assumption compared to the five arrays of 24 fibers demonstrated Hosseini
+et al. [2021]. Each fiber supports 64 wavelengths (channels) of 25 Gbps each for a 6400 GBps escape bandwidth per
+MCM. We vary the number of chips per MCM such that each chip enjoys the same escape bandwidth as in our baseline
+rack Per. Therefore, our photonic architecture does not restrict chip escape bandwidth. Table 3 shows the number of
+chips per MCM and the total number of MCMs containing chips of that type to satisfy chip escape bandwidth. Each
+MCM contains a controller chip that interfaces the native protocol of the disaggregated resource to the CXL protocol
+over the photonic links. CXL’s overhead and its associated FEC is included in our model of the overall architecture.
+5.2
+Optical Switches
+The radix and wavelengths per port of optical switches dictate number of MCMs we can fully connect optically with
+a single switch as well as the amount of direct (single-hop) bandwidth. From Section 3.4, we pick state-of-the-art
+representatives of wave-selective, cascaded AWGRs, and spatial optical switches. Their parameters are shown in
+Table 4. Even though spatial Seok et al. [2019b] and wave-selective switches Huang et al. [2020] are capable of 100
+10
+
+arXiv Template
+A PREPRINT
+Chip type
+Chips per MCM
+# MCMs per rack
+CPU
+14
+10
+GPU
+3
+171
+NIC
+203
+3
+HBM
+4
+128
+DDR4
+27
+38
+Total
+350
+Table 3: The number of chips ((CPU, GPU, NIC, HBM, or DDR4 module) per MCM and MCMs in a rack assuming
+32 fibers per MCM, 64 wavelengths of 25 Gbps per fiber. The target BER to and from memory is 10−18 (Section 3.1).
+Switch type
+State of the art
+Switch radix
+Cascaded AWGRs Sato [2018]
+370
+Spatial Seok et al. [2019b]
+240
+Wave-Selective Huang et al. [2020]
+256
+Gbps per wavelength
+All switches
+25
+Wavelengths per port
+Cascaded AWGRs Sato [2018]
+370
+Spatial Seok et al. [2019b]
+240
+Wave-Selective Huang et al. [2020]
+256
+Table 4: Switch configuration for our study.
+Gbps per wavelength, most links available widely today do not support that (Table 1). In addition, we show that we can
+still satisfy bandwidth demands with the conservative assumption of 25 Gbps per wavelength.
+To connect our 350 MCMs using 370×370 AWGRs, we can combine MCM fibers in five groups of six and connect each
+group to one port of five parallel AWGRs. However, this would require each AWGR port to handle 384 wavelengths.
+To respect the per port 370 wavelength limitation of our AWGR configuration but still satisfy the full escape bandwidth
+of MCMs, we combine the remaining 14 wavelengths along with the remaining two fibers per MCM (128 + 14 = 142
+wavelengths total) that were left unconnected into an extra parallel AWGR, for a total of six parallel AWGRs. We then
+connect MCM fibers to AWGRs in a staggered manner such that each MCM connects to each other MCM using at least
+five 25 Gbps direct-path wavelengths, for a direct MCM–MCM bandwidth of 25 × 5 = 125 Gbps.
+For simplicity, because of their relative small difference and because wave-selective switches can also achieve configu-
+rations that spatial switches can, we treat both wave-selective and spatial switches as 256 ports with 256 wavelengths
+per port. Each MCM can connect to 2048
+256 = 8 parallel switches. However, because the radix of optical switches is lower
+than the number of MCMs, we instantiate 11 optical switches and connect MCMs in a staggered manner such that
+optical switch with an index I connects to MCMs that have an index starting from (32 × I) mod 350 until (I + 255)
+mod 350. This way, a small number of optical switch ports are left unconnected in order to not exceed the 32 fibers
+per MCM. Similar to AWGRs, these ports can support future larger racks. If the switches configure appropriately,
+each MCM has at least three direct paths to any other MCM. Each path has 256 wavelengths, thus the direct MCM
+bandwidth is 256 × 3 × 25 = 2304 Gbps.
+6
+Evaluation
+Having previously evaluated in Section 3.3.3 that photonic switches satisfy BER requirements, in this Section we
+analyze the impact of photonic-based intra-rack resource disaggregation to bandwidth, latency, and power. We then
+compare against electronic switches and estimate system-wide savings.
+6.1
+Bandwidth Evaluation
+We distinguish two test cases based on Section 5.2: (A) Six parallel AWGRs and (B) 11 parallel wave-selective switches.
+6.1.1
+Available Bandwidth
+Using indirect routing and switch reconfiguration, any one particular MCM can use its full escape bandwidth to reach a
+single destination MCM. In test case (A), all wavelengths escaping an MCM can reach the same destination MCM
+using indirect routing. In test case (B), 768 wavelengths can be configured to route directly to a destination MCM
+11
+
+arXiv Template
+A PREPRINT
+and the other 2048 − 768 = 1280 wavelengths can be configured to route indirectly through intermediate MCMs.
+This assumes that other MCMs will not contend for bandwidth that may disrupt indirect routing or complicate switch
+reconfiguration. While the direct (single-hop) bandwidth between cases (A) and (B) has a large difference, case (A)
+always provides that direct bandwidth between MCMs whereas a spatial or wave-selective switch requires a scheduler
+and leaves the majority of input–output combinations unconnected at any one time, thus also has to use indirect routing
+to compensate.
+Based on system profiling data of a production open-science HPC system Michelogiannakis et al. [2022], the 125
+Gbps direct bandwidth between MCMs in test case (A) suffices over 99.5% of the time between CPUs and main
+memory (DDR4) and virtually all the time between memory and NICs. In addition, the bandwidth of a single AWGR
+wavelength of 25 Gbps suffices 97% of the time between CPUs and memory as well as between memory and NICs.
+This means that with a 97% probability, four of the five wavelengths between a memory and CPUs or NICs and
+memory pair are available to use for indirect routing in case the direct 125 Gbps bandwidth does not suffice between
+another memory–CPU or NIC–memory pair. Therefore, the probability at any one time that the direct bandwidth does
+not suffice for a number of CPU–memory and NIC–memory pairs large enough such that they cannot find unused
+bandwidth in other pairs to use for indirect routing is multiple orders of magnitude less than 0.1% and thus negligible.
+To further reduce the probability, congested pairs can use direct paths from CPUs to CPUs that communicate minimally
+and NICs to other NICs that do not communicate at all Michelogiannakis et al. [2022]. Therefore, test case (A) satisfies
+bandwidth between CPUs, NICs, and main memory (DDR4).
+Figure 6: Average and maximum slowdown for each suite and input set size. The slowdown is for an additional 35ns of
+latency between the LLC and main memory from the additional photonic components. Left: in-order pipeline compute
+cores. Right: Out of order (OOO) compute cores.
+For GPUs, in test case (A) with indirect routing a single GPU can use a total of 125 × 512 = 8000 GBps to access any
+one HBM or more in case a GPU is allocated more than one HBMs. This well satisfies the 1555.2 GBps that NVIDIA
+Ampere A100 GPUs in our model rack Per access HBMs with today and leaves 8000 − 1555.2 = 6444.8 GBps unused
+per GPU. In addition, in the worst case, an MCM containing three GPUs will communicate at full bandwidth (12
+NVLink links of 25 GBps per each of the three GPU equals 900 GBps) to other MCMs containing GPUs. Here, if
+all GPUs in the rack acts similarly, we cannot rely on indirect routing from a GPU through an intermediate GPU to
+reach a destination GPU. The direct 125 Gbps bandwidth between GPU MCMs do not suffice. Therefore, each GPU
+can use the 6444.8 GBps of unused bandwidth to and from HBMs for indirect routing to well cover the 900 GBps
+bandwidth that would otherwise use NVLink GPU–GPU links. This leaves 6444.8 − 900 = 5544.8 GBps per GPU that
+can support direct HBM–HBM communication such as due to GPUDirect RDMA, indirect routing for other MCMs, or
+simply increase available bandwidth to memory. Of note, our analysis does not use direct optical paths from GPUs to
+main memory (DDR4). Future protocols may use for these paths or they can be used to provide even more indirect
+routing bandwidth.
+Our analysis shows that test case (A) with AWGRs more than satisfies bandwidth demands and avoids the need for a
+scheduler to reconfigure spatial and wave-selective switches that would otherwise add overhead and reduce reaction
+time.
+6.2
+Latency Evaluation
+Intra-rack resource disaggregation based on modern photonics increases the latency significantly less than full system
+disaggregation. For intra-rack disaggregation we assume an additional latency between MCMs of 35 ns. That additional
+latency covers 15 ns for electrical–optical–electrical conversion and 4 meters of photonic propagation at 5 ns per meter,
+which covers round-trip distance of typical two-meter tall racks (Section 3.3.2). The small impact of distance to latency
+12
+
+Average
+Maximum
+100
+1
+(%)
+75
+umopmos
+50
+Percentage
+25
+0
+Parsec
+Parsec
+Parsec
+NAS A
+NAS B
+NAS C
+Rodinia
+small
+medium
+largeAverage
+Maximum
+125
+100
+(%)
+slowdown
+75
+Percentage s
+50
+25
+0
+Parsec
+Parsec
+Parsec
+NAS A
+NAS B
+NAS C
+Rodinia
+small
+medium
+largearXiv Template
+A PREPRINT
+with photonics practically makes MCMs in a rack equi-distant, thus mitigating a traditional queuing delay versus
+locality tradeoff in job scheduling Jeon et al. [2019]. Indirect routing would increase latency by a few extra ns, but the
+probability of routing indirectly is low. Because 35 ns is orders of magnitude lower than system-wide network latency,
+we do not consider the effect of the additional 35 ns to inter-rack communication through NICs.
+6.2.1
+CPU Evaluation
+We experimentally quantify the impact to application performance with in-order pipeline and out-of-order (OOO)
+compute cores. In-order cores provide insight of the impact of memory latency when the compute core does not mask
+latency, whereas OOO cores are representative of modern cores. We use full system simulation in Gem5 Binkert et al.
+[2011] of x86 compute cores running an Ubuntu 18.4 guest OS. We configure the cache hierarchy to match the CPUs
+of our model HPC rack Per. We calculate the slowdown of application execution time when we add 35 ns of latency
+between the LLC and main memory, compared to a baseline system with no additional latency to memory. Latency is
+the only potential source of application slowdown since our architecture satisfies the full escape bandwidth for each
+chip.
+We evaluate the impact in three benchmark suites: PARSEC 3.1 Bienia et al. [2008], NAS parallel benchmarks
+3.4.1 Bailey et al. [1992], and Rodinia Che et al. [2009]. For PARSEC we evaluate small, medium, and large input sets.
+For NAS, we evaluate input sizes “A”, “B”, and “C”. For Rodinia we use the single default input set. These benchmark
+suites have been widely used and contain a large variety of computation kernels that are representative of key HPC
+applications such as stencils, graph processing, linear algebra, computational mathematics, grid, sorting, and many
+others that have been observed to be important workloads in NERSC’s systems ?. Overall, we use 58 benchmarks to
+provide a wide representation. We use a single compute core to better focus on the effect of the additional latency to
+memory.
+Figure 6 shows slowdown percentages for benchmarks across our three suites for an in-order core on the left and OOO
+core on the right. As shown, NAS benchmarks are negligibly affected by the increased latency. Rodinia benchmarks
+have an average slowdown of 15% with in-order cores and 13% for OOO cores. However, a single benchmark (NW)
+has a slowdown of approximately 76% for in-order cores. The largest slowdown for the rest of Rodinia benchmarks
+across in-order and OOO cores is 12%. Finally, PARSEC benchmarks are impacted the most, but the average slowdown
+remains below 25% except for large inputs using OOO cores. OOO cores typically tolerate memory access latency
+better, but they also produce more memory accesses per unit of time compared to in-order cores.
+Figure 7 shows slowdown for individual PARSEC benchmarks for large inputs and in-order cores. As shown, only
+three benchmarks exceed a 25% slowdown, while eight benchmarks have a slowdown of no more approximately 3.5%.
+Therefore, our experiments show that while some benchmarks (three in PARSEC and one in Rodinia) experience
+important slowdowns, the majority of benchmarks are impacted minimally even without mitigation strategies. This is
+the case with all of the NAS benchmarks we used, eight PARSEC, and all but one Rodinia benchmarks. For benchmarks
+that are more affected, there is a range of hardware and software techniques Mutlu et al. [2006], Parcerisa and Gonzalez
+[2001], Mowry et al. [1998], Nekkalapu et al. [2008] to increase memory tolerance that we can apply to further reduce
+application slowdown.
+6.2.2
+Recovering Performance
+To gauge the effectiveness of strategies to recover application performance, we test the impact of the following remedies
+applied one at a time: (i) 256 miss status handling registers (MSHRs) in the LLC, (ii) doubling the LLC size with the
+default number of 16 MSHRs, and (iii) default LLC configuration but a strided prefetcher with a larger stride than the
+default four. Figure 8 shows the slowdown percentage that we were able to recover for PARSEC benchmarks through
+these three techniques at the best, average, and worst case. This figure is the only one that includes these remedies in
+this results of this section. As shown, about a 20% performance loss for small and large inputs is recovered by average.
+The most effective remedy is doubling the LLC size. The reason for the smaller speedup for medium is that due the
+particular LLC size, memory access patterns, and input sizes in PARSEC, medium experienced a smaller benefit from a
+larger LLC. These findings motivate future work to mitigate the latency impact of the disaggregation hardware, similar
+to mitigating the increased latency to access emerging memory technologies Mittal and Vetter [2016].
+6.2.3
+Sensitivity to Latency
+To show the sensitivity of application performance to the amount of additional latency, Figure 9 shows application
+slowdown for 25 ns, 30 ns, and 35 ns for in-order cores (OOO cores show comparable trends). As shown, reducing
+the additional latency to 25 ns from 35 ns reduces application slowdown by as much as half. This motivates latency
+improvements in photonic components or shorter rack distances.
+13
+
+arXiv Template
+A PREPRINT
+Figure 7: Results for individual PARSEC benchmarks with large inputs.
+Figure 8: Percentage of slowdown that we can recover with LLC modifications.
+14
+
+Parsec slowdown: Large inputs. Single in-order core
+100
+75
+(%) easad
+50
+25Best case
+Average
+Worst case
+(%)
+50
+recovered
+40
+30
+slowdown
+20
+Percentage
+10
+0
+Parsec small
+Parsec medium
+Parsec largearXiv Template
+A PREPRINT
+Figure 9: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for in order cores.
+The overall average slowdown across all benchmarks is approximately 13% for both in-order cores and OOO cores
+without architectural remedies, for large PARSEC inputs, and “B” size NAS inputs. This considerably less than
+slowdowns quoted in past work for full-system disaggregation, furthering the case for intra-rack disaggregation.
+6.2.4
+GPU Evaluation
+To evaluate the impact of the additional latency between GPUs and HBMs or DDR4 main memory, we extend the
+publicly available version of PPT-GPU Arafa et al. [2021] toolkit to account for the additional latency between the
+main memory of the GPU and the LLC. In our evaluation, we modeled one NVIDIA A100 GPU Choquette and Gandhi
+[2020] running a total of 27 applications that have a total of 2133 kernels from different benchmark suites. We run 13
+applications from Rodinia Che et al. [2009] and 10 applications from Polybench Grauer-Gray et al. [2012]. Polybench
+applications are linear algebra applications that stress the GPU cache and main memory. Furthermore, we run AlexNet,
+CifarNet, GRU, and LSTM from the Tango deep network Karki et al. [2019] benchmark suite. We use the default input
+sizes and configuration that came with the benchmarks, detailed in Arafa et al. [2021]. We run applications using the
+“SASS” model, where we extract memory and instruction traces for each application.
+Figure 10 shows the effect of different latencies on the performance of our GPU benchmarks. We compare performance
+in terms of the total predicted cycles. As shown, the highest average slowdown is 24% for Polybench. The overall
+average slowdowns across the 27 applications is only 8%, 10%, and 12% for the 25 ns, 30 ns, and 35 ns additional
+latency, respectively. For these benchmarks, doubling the LLC size recovers an average of 8% of the performance loss.
+6.2.5
+CPU–GPU Comparison
+We illustrate the difference in memory latency tolerance of in-order CPUs, OOO CPUs, and GPUs in Figure 11 for the
+intersection of Rodinia benchmarks that correctly ran on both CPU and GPU with their default input sets. As shown,
+GPUs tolerate the additional 35 ns latency significantly better with a maximum slowdown of 3.3%. This is promising
+for resource disaggregation given the steady growth of GPU presence in HPC systems.
+6.3
+Power Overhead
+We calculate the per-rack power overhead of our photonic solution for 350 MCMs with 2048 escape wavelengths from
+each MCM and 25 Gbps per wavelength. If we use a DFB laser array demonstrated in Rahimi et al. [2022] with a 11%
+wall plug efficiency (WPE) at 10 dDm, a total of 256 × 256 such lasers consumes 64.5 kW. For the components and
+distances in our study, the required optical power per wavelength is 10 dBm. Furthermore, 350×2048 of the modulators
+15
+
+25ns
+30ns
+35ns
+100
+75
+50
+25
+0
+Parsec
+par
+ROarXiv Template
+A PREPRINT
+Figure 10: Percentage slowdown for 25ns, 30ns, and 35ns of additional LLC–memory latency for different GPU
+benchmark suites.
+Figure 11: Percentage slowdown for CPU and GPU Rodinia benchmarks.
+16
+
+25ns
+30ns
+35ns
+80.00%
+%
+60.00%
+S
+40.00%
+20.00%
+0.00%
+Rodinia-avg Rodinia-max PBench-avg PBench-max Tango-avg
+Tango-max35ns in-order CPU
+35nsO0OCPU
+35ns GPU
+80
+(%)
+60
+slowdown (
+40
+ercentage
+20
+ParXiv Template
+A PREPRINT
+and receivers of Sun et al. [2020] that consume 0.8 and 2.12 pJ/bit at 25 Gbps respectively result in a total additional
+power of 52.5 kW. Finally, the switches of Table 2 consume no more than 1 kW at the worst case. In summary, the total
+power overhead taking into account parallel switches is no more than 150 kW. Our analysis assumes the components
+are constantly on. Considering that the maximum power consumption of a single A100 GPU is a few hundreds of Ws
+and our modelled rack contains 512 such GPUs, the power overhead for our photonic solution is negligible.
+6.4
+Comparison With Electronic Switches
+Electronic SERDES signalling rate per wire is only 112 Gbps for a short reach. Also, typical CXL or PCIe signaling
+rates top out at 35 GHz/wire. In fact, as SERDES rates increase, the distance that those signals can reach reduces down
+to even a few millimeters due to the resistance and capacitance of copper wires. Photonics break the reach limitations
+of copper and with co-packaging can achieve 4 Tbps per mm of shoreline on the chip die.
+Focusing on electronic switches, Rosetta De Sensi et al. [2020b] and Infiniband Katebzadeh et al. [2020] have a
+measured per hop latency of no less than approximately 200 ns. Emerging PCIe Gen5 switches add just 10 ns per
+hop Vasa et al. [2020], but only support 100 lanes per switch. To fully connect our disaggregated rack, we consider a
+two-level tree network with four hops (the top level is composed of an internal two-hop subnetwork). These four hops
+will be in addition to the 35 ns we previously evaluated for FEC and propagation (propagation delay is comparable
+between copper and photonic for rack distances), since our photonic solution uses switches with negligible traversal
+latency. Therefore, the additional latency for disaggregation in the PCIe case becomes 85 ns compared to 35 ns for
+our photonic architecture. Finally, we also consider the latency through one hop of an Anton 3 network, which is
+approximately 90 ns by average Shim et al. [2022], though scaling up to match our rack size would require multiple
+hops. These latencies represent the best case for electronic packet switches because scheduler decisions or congestion
+can cause higher worst-case (tail) latencies that may further penalize application performance. This assumes that we
+connect only one lane per endpoint which carries 32 Gbps for PCIe Gen5 and 29 Gbps for Anton 3. This is multiple
+times less than the per-chip bandwidth of photonics our photonic architecture.
+Figure 12 shows the speedup of a system that implements intra-rack disaggregation with emerging photonics with an
+additional 35 ns latency to and from DDR4 and HBM memory compared to a similar system that uses modern electronic
+switches instead. 85 ns is the lowest case for electronic switches and corresponds to a four-hop PCIe Gen5 network or a
+single-hop Anton 3 network. As shown, for CPU benchmarks if we only take into account “medium” from PARSEC to
+avoid counting PARSEC benchmarks three times, the average speedup for in-order CPUs is 12.7% and the maximum
+76%. For OOO compute cores, the average is 23.9% and maximum 78.3%. For GPUs, the average and maximum are
+both 61.5%. This analysis clearly shows the adverse impact of the additional latency of electronic switches and further
+motivates the use of photonics for intra-rack resource disaggregation. Furthermore, the four electronic switches of this
+analysis consume at least many tens of Watts of power, which is multiple orders of magnitude higher than our photonic
+solution.
+6.5
+Iso-Performance Comparison
+Based on our performance evaluations, in order to preserve system-wide average computational throughput as our
+baseline GPU-accelerated HPE/Cray EX system Per, our photonically-disaggregated system requires 13% more CPUs
+and 8% more GPUs. However, intra-rack resource disaggregation allows our rack to have an average 4× fewer memory
+modules and 2× fewer NICs Michelogiannakis et al. [2022]. Combining the two effects, our disaggregated rack has
+1082 total modules compared to 1920 in the baseline system, a 43% reduction. Alternatively, we can preserve all
+rack resources and instead add 128 of a combination of CPUs and GPUs (with their HBMs), which is only a 7% chip
+increase across the rack. Doing so doubles computational throughput.
+7
+Conclusion
+We have designed a resource disaggregated HPC rack that uses modern photonic links and switches to meet BER and
+bandwidth requirements of HPC applications, has a negligible power impact, uses distributed indirect routing instead of
+complex switch reconfiguration, and provides a 23.9% for CPUs or 61.5% for GPUs speedup compared to a similar
+disaggregated rack implemented with modern electronic switches. Our architecture enables a disaggregated system to
+preserve its performance but use 43% fewer overall chips.
+17
+
+arXiv Template
+A PREPRINT
+Figure 12: Speedup of a system that uses emerging photonics to implement intra-rack resource disaggregation that adds
+35 ns of additional latency to and from memory compared to a similar system that uses modern electronic switches and
+adds 85 ns of memory latency instead.
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+page_content=' Engineering Electrical and Computer Engineering  University of Ottawa  Cairo, Egypt  ahass202@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content=' biomedical engineering department  Minai university  Minya, Egypt  ahmd654@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='com  Alaa Nagy  dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Engineering Electrical and Computer Engineering  University of Ottawa  Cairo, Egypt  aelba046@uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='ca  Daila Farghl  dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' biomedical engineering department  Minai university  Minya, Egypt  dolly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='mostafa93@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='com  Abstract— In this paper we discuss a new method for  detecting leukemia in microscopic blood smear images using  deep neural networks to diagnose leukemia early in blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  leukemia  is  considered one  of  the most  dangerous mortality  causes for a human  being,  the  traditional process  of  diagnosis of leukemia in blood   is complex, costly, and time-  consuming, so patients could not receive medical treatment  on  time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Computer vision  classification  technique using  deep  learning can overcome the problems of traditional  analysis of  blood smears,   our  system for  leukemia  detection provides  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3  % accuracy in classifying samples as cancerous or normal  samples by taking a  shot of blood smear and passing it as an  input to the system that will check whether it contains cancer  or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' In case of containing cancer cells, then the  hematological expert passes the sample to a more complex  device such as flow cytometry to generate complete  information about the progress of cancer in the blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Keywords— Leukemia cells, leukemia detection, deep  neural networks, deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  INTRODUCTION  Leukemia is a type of cancer affecting blood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' if it is  detected late, it will result in death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Leukemia develops when  the bone marrow produces an excessive number of aberrant  white blood cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The normal of the blood system will be  disrupted when aberrant white blood cells are in excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Hematologists can identify abnormal blood when they draw  a blood sample and study it[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' However, hematologists will  inspect microscopic images visually, and the process is time-  consuming and tiring [1 - 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Moreover, the process requires  human experts and is prone to errors due to emotional  disturbance and human physical capability, which has its  limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Moreover, it is not easy to get consistent results from  visual inspection [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Visual inspection can only give  qualitative results for further research [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Studies indicate  that the majority of modern methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Use all blood-related  data, such as the number of red blood cells, hemoglobin  level, hematocrit level, mean corpuscular volume, and much  more, as the criterion for categorizing disorders like cancer,  thalassemia, Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Expensive testing and equipment labs are  required to know all information about blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' An automatic  image processing system is urgently needed and can  overcome related constraints in visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The system  to be developed will be based on microscopic images to  recognize leukemia cells in blood smears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The early and fast  identification of the leukemia type greatly aids in providing  the appropriate treatment for a particular type of leukemia  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The currently used diagnostic methods rely on analyzing  immuno- phenotyping, fluorescence in situ hybridization,  cytogenetic analysis, and cytochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Sophisticated and  expensive laboratories are required in order to run the  diagnostic methods, and it has been reported to provide a  high ratio of misidentification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' with this system, more images  can be processed, reduce analyzing time, exclude the  influence of subjective factors, and increase the accuracy of  identification process at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' In machine learning,  the inspection and classification of leukemia will be based on  the texture, shape, size, color, and statistical analysis of white  blood cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content="  In contrast, deep learning makes it much more profound  and gets the whole image's exclusive features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This project is  applied  to  increase  efficiency  globally  and  can  simultaneously benefit and be a massive contribution to the  medical and pattern recognition field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The main objective is  to enhance algorithms that can extract data from human  blood where human blood is the primary source to detect  diseases at an earlier stage and can prevent it quickly [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  This system should be robust towards diversity among  individuals, sample collection protocols, time, Etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This  automated system can produce lab results quickly, easily,  and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' DATASET  Images that were used in this project were downloaded  from the internet and are available in ALL IDB[6], ASH  Image Bank Hematology [7], Stock photo, vectors and  Royalty-free  Images[8],  Shutter  stock[9],  Atlas  of  Hematology [10], Atlas of blood smear analysis[11], Blue  Histology and American Society of Hematology [12], This  dataset is composed of 630 images, contains 480 cancer  images and 150 normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=" Data Preprocessing  1) Remove duplication  As the dataset is collected from various resources,  had found that there are some repetitions, some images  contain a watermark, and other contains websites' logo  totally about 43 images, so now the data set has become  587 images." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  2) Resizing of images  As the dataset has a different distribution of size,  and for training the CNN model, it was needed to make  all images in the dataset has the same size, so we applied  a resizing technique and make all image 256 x 256  pixels to reduce the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' as shown in figure  [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1]  3) Filtering images  Before the processing stage, we need to remove noise  and enhance line structures in images [13], and this is  available by applying a median filter (3 x3) and  sharpening the image (3 x3) ,as shown in Fig[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content="  Figure 1:(a) original image,(b)  image  resized  by  256*256 and  filtered by median and sharpen filters  4) Data augmentation  Image data augmentation is a method for artificially  increasing the size of a training dataset by producing  altered copies of the dataset's images [14]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The capacity  of fit models to generalize what they have learned to  new pictures may be improved by training deep-learning  neural network models on more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Additionally,  augmentation techniques can provide variants of the  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Through the ImageDataGenerator class, the  Keras deep learning neural network framework can fit  models by adding picture data [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' There are many  different types of augmentation techniques, some of  them as:  a) Flipping  An image flip means reversing the rows or columns  of pixels in the case of a vertical or horizontal flip [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  b) Horizontal and Vertical Shift Augmentation  A shift to an image means moving all pixels of the  image in one direction, such as horizontally or  vertically, while keeping the image dimensions the  same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' this means that some of the pixels will be  clipped off the image, and there will be a region of the  image where new pixel values will have to be  specified [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  c) Random Zoom Augmentation  A zoom augmentation randomly enlarges the image  and either interpolate or adds new pixel values around  the image [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  d) Shearing  Shearing will automatically crop the correct area  from the sheared image so that we have an image with  no black space or padding [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  e) Interpolation (Nearest)  A technique for creating new data points within the  range of a discrete set of existing data points is  interpolation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Nearest neighbor interpolation is  the most straightforward approach to interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Rather than calculate an average value by some  weighting criteria or generate an intermediate value  based on complicated rules, this method simply  determines the "nearest" neighboring pixel and  assumes its intensity value of it [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' And Fig[2]  indicates a sample image with its augmented one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  (a)                                          (b)  Fig 2: (a) original image   and (b) augmented image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Processing stage  After augmentation processes, our data become 1550  images for cancer and 1480 for normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' To fit data to  models, we divided it through coding into three data sets:  training set, validation set, and test set by ratios 60%, 20%,  and 20%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Then the next stage is to train the  model that can be able to classify the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Our optimizing parameters are accuracy and validation  accuracy: to get the best of them as possible, we trained  three networks with different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  a) BasicCNN model  In this model, the input images were (RGB) color  images with a resolution of 128x128 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' It consists  of 3 convolutional layers with max pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' A  rectified linear unit follows each convolutional layer  (relu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' We used a constant filter size (3x3), and the  number of  (a)  (b)Filters (128), the stride of ones (equal 1), and fully  connected layers trained for two categories  classification using the sigmoid activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Where we classified the data set into leukemia cells or  normal cells, this architect achieved 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='99% accuracy  and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='97 % validation accuracy after 17 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3:  Indicates the block diagram of the basic CNN model  b) Alexnet architecture  In this study, we deployed the pre-trained AlexNet to  detect ALL and classify its subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This architecture  was proposed by Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=', nine who  deployed this architecture for the ImageNet Large  Scale Visual Recognition Challenge 2012,20 and won  the challenge in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Input images were  Red, Green, and Blue (RGB) color images with a  resolution of 227 x 227 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' It consists of 5  convolutional layers with three max polling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Each convolutional layer in AlexNet architecture is  followed by a rectified linear unit (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' All the  parameters, including the filter size, the number of  filters, and the stride for each layer, are illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' we replaced the SoftMax layer with a sigmoid  layer as we want to classify the input image into only  two types of this architect achieved 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='35% accuracy  and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='76 % validation accuracy after 12 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Figure 4: AlexNet architecture for acute lymphoblastic leukemia  subtype classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Last 2 layers are newly added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  c) Modification of  model used in published paper  This used a retrained model that had been used in a  published paper [20], shown in figure 5, and we  changed the values of the hyperparameter to become  as shown in figure 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This network contains five  layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The first three layers perform feature  extraction, and the other two layers (fully connected  and SoftMax) classify the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The  input image has a size of 128x128x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' In convolution  layer 1, we used a constant filter size of 5x5 and a  total of 16 different filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The stride is one, and no  zero-padding was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The second and third  convolution layers have the same structure as the  first one but a different number of filters, 32 and 64,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' We used a pooling layer with filter size  two and stride 2 to decrease the volume spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content="  During the model we learned, the mini-batch's  chosen size was 128, and ReLu was used as the  activation function." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This architect gives: accuracy =  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='73 % validation accuracy = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='64 %  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 5: The original architecture of CNN in the mentioned paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 6: Architecture of CNN after changes in hyperparameter  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' EXPREMENTL RESULT  Our experiments were conducted on Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7 with  3030 images, 60% (1818 images) of them for training, 20%  (606 images) for validation, and the remaining 20% (606  images) for testing our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' In order to evaluate each  model and clarify the best one, we compare them by some  statistically measured parameters:  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Accuracy  Train accuracy  For the basic CNN model, train accuracy comes to  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='99% after 17 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' our leukemia classifier is doing an  excellent classification, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' For AlexNet  architecture, the accuracy achieved its maximum accuracy of  56% after 11 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' that means our model is terrible on  leukemia classification as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1 b, but the  Modification of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' paper [18]  achieved the maximum accuracy over all models 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='73 %  after ten epochs as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3  FC  Max  Max  Max  Conv layer  Conv layer  Conv layer  Input image  pooling  pooling  sigmoid  pooling  128*128*3  63*63*128  61*61*128  30*30*128  28*28*12814*14*128  126*126*128  No padding  No paddingFullyConnected  Layer  Fully Connected  4096  Layer  Follo  wedbyRelu  1024  L1  L2  L3  256  Norn  Convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='.ReLu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='.MaxPolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='soon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='384  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='Convolution5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='Convolution2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='ImageSize=13*13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='No padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='NopaddingValidation accuracy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='Basic CNN Model validation accuracy reaches 85% after  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='17 epochs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Therefore, we expect our  model to perform with ~85% accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' For  AlexNet architecture, the accuracy achieved its maximum  accuracy of 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6% after 11 epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' that means our model is  terrible on leukemia classification, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1 b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  This means that we expect our model to perform with  ~53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6% accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Nevertheless, in Modification  of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' paper [94] achieved the  maximum validation accuracy over all models at 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3 %  after ten epochs, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Therefore, we expect  our model to perform with ~94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3 % accuracy on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  We notice that our train metric increases as epochs increase  while the validation accuracy metric decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' That means  that our model fits the training set better but slightly loses its  ability to predict new data, indicating that our models are  beginning to overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1a, curve of val acc & train acc for basic CNN model  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='2b, curve of val acc & train acc for AlexNet architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='2c, curve of validation accuracy & train accuracy for  Modification of model used in Thanh et al paper [18]  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=" Confusion Matrix  A classification problem's predicted outcomes are compiled  in a confusion matrix." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The count values describe the number  of accurate and inaccurate predictions for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Because it is feasible to see the relationships between the  classifier outputs and the real ones, this is a great alternative  for reporting results in M-class classification issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' For the  basic CNN model, the number of leukemia images that are  predicted as leukemia is 372, the number of leukemia  images that are predicted as normal is 8, the number of  normal images predicted as normal is 269, and the number  of normal images that are predicted as leukemia is 51, as  shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' These accuracies show that this model is  good at predicting leukemia images but bad at predicting  normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' For AlexNet architecture, the number of  leukemia images that are predicted as leukemia is 0, the  number of leukemia images that are predicted as normal is  380, the number of normal images predicted as normal is  157, and the number of normal images that are predicted as  leukemia is 163, as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' These accuracies  show that this model is terrible at predicting normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  The number of leukemia images predicted as leukemia for  the modified model used in the published paper [18] is 369,  the number of leukemia images predicted as normal is 11,  the number of normal images predicted as normal is 301,  and the number of normal images predicted as leukemia is  19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' as shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' These accuracies show that this  model has done a great job of predicting normal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3a, Confusion matrix of basic CNN model  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='3b, Confusion matrix of AlexNet Architecture  trainaccvsval acc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='548  accuracy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='538  val  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='536  0  2  4  6  8  10  12  numofEpochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='98  trainaccvsval acc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='92  train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='90  val  0  1  2  3  4  5  6  7  8  numofEpochsConfusionmatrix  320  372  280  class o(cancer)  240  True label  200  160  120  class 1(normal)  51  269  80  40  Predicted labelConfusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='matrix  320  0  380  280  class O(cancer)  240  Truelabel  200  160  120  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='1(normal)  163  157  80  40  Predicted label  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3c, Confusion matrix for Modification of model used in  Thanh et al paper [18]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Percsision  It is calculated as the proportion of accurate positive results  to those that the classifier predicted to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Our CNN  model has medium precision, AlexNet architecture has very  low precision, and the modified version of the model used in  Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=" 's [18] paper has good precision due to its  goodness method, as shown in fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Recall  It is determined by dividing the total number of pertinent  samples (all samples that should have been labeled as  positive) by the total number of reliable positive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  As illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4a for our CNN model, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4b for  the AlexNet architecture, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4c for the Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The perfect model regarded recall is the third  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  The first CNN model in class 1 has a high recall but low  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' This means that most of the positive examples are  correctly recognized (low FN), but there are a lot of false  positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Nevertheless, in class 0, low recall and high  precision show that we miss a lot of positive examples (high  FN), but those we predict as positive are indeed positive  (low FP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' F1 Score  The harmonic mean of recall and accuracy is the F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  The F1 score has a range of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' It tells how accurate the  classifier is (how many instances it classifies correctly) and  how robust it is (it recognizes a significant number of  instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' As illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4a for our CNN model,  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4b for the AlexNet architecture, and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4c for the  Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' paper [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' These figures show that the  modification of the model used in Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=" 's paper [18]  is precise and robust." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Support  Support is the number of samples accurately representing  the response within that category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  It provides information on the precise numbers of each class  in the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Figures 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4a and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4b for the fundamental CNN model, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4b  for the AlexNet architecture, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='8c for a modified version  of the model from the Thanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' work [18] serve as  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4a, values of precision, recall, f1 score and support for our  CNN model  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4b, values of precision, recall, f1 score and support for  AlexNet architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='4c, values of precision, recall, f1 score and support for  Modification of model used in Thanh et al paper [18]  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content="  DISCUSSION  Leukemia is a malignancy that affects the body's blood-  forming tissues, including the lymphatic system and bone  marrow." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' To get the most effective treatment, the patient  needs early Diagnosis, so we deploy three models using the  power of CNN to classify blood smears into normal and  abnormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Our dataset had not been taken under the same conditions as  it was collected from various resources, and it needed to be  bigger to use with DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' To overcome this problem, we used  the power of data augmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' this solution was suitable  for us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' our data before augmentation was 260 images, and  after augmentation became 3030 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  Our optimizing parameters were accuracy and validation  accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' by using CNN, we trained the model: the First model consists of 3 convolutional layers  with max pooling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' Its accuracy was 90%  and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='97 % validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' It was terrible  with our dataset due to its few layers, so we  trained another model the Second model was AlexNet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' this architecture  proved its efficiency in CNN models, so we  trained it with our data, input is (RGB) color  images with a resolution of 227 x 227 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' It  consists of 5 convolutional layers with three max  polling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' These models achieved 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='35%  accuracy and 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='76 % validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' We  found that it does not fit our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' So we still  have the same problem of low accuracy and keep  looking for another model In the last model, we used a retrained model that  had been used in a published paper [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' it  contain7 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The first five layers perform  feature extraction, and the other two layers (fully  connected and SoftMax) classify the extracted  Confusionmatrix  320  369  11  280  class o(cancer)  240  True label  200  160  120  class 1(normal  19  301  80  40  cer)  Predicted labelprecision  recall  f1-score  support  class 0(cancerous)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='76  78  class 1(Normal)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content=' The input image has a size of 128x128x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  This architect has an accuracy of 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='73 %  validation accuracy is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='64 %, finally, we found  that this model fit our data VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' CONCLUSIONS  In this system, we investigated the application of deep  CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' We deployed a pre-trained model for detecting and  classifying the blood sample into normal and abnormal  samples using microscopic blood sample images and  convolutional neural network classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The  system was built by deep learning, which uses all features in  microscopic images, not only examining changes of specific  features as a classifier input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=" We have performed the pre-  trained model in a largely augmented dataset to confirm the  system's accuracy and reliability." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' By performing data  augmentation, we can achieve 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='3% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' The system  has high accuracy, and less processing time (show results in  less than 30 seconds)  , minor errors, and early identification of leukemia  successful in giving the patient the proper care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' And cheaper  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  The detection system was built in three parts:  1) the acquisition part, which consists of a digital camera  that has been installed at the top of the eyepiece of the  microscope,  2) pre-trained CNN model responsible for the  classification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  3)  a graphical user interface to display the image obtained  from the camera and show the classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' FUTURE WORK  Expanding the focus on classifying the subtypes of leukemia  cells such as Acute Myeloid Leukemia or AML, Chronic  Myeloid Leukemia or CML, Acute Lymphoid Leukemia or  ALL, and Chronic Lymphoid Leukemia or CLL not only  separating between cancerous and non-cancerous cells and  developing a convenient environment to construct an  extensive leukemia dataset as this topic of research suffer  from leaks in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' REFERENCES  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=',  Ritter,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=',  Cooper,  “Segmentation  and  Border  Identification  of Cells in Images of Peripheral Blood Smear Slides”, 30th  Australasian Computer Science Conference, Conference in Research  and Practice in Information Technology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 62, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 161-169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=',  Costa,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=',  Rizzatti,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=',  Zago, “A   Texture   Approach   to   Leukocyte   Recognition”,   Real  Time Imaging,            Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=' 10,            2004,            pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  205-206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  [3] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=', Colunga, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
+page_content=', Siordia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content=', Maybank, “Leukocyte  Recognition Using  EMAlgorithm”,  MICAI  2009,  LNAI  5845,  Springer  Verlag Berlin Heidelberg, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+page_content='      Walter,”      understanding      leukemia”,      2009,      pp-  8-18  [5] Warren    S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE1T4oBgHgl3EQfsgWq/content/2301.03367v1.pdf'}
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+1
+Learning Optimal Phase-Shifts of Holographic
+Metasurface Transceivers
+Debamita Ghosh, IITB-Monash Research Academy, IIT Bombay, India
+Manjesh K. Hanawal, MLioNS Lab, IEOR, IIT Bombay, India
+Nikola Zlatanov, Innopolis University, Russia
+Abstract—Holographic metasurface transceivers (HMT) is
+an
+emerging
+technology
+for
+enhancing
+the
+coverage
+and
+rate of wireless communication systems. However, acquiring
+accurate channel state information in HMT-assisted wireless
+communication systems is critical for achieving these goals.
+In this paper, we propose an algorithm for learning the
+optimal
+phase-shifts
+at
+a
+HMT
+for
+the
+far-field
+channel
+model.
+Our
+proposed
+algorithm
+exploits
+the
+structure
+of
+the channel gains in the far-field regions and learns the
+optimal
+phase-shifts
+in
+presence
+of
+noise
+in
+the
+received
+signals.
+We
+prove
+that
+the
+probability
+that
+the
+optimal
+phase-shifts estimated by our proposed algorithm deviate from
+the true values decays exponentially in the number of pilot
+signals. Extensive numerical simulations validate the theoretical
+guarantees and also demonstrate significant gains as compared
+to the state-of-the-art policies.
+Index Terms—Holographic Metasurface Transceivers, Channel
+State Information, Uniform Exploration
+I. INTRODUCTION
+Future wireless network technologies, namely beyond-5G
+and 6G, have been focused on millimeter wave (mmWave)
+and TeraHertz (THz) communications technologies as possible
+solutions to the ever growing demands for higher data rates and
+lower latency. However, mmWave and THz communications
+have challenges that need to be addressed before this technology
+is adopted [1], [2]. One such major challenge is signal
+deterioration due to reflections and absorption.
+A possible solution for the signal deterioration are base
+stations (BSs) with massive antennas arrays that can provide
+large beamforming gains and thereby compensate for the
+signal deterioration [3]. However, implementing a BS with
+a massive antenna array is itself challenging due to the high
+hardware costs. Holographic Metasurface Transceivers (HMTs)
+are introduced as a promising solution for building a massive
+antenna array [4], [5]. A HMT is comprised of a large number
+of metamaterial elements densely deployed into a limited
+surface area in order to form a spatially continuous transceiver
+aperture. These metamaterial elements at the HMT acts as
+phase-shifting antennas, where each phase-shifting element
+of the HMT can change the phase of transmiting/receiving
+signal and thereby beamform towards desired directions where
+the users are allocated [6]. Due to these continuous apertures,
+HMTs can be represented as an extension of the traditional
+massive antenna arrays with discrete antennas to continuous
+reflecting surfaces [6].
+In this paper, we consider the HMT-assisted wireless systems
+illustrated in Fig. 1, where a HMT acts as a BS that serves
+multiple users. The performance of this system is dependent on
+channel state information (CSI) estimates at the HMT, which
+are used for accurate beamforming towards the users. The
+authors in [7] and [8] have studied the effect of HMT-assisted
+systems on enhancing the communication performance under
+the assumption of perfect CSI. However, perfect CSI is not
+available in practice. In practice, the CSI has to be estimated
+via pilot signals, which results in inaccurate CSI estimates at
+the HMT.
+The aim of this paper is to obtain accurate CSI estimates at
+the HMT, which in turn is used to set the optimal phase-shifts
+at the HMT that maximize the data rate to the users when
+the users are located in the far-field. To this end, we exploit
+the structure of the far-field channel model between the HMT
+and the users to show that the optimal phase-shifts at the
+HMT can be obtained from five samples of the received pilot
+signals at the HMT in a noiseless environment. We then use this
+approach to develop a learning algorithm that learns the optimal
+phase-shifts from the received pilot signals at the HMT in a
+noisy environment. Finally, we provide theoretical guarantees
+for our learning algorithm. Specifically, we prove that the
+probability of the phase-shifts generated by our algorithm to
+deviate by more than 𝜖 from the optimal phase-shifts is small
+and decays as the number of pilot symbols increases. The
+error analysis is based on tail probabilities of the non-central
+Chi-squared distribution.
+In summary, our main contributions are as follows:
+• We propose an efficient learning algorithm for estimating
+the optimal phase-shifts at an HMT in the presence of
+noise for the case when the users that the HMT is serving
+are located at the far-field region.
+• We prove that the probability of the phase-shifts generated
+by our algorithm to deviate by more than 𝜖 from the
+optimal phase-shifts is small and decays exponentially as
+the number of pilots used for estimation increases.
+• We show numerically that the performance of the
+proposed algorithm significantly outperforms existing CSI
+estimation algorithms.
+A. Related Works
+Several channel estimation schemes, which are proposed
+for the massive antenna arrays, are also applicable to the
+considered HMT including exhaustive search [9], hierarchical
+search [10], [11], and compressed sensing (CS) [11]. As the
+exhaustive search in [9] significantly increases the training
+arXiv:2301.03371v1  [eess.SP]  12 Dec 2022
+
+2
+overhead, the authors in [10] and [11] proposed the hierarchical
+search based on a predefined codebook as an improvement over
+the exhaustive search. The hierarchical schemes, in general,
+may incur high training overhead and system latency since they
+require non-trivial coordination among the transmitter and the
+user [11]. On the other hand, the proposed CS-based channel
+estimation scheme in [11] provides trade-offs between accuracy
+of estimation and training overhead at different computational
+costs.
+On the other hand, CSI estimation schemes developed
+specifically for HMTs can be found in [12] and [13]. The
+authors in [12] proposed the least-square estimation based
+approach to study the channel estimation problem for the
+uplink between a single user and the BS equipped with the
+holographic surface with a large number of antennas. However,
+the authors require an additional knowledge of antennas array
+geometry to reduce the pilot overhead required by the channel
+estimation, and hence the computational complexity scales
+up with the number of antennas at the BS. In [13], the
+authors proposed a scheme for the estimation of the far-field
+channel between a HMT and a user that requires only five
+pilots for perfect estimation in the noise-free environment.
+In the noisy case, the authors of [13] proposed an iterative
+algorithm that efficiently estimates the far-field channel. Unlike
+the existing works, the training overhead and the computational
+cost of the proposed scheme in [13] does not scale with the
+number of phase-shifting elements at the HMT. The iterative
+algorithm in [13] significantly outperforms the hierarchical
+and CS based schemes. However, the authors in [13] did not
+provide any theoretical guarantees on their proposed algorithm.
+Motivated by [13], in this work, we propose an algorithm
+which outperforms the one in [13], and, in addition, we also
+provide theoretical guarantees for our proposed algorithm.
+This paper is organized as follows. The system and channel
+models for the HMT communication system are given in Sec.
+II. The proposed algorithm for learning the optimal phase-shifts
+is given in Sec. III and its theoretical guarantee is provided
+in Sec. IV. Numerical evaluation of the proposed algorithm is
+provided in Sec. V. Finally, Sec. VI concludes the paper.
+II. SYSTEM AND CHANNEL MODELS
+We consider a HMT-assisted wireless communication system,
+shown in Fig. 1, where an HMT communicates with multiple
+users in the mmWave band. We assume that there is a Line
+of Sight (LoS) between the HMT and each user. As a result,
+when modeling the far-field channel, we only take into account
+the LoS path since its power is order of magnitude higher than
+non-line-of-sight (NLoS) paths [14]. The NLoS components
+are incorporated in the noise. We assume that the users send
+orthogonal pilots to the HMT for channel estimation. Based
+on the estimated CSI at the HMT to each user, the HMT sends
+data to the users. Hence, the data rate from the HMT to the
+users is directly dependent on the accuracy of the CSI estimates
+at the HMT. Since in this paper our main goal is the accurate
+CSI estimation at the HMT to each user, which in turn send
+orthogonal pilots to the HMT, in the rest of the paper, we will
+focus on the CSI estimation between the HMT and a typical
+user.
+RF Generator
+Phase-shifting
+Element
+User 1
+User 2
+User 3
+Fig. 1: The HMT-assisted wireless communication system [13].
+A. HMT Model
+The HMT has a rectangular surface of size 𝐿𝑥 × 𝐿𝑦, where
+𝐿𝑥 and 𝐿𝑦 are the width and the length of the surface,
+respectively. The HMT’s surface is comprised of a large
+number of sub-wavelength phase-shifting elements, where
+each elements is assumed to be a square of size 𝐿𝑒 × 𝐿𝑒
+and can change the phase of the transmit/receive signal
+independently from rest of the elements. Let 𝑑𝑟 be the
+distance between two neighboring phase-shifting elements.
+The total number of phase-shifting elements of the HMT is
+given by 𝑀 = 𝑀𝑥 × 𝑀𝑦, where 𝑀𝑥 = 𝐿𝑥/𝑑𝑟 and 𝑀𝑦 = 𝐿𝑦/𝑑𝑟.
+Without loss of generality, we assume that the HMT lies
+in the 𝑥 − 𝑦 plane of a Cartesian coordinate system, where
+the center of the surface is at the origin of the coordinate
+system. Assuming 𝑀𝑥 and 𝑀𝑦 are odd numbers, the position
+of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element in the Cartesian
+coordinate system is given as (𝑥, 𝑦) = (𝑚𝑥𝑑𝑟,𝑚𝑦𝑑𝑟), where
+𝑚𝑥 ∈
+�
+− 𝑀𝑥−1
+2
+,..., 𝑀𝑥−1
+2
+�
+and 𝑚𝑦 ∈
+�
+− 𝑀𝑦−1
+2
+,..., 𝑀𝑦−1
+2
+�
+. When
+𝑀𝑥 or 𝑀𝑦 is even, the position of the (𝑚𝑥,𝑚𝑦)𝑡ℎ element can
+be appropriately defined.
+B. Channel Model
+Consider the channel between the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting
+element at the HMT and the typical user. Let the beamforming
+weight imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element at
+the HMT be Γ𝑚𝑥𝑚𝑦 = 𝑒 𝑗𝛽𝑚𝑥 𝑚𝑦 , where 𝛽𝑚𝑥𝑚𝑦 is the phase shift
+at the (𝑚𝑥,𝑚𝑦)𝑡ℎ element. Let 𝜆 denote the wavelength of
+the carrier frequency, 𝑘0 = 2𝜋
+𝜆 be the wave number, 𝑑0 be the
+distance between the user and the center of the HMT and
+let 𝐹𝑚𝑥𝑚𝑦 denote the effect of the size and power radiation
+pattern of the (𝑚𝑥,𝑚𝑦)𝑡ℎ phase-shifting element on the channel
+coefficient [15]. Due to the far-field assumptions, the radiation
+pattern of all the phase-shifting elements of the HMT are
+identical, i.e., 𝐹𝑚𝑥𝑚𝑦 = 𝐹, ∀𝑚𝑥,𝑚𝑦 holds. Finally, let 𝜃 and
+𝜙 denote the elevation and azimuth angles of the impinging
+wave from the user to the center of the HMT, see Fig. 2.
+Now, if the phase-shift imposed by the (𝑚𝑥,𝑚𝑦)𝑡ℎ element,
+𝛽𝑚𝑥,𝑚𝑦, is set to
+𝛽𝑚𝑥𝑚𝑦 = −
+mod (𝑘0𝑑𝑟 (𝑚𝑥𝛽1 +𝑚𝑦𝛽2),2𝜋),∀𝑚𝑥,𝑚𝑦,
+
+3
+User
+HMT
+Fig. 2: Distance between the (𝑚𝑥,𝑚𝑦)-th phase-shifting element at
+the HMT and the user [13].
+where 𝛽1 and 𝛽2 are the phase-shift parameters [13], [16],
+[17], which are the only degrees of freedom within the
+phase-shift 𝛽𝑚𝑥𝑚𝑦, then the HMT-user channel in the far-field
+is approximated accurately by [13], [16], [17]
+𝐻(𝛽1, 𝛽2) =
+�√
+𝐹𝜆𝑒−𝑗𝑘0𝑑0
+4𝜋𝑑0
+�
+𝐿𝑥𝐿𝑦 ×sinc
+�
+𝐾𝑥𝜋(𝛼1 − 𝛽1)
+�
+×sinc
+�
+𝐾𝑦𝜋(𝛼2 − 𝛽2)
+�
+,
+(1)
+where 𝐾𝑥 = 𝐿𝑥
+𝜆 ,𝐾𝑦 = 𝐿𝑦
+𝜆 ,𝛼1 = sin(𝜃) cos(𝜙),𝛼2 = sin(𝜃) sin(𝜙),
+and sinc(𝑥) = sin(𝑥)
+𝑥
+. Please note that 𝛼1 ∈ [−1,1] and 𝛼2 ∈
+[−1,1], and their values depend on the location of the user,
+i.e., on 𝜃 and 𝜙.
+From (1), it is clear that the absolute value of the HMT-user
+channel is maximized when the two sinc functions attain
+their maximum values, which occurs when the phase-shifting
+parameters, 𝛽1 and 𝛽2, are set to 𝛽1 = 𝛼1 and 𝛽2 = 𝛼2, where
+(𝛼1,𝛼2) are unknown to the HMT since they depend on the
+location of the user. Therefore, in the far-field case, the problem
+of finding the optimal phase-shifts of the elements at the HMT
+reduces to estimating the two parameters, 𝛼1 and 𝛼2 at the
+HMT.
+Remark 1. Fig. 3 shows an example of |𝐻(𝛽1, 𝛽2)| as a
+function of (𝛽1, 𝛽2). As can be seen from Fig. 3, the graph
+of |𝐻(𝛽1, 𝛽2)| hits zero periodically and has several lobes.
+The optimal value (𝛼1,𝛼2) = (0.68,−0.45) is attained at the
+central lobe which has the highest peak and is attained for
+(𝛽∗
+1, 𝛽∗
+2) = (𝛼1,𝛼2) = (0.68,−0.45).
+III. PROPOSED CHANNEL ESTIMATION STRATEGY
+In this section, we propose an algorithm that estimates the
+optimal phase-shifting parameters 𝛽1 and 𝛽2 that maximize
+|𝐻(𝛽1, 𝛽2)| in (1) in the presence of noise.
+A. Problem Formulation
+In the channel estimation procedure, the user sends a
+pilot symbol 𝑥𝑝 =
+√
+𝑃 to the HMT, where 𝑃 is the pilot
+transmit power. Then, the received signal at the HMT for
+fixed phase-shifting parameters (𝛽1, 𝛽2), denoted by 𝑦(𝛽1, 𝛽2),
+is given by
+𝑦(𝛽1, 𝛽2) =
+√
+𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁,
+(2)
+Fig. 3: |𝐻(𝛽1, 𝛽2)| v/s (𝛽1, 𝛽2) for values of (𝛼1,𝛼2) = (0.68,−0.45).
+where 𝜁 is the complex-valued additive white Gaussian noise
+(AWGN) with zero mean and variance 𝜎2 at the HMT. The
+received signal in (2) is then squared in order to obtain the
+received signal squared, denoted by 𝑟(𝛽1, 𝛽2), and given by
+𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 =
+���
+√
+𝑃 × 𝐻(𝛽1, 𝛽2) + 𝜁
+���
+2
+.
+(3)
+Objective: Our goal is to identify the optimal phase-shifting
+parameters, denoted by (𝛽∗
+1, 𝛽∗
+2), at the HMT that maximizes
+𝑟(𝛽1, 𝛽2) given by (3). Specifically, we aim to solve the
+following optimisation problem
+(𝛽∗
+1, 𝛽∗
+2) = argmax
+𝛽1∈[−1,1]
+𝛽2∈[−1,1]
+𝑟(𝛽1, 𝛽2).
+(4)
+The expected value of 𝑟(𝛽1, 𝛽2), denoted by 𝜇(𝛽1, 𝛽2), is given
+by
+𝜇(𝛽1, 𝛽2) = E [𝑟(𝛽1, 𝛽2)]
+=
+���
+√
+𝑃 × 𝐻(𝛽1, 𝛽2)
+���
+2
++ 𝜎2.
+(5)
+Using (5), the optimization problem in (4) can be written
+equivalently as
+(𝛽∗
+1, 𝛽∗
+2) = argmax
+𝛽1∈[−1,1]
+𝛽2∈[−1,1]
+𝜇(𝛽1, 𝛽2).
+(6)
+In order to obtain an intuition on how to solve (6), we first
+assume that 𝜇(𝛽1, 𝛽2) in (5) is known perfectly at the HMT for
+five specific values of the pair (𝛽1, 𝛽2). Later, we use the same
+intuition to solve (6) when 𝜇(𝛽1, 𝛽2) are not known perfectly
+but can be estimated.
+B. The Optimal Phase-Shifting Parameters When 𝜇(𝛽1, 𝛽2)
+Are Known In Advance
+For notational convenience, let us define the set B as
+B =
+�
+(𝛽0
+1, 𝛽0
+2), (𝛽0
+1 +𝑣, 𝛽0
+2), (𝛽0
+1 −𝑣, 𝛽0
+2),
+(𝛽0
+1, 𝛽0
+2 + 𝑤), (𝛽0
+1, 𝛽0
+2 − 𝑤)
+�
+.
+(7)
+The set B is comprised of five pairs of the phase-shifting
+parameters (𝛽1, 𝛽2), where 𝛽0
+1 and 𝛽0
+2 are some initial arbitrarily
+selected phase-shifting parameters, 𝑣 and 𝑤 are numbers chosen
+such that 𝐾𝑥𝑣 ∈ N and 𝐾𝑦𝑤 ∈ N hold, where N is the set of
+natural numbers. Please note that for a selected (𝛽0
+1, 𝛽0
+2) and a
+
+1
+(α_1,α2)=(0.68,-0.45)
+0.8
+[H(β1, β2) /2
+0.5
+0.6
+0.4
+0.4
+β1
+01
+0.8
+0.6
+-0.6
+0.2
+0.4
+0.8
+0.2
+2
+β2
+0.4
+14
+chosen 𝑣 and 𝑤, if
+��𝛽0
+1 ±𝑣
+�� ≥ 1 then we set
+��𝛽0
+1 ±𝑣
+�� = 1. In the
+same way, if
+��𝛽0
+2 ± 𝑤
+�� ≥ 1 then we set
+��𝛽0
+2 ± 𝑤
+�� = 1.
+Theorem 1. If the HMT can obtain 𝜇(𝛽0
+1, 𝛽0
+2), 𝜇(𝛽0
+1 + 𝑣, 𝛽0
+2),
+𝜇(𝛽0
+1 − 𝑣, 𝛽0
+2), 𝜇(𝛽0
+1, 𝛽0
+2 + 𝑤) and 𝜇(𝛽0
+1, 𝛽0
+2 − 𝑤), i.e., obtain
+𝜇(𝛽1, 𝛽2) for the five phase-shifting parameters in (𝛽1, 𝛽2) ∈ B
+given in (7), then the optimal phase-shifting parameters 𝛽∗
+1
+and 𝛽∗
+2, which are the solutions of (6), are given by
+𝛽∗
+1 =
+�
+𝛼(𝑖)
+1 +𝛼( 𝑗)
+1
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+���𝛼(𝑖)
+1 −𝛼( 𝑗)
+1
+���
+�
+,
+(8)
+where
+𝛼(1)/(2)
+1
+= 𝛽0
+1 +
+𝑣
+1±
+√︄����
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1+𝑣,𝛽0
+2)−𝜎2
+����
+𝛼(3)/(4)
+1
+= 𝛽0
+1 −
+𝑣
+1±
+√︄����
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1−𝑣,𝛽0
+2)−𝜎2
+����
+and
+𝛽∗
+2 =
+�
+𝛼(𝑖)
+2 +𝛼( 𝑗)
+2
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+���𝛼(𝑖)
+2 −𝛼( 𝑗)
+2
+���
+�
+,
+(9)
+where
+𝛼(1)/(2)
+2
+= 𝛽0
+2 +
+𝑣
+1±
+√︄����
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1,𝛽0
+2+𝑤)−𝜎2
+����
+𝛼(3)/(4)
+2
+= 𝛽0
+2 −
+𝑣
+1±
+√︄����
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1,𝛽0
+2−𝑤)−𝜎2
+����
+Proof. By using (5) and (1) for any (𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2), we
+have the following
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2 =
+�����
+√
+𝑃
+�√
+𝐹𝜆𝑒− 𝑗𝑘0𝑑0
+4𝜋𝑑0
+�
+𝐿𝑥𝐿𝑦sinc
+�
+𝐾𝑥(𝛼1 − 𝛽0
+1)
+�
+×sinc
+�
+𝐾𝑦(𝛼2 − 𝛽0
+2)
+������
+2
+.
+(10)
+For (𝛽1, 𝛽2) = (𝛽0
+1 +𝑣, 𝛽0
+2), where 𝑣 is any arbitrary parameter
+such that 𝐾𝑥𝑣 ∈ N and
+��𝛽0
+1 ±𝑣
+�� ≤ 1 holds, we have
+𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) − 𝜎2 =
+�����
+√
+𝑃
+�√
+𝐹𝜆𝑒−𝑗𝑘0𝑑0
+4𝜋𝑑0
+�
+𝐿𝑥𝐿𝑦
+×sinc
+�
+𝐾𝑦(𝛼2 − 𝛽0
+2)
+������
+2
+.
+(11)
+Dividing (10) by (11), we obtain
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) − 𝜎2 =
+����sinc
+�
+𝐾𝑥(𝛼1 − 𝛽0
+1)
+�����
+2
+����sinc
+�
+𝐾𝑥(𝛼1 − 𝛽0
+1 −𝑣)
+�����
+2
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) − 𝜎2 =
+����
+sin(𝐾𝑥 𝜋(𝛼1−𝛽0
+1))
+𝐾𝑥 𝜋(𝛼1−𝛽0
+1)
+����
+2
+����
+sin(𝐾𝑥 𝜋(𝛼1−𝛽0
+1−𝑣))
+𝐾𝑥 𝜋(𝛼1−𝛽0
+1−𝑣)
+����
+2 .
+(12)
+If
+𝑣
+is
+selected
+such
+that
+𝐾𝑥𝑣 ∈ N,
+then
+we
+have
+����sin
+�
+𝐾𝑥𝜋(𝛼1 − 𝛽0
+1 ±𝑣)
+����� =
+����sin
+�
+𝐾𝑥𝜋(𝛼1 − 𝛽0
+1)
+�����. As a result,
+(12) is simplified to
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) − 𝜎2 =
+�����
+𝛼1 − 𝛽0
+1 −𝑣
+𝛼1 − 𝛽0
+1
+�����
+2
+.
+(13)
+Since
+𝜇(𝛽1, 𝛽2) ≥ 𝜎2, it follows that
+𝜇(𝛽1, 𝛽2) − 𝜎2 =
+��𝜇(𝛽1, 𝛽2) − 𝜎2�� always holds, for all (𝛽1, 𝛽2) ∈ B. Using this
+fact, (13) can be written equivalently as
+�
+�
+������
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) − 𝜎2
+����� =
+�����
+𝛼1 − 𝛽0
+1 −𝑣
+𝛼1 − 𝛽0
+1
+�����.
+(14)
+By solving the nonlinear equation in (14) w.r.t. the unknown
+𝛼1, we obtain two solutions for 𝛼1, denoted by 𝛼(1)
+1
+and 𝛼(2)
+1 ,
+given by
+𝛼(1)/(2)
+1
+= 𝛽0
+1 +
+𝑣
+1±
+√︂���
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1+𝑣,𝛽0
+2)−𝜎2
+���
+.
+(15)
+It is not known which of the two values 𝛼(1)
+1
+and 𝛼(2)
+1
+is
+equal to 𝛼1. To identify the correct solution for 𝛼1 of the
+two solutions given by (15), we need the value of 𝜇(𝛽1, 𝛽2)
+for (𝛽1, 𝛽2) = (𝛽0
+1 − 𝑣, 𝛽0
+2). Following the same procedure as
+for (10)-(15), but now by using the values of 𝜇(𝛽1, 𝛽2) for
+(𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2) and (𝛽1, 𝛽2) = (𝛽0
+1 −𝑣, 𝛽0
+2), we obtain
+�
+�
+������
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1 −𝑣, 𝛽0
+2) − 𝜎2
+����� =
+�����
+𝛼1 − 𝛽0
+1 +𝑣
+𝛼1 − 𝛽0
+1
+�����.
+(16)
+By solving (16), we obtain
+𝛼(3)/(4)
+1
+= 𝛽0
+1 −
+𝑣
+1±
+√︂���
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1−𝑣,𝛽0
+2)−𝜎2
+���
+.
+(17)
+One of the solutions in (15) is identical to one of the solutions
+in (17). Therefore, using (15) and (17), the correct solution of
+𝛼1 can be obtained as1
+𝛼1 =
+�
+𝛼(𝑖)
+1 +𝛼( 𝑗)
+1
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+���𝛼(𝑖)
+1 −𝛼( 𝑗)
+1
+���
+�
+.
+(18)
+In order to obtain 𝛼2, we need the value of 𝜇(𝛽1, 𝛽2) for
+(𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2), which we already have, and for (𝛽1, 𝛽2) =
+(𝛽0
+1, 𝛽0
+2 +𝑤), where 𝑤 is selected such that 𝐾𝑦𝑤 ∈ N,
+��𝛽0
+2 ± 𝑤
+�� ≤
+1 and
+��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0
+2 ± 𝑤))
+�� =
+��sin(𝐾𝑦𝜋(𝛼2 − 𝛽0
+2))
+��. Then,
+similar to (10)-(14), we use the values of 𝜇(𝛽1, 𝛽2) for
+(𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2) and (𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2 + 𝑤) to obtain
+1Note
+that
+𝛼1
+can
+also
+be
+written
+equivalently
+as
+𝛼1 =
+�
+𝛼(1)
+1
+, 𝛼(2)
+1
+� � �
+𝛼(3)
+1
+, 𝛼(4)
+1
+�
+. However, the expression in (18) is more
+convenient for the case when the values of 𝜇(𝛽1, 𝛽2) need to be estimated.
+
+5
+�
+�
+������
+𝜇(𝛽0
+1, 𝛽0
+2) − 𝜎2
+𝜇(𝛽0
+1, 𝛽0
+2 + 𝑤) − 𝜎2
+����� =
+�����
+𝛼2 − 𝛽0
+2 − 𝑤
+𝛼2 − 𝛽0
+2
+�����.
+(19)
+By solving the nonlinear equation (19), we obtain two solutions
+for 𝛼2, denoted by 𝛼(1)
+2
+and 𝛼(2)
+2 , given by
+𝛼(1)/(2)
+2
+= 𝛽0
+2 +
+𝑤
+1±
+√︂���
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1,𝛽0
+2+𝑤)−𝜎2
+���
+.
+(20)
+To identify the correct solution for 𝛼2 of the two given in
+(20), we need the value of 𝜇(𝛽1, 𝛽2) for (𝛽1, 𝛽2) = (𝛽0
+1, 𝛽0
+2 −𝑤).
+Again, following the procedure from (10)-(15), by using the
+values of 𝜇(𝛽1, 𝛽2) for (𝛽0
+1, 𝛽0
+2) and (𝛽0
+1, 𝛽0
+2 − 𝑤), we obtain
+𝛼(3)/(4)
+2
+= 𝛽0
+2 −
+𝑤
+1±
+√︂���
+𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+𝜇(𝛽0
+1,𝛽0
+2−𝑤)−𝜎2
+���
+.
+(21)
+One of the solutions in (20) is exactly same as the solutions
+of (21). Therefore, using (20) and (21), the correct solution of
+𝛼2 can be obtained as
+𝛼2 =
+�
+𝛼(𝑖)
+2 +𝛼( 𝑗)
+2
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+���𝛼(𝑖)
+2 −𝛼( 𝑗)
+2
+���
+�
+.
+(22)
+Finally, by setting 𝛽∗
+1 = 𝛼1 and 𝛽∗
+2 = 𝛼2, where 𝛼1 and 𝛼2 are
+given by (18) and (22), respectively, we obtain (8) and (9).
+■
+Remark 2. In [13, Sec. IV.A], the authors proposed the channel
+estimation strategy under the assumption that there is no noise
+in the system. However, in the noisy case, we proposed an
+estimation scheme based on the assumption that 𝜇(𝛽1, 𝛽2) for
+any of the phase-shifting parameters (𝛽1, 𝛽2) ∈ B are perfectly
+known at the HMT.
+However, in practice the exact values of 𝜇(𝛽1, 𝛽2) for any of
+the phase-shifting parameters (𝛽1, 𝛽2) ∈ B cannot be known in
+advance at the HMT, and therefore they need to be estimated
+using pilot symbols. In the following, we propose an algorithm
+that estimates 𝜇(𝛽1, 𝛽2) for the phase-shifting parameters in
+B and then uses the estimated values of 𝜇(𝛽1, 𝛽2) to find the
+optimal phase-shifting parameters (𝛽∗
+1, 𝛽∗
+2) in the presence of
+noise.
+C. Estimation Of The Optimal Phase-Shifting Parameters In
+The Noisy Case
+The user sends in total 𝑁 number of pilot signals to the
+HMT for the estimation of the five values of 𝜇(𝛽1, 𝛽2) for the
+five pairs of (𝛽1, 𝛽2) ∈ B. As a result, the proposed algorithm
+works in five epochs. In the 𝑘𝑡ℎ epoch, for 𝑘 = 1,2,..., the user
+transmits
+� 𝑁
+5
+� number of pilots to the HMT. The HMT sets
+(𝛽1, 𝛽2) to the 𝑘𝑡ℎ element in B, and collects
+� 𝑁
+5
+� samples
+of the received signal squared, given by (3). Then 𝜇(𝛽1, 𝛽2),
+for (𝛽1, 𝛽2) being the 𝑘𝑡ℎ elements in B, is estimated as
+ˆ𝜇(𝛽1, 𝛽2) =
+1
+⌊𝑁/5⌋
+⌊𝑁 /5⌋
+∑︁
+𝑖=1
+𝑟𝑖(𝛽1, 𝛽2),
+(23)
+where 𝑟𝑖(𝛽1, 𝛽2) is the 𝑖𝑡ℎ sample of 𝑟(𝛽1, 𝛽2) in (3).
+Next, we replace 𝜇(𝛽1, 𝛽2) in (15), (17), (20), and (21)
+by ˆ𝜇(𝛽1, 𝛽2), ∀(𝛽1, 𝛽2) ∈ B, and thereby obtain our estimates
+for 𝛽∗
+1 and 𝛽∗
+2, denoted by ˆ𝛽∗
+1 and ˆ𝛽∗
+2. The pseudo-code of
+the proposed algorithm is given in Two-Stage Phase-Shifts
+Estimation Algorithm below. We note that the choice of the
+Two-Stage Phase-Shifts Estimation Algorithm
+1: Input: 𝑁,B,𝜎2.
+2: ***Stage 1: Uniform Exploration ***
+3: for 𝑘 = 1 to 5 do
+4:
+HMT sets (𝛽1, 𝛽2) to the 𝑘𝑡ℎ pair in B.
+5:
+User sends ⌊𝑁/5⌋ number of pilots to the HMT.
+6:
+For the 𝑖𝑡ℎ pilot, the HMT receives 𝑟𝑖(𝛽1, 𝛽2), given by
+(3), for 𝑖 = 1,2,..., ⌊𝑁/5⌋ .
+7:
+The HMT computes ˆ𝜇𝑘 (𝛽1, 𝛽2) using (23).
+8: end for
+9: ***Stage
+2:
+Estimate
+Optimal
+Phase-Shifting
+Parameters***
+10: Obtain ˆ𝛽∗
+1 as
+ˆ𝛽∗
+1 =
+�
+ˆ𝛼(𝑖)
+1 + ˆ𝛼( 𝑗)
+1
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+��� ˆ𝛼(𝑖)
+1 − ˆ𝛼( 𝑗)
+1
+���
+�
+,
+(24)
+where ˆ𝛼(1)/(2)
+1
+is obtained by replacing the value of
+𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (15), and ˆ𝛼(3)/(4)
+1
+is obtained
+by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (17).
+11: Obtain ˆ𝛽∗
+2 as
+ˆ𝛽∗
+2 =
+�
+ˆ𝛼(𝑖)
+2 + ˆ𝛼( 𝑗)
+2
+2
+:
+min
+𝑖∈{1,2}, 𝑗 ∈{3,4}
+��� ˆ𝛼(𝑖)
+2 − ˆ𝛼( 𝑗)
+2
+���
+�
+,
+(25)
+where ˆ𝛼(1)/(2)
+2
+is obtained by replacing the value of
+𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (20), and ˆ𝛼(3)/(4)
+2
+is obtained
+by replacing the value of 𝜇(𝛽1, 𝛽2) by ˆ𝜇(𝛽1, 𝛽2) in (21).
+12: Output: ˆ𝛽∗
+1 and ˆ𝛽∗
+2.
+13: Phase-shifts at HMT Set the phase-shift of the (𝑚𝑥,𝑚𝑦)𝑡ℎ
+element at the HMT to
+𝛽𝑚𝑥𝑚𝑦 = −
+mod (𝑘0𝑑𝑟 (𝑚𝑥 ˆ𝛽∗
+1 +𝑚𝑦 ˆ𝛽∗
+2),2𝜋).
+initial (𝛽0
+1, 𝛽0
+2) in the set B was arbitrary. The values of (𝛽0
+1, 𝛽0
+2)
+can effect the estimation error. In general, if the values (𝛽0
+1, 𝛽0
+2)
+are closer to the (𝛼1,𝛼2), the better the estimation will be. A
+good choice for (𝛽0
+1, 𝛽0
+2) is given in [13, Sec. V.C], which leads
+to faster learning of (𝛼1,𝛼2).
+IV. THEORETICAL GUARANTEES FOR THE PROPOSED
+ALGORITHM
+In the section, we bound the probability that the estimates,
+obtained from the proposed Two-Stage Phase-Shifts Estimation
+Algorithm Algorithm, deviate from the true values of (𝛼1,𝛼2)
+by an amount 0 ≤ 𝜖 ≤ 1. In particular, we upper bound the
+following error probability
+P
+��
+ˆ𝛽∗
+1 −𝛼1
+�2
++
+�
+ˆ𝛽∗
+2 −𝛼2
+�2
+≥ 𝜖
+�
+.
+(26)
+
+6
+We use the following results to upper bound the error probability
+in (26).
+Lemma 1. Let {𝑋𝑛} be a sequence of random variables (RVs)
+on a probability space. Let 𝑋 be a RV defined on the same
+probability space. Then, the following holds
+P{|𝑋𝑛 − 𝑋𝑚| ≥ 𝜖} ≤ P
+�
+|𝑋𝑛 − 𝑋| ≥ 𝜖
+2
+�
++P
+�
+|𝑋𝑚 − 𝑋| ≥ 𝜖
+2
+�
+.
+Proof. The proof is given in the Appendix A.
+■
+Let 𝜒2
+𝑝(𝜆) denote a non-central Chi-squared distribution with
+𝑝 degrees of freedom and non-centrality parameter 𝜆.
+Lemma 2. Let 𝑋 =
+2
+𝜎2 𝑟(𝛽1, 𝛽2), where 𝑟(𝛽1, 𝛽2) is given by
+(3), and let 𝜆1 =
+2
+𝜎2
+���
+√
+𝑃𝐻(𝛽1, 𝛽2)
+���
+2
+. Then, 𝑋 is distributed as
+𝜒2
+2(𝜆1), i.e., 𝑋 ∼ 𝜒2
+2(𝜆1). Furthermore, if 𝑋𝑖 for 𝑖 = 1,2,...,𝑛
+are 𝑛 independently and identically distributed (i.i.d.) RVs of
+𝜒2
+2(𝜆1), then
+𝑛
+∑︁
+𝑖=1
+𝑋𝑖 ∼ 𝜒2
+2𝑛(𝑛𝜆1).
+Proof. The proof is given in the Appendix B.
+■
+The following theorem provides an upper bound on the error
+probability in (26).
+Theorem 2. Let us perform uniform exploration on the set B
+given in (7). For any 0 ≤ 𝜖 ≤ 1, the error probability in (26)
+is upper bounded as
+P
+��
+ˆ𝛽∗
+1 −𝛼1
+�2
++
+�
+ˆ𝛽∗
+2 −𝛼2
+�2
+≥ 𝜖
+�
+≤ 4
+�
+𝑒− 𝑛
+32
+� 𝜖 𝜆2
+1+𝜆2
+�2
++ 𝑒− 𝑛
+32
+� 𝜖 𝜆3
+1+𝜆3
+�2
++ 𝑒− 𝑛
+32
+� 𝜖 𝜆4
+1+𝜆4
+�2
++ 𝑒− 𝑛
+32
+� 𝜖 𝜆5
+1+𝜆5
+�2�
+,
+(27)
+where
+𝜆1 =
+2
+���
+√
+𝑃𝐻(𝛽0
+1, 𝛽0
+2)
+���
+2
+𝜎2
+,
+𝜆2 =
+2
+���
+√
+𝑃𝐻(𝛽0
+1 +𝑣, 𝛽0
+2)
+���
+2
+𝜎2
+,
+𝜆3 =
+2
+���
+√
+𝑃𝐻(𝛽0
+1 −𝑣, 𝛽0
+2)
+���
+2
+𝜎2
+,
+𝜆4 =
+2
+���
+√
+𝑃𝐻(𝛽0
+1, 𝛽0
+2 + 𝑤)
+���
+2
+𝜎2
+,
+𝜆5 =
+2
+���
+√
+𝑃𝐻(𝛽0
+1, 𝛽0
+2 − 𝑤)
+���
+2
+𝜎2
+.
+Proof. Let us denote the estimate of 𝜇(𝛽0
+1, 𝛽0
+2) by ˆ𝜇(𝛽0
+1, 𝛽0
+2)
+which is given by
+ˆ𝜇(𝛽0
+1, 𝛽0
+2) = 1
+𝑛
+𝑛
+∑︁
+𝑖=1
+𝑟𝑖(𝛽0
+1, 𝛽0
+2) = 𝜎2
+2𝑛
+𝑛
+∑︁
+𝑖=1
+𝑋𝑖.
+Using Lemma 2, we have
+ˆ𝜇1 := 2𝑛
+𝜎2 ˆ𝜇(𝛽0
+1, 𝛽0
+2) ∼ 𝜒2
+2𝑛(𝑛𝜆1)
+(28)
+ˆ𝜇2 := 2𝑛
+𝜎2 ˆ𝜇(𝛽0
+1 +𝑣, 𝛽0
+2) ∼ 𝜒2
+2𝑛(𝑛𝜆2)
+(29)
+ˆ𝜇3 := 2𝑛
+𝜎2 ˆ𝜇(𝛽0
+1 −𝑣, 𝛽0
+2) ∼ 𝜒2
+2𝑛(𝑛𝜆3)
+(30)
+ˆ𝜇4 := 2𝑛
+𝜎2 ˆ𝜇(𝛽0
+1, 𝛽0
+2 + 𝑤) ∼ 𝜒2
+2𝑛(𝑛𝜆4)
+(31)
+ˆ𝜇5 := 2𝑛
+𝜎2 ˆ𝜇(𝛽0
+1, 𝛽0
+2 − 𝑤) ∼ 𝜒2
+2𝑛(𝑛𝜆5)
+(32)
+where 𝜆1,𝜆2,𝜆3,𝜆4 and 𝜆5 is given in Theorem 2.
+The random variables ˆ𝜇1, ˆ𝜇2, ˆ𝜇3, ˆ𝜇4, and ˆ𝜇5 are mutually
+independent, since they are sampled at different epochs. The
+estimated optimal phase-shifting parameters ( ˆ𝛽∗
+1, ˆ𝛽∗
+2), are given
+by (24) and (25), where the values of ˆ𝛼(1)
+1 , ˆ𝛼(2)
+1 , ˆ𝛼(3)
+1 , ˆ𝛼(4)
+1 ,
+and, ˆ𝛼(1)
+2 , ˆ𝛼(2)
+2 , ˆ𝛼(3)
+2 , and ˆ𝛼(4)
+2
+are given by
+ˆ𝛼(1)/(2)
+1
+= 𝛽0
+1 +
+𝑣
+1±
+√︄����
+ˆ𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+ˆ𝜇(𝛽0
+1+𝑣,𝛽0
+2)−𝜎2
+����
+(33)
+ˆ𝛼(3)/(4)
+1
+= 𝛽0
+1 −
+𝑣
+1±
+√︄����
+ˆ𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+ˆ𝜇(𝛽0
+1−𝑣,𝛽0
+2)−𝜎2
+����
+(34)
+ˆ𝛼(1)/(2)
+2
+= 𝛽0
+2 +
+𝑤
+1±
+√︄����
+ˆ𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+ˆ𝜇(𝛽0
+1,𝛽0
+2+𝑤)−𝜎2
+����
+(35)
+ˆ𝛼(3)/(4)
+2
+= 𝛽0
+2 −
+𝑤
+1±
+√︄����
+ˆ𝜇(𝛽0
+1,𝛽0
+2)−𝜎2
+ˆ𝜇(𝛽0
+1,𝛽0
+2−𝑤)−𝜎2
+����
+.
+(36)
+By inserting (28), (29), (30), (31), and (32) into (33), (34),
+(35) and (36), we obtain
+ˆ𝛼(1)/(2)
+1
+= 𝛽0
+1 +
+𝑣
+1±
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+(37)
+ˆ𝛼(3)/(4)
+1
+= 𝛽0
+1 −
+𝑣
+1±
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇3−2𝑛
+���
+(38)
+ˆ𝛼(1)/(2)
+2
+= 𝛽0
+2 +
+𝑤
+1±
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇4−2𝑛
+���
+(39)
+ˆ𝛼(3)/(4)
+2
+= 𝛽0
+2 −
+𝑤
+1±
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇5−2𝑛
+���
+.
+(40)
+Let us denote
+𝐼 := P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥
+√︂
+𝜖
+2
+�
+𝐼𝐼 := P
+��� ˆ𝛽∗
+2 −𝛼2
+�� ≥
+√︂
+𝜖
+2
+�
+.
+Now, applying Lemma 1 in (26), we obtain
+P
+��
+ˆ𝛽∗
+1 −𝛼1
+�2
++
+�
+ˆ𝛽∗
+2 −𝛼2
+�2
+≥ 𝜖
+�
+≤ P
+��
+ˆ𝛽∗
+1 −𝛼1
+�2
+≥ 𝜖
+2
+�
++P
+��
+ˆ𝛽∗
+2 −𝛼2
+�2
+≥ 𝜖
+2
+�
+≤ 𝐼 + 𝐼𝐼.
+(41)
+We upper bound each of the term in right-hand side of (41).
+We begin with the first term P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥ √︁ 𝜖
+2
+�
+, denoted as I.
+G Step 1: Upper bound on I
+From (24), we have
+
+7
+P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥
+√︂
+𝜖
+2
+�
+= P
+� ������
+ˆ𝛼(1)
+1
++ ˆ𝛼(3)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
+� ������
+ˆ𝛼(1)
+1
++ ˆ𝛼(4)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
+� ������
+ˆ𝛼(2)
+1
++ ˆ𝛼(3)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
+� ������
+ˆ𝛼(2)
+1
++ ˆ𝛼(4)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+� �
+≤ P
+������
+ˆ𝛼(1)
+1
++ ˆ𝛼(3)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
++P
+������
+ˆ𝛼(1)
+1
++ ˆ𝛼(4)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
++P
+������
+ˆ𝛼(2)
+1
++ ˆ𝛼(3)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
++P
+������
+ˆ𝛼(2)
+1
++ ˆ𝛼(4)
+1
+2
+−𝛼1
+����� ≥
+√︂
+𝜖
+2
+�
+=
+∑︁
+𝑖=1,2
+𝑗=3,4
+P
+�����
+�
+ˆ𝛼(𝑖)
+1 −𝛼1
+�
++
+�
+ˆ𝛼( 𝑗)
+1
+−𝛼1
+����� ≥ 2
+√︂
+𝜖
+2
+�
+≤
+∑︁
+𝑖=1,2
+𝑗=3,4
+�
+P
+���� ˆ𝛼(𝑖)
+1 −𝛼1
+��� ≥
+√︂
+𝜖
+2
+�
++P
+���� ˆ𝛼( 𝑗)
+1
+−𝛼1
+��� ≥
+√︂
+𝜖
+2
+� �
+= 2
+�
+P
+���� ˆ𝛼(1)
+1
+−𝛼1
+��� ≥
+√︂
+𝜖
+2
+�
++P
+���� ˆ𝛼(2)
+1
+−𝛼1
+��� ≥
+√︂
+𝜖
+2
+�
++P
+���� ˆ𝛼(3)
+1
+−𝛼1
+��� ≥
+√︂
+𝜖
+2
+�
++P
+���� ˆ𝛼(4)
+1
+−𝛼1
+��� ≥
+√︂
+𝜖
+2
+� �
+,
+(42)
+where we applied the union bound to get the first inequality
+and applied Lemma 1 for the second inequality. We now
+bound each term in (42) separately.
+® Upper bound of P
+���� ˆ𝜶(1)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+: Substituting
+the
+values
+of
+ˆ𝛼(1)
+1 ,
+as
+given
+by
+(37),
+in
+P
+���� ˆ𝜶(1)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+, we obtain
+P
+���� ˆ𝛼(1)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+= P
+������
+������
+���������
+𝑣
+1+
+√︂���� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+− (𝛼1 − 𝛽0
+1)
+���������
+≥
+√︂
+𝜖
+2
+������
+������
+.
+(43)
+Note that the following holds.
+���������
+𝑣
+1+
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+− (𝛼1 − 𝛽0
+1)
+���������
+≤
+���������
+𝑣
+1+
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+���������
++
+�����𝛼1 − 𝛽0
+1
+�����.
+(44)
+By applying (44) in (43), we obtain
+P
+���� ˆ𝛼(1)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+������
+������
+���������
+1
+1+
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+���������
+≥ 1
+𝑣
+�√︂
+𝜖
+2 −
+�����𝛼1 − 𝛽0
+1
+�����
+�������
+������
+.
+(45)
+For the RVs ˆ𝜇1 and ˆ𝜇2,
+1
+1+
+√︂���
+ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+is always positive.
+Using this fact in (45), we obtain
+P
+���� ˆ𝛼(1)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+������
+������
+1
+1+
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+≥ 1
+𝑣
+�√︂
+𝜖
+2 −
+�����𝛼1 − 𝛽0
+1
+�����
+�������
+������
+.
+Let 𝑎 = 1
+𝑣
+�√︁ 𝜖
+2 −
+��𝛼1 − 𝛽0
+1
+��
+�
+. We have
+P
+���� ˆ𝛼(1)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+��
+��
+1+
+√︄����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+���� ≤ 1
+𝑎
+��
+��
+= P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+���� ≤
+�
+1− 1
+𝑎
+�2�
+.
+(46)
+® Upper bound of P
+���� ˆ𝜶(2)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+: Substituting
+the
+values
+of
+ˆ𝛼(2)
+1 ,
+as
+given
+by
+(37),
+in
+P
+���� ˆ𝜶(2)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+, we obtain
+P
+���� ˆ𝛼(2)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+= P
+������
+������
+���������
+𝑣
+1−
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+− (𝛼1 − 𝛽0
+1)
+���������
+≥
+√︂
+𝜖
+2
+������
+������
+.
+(47)
+Note that the following holds.
+���������
+𝑣
+1−
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+− (𝛼1 − 𝛽0
+1)
+���������
+≤
+���������
+𝑣
+1−
+√︂��� ˆ𝜇1−2𝑛
+ˆ𝜇2−2𝑛
+���
+���������
++
+�����𝛼1 − 𝛽0
+1
+�����.
+(48)
+By applying (48) in the right-hand side of (47), we
+
+8
+obtain
+P
+���� ˆ𝛼(2)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+���
+���
+������
+1−
+√︄����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+����
+������
+≤ 1
+𝑎
+���
+���
+= P
+��
+1− 1
+𝑎
+�2
+≤
+����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+���� ≤
+�
+1+ 1
+𝑎
+�2�
+≤ P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+���� ≤
+�
+1+ 1
+𝑎
+�2�
+−P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇2 −2𝑛
+���� ≤
+�
+1− 1
+𝑎
+�2�
+.
+(49)
+® Upper bound of P
+���� ˆ𝜶(3)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+: Substituting
+the
+values
+of
+ˆ𝛼(3)
+1 ,
+as
+given
+by
+(38),
+in
+P
+���� ˆ𝜶(3)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+and
+following
+similar
+steps
+to bound P
+���� ˆ𝜶(1)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+, we obtain
+P
+���� ˆ𝛼(3)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇3 −2𝑛
+���� ≤
+�
+1− 1
+𝑎
+�2�
+.
+(50)
+® Upper bound of P
+���� ˆ𝜶(4)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+: Substituting
+the
+values
+of
+ˆ𝛼(4)
+1 ,
+as
+given
+by
+(38),
+in
+P
+���� ˆ𝜶(4)
+1
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+and
+following
+similar
+steps
+to bound P
+���� ˆ𝜶(2)
+2
+−𝜶1
+��� ≥
+√︁ 𝝐
+2
+�
+, we obtain
+P
+���� ˆ𝛼(4)
+1
+−𝛼1)
+��� ≥
+√︂
+𝜖
+2
+�
+≤ P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇3 −2𝑛
+���� ≤
+�
+1+ 1
+𝑎
+�2�
+−P
+�����
+ˆ𝜇1 −2𝑛
+ˆ𝜇3 −2𝑛
+���� ≤
+�
+1− 1
+𝑎
+�2�
+.
+(51)
+By inserting the bounds (46), (49), (50) and (51) to (42)
+we obtain
+P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥
+√︂
+𝜖
+2
+�
+≤ 2
+�
+P
+�����
+ˆ𝜇2 −2𝑛
+ˆ𝜇1 −2𝑛
+���� ≥ 𝛾1
+�
++P
+�����
+ˆ𝜇3 −2𝑛
+ˆ𝜇1 −2𝑛
+���� ≥ 𝛾1
+��
+,
+(52)
+where we set 𝛾1 =
+�
+1
+1+(1/𝑎)
+�2
+.
+We next upper bound each term on the right-hand side of
+(52). The bounds are derived using the properties of the
+sub-exponential distributions which we introduce below.
+G Step 2: Sub-exponential Distributions and its Tail
+Bound
+Definition IV.1 (sub-exponential distribution). A RV 𝑋
+with mean 𝜇 is said to be sub-exponential with parameters
+(𝜈,𝛼), for 𝛼 > 0, if
+E
+�
+exp
+�
+𝑡(𝑋 − 𝜇)
+��
+≤ exp
+�𝑡2𝜈2
+2
+�
+, for |𝑡| < 1
+𝛼 .
+Theorem
+3
+([18]). Let
+𝑋𝑘
+for
+𝑘 = 1,2,...,𝑛 be
+independent RVs where 𝑋𝑘 is sub-exponential with
+parameters
+(𝜈𝑘,𝑏𝑘), and mean
+𝜇𝑘 = E [𝑋𝑘]. Then
+𝑛�
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘) is a sub-exponential RV with parameters
+(𝜈∗,𝑏∗) where
+𝑏∗ =
+max
+𝑘=1,2...,𝑛𝑏𝑘,
+and
+𝜈∗ =
+�
+� 𝑛
+∑︁
+𝑘=1
+𝜈2
+𝑘.
+Furthermore, its tail probability can be bounded as
+P
+������
+1
+𝑛
+𝑛
+∑︁
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘)
+����� ≥ 𝑡
+�
+≤
+���
+���
+2𝑒
+−
+𝑛𝑡2
+2(𝜈2∗ /𝑛) ,
+for 0 ≤ 𝑡 ≤
+𝜈2
+∗
+𝑛𝑏∗
+2𝑒− 𝑛𝑡
+2𝑏∗ ,
+for 𝑡 ≥
+𝜈2
+∗
+𝑛𝑏∗ .
+Proof. The proof is given in Appendix C.
+■
+Corollary
+1.
+Let
+𝑋𝑘
+for
+𝑘 = 1,2...,𝑛
+be
+i.i.d.
+sub-exponential RVs with parameters (2(2+2𝑎),4) each
+with mean 2+ 𝑎. Then,
+P
+������
+1
+𝑛
+𝑛
+∑︁
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘)
+����� ≥ 𝑡
+�
+≤ 2𝑒
+−
+𝑛𝑡2
+8(2+2𝑎)2 ,
+for 𝑡 > 0.
+Proof. The proof is given in then Appendix D.
+■
+We use Corollary 1 to upper bound of the right-hand
+side terms in (52). The following lemma establishes
+the connection between the non-central chi-squared
+distribution and the sub-exponential distributions.
+Lemma 3. Let 𝑋 ∼ 𝜒2
+𝑝(𝑎). Then, 𝑋 is sub-exponential
+with parameters �2(𝑝 +2𝑎),4�.
+Proof. The proof is given in then Appendix E.
+■
+G Step 3: Upper Bounding Eq. (52)
+– Recall that ˆ𝜇1 ∼ 𝜒2
+2𝑛(𝑛𝜆1) and ˆ𝜇2 ∼ 𝜒2
+2𝑛(𝑛𝜆2). Let
+𝑓 ˆ𝜇1 denote the pdf of ˆ𝜇1. We upper bound the term
+P
+���� ˆ𝜇2−2𝑛
+ˆ𝜇1−2𝑛
+��� ≥ 𝛾1
+�
+as follows
+P
+�����
+ˆ𝜇2 −2𝑛
+ˆ𝜇1 −2𝑛
+���� ≥ 𝛾1
+�
+=
+∞
+∫
+0
+P
+����� ˆ𝜇2 −2𝑛
+���� ≥ 𝛾1
+����𝑢 −2𝑛
+����
+�
+𝑓 ˆ𝜇1(𝑢)𝑑𝑢
+=
+∞
+∫
+0
+P
+����� ˆ𝜇2 −2𝑛 −𝑛𝜆2 +𝑛𝜆2
+���� ≥ 𝛾1
+����𝑢 −2𝑛
+����
+�
+𝑓 ˆ𝜇1(𝑢)𝑑𝑢
+≤
+∞
+∫
+0
+P
+�1
+𝑛
+���� ˆ𝜇2 −𝑛(2+𝜆2)
+���� ≥ 𝛾1|𝑢 −2𝑛| −𝑛𝜆2
+𝑛
+�
+𝑓 ˆ𝜇1(𝑢)𝑑𝑢
+(53)
+Note that, if 𝛾1 |𝑢−2𝑛|−𝑛𝜆2
+𝑛
+< 0, then P
+���� ˆ𝜇2−2𝑛
+ˆ𝜇1−2𝑛
+��� ≥ 𝛾1
+�
+≤
+1 as P
+�
+1
+𝑛
+���� ˆ𝜇2 −𝑛(2+𝜆2)
+���� ≥ 𝛾1 |𝑢−2𝑛|−𝑛𝜆2
+𝑛
+�
+= 1, which is
+trivial.
+
+9
+For 𝛾1 |𝑢−2𝑛|−𝑛𝜆2
+𝑛
+≥ 0, using the assumption 0 ≤ 𝜖 ≤ 1 in
+(53), we have
+P
+�����
+ˆ𝜇2 −2𝑛
+ˆ𝜇1 −2𝑛
+���� ≥ 𝛾1
+�
+≤
+∞
+∫
+0
+P
+�1
+𝑛
+���� ˆ𝜇2 −𝑛(2+𝜆2)
+���� ≥ 𝜖
+� 𝛾1|𝑢 −2𝑛| −𝑛𝜆2
+𝑛
+� �
+× 𝑓 ˆ𝜇1(𝑢)𝑑𝑢.
+(54)
+The last inequality follows from Lemma 1. Let
+𝑡1 := 𝑡1(𝑢) = 𝜖
+�
+𝛾1 |𝑢−2𝑛|−𝑛𝜆2
+𝑛
+�
+. As E [ ˆ𝜇2] = 2𝑛 +𝑛𝜆2, by
+applying Corollary 1, we obtain
+P
+�1
+𝑛
+���� ˆ𝜇2 −𝑛(2+𝜆2)
+���� ≥ 𝑡1
+�
+≤ 2𝑒
+−
+𝑛𝑡2
+1
+8(2+2𝜆2)2 ,
+𝑡1 ≥ 0.
+(55)
+By applying (55) to (54), we obtain
+P
+����� ˆ𝜇2 −2𝑛
+���� ≥ 𝛾1
+���� ˆ𝜇1 −2𝑛
+����
+�
+≤
+∞
+∫
+0
+2𝑒
+−
+𝑛𝑡2
+1
+8(2+2𝜆2)2 𝑓 ˆ𝜇1(𝑢)𝑑𝑢,
+=
+2𝑛
+∫
+0
+2𝑒
+−
+𝑛
+�
+𝜖𝑛
+�
+𝛾1 (2𝑛−𝑢)−𝑛𝜆2
+��2
+8(2+2𝜆2)2
+𝑓 ˆ𝜇1(𝑢)𝑑𝑢
++
+∞
+∫
+2𝑛
+2𝑒
+−
+𝑛
+�
+𝜖𝑛
+�
+𝛾1 (𝑢−2𝑛)−𝑛𝜆2
+��2
+8(2+2𝜆2)2
+𝑓 ˆ𝜇1 (𝑢)𝑑𝑢.
+(56)
+For 0 ≤ 𝑢 ≤ 2𝑛, we have
+2𝑒
+−
+𝑛
+�
+𝜖𝑛
+�
+𝛾1 (2𝑛−𝑢)−𝑛𝜆2
+��2
+8(2+2𝜆2)2
+≤ 2𝑒
+−
+𝑛�
+𝜖 𝜆2
+�2
+8(2+2𝜆2)2 .
+(57)
+For 2𝑛 ≤ 𝑢 ≤ ∞, we have
+2𝑒
+−
+𝑛
+�
+𝜖𝑛
+�
+𝛾1 (𝑢−2𝑛)−𝑛𝜆2
+��2
+8(2+2𝜆2)2
+≤ 2𝑒
+−
+𝑛�
+𝜖 𝜆2
+�2
+8(2+2𝜆2)2 .
+(58)
+Using (57) and (58) in (56), we obtain
+P
+����� ˆ𝜇2 −2𝑛
+���� ≥ 𝛾1
+���� ˆ𝜇1 −2𝑛
+����
+�
+≤ 2𝑒
+−
+𝑛�
+𝜖 𝜆2
+�2
+8(2+2𝜆2)2 P{0 < ˆ𝜇1 < 2𝑛}
++2𝑒
+−
+𝑛�
+𝜖 𝜆2
+�2
+8(2+2𝜆2)2 P{ ˆ𝜇1 > 2𝑛}
+P
+����� ˆ𝜇2 −2𝑛
+���� ≥ 𝛾1
+���� ˆ𝜇1 −2𝑛
+����
+�
+≤ 2𝑒
+−
+𝑛�
+𝜖 𝜆2
+�2
+8(2+2𝜆2)2 .
+(59)
+– We next upper bound P
+���� ˆ𝜇3−2𝑛
+ˆ𝜇1−2𝑛
+��� ≥ 𝛾1
+�
+. Set 𝑡2 =
+𝜖
+�
+𝛾1 |𝑢−2𝑛|−𝑛𝜆3
+𝑛
+�
+. Recall that ˆ𝜇3 ∼ 𝜒2
+2𝑛(𝑛𝜆3). Following
+the steps similar to the derivation of the bound in (59),
+we obtain
+P
+����� ˆ𝜇3 −2𝑛
+���� ≥ 𝛾1
+���� ˆ𝜇1 −2𝑛
+����
+�
+≤ 2𝑒
+−
+𝑛�
+𝜖 𝜆3
+�2
+8(2+2𝜆3)2 .
+(60)
+Combining (59) and (60) we obtain the following upper
+bound on (52)
+P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥
+√︂
+𝜖
+2
+�
+≤ 4
+�
+𝑒− 𝑛
+32
+� 𝜖 𝜆2
+1+𝜆2
+�2
++ 𝑒− 𝑛
+32
+� 𝜖 𝜆3
+1+𝜆3
+�2�
+. (61)
+G Step 4: Upper bound on II
+By following the same steps for deriving the upper bound
+of P
+��� ˆ𝛽∗
+1 −𝛼1
+�� ≥ √︁ 𝜖
+2
+�
+, we can obtain the following bound
+P
+��� ˆ𝛽∗
+2 −𝛼2
+�� ≥
+√︂
+𝜖
+2
+�
+≤ 4
+�
+𝑒− 𝑛
+32
+� 𝜖 𝜆4
+1+𝜆4
+�2
++ 𝑒− 𝑛
+32
+� 𝜖 𝜆5
+1+𝜆5
+�2�
+. (62)
+Combining (61) and (62), we obtain the required upper
+bound in (27).
+■
+V. NUMERICAL SIMULATIONS
+We estimate the initial value of (𝛽0
+1, 𝛽0
+2) as given in [13, Sec.
+V.C]. Based on the the initial value of (𝛽0
+1, 𝛽0
+2) we set B as
+given in (7), where 𝑣 and 𝑤 are selected such that 𝐾𝑥𝑣 ∈ N and
+𝐾𝑦𝑤 ∈ N respectively. In addition to the LoS path, we assume
+that there are 4 NLoS path components due to scatters between
+the user and the HMT. The elevation and azimuth angles of
+each NLoS path from these scatters to the center of HMT follow
+the uniform distribution, i.e., 𝑈(0,2𝜋). Moreover, we consider
+the path coefficient of each NLoS path as a complex Gaussian
+distribution, i.e., 𝐶𝑁(0,𝜎2
+𝑠 ), where 𝜎2
+𝑠 is 20 dB weaker than
+the power of the LoS component [19]. The system parameters
+for numerical simulations are listed in Table I.
+TABLE I: A list of system parameters for numerical simulations
+Parameters
+Values
+Description
+𝑓𝑐
+30 GHz
+Carrier frequency
+𝜆
+1 cm
+Wavelength
+𝐿𝑥
+1 m
+Width of the HMT
+𝐿𝑦
+1 m
+Length of the HMT
+𝑑𝑟
+𝜆/4
+Unit element spacing
+𝐿𝑒
+𝑑𝑟
+Width
+and
+length
+of
+each
+phase-shifting element
+𝑃
+20 dBm
+Transmission power of the HMT
+during data transmission
+𝜎2
+-115 dBm
+Noise power for 200 KHz 2
+A. Comparison
+Between
+the
+Proposed
+Algorithm
+and
+Benchmark Scheme
+According to the approximated channel model, where the
+phase-shift parameters at the HMT are given by 𝛽1 and 𝛽2,
+the achieved data rate at the user of the HMT-assisted wireless
+communication system is given by
+𝑅(𝛽1, 𝛽2) = 𝑙𝑜𝑔2
+�
+1+ 𝑃|𝐻(𝛽1, 𝛽2)|2
+𝜎2
+�
+,
+(63)
+
+10
+where 𝑃 is the transmission power at the HMT. The HMT
+uses the acquired CSI during the channel estimation period to
+maximize the received data rate by the user. Hence, we consider
+the achieved data rate by the user, using the acquired CSI as
+a performance metric. We applied the proposed algorithm in
+two different cases when the distance between the user and
+the center of the HMT (𝑑0 = 200 m and when 𝑑0 = 10 m. We
+compared our proposed algorithm with two benchmarks, the
+proposed algorithm in [13] and the oracle scheme where 𝛼1
+and 𝛼2 are estimated perfectly and thereby the maximum rate
+is achieved.
+In Fig. 4, we compared the achievable rates, given by (63),
+of the proposed scheme and the benchmark schemes. We
+considered both 𝑑0 = 200 m and 𝑑0 = 10 m regions of the
+HMT with respect to the transmit power of the pilot signals
+when the number of the pilot signals is fixed to 23. For all
+the algorithms we use the same number of pilots, i.e. 23,
+for both the cases. Our proposed algorithm uses four pilots
+in each epoch and there are five epochs, which makes the
+total number of pilots equals to 20. We require additional three
+number of pilots to estimate (𝛽0
+1, 𝛽0
+2). We run the simulation for
+1000 times. We see that in both cases, the proposed Two-Stage
+Phase-Shifts Estimation Algorithm gives higher rates than other
+two benchmark schemes.
+20
+10
+0
+10
+20
+30
+40
+Power of pilot signals (in dBm)
+15
+20
+25
+30
+35
+40
+45
+50
+Achievable Rate (in bits/symbol)
+Maximum Rate d0=200
+Proposed Algorithm d0=200
+Algorithm from [13] d0=200
+Maximum Rate d0=10
+Proposed Algorithm d0=10
+Algorithm from [13] d0=10
+Fig. 4: Achievable rate vs. the transmit power of the pilot signals
+(in dBm).
+B. Convergence of The Proposed Algorithm
+We now numerically evaluate the convergence of the upper
+bound of the proposed algorithm, given by (27). We also
+compare the actual probability, given by (26), that we obtain
+by simulations.
+In Fig. 5, we show the convergence property of the error
+probability and its upper bound of the proposed algorithm
+for increasing values of 𝜖 = {0.01,0.05,0.1} when the power
+of the pilot signal is 𝑃 = 10 dBm and 𝑑0 = 200 m. We run
+the simulation for 1000 times. We see that for each value
+of 𝜖, the proposed algorithm converges towards zero as we
+increase the number of pilots. Moreover, the upper bound of
+the error probability also converges to the error probability as
+the number of pilot signals increases.
+2This setting corresponds to the noise power spectrum density at the HMT
+is −174 dBm/Hz and signal bandwidth is 200 KHz, assuming the noise figure
+of each user to be 6 dB [8].
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of Pilots
+10
+1
+100
+Error Probability
+Proposed Scheme,  = 0.01
+Upper Bound,  = 0.01
+Proposed Scheme,  = 0.05
+Upper Bound,  = 0.05
+Proposed Scheme,  = 0.1
+Upper Bound,  = 0.1
+Fig. 5: Error Probability Bound v/s Number of Pilots for 𝜖 =
+{0.01,0.05,0.1} for 𝑃 = 10 dBm.
+In Fig. 6, we compare the convergence property of the error
+probability of the proposed algorithm with respect to 𝜖 = 0.05
+and 𝑑0 = 200 m for different levels of power of the pilot signals,
+𝑃 = {5,10,20} dBm. As we increase the power of the pilot
+signals, the estimation accuracy of 𝛼1 and 𝛼2 increases and
+hence the error probability decreases. This is so because, as
+we increase the power of pilot signals the received signals will
+be less noisy which increases the chances of estimating the 𝛼1
+and 𝛼2 more accurately.
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of Pilots
+10
+2
+10
+1
+100
+Error Probability
+Proposed Scheme, Pilot Power = 5 dBm
+Upper Bound,  Pilot Power = 5 dBm
+Proposed Scheme, Pilot Power = 10 dBm
+Upper Bound,  Pilot Power = 10 dBm
+Proposed Scheme,  Pilot Power = 20 dBm
+Upper Bound,  Pilot Power = 20 dBm
+Fig. 6: Error Probability Bound v/s Number of Pilots for 𝜖 = 0.05
+for 𝑃 = {5,10,20} dBm.
+VI. CONCLUSION
+We investigated the problem of estimation of the optimal
+phase-shift at the HMT-assisted wireless communication system
+in a noisy environment. We proposed a learning algorithm to
+estimate the optimal phase-shifting parameters and showed that
+the probability that the phase-shifting parameters generated by
+the proposed algorithm to deviate by more than 𝜖 from the
+optimal values decay exponentially fast as the number of pilots
+grows. Our proposed algorithm exploited structural properties
+of the channel gains in the far-field regions.
+
+11
+APPENDIX
+A. Proof of Proposition 1
+Proof. Let us define the following events.
+𝐴𝑛,𝑚 = |𝑋𝑛 − 𝑋𝑚| > 𝜖,
+𝐴𝑛 = |𝑋𝑛 − 𝑋| > 𝜖
+2,
+and
+𝐴𝑚 = |𝑋𝑚 − 𝑋| > 𝜖
+2
+By the triangle inequality, we have
+|𝑋𝑛 − 𝑋𝑚| ≤ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋|.
+(64)
+Using (64), the event 𝐴𝑛,𝑚 can be written as
+|𝑋𝑛 − 𝑋𝑚| ≥ 𝜖 =⇒ |𝑋𝑛 − 𝑋| + |𝑋𝑚 − 𝑋| ≥ 𝜖
+Therefore, we have
+𝐴𝑛,𝑚 ⊂ {|𝑋𝑛 − 𝑋| + |𝑋 − 𝑋𝑚| > 𝜖}
+⊂
+�
+|𝑋𝑛 − 𝑋| > 𝜖
+2
+�
+|𝑋 − 𝑋𝑚| > 𝜖
+2
+�
+(65)
+Note that for any two events 𝐴 and 𝐵 where 𝐴 ⊂ 𝐵, then
+P{𝐴} ≤ P{𝐵}. We use this fact in (65), and we get
+P{|𝑋𝑛 − 𝑋𝑚| > 𝜖} ≤ P
+�
+|𝑋𝑛 − 𝑋| > 𝜖
+2
+�
++P
+�
+|𝑋𝑚 − 𝑋| > 𝜖
+2
+�
+■
+B. Proof of Lemma 2
+Proof. We consider 𝑟(𝛽1, 𝛽2) as given in (3) which comprises
+of two complex-valued factors
+√
+𝑃 × 𝐻(𝛽1, 𝛽2) (see (1)) and 𝜁.
+Write 𝜁 = 𝑛1 + 𝑗𝑛2, where 𝑛1 and 𝑛2 follows 𝑁
+�
+0, 𝜎2
+2
+�
+and
+are independent, and write
+√
+𝑃 × 𝐻(𝛽1, 𝛽2) = 𝑎 + 𝑗𝑏, where 𝑎
+and 𝑏 are real values. Therefore,
+𝑟(𝛽1, 𝛽2) = |𝑦(𝛽1, 𝛽2)|2 = (𝑎 +𝑛1)2 + (𝑏 +𝑛2)2.
+(66)
+Note that 𝑎+𝑛1
+𝜎/
+√
+2 ∼ 𝑁
+�
+𝑎
+𝜎/
+√
+2,1
+�
+and 𝑏+𝑛2
+𝜎/
+√
+2 ∼ 𝑁
+�
+𝑏
+𝜎/
+√
+2,1
+�
+and they
+are independent. Therefore,
+2
+𝜎2
+�
+(𝑎 +𝑛1)2 + (𝑏 +𝑛2)2
+�
+∼ 𝜒2
+2
+� 2
+𝜎2
+�
+𝑎2 + 𝑏2��
+.
+(67)
+Applying (67) in (66), we get 𝑋 =
+2
+𝜎2 𝑟(𝛽1, 𝛽2) ∼ 𝜒2
+2 (𝜆1) ,
+where 𝜆1 =
+2
+𝜎2
+���
+√
+𝑃 × 𝐻(𝛽1, 𝛽2)
+���
+2
+. The second part of the lemma
+follows from the additive property of non-central Chi-squared
+distribution of the sum of 𝑛 i.i.d. RVs of 𝜒2
+2 (𝜆1) .
+■
+C. Proof of Theorem 3
+As 𝑋𝑘,∀𝑘 are independent, applying the definition IV.1, the
+moment generating function of
+𝑛�
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘) is given by
+E
+�
+𝑒
+𝑡
+𝑛�
+𝑘=1
+(𝑋𝑘−𝜇𝑘)
+�
+≤ 𝑒
+𝜆2
+2
+𝑛�
+𝑘=1
+𝜈2
+𝑘,
+∀|𝑡| < ��
+�
+1
+max
+𝑘=1,2,...,𝑛𝑏𝑘
+��
+�
+.
+Since the moment generating functions uniquely determines
+the distribution, comparing with the definition IV.1, it follows
+that
+𝑛�
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘) is a sub-exponential (𝜈∗,𝑏∗) random variable,
+where
+𝑏∗ =
+max
+𝑘=1,2...,𝑛𝑏𝑘
+and
+𝜈∗ =
+�
+� 𝑛
+∑︁
+𝑘=1
+𝜈2
+𝑘.
+To prove the second part of the Theorem we use the following
+tail bound on a sub-exponential distribution proved in [18].
+Proposition 1 ([18] Proposition 2.9). Let 𝑋 is sub-exponential
+random variable with parameters (𝜈,𝑏) and E [𝑋] = 𝜇. Then
+P{|𝑋 − 𝜇| ≥ 𝑡} ≤
+�
+2𝑒− 𝑡2
+2𝜈2 ,
+if 0 ≤ 𝑡 ≤ 𝜈2
+𝑏
+2𝑒− 𝑡
+2𝑏 ,
+if 𝑡 ≥ 𝜈2
+𝑏 .
+The claim immediately follows by applying the above result on
+𝑍𝑛 :=
+𝑛�
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘), which is sub-exponential (𝜈∗,𝑏∗), where
+𝑏∗ =
+max
+𝑘=1,2...,𝑛𝑏𝑘 and 𝜈∗ =
+√︂ 𝑛�
+𝑘=1
+𝜈2
+𝑘.
+D. Proof of Corollary 1
+From Theorem 3, �𝑛
+𝑘=1(𝑋𝑘 −𝜇𝑘) is sub-exponential (𝜈∗,𝑏∗),
+where 𝑏∗ = 4 and 𝜈∗ = 2√𝑛(2 + 2𝑎). Using the parameters
+(𝜈∗,𝑏∗) = (2√𝑛(2+2𝑎),4) in Proposition 1, we get the required
+upper bound as
+P
+������
+1
+𝑛
+𝑛
+∑︁
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘)
+����� ≥ 𝑡
+�
+≤
+��
+��
+2𝑒
+−
+𝑛𝑡2
+8(2+2𝑎)2 ,
+0 ≤ 𝑡 ≤ (2+2𝑎)2
+2𝑒− 𝑛𝑡
+8 ,
+𝑡 ≥ (2+2𝑎)2
+P
+������
+1
+𝑛
+𝑛
+∑︁
+𝑘=1
+(𝑋𝑘 − 𝜇𝑘)
+����� ≥ 𝑡
+�
+≤ 2𝑒
+−
+𝑛𝑡2
+8(2+2𝑎)2 ,
+𝑡 > 0.
+E. Proof of Lemma 3
+If 𝑋 ∼ 𝜒2
+𝑝(𝑎), then according to [20], the moment-generating
+function (MGF) of 𝑋 is given by
+E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] = 𝑒−𝑡 ( 𝑝+𝑎)E
+�
+𝑒𝑡𝑋�
+= 𝑒−𝑡 ( 𝑝+𝑎)𝑒
+𝑎𝑡
+1−2𝑡
+(1−2𝑡) 𝑝/2
+= 𝑒
+2𝑎𝑡2
+1−2𝑡
+𝑒−𝑝𝑡
+(1−2𝑡) 𝑝/2 ,
+for 𝑡 < 1
+2.
+(68)
+By following some calculus, refer [21], [18, Example 2.8], we
+obtain
+𝑒−𝑝𝑡
+(1−2𝑡) 𝑝/2 ≤ 𝑒2𝑝𝑡2,
+for |𝑡| ≤ 1
+4.
+(69)
+For |𝑡| ≤ 1
+4, we have
+𝑒
+2𝑎𝑡2
+1−2𝑡 ≤ 𝑒4𝑎𝑡2.
+(70)
+Applying (69) and (70) to (68), we obtain
+E [exp{𝑡(𝑋 − (𝑝 + 𝑎)}] ≤ 𝑒2( 𝑝+2𝑎)𝑡2,
+∀|𝑡| ≤ 1
+4.
+(71)
+Therefore, by (71), 𝑋 is Sub-exponential distribution with
+parameters �2(𝑝 +2𝑎),4�.
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+

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+Realization of valley-spin polarized current via parametric pump in monolayer MoS2
+Kai-Tong Wang,1, 2 Hui Wang,2 Fuming Xu,1, ∗ Yunjin Yu,1 and Yadong Wei1
+1College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
+2School of Physics and Engineering, Henan University of Science and Technology, Luoyang 471023, China
+Monolayer MoS2 is a typical valleytronic material with valley-spin locked valence bands.
+We
+numerically investigate the valley-spin polarized current in monolayer MoS2 via adiabatic electron
+pumping. By introducing an exchange field to break the energy degeneracy of monolayer MoS2, the
+top of its valence bands is valley-spin polarized and tunable by the exchange field. A device with
+spin-up polarized left lead, spin-down polarized right lead, and untuned central region is constructed
+through applying different exchange fields in the corresponding regions. Then, equal amount of
+pumped currents with opposite valley-spin polarization are simultaneously generated in the left and
+right leads when periodically varying two pumping potentials. Numerical results show that the phase
+difference between the pumping potentials can change the direction and hence polarization of the
+pumped currents. It is found that the pumped current exhibits resonant behavior in the valley-spin
+locked energy window, which depends strongly on the system size and is enhanced to resonant current
+peaks at certain system lengths. More importantly, the pumped current periodically oscillates as
+a function of the system length, which is closely related to the oscillation of transmission. The
+effects of other system parameters, such as the pumping amplitude and the static potential, are also
+thoroughly discussed.
+I.
+INTRODUCTION
+Valleytronics has attracted enormous attention on ac-
+count of its potential for information processing1–16. In
+many crystalline materials, there are two or more min-
+ima(maxima) at the conduction(valence) band in the mo-
+mentum space, known as valleys.
+The degenerate but
+inequivalent valley states constitute new pseudospin de-
+gree of freedom for low energy carriers. Similar to spin-
+tronics, the essential of valleytronics is to generate and
+manipulate valley polarization to encode and store infor-
+mation. Various materials have been explored to real-
+ize valley polarization, including silicon17,18, bismuth19,
+diamond20,21, carbon nanotube22,23, etc. In particular,
+two-dimensional(2D) honeycomb lattice materials such
+as graphene or transition metal dichalcogenides (TMDs)
+provide a perfect platform to investigate valleytronics.
+Compared to graphene, TMDs labeled as MX2 (M =
+Mo, W, X = S, Se, Te), also have two well-separated val-
+leys in the Brillouin zone2,24. However, due to inversion
+symmetry breaking, TMDs are natural gapped semicon-
+ductors, which makes TMDs the promising candidates of
+valleytronic materials25–31.
+As a typical TMDs material, monolayer MoS2 has a
+strong spin-orbit coupling(SOC) interaction26,32, which
+leads to the locking between valley and spin at the top of
+its valance band. The valley-spin locking means that the
+valley and spin can be polarized together, and the lifetime
+of polarization can be enhanced due to the large spacing
+between K and K′ valleys. In the presence of an exchange
+field, TMDs exhibit interesting phenomena, such as the
+quantum anomalous Hall effect33,34, spin and valley Hall
+effects35, and unconventional superconductivity36.
+Be-
+sides, an exchange field can induce polarized valleys,
+which can be inverted by tuning the spin polarization.
+Through the ferromagnetic proximity effect37,38 or mag-
+netic doping39,40, the exchange field can be introduced
+into TMDs materials, which provides an effective way to
+manipulate its valley/spin degree of freedom. In exper-
+iments, the exchange field for valley splitting has been
+realized in Fe-doped41 or Co-doped monolayer MoS2.42
+EuS as a ferromagnetic substrate can efficiently induce
+the magnetic exchange field in monolayer TMDs.43,44
+Based on the valley optical selection rules, the opti-
+cal pumping of valley polarization has been experimen-
+tally realized by circular polarized light in 2D TMDs45–47.
+Very recently, the spin-valley coupled dynamics at the
+MoS2-MoSe2 interface is experimentally studied using
+optical pumping48; photoinduced valley-selective polar-
+ization in monolayer WS2 has been realized with cir-
+cularly polarized light pumping.49 Besides, The line
+defects50, nonmagnetic disorders51, and spatially vary-
+ing potentials16 were predicted to achieve the valley po-
+larization in monolayer MoS2. In terms of applications,
+it is desirable to obtain pure valley polarized current
+by electrical methods.
+Accordingly, we propose that
+quantum parametric pump can drive valley and spin po-
+larized currents in monolayer MoS2 through adiabati-
+cally varying two gate voltages. The parametric pump
+can produce dc current by periodically varying system
+parameters, which has been generalized to various 2D
+materials52–55. Specially, spin pump has been reported
+in several nanostructures56–58, where pure spin current
+and zero charge current are obtained.
+In this paper, we numerically study the generation and
+manipulation of valley-spin polarized currents via adia-
+batic pump in monolayer MoS2.
+The system setup is
+shown in Fig.1. By magnetic doping, an exchange inter-
+action is introduced in the left and right leads, which in-
+duces locked valley-spin polarization at the top of valence
+band as shown in Fig.1(a) and 1(c). When the pumping
+potentials periodically change, fully valley-spin polarized
+dc currents are driven into the leads. At one moment,
+the current with K valley and spin up is pumped into the
+arXiv:2301.11644v1  [cond-mat.mes-hall]  27 Jan 2023
+
+2
+FIG. 1: Schematics of the band structures of monolayer MoS2
+for (a) with exchange field M, (b) without exchange field, (c)
+with exchange field −M. The red and blue valance bands de-
+note valley K with spin up and K′ with spin down. (d) The
+pump setup based on monolayer MoS2 consisting of left/right
+leads and the scattering region, whose band structures are
+correspondingly shown in (a) to (c), respectively. The MoS2
+lattice is represented by the simple honeycomb lattice. The
+pumping potentials V1 and V2 are added in the scattering re-
+gion, adjacent to the leads. As V1 and V2 periodically change,
+electric currents with opposite valley-spin polarizations are si-
+multaneously pumped into the left and right leads, as shown
+by the block arrows.
+left lead while the current with opposite valley-spin po-
+larization flows into the right lead. The polarized current
+exhibits resonant behavior in the valley-spin locked en-
+ergy window, which mainly depends on the system size.
+With the increasing of the system length, the pumped
+currents show periodic oscillation behavior and robust
+resonant current peaks can be observed. We also inves-
+tigate the influence of other system parameters, includ-
+ing the phase difference, the Rashba SOC strength, the
+static potential, and the Fermi energy. It is found that
+the phase difference and static potentials can invert the
+direction and hence polarization of the pumped current.
+The paper is organized as follows. In Sec. II, we in-
+troduce the Hamiltonian of monolayer MoS2 and the for-
+malism of adiabatic parametric pumping. In Sec. III,
+numerical results and relevant discussions are presented.
+Finally, a brief summary is given in Sec. IV.
+II.
+MODEL AND FORMALISM
+In monolayer MoS2, the low-energy spectrum at K
+and K′ valleys consists of three d orbitals of Mo, i.e.,
+dz2, dx2−y2, dxy.
+The relations between these orbitals
+and basis wave functions satisfy: |ϕc⟩ = |dz2⟩, |ϕλ
+υ⟩ =
+(|dx2−y2⟩+iλ|dxy⟩)/
+√
+2, where the subscript c/υ denotes
+the conduction/valence band and λ = ±1 corresponds
+to different valleys K and K′. Based on above low-lying
+states, the effective Hamiltonian of monolayer MoS2 has
+the following form26,59
+H0(k) = at(λkxσx + kyσy) + ∆σz − tSOλσz − 1
+2
+τz, (1)
+where a and t are the lattice constant and hopping
+strength, respectively. σx,y,z and τz represent the Pauli
+matrices of basis functions(|ϕc⟩ and |ϕυ⟩) and spin(↑ and
+↓). ∆ is the mass term and the last term is the intrinsic
+SOC with strength tSO.
+We employ the tight-binding model of MoS2, which
+treats monolayer MoS2 as a simplified honeycomb lattice.
+The lattice includes A and B sublattices, corresponding
+to the dz2 orbit and dx2−y2 + iλdxy orbits of Mo, respec-
+tively. In the tight-binding approximation, the Hamilto-
+nian can be expressed as60,61
+H0 =
+�
+i
+ϵic†
+iαciα + t
+�
+<i,j>
+c†
+iαciα + HSO,
+(2)
+with
+HSO = 2itSO
+3
+√
+3
+�
+≪i,j≫,α,α′
+υijc†
+iατz,αα′cjα′,
+(3)
+where c†
+iα(ciα) is the creation(annihilation) operator at
+site i with spin α = ±1, ϵi is the on-site energy. HSO
+denotes the intrinsic SOC term and the summation over
+the second nearest-neighbor sites only involves B sublat-
+tice. Besides, υij = +1(−1) if an electron moves from
+site j to site i with taking a left(right) turn62.
+Based on this model, we consider a monolayer MoS2
+setup, which contains three parts: the central scattering
+region, the left and right leads, as shown in Fig.1(d).
+By magnetic doping, different valley-spin polarizations
+can be induced in the left and right leads due to the
+exchange field39,63. The schematic band structures with
+or without the exchange field are depicted in Fig.1(a)-(c).
+In the presence of Rashba spin-orbit coupling (RSOC),
+the Hamiltonians for the scattering region with pumping
+potentials and the leads can be written as
+HC = H0+ 3itR
+4
+�
+<i,j>,α,α′
+(ταα′×dij)zc†
+iαcjα′+V (x, y, t),
+(4)
+HL/R = H0 ± M
+�
+i,α,α′
+τz,αα′c†
+iαciα′,
+(5)
+where M and tR denote the strengths of the exchange
+field and RSOC, respectively. ταα′ = (τx, τy, τz) is the
+Pauli matrix for spin, and dij is the lattice vector con-
+necting sites i and j.
+The potential term V (x, y, t) =
+Vs(x, y) + Vt(x, y, t), where Vs = V0
+�
+i Πi(x, y) corre-
+sponds to the static potential defining the shape of the
+pumping region. Vt = Vp
+�
+i Πi(x, y)cos(ωt + ϕi) is the
+periodic pumping potential. V0 and Vp are the ampli-
+tudes of Vs and Vt. i = 1, 2 are the indices of the po-
+tential and Πi represents the potential profile, which is
+
+(a)
+(b)
+(c)
+:
+spin up
+spin down
+K
+K'
+K
+K'
+K
+K
+(d)
+个,K
+*,K'
+Lead-L
+Vi(t)
+Scattering region
+V2(t)
+Lead-R3
+highlighted in green in Fig.1(d). ϕi is the initial phase of
+the pumping potential.
+To evaluate the adiabatic valley-spin pump, we need to
+calculate the average current flowing into lead β. Con-
+sider a slowly varying time-dependent pumping potential
+Vt,i, the average current in one period is expressed as64
+Iβ = qω
+2π
+� T
+0
+dt[ dNβ
+dVt,1
+dVt,1
+dt
++ dNβ
+dVt,2
+dVt,2
+dt ],
+(6)
+where the period of Vt,i is T = 2π/ω with frequency ω
+and β = L/R labels the lead.
+The emissivity
+dNβ
+dVi
+is
+defined in terms of the scattering matrix Sββ′ as65,66
+dNβ
+dVi
+=
+� dE
+2π (−∂Ef)
+�
+β′
+Im∂Sββ′
+∂Vi
+S∗
+ββ′,
+(7)
+with f the Fermi distribution function. Under the adi-
+abatic condition, the pumped current is independent of
+the pumping frequency ω, hence we set ω = 1 in the
+calculation.
+In the language of nonequilibrium Green’s functions,
+the pumped current is expressed as67–69
+Iβ = − q
+2π
+� 2π
+0
+dt
+�
+dE(∂Ef)Tr[ΓβGr dVt
+dt Ga].
+(8)
+Here Gr/Ga is the retarded/advanced Green’s function
+of the central scattering region, which is defined as Gr =
+Ga,† = [E − HC − �
+β Σr
+β]−1. HC is the corresponding
+Hamiltonian. Σr
+β is the retarded self-energy of lead β,
+which can be calculated by surface Green’s function70,71.
+Γβ = i(Σr
+β − Σa
+β) denotes the linewidth function.
+As shown by block arrows in Fig.1(d), at one moment
+of the pumping period, polarized current with K valley
+and spin up (pink arrow) is driven into the left lead, and
+equal amount current with K′ valley and spin down (blue
+arrow) flows in the right lead. Detailed numerical results
+are shown in the following section.
+We use the short
+term, the pumped current, to stand for the valley-spin
+polarized currents in the leads.
+III.
+RESULTS AND DISCUSSION
+In the calculations, the on-site energy is ϵi = ±0.83 eV
+for A and B sublattices. Other parameters26,60 are set
+as t = 1.27 eV, tSO = 0.038 eV, and the lattice constant
+a=0.32 nm. Without loss of generality, we set the ex-
+change field strength M = 0.06 eV, and eV is taken as
+the energy unit throughout the calculation. The periodic
+boundary condition (PBC) is considered for monolayer
+MoS2, and thus the edge effect is removed. To realize
+PBC, the upper and lower edges of a zigzag MoS2 rib-
+bon shown in Fig.1(d) are connected with appropriate
+hopping interactions, which is also called the cylinder
+boundary.
+FIG. 2: The dispersion relation of monolayer MoS2 ribbon
+with an exchange field: (a) M = 0.06, (b) M = −0.06. Both
+valley and spin are polarized together, where ∆E is defined
+as the valley-spin locked energy window.
+A.
+Valley-spin polarized current
+The dispersion of monolayer MoS2 with an exchange
+field is plotted in Fig.2. From Fig.2(a), it is clear that
+the valley K with spin up is polarized at the valence band
+top. However, as the exchange field changes, the polar-
+ization of both valley and spin is inverted as shown in
+Fig.2(b). In the valley-spin locked window ∆E, perfect
+polarization can be realized. To investigate the valley-
+spin polarized current, we consider only the ∆E energy
+range. In Fig.3, We study the dependence of the pumped
+current on the phase difference ϕ12 between V1 and V2.
+Due to the inverse valley-spin polarization of the left and
+right leads, the holes are forbidden to propagate through
+the scattering region, so there is no pumped current gen-
+erated in this setup at tR = 0.
+Introducing RSOC in
+the scattering region, it is found that the pumped cur-
+rent arises and flows into left or right lead. We attribute
+such a dc current to the spin flip process induced by the
+RSOC. Importantly, nonzero valley-spin polarized cur-
+rents are pumped into different leads, which depends on
+the exchange field in leads. Our results show that IL,R is
+an odd function about ϕ12, i.e., I(ϕ12) = −I(−ϕ12). The
+maximum of the pumped current appears at ϕ12 = ±π/2.
+From Fig.3, when the phase difference ϕ12 shifts from
+−π to 0, the valley-polarized holes with spin up will be
+pumped out of the scattering region and flow into the
+left lead. On the contrary, the opposite valley-polarized
+holes with spin down will spread into the right lead when
+ϕ12 shifts from 0 to π. It means that the direction of the
+pumped current can be tuned by the phase difference ϕ12,
+then different valley-spin polarized currents are pumped
+into different leads. We calculate the pumped current as
+a function of the phase difference ϕ12 for different RSOC
+strengths tR. Consequently, with the increasing of tR,
+the pumped current increases. IL versus tR at ϕ12 = π
+2 is
+plotted in the inset. The result is understandable: since
+the spin-flip efficiency increases when tR is increased, the
+spin-up carriers from one lead can be more easily flipped
+as spin-down carriers and flow into the other lead, which
+
+1.5
+(a)
+(b)
+1
+0.5
+spin up
+E(eV)
+spin down
+0
+-0.5
+K
+K'
+K
+K'
+△E
+-1
+-1.5
+0
+0.5
+1
+1.5
+2 0
+0.5
+1
+1.5
+2
+k(π/a)
+k (π/a)
+X4
+-3
+-2
+-1
+0
+1
+2
+3
+-0.003
+-0.002
+-0.001
+0.000
+0.001
+0.002
+0.003
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.000
+0.002
+0.004
+0.006
+0.008
+ϕ12
+Pumped current
+ tR=0
+ tR=0.02
+ tR=0.04
+ tR=0.05
+Pumped current
+tR
+FIG. 3: The pumped current IL as a function of the phase
+difference ϕ12 between V1 and V2. Inset: the pumped current
+versus RSOC strength tR at ϕ12 = π/2. Other parameters:
+Ef = −0.74, L = 10a, V0 = 0.1, Vp = 0.05.
+-0.76
+-0.75
+-0.74
+-0.73
+-0.72
+-0.71
+-0.70
+0.000
+0.002
+0.004
+0.006
+0.00
+0.02
+0.04
+0.06
+0.08
+-0.006
+-0.003
+0.000
+0.003
+0.006
+ IL
+(b)
+Transmission
+Pumped current
+Ef
+(a)
+0.00
+0.03
+0.06
+0.09
+ T
+ 
+Pumped current
+Vp
+ IL,V0=0.05
+ IR,V0=0.05
+ IL,V0=0.07
+ IR,V0=0.07
+ IL,V0=0.1
+ IR,V0=0.1
+FIG. 4:
+(a) The pumped current and transmission versus
+Fermi energy at V0 = 0.1 and Vp = 0.05. (b) The polarized
+current IL and IR versus the pumping potential Vp for dif-
+ferent static potentials V0 at Ef = −0.74. Other parameters:
+ϕ12 = π/2, L = 10a, tR = 0.05.
+is required by the conservation of charge.
+In Fig.4(a), the pumped current and the transmission
+versus Fermi energy Ef are plotted at ϕ12 =
+π
+2 .
+Ob-
+viously, a broad transmission peak arises in the locked
+window ∆E, and the pumped current exhibits similar
+behavior as the transmission. This result indicates that
+the transport of polarized holes is dominated by quan-
+tum resonance, which originates from quantum interfer-
+ence effect. In fact, the resonance assisted transport is a
+common property of electron pump66. Besides, an im-
+passable interval for the pumped current is generated
+as the Fermi energy is away from the resonant peak as
+10
+20
+30
+40
+50
+0.00
+0.02
+0.04
+0.06
+10
+20
+30
+40
+50
+0.000
+0.002
+0.004
+0.006
+ 
+(a)
+Transmission
+46
+34
+22
+10
+(b)
+Pumped current
+L/a
+ tR=0.02
+ tR=0.05
+ tR=0.08
+FIG. 5: (a) The transmission versus the length L of scattering
+region for tR = 0.05. (b) The pumped current IL as a function
+of system length L for different RSOC strengths at ϕ12 =
+π/2. The numbers label the lengths of the scattering region
+where current peaks emerge. Other parameters are the same
+as Fig.4.
+shown in Fig.4(a).
+The variation of polarized current
+is well correlated to the transmission. The pumped cur-
+rent IL,R versus the pumping potential for different static
+potentials is plotted in Fig.4(b). Due to the particle con-
+servation, the current flowing into the scattering region
+must satisfy IL = −IR, which is confirmed through the
+symmetric curves of IL and IR. Furthermore, the results
+show that the valley-spin polarized current linearly in-
+creases with the increasing of pumping potential. For a
+relatively small Vp, the static potential V0 can enhance
+the magnitude of pumped currents.
+B.
+Size effect on the pumped current
+In the following, we study the influence of the sys-
+tem size on the pumped current. Transmission and the
+pumped current versus the length of the scattering re-
+gion are plotted in Fig.5.
+It is interesting that some
+robust peaks of the pumped current appear at certain
+lengths, but the currents for other lengths are almost
+zero.
+Moreover, these peaks show periodic oscillation
+behavior with the period length 12a.
+In Fig.5(a), we
+plot the transmission as a function of the length L for
+tR = 0.05.
+The transmission exhibits a similar be-
+havior as the pumped current, where the transmission
+peaks also appear at certain system lengths. It is clear
+that these transmission peaks correspond exactly to the
+pumped current peaks. It is reasonable that the peri-
+odic behavior of the pumped current originates from the
+spin precession72–74 induced by Rashba SOC. When the
+current carriers travel through the central scattering re-
+
+5
+0.000
+0.002
+0.004
+0.000
+0.001
+0.002
+10
+20
+30
+40
+50
+0.000
+0.001
+0.002
+ IL
+(c)
+(b)
+ 
+Pumped current
+(a)
+0.00
+0.05
+0.10
+ T
+ 
+ 
+Pumped current
+0.00
+0.05
+0.10
+ 
+Transmission
+Transmission
+Transmission
+Pumped current
+L/a
+0.00
+0.05
+0.10
+ 
+FIG. 6: The pumped current and transmission versus the
+length L of the scattering region for different Fermi energies.
+(a): Ef = −0.72, (b): Ef = −0.735, (c): Ef = −0.75. Other
+parameters are the same as Fig.4.
+gion with RSOC interaction, carrier spin keeps precess-
+ing and spin flip occurs, which results in the periodic
+oscillating behavior of the pumped current. Our calcu-
+lation further demonstrates that the width of scattering
+region has almost no influence on the periodic behavior
+of polarized current as long as Fermi energy lies in the
+valley-spin locked energy window. To evaluate the influ-
+ence of RSOC, We show in Fig.5(b) the pumped currents
+for different RSOC strengths. It is clear that the RSOC
+strength has less influence on the resonant period, which
+is L = 12a. However, with the increasing of tR, the reso-
+nant current peaks become significant. The peak current
+value grows larger as tR increases, which is consistent
+with the results shown in Fig.3.
+Notice that this periodic oscillation behavior of the
+pumped current is different from the even-odd conduc-
+tance oscillation of carbon-atom chains.75,76 When two
+metallic electrodes are attached to a carbon atomic chain,
+the electronic structure of the carbon chain is modified,
+which leads to the difference in the density of states be-
+tween odd- and even-number carbon chains.75,76 This
+leads to the even-odd conductance oscillation driven by
+dc bias.
+However, for the periodic oscillation of the
+valley-spin polarized currents, it is due to the spin-
+flipping process induced by Rashba SOC and driven by
+periodic pumping potentials.
+In Fig.6, we plot the pumped current and transmission
+versus the system length for different Fermi energies. Ap-
+parently, the periodic oscillation behavior of the pumped
+current persists as the Fermi energy changes.
+The re-
+sult shows that, with the increasing of Ef, the number of
+peaks increases while the corresponding periodic length
+FIG. 7: The pumped current with respect to both the system
+length L and the Fermi energy Ef.
+decreases. Besides, the magnitude of current peaks will
+decrease as the Fermi energy increases. By calculating
+the transmission, it can be seen that the behavior of the
+pumped current is still consistent with the transmission,
+which means the periodic oscillation is a universal phe-
+nomenon in the setup.
+To exhibit an overall view of the pumped current,
+in Fig.7, we provide a two-dimensional diagram of the
+valley-spin polarized current as a function of both the
+system length L and the Fermi energy Ef.
+By vary-
+ing L and Ef, we find that there are five curves with
+discrete extrema of currents. The periodic dependence
+of the pumped current on the system length L is clearly
+shown. When Ef approaches the top of valley, the largest
+pumped current appears and becomes more sharp as
+shown by the orange regions.
+Moreover, it is further
+confirmed that, multiple resonant peaks of the pumped
+current arise under an appropriate system length L. Our
+numerical results reveal that it is possible to design a
+high-efficiency device setup for generating valley-spin po-
+larized currents.
+C.
+Influence of the static potential
+In this section, we focus on the dependence of IL on
+the static potential V0.
+For this purpose, the system
+length and the RSOC strength are fixed at L = 10a and
+tR = 0.05. In Fig.8(a), we plot the pumped current as
+well as transmission coefficient versus the static potential
+V0. With the increasing of the static potential, a current
+peak first emerges at V0 = 0.126 labeled by the blue point
+γ3. As V0 scans the critical point γ1 at V0 = 0.151, the
+direction of IL is reversed. Continuing to increase V0, we
+can see a negative current peak. The result shows that
+the static potential can also change the direction and
+hence the polarization of the pumped current. Besides,
+in the vicinity of the critical point γ1, a transmission
+
+0.008
+50
+0.006
+0.004
+40
+0.002
+a
+30
+0
+20
+10.
+-0.76
+-0.74
+-0.72
+-0.706
+0.00
+0.06
+0.12
+0.18
+0.24
+-0.002
+0.000
+0.002
+0.004
+-0.750
+-0.745
+-0.740
+-0.735
+-0.730
+0.1
+0.2
+0.3
+0.4
+γ3
+γ2
+Pumped current
+V0
+ IL
+ T/50
+(a)
+γ1
+(b)
+V0&T
+Ef
+ γ1 transition  V0 
+ γ2 maximum T   
+0.000
+0.002
+0.004
+0.006
+0.008
+ Maximum IL
+ γ3 maximum  IL   
+FIG. 8: (a) The pumped current as well as the transmission
+coefficient T as a function of the static potential V0. A factor
+of 1/50 is multiplied to T for better illustration. γ1, γ2, γ3
+label the critical points. (b) The maximum of IL, transition
+point of V0 and corresponding maximum of T versus the Fermi
+energy. Other parameters: Vp = 0.03, ϕ12 = π/2, L = 10a.
+peak is clear, which suggests the influence of the static
+potential also results from quantum resonance.
+In Fig.8(b), The critical point of V0 and the maxima
+of both T and IL versus Fermi energy Ef are plotted.
+It is found that the critical value of V0 increases linearly
+with the increasing of Ef, which indicates the resonant
+energy level depends on the static potential. Besides, the
+curves of the peak values for transmission and pumped
+current grow with the Fermi energy. The variation of the
+pumped current with the Fermi energy is consistent with
+the results in Fig.6.
+In this work, the valley-spin polarized current is gener-
+ated by electric pumping in adiabatic regime, which re-
+quires two independently varying system parameters. We
+emphasize that similar mechanism can also be achieved
+with optical pumping, which is in the non-adiabatic
+regime due to the high frequency of light wave. In this
+case, the light frequency can serve as a pumping param-
+eter. Therefore, non-adiabatic parametric pumping us-
+ing electric or optical ways will certainly bring in more
+physics.
+IV.
+CONCLUSIONS
+In conclusion, we study the valley-spin polarized cur-
+rent in monolayer MoS2 ribbon via parametric electron
+pump. In the proposed setup, different valley-spin polar-
+ized currents can be controlled to flow into different leads
+in the valley-spin locked energy window. The phase dif-
+ference between the pumping potentials can change the
+direction and hence polarization of the pumped current,
+where quantum resonance dominates the transport pro-
+cess. Furthermore, the size effect on the valley-spin po-
+larized current is numerically investigated. As the length
+of the scattering region changes, resonant peaks of the
+pumped current arise and show periodic oscillation due
+to the spin precession.
+With the increasing of Fermi
+energy, the number of peaks decreases while the peak
+height increases within a fixed length range. The depen-
+dence of the resonant pumped current on the scattering
+region length and the Fermi energy is numerically re-
+vealed in a two-dimensional diagram.
+It is also found
+that the direction of valley-spin polarized currents can
+be inverted by the static pumping potential. As a po-
+tential valleytronic device, the pump setup proposed in
+this work can serve as a valley-spin polarization source,
+which can simultaneously generate opposite polarization
+in one device.
+The polarized signals can be efficiently
+manipulated by many system parameters, and significant
+resonant enhancement has been demonstrated.
+ACKNOWLEDGMENTS
+This
+work
+was
+supported
+by
+the
+National
+Natural
+Science
+Foundation
+of
+China
+(Grant
+Nos.
+12034014
+and
+61674052)
+and
+the
+Natu-
+ral
+Science
+Foundation
+of
+Shenzhen
+(Grant
+Nos.
+20200812092737002,
+JCYJ20190808115415679,
+and
+JCYJ20190808152801642). Hui Wang also acknowledges
+supports from the Outstanding Youth Foundation of
+Henan Scientific Committee (212300410041) and the
+Key Scientific and Technological Projects in Henan
+Province (212102210223).
+∗ [email protected]
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+

OdAzT4oBgHgl3EQfzf5C/content/tmp_files/2301.01769v1.pdf.txt

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+Comprehensive analysis of gene expression profiles to radiation exposure reveals
+molecular signatures of low-dose radiation response
+Xihaier Luo∗, Sean McCorkle∗, Gilchan Park∗, Vanessa L´opez-Marrero∗, Shinjae Yoo∗,
+Edward R. Dougherty†, Xiaoning Qian∗†, Francis J. Alexander∗, Byung-Jun Yoon∗†
+∗ Computational Science Initiative, Brookhaven National Laboratory, Upton, NY
+† Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX
+Abstract—There are various sources of ionizing radiation ex-
+posure, where medical exposure for radiation therapy or diag-
+nosis is the most common human-made source. Understanding
+how gene expression is modulated after ionizing radiation
+exposure and investigating the presence of any dose-dependent
+gene expression patterns have broad implications for health
+risks from radiotherapy, medical radiation diagnostic proce-
+dures, as well as other environmental exposure. In this paper,
+we perform a comprehensive pathway-based analysis of gene
+expression profiles in response to low-dose radiation exposure,
+in order to examine the potential mechanism of gene regulation
+underlying such responses. To accomplish this goal, we employ
+a statistical framework to determine whether a specific group
+of genes belonging to a known pathway display coordinated
+expression patterns that are modulated in a manner consistent
+with the radiation level. Findings in our study suggest that
+there exist complex yet consistent signatures that reflect the
+molecular response to radiation exposure, which differ between
+low-dose and high-dose radiation.
+Index Terms—Gene expression analysis, radiation biology, low-
+dose radiation response, pathway analysis.
+1. Introduction
+Environmental threats constitute a major factor in deter-
+mining a person’s susceptibility to disease. With the progress
+of industrialization and modernization, radiation exposure
+has become one of the most serious environmental threats
+in today’s world. Mounting evidence suggests that ionizing
+radiation is linked to the development of thyroid cancers,
+multiple myeloma, and myeloid leukemia in children and
+adults [1]. It is well documented that the biological effects of
+ionizing radiation on mammalian cells are closely related to
+radiation doses and dose rates. In general, low-dose radiation
+exposure is far more common than high-dose radiation ex-
+posure because low-dose radiation can come from a variety
+of sources, including natural sources, cosmic rays, nuclear
+power, and various types of radioactive waste. However,
+in contrast to the more well-defined effects of high-dose
+radiation exposure, the biological effects and consequences
+of low-dose radiation and mixed exposures remain poorly
+understood [2], [3].
+Historically, the health risks associated with low-dose
+ionizing radiation exposure have been estimated by extrap-
+olating from available high-dose radiation exposure data.
+However, the majority of the data come from experiments
+that used extremely high, even supra-lethal, doses. Extrapo-
+lating the results of such studies to physiologically relevant
+doses can thus be difficult [4]. Furthermore, an increasing
+number of studies show that the biological reactions to
+high and low doses of radiation are qualitatively distinct,
+necessitating a direct examination of low-dose responses to
+better understand potential risks [5].
+Genome-wide expression assays using microarrays or
+RNA sequencing can provide snapshots of transcriptional
+activities in a biological sample, hence studying the gene
+expression profiles under low doses of ionizing radiation
+can provide novel insights into the biological reactions to
+such radiation exposure. In fact, mining gene expression
+profiles has proven useful in understanding pathophysiolog-
+ical mechanisms, diagnosis and prognosis of complex dis-
+eases, and deciding on treatment plans. Several studies have
+demonstrated the effectiveness of using gene expression
+profiles for traditionally challenging problems, for instance,
+discriminating between different subtypes of a complex
+disease, such as cancer [6], [7]. Despite these successful
+applications, quantification and interpretation at the genetic
+level of the impact from radiation exposure on the risk of
+developing such diseases are still challenging. Especially,
+the small sample size of typical clinical data, on the other
+hand, frequently impedes meaningful analysis, making pat-
+tern discovery, disease marker identification, risk prediction,
+reproducibility, and validation extremely difficult [8], [9].
+Adjusting for multiple hypothesis testing is another critical
+issue for all microarray analysis methods. The similarities
+of such signatures across different sample types have not
+been demonstrated to be strong enough to conclude that
+they represent a universal biological mechanism shared by
+different sample types [10]–[12].
+In recent years, scientists have gained a better under-
+standing of the transcriptional response in cells to radiation
+exposure [13]. When cells are exposed to ionizing radiation,
+multiple signal transduction pathways are activated, mak-
+ing pathway activity a potentially powerful and informa-
+tive approach for determining disease states. Furthermore,
+pathways, the most well-documented protein interactions,
+arXiv:2301.01769v1  [q-bio.GN]  3 Jan 2023
+
+are known to closely reflect functional relationships related
+to molecular biological activities such as metabolic, sig-
+naling, protein interaction, and gene regulation processes.
+A growing body of research indicates that tasks such as
+class distinction based on differences in pathway activity
+can be more stable than distinction based solely on genes.
+For example, [14] incorporated pathway information into
+expression-based disease diagnosis and proposed a classifi-
+cation method based on pathway activities inferred for each
+patient. Later in [15], pathway activity patterns are used to
+describe a classification scheme for human breast cancer
+and to reveal complexity in intrinsic breast cancer subtypes.
+The probabilistic inference of differential pathway activity
+across different classes (e.g., disease states or phenotypes)
+using probabilistic graphical models [16] was shown to
+identify molecular signatures that can be used as robust
+and reproducible disease markers. The marker identification
+method in [16] was further extended in [17], where a novel
+algorithm for discovering robust and effective subnetwork
+markers in a human protein-protein interaction network that
+can accurately predict cancer prognosis and simultaneously
+discover multiple synergistic subnetwork markers. It should
+be noted that at the heart of these pathway-based analyses
+is determining the activity of a given pathway based on the
+expression levels of the constituent genes.
+The primary goal of this paper is to perform a compre-
+hensive pathway-based analysis of gene expression profiles
+to investigate the differential time and dose effects, primarily
+in low-dose experiments, in order to uncover molecular
+signatures of low-dose radiation response. Towards this goal,
+we adopt the probabilistic pathway activity inference scheme
+in [16], where the pathway activity level is estimated from
+gene expression data via the use of a simple probabilistic
+graphical model. More specifically, the scheme estimates the
+log-likelihood ratio between different classes (e.g., differ-
+ent levels of radiation exposure) based on the expression
+level of each member gene. The log-likelihood ratios of
+the member genes in a given pathway are then aggregated
+for probabilistic inference of differential pathway activity.
+Through this analysis, we identify the most significantly
+differentially activated pathways in response to low-dose
+radiation. These pathways are investigated to determine
+the presence of consistent dose-dependent gene expression
+patterns. Our cross-validation experiments demonstrate that
+the proposed method can generate reliable and consistent
+pathway analysis results even with limited data.
+2. Data
+2.1. Low-dose radiation gene expression data
+The goal of the current study is to identify poten-
+tial molecular signatures underlying the biological response
+to low-dose ionizing radiation exposure through pathway-
+based analysis of gene expression profiles. For this purpose,
+we conducted a thorough literature search and preliminary
+analysis to identify human gene expression data suitable
+for studying the low-dose radiation response. The gene
+Dose Level
+Number of Samples
+0 Gy
+18
+0.005 Gy
+16
+0.01 Gy
+18
+0.025 Gy
+18
+0.05 Gy
+17
+0.1 Gy
+18
+0.5 Gy
+16
+TABLE 1. DESCRIPTION OF THE GENE EXPRESSION DATASET
+GSE43151 THAT WAS USED TO INVESTIGATE THE MOLECULAR
+SIGNATURES OF LOW-DOSE RADIATION RESPONSE IN THIS STUDY.
+expression dataset GSE431511 was identified to be the most
+suitable for our study, in terms of sample size and the range
+of radiation levels that were considered. Overall, GSE43151
+contains gene expression measurements from 121 blood
+samples, where five healthy male donors provided 400 mL
+venous peripheral blood samples each [18]. A complete
+blood count was performed on each whole blood sample
+using an ADVIA Hematology System (Bayer HealthCare).
+The standard lymphocyte proportion of 16-45 percent was
+met by all samples. Heparin at a final concentration of 34
+U ml−1 was added to whole blood samples. The blood
+was then diluted 1:10 with Iscove’s Modified Dulbecco’s
+Medium (IMDM, Life Technologies). Finally, blood samples
+were incubated overnight at 37 Cina 5% CO2 concentration.
+For the ex vivo irradiation, whole blood exposures were
+performed at the ICO-4000 facility (Fontenay-aux-Roses,
+France) with a Co source at a low dose rate (50 mGy
+min−1). Exposures were carried out independently on each
+donor’s blood sample. The kerma rate was calculated us-
+ing a Physikalisch-Technische Werkst¨atten (PTW) ionization
+chamber that was irradiated under the same conditions as the
+samples. Doses of 5, 10, 25, 50, 100, and 500 mGy were
+tested (See Table. 1), as well as sham irradiated conditions.
+Following ex vivo irradiation, blood samples were incubated
+at 37 degrees Celsius for 150, 300, 450, and 600 minutes
+in a 5% CO2 atmosphere.
+A density medium was used to collect CD4+ T lym-
+phocytes for cell sorting. Following that, total RNA was
+extracted from CD4+ T lymphocytes using RNeasy Mini
+columns from the RNeasy Mini Kit (Qiagen) as directed by
+the manufacturer. For all RNA samples, the RIN (RNA in-
+tegrity number) was calculated for assigning integrity values
+to RNA measurements. For gene expression assays, all RIN
+values were greater than the recommended value of 7.
+Before performing the pathway analysis based on the
+GSE43151 gene expression dataset, all 121 samples in the
+dataset were normalized, filtered, and analyzed using GAGE
+in R software [19]. Following the filtering step, a total of
+10,875 probes were chosen, where the basic filtering criteria
+consisted of removing a probe when it was undetected in at
+least 75% of the replicates considered.
+1. https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE43151
+
+(D.1) Overall algorithm - pseudo code
+(D.2) Rank pathways
+Rank pathways using computed t-scores
+for   pathway   in   KEGG database
+for   gene   in   selected pathway
+compute the active score
+compute the t-score
+•
+hsa00010 
+•
+hsa00020 
+•
+hsa00030 
+•
+hsa00040
+•
+…  
+•
+hsa00400 
+•
+hsa00560 
+•
+hsa04907
+•
+hsa00790
+•
+… 
+Task 1: low-dose                      Task 2: high-dose
+(C.1) Label samples
+(C.2) Build conditional distributions
+(C.3) Estimate activity score 
+Task 1: zero-dose vs low-dose                Task 2: zero-dose vs high-dose 
+. . .
+•
+Zero-dose
+•
+Low-dose
+•
+High-dose
+Compute log-likelihood ratio by Equation 1
+• Task 1
+• Task 2
+<latexit sha1_base64="5FlLgQ/XWm6UEAeV2onZM/k+9I=">ACHicbVDLSsNAFJ3UV62vqEs3g0VoNzXRom6EghsFxXsA9oQJpNJO3SiTMTpYR+iBt/xY0LRdy4EPwbp20W2nrgXg7n3MvMPV7MqFSW9W3kFhaXlfyq4W19Y3NLXN7pyl5IjBpYM64aHtIEkY
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+)/f(
+))
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+(B.1) GSE Database
+(B.2) Identify the low-dose radiation data set
+(B.3) Sample classification
+Zero radiation samples 
+Low-dose radiation samples 
+ERR127303
+ERR127302
+ERR127305
+ERR127304
+ERR127309
+ERR127307
+ERR127306
+ERR127308
+26472
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+1028
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+7079
+124790
+89932
+−2
+0
+2
+Value
+Color Key
+High-dose radiation samples 
+(D.3) Expert interpretation
+(A.1) KEGG Pathway Database
+(A.2) Extract the pathway information
+(A.3) Gene list 
+gene 1 
+gene 2 
+gene k 
+. . .
+A. Build a gene list from a selected pathway from the KEGG database
+B. Identify low-dose radiation data set from GSE database
+C. Probabilistic inference of pathway activity
+D. Rank pathways based on their discriminative powers
+Figure 1. Overview of the pathway-based analysis of gene expression profiles in response to low-dose radiation exposure.
+2.2. Pathway database
+We used the KEGG (Kyoto Encyclopedia of Genes
+and Genomes) database to obtain a reliable set of known
+biological pathways [20]. KEGG is a collection of manually
+drawn pathway maps for understanding high-level functions
+and utilities of the biological system. The genomic infor-
+mation is maintained in the GENES database, which is a
+collection of gene catalogs for all fully sequenced genomes
+and some partially sequenced genomes with current annota-
+tions of gene functions. The PATHWAY database’s higher-
+order functional information is augmented with a collection
+of ortholog group tables for information about conserved
+subpathways, which are frequently encoded by positionally
+related genes on the chromosome and are especially valuable
+in predicting gene functions. In our case, we identified 343
+pathways relevant to the gene expression dataset GSE43151
+from the available 548 KEGG pathway maps by discarding
+the pathways that did not contain any gene whose measure-
+ment was included in GSE43151.
+3. Methods
+In this section, we describe the technical details of the
+pathway-based gene expression data analysis procedure that
+was used to detect potential molecular signatures underlying
+low-dose radiation response. Figure 1 provides an overview
+of the overall procedure.
+3.1. Pathway activity inference
+To perform the pathway analysis, we first identified the
+genes whose measurements were included in the gene ex-
+pression dataset GSE43151 for the pathways of our interest.
+
+P53 SIGNALING PATHWAY
+Target genes
+Cyclin D
+CDK416
+Response
+G1 arest
+p21
+Cyclin E
+(sustaine d)
+-irradlia
+143-3-
+CDK2
+Cell cyc le arrest
+UV
+/Rerrima
+Genotoxic
+Cyelin E
+ATM
+CHK2
+G2 arrest
+Cell cyc le
+Cellular se rnescence
+drugs
+ DNA damage
+Gadd45
+Cdc2
+(sustained)
+Nutrition
+ATR
+CHK1
+B99
+deprivation
+Hypoxia
+Heaticold .
+Fas
+shock
+Nitric oxide
+DRS
+CASP8
+PIDD
+Bil
+A poptosis
+Stress signak
+Noxa
+PUIMAP53AIP
+tBid
+Cytc
+Jncogene
+Bcl-xL
+Ras, BCR-ABL)
+7Sival
+-CASP9
+CASP3
+SCYL1EPI
+Bc12
+Araf-1
++p
+ROS
+ PIGs
+P14ARF
+MDM2
+r53
+DNA
+IVitoc hordrior
+ScotinPERPPAG608Siah
+Apoptosis
+AAIFM2
+MDMX
+IGF-BF3
+HIGF
+Cell cy le
+PAIBAI-1KAIGDAiFTSP1Maspin
+Irhibition ofangiogene sis
+ar retastasis
+P48p53R2Gadd45Sestins
+DNA re pair and
+PTENTSC2IGF-BP3
+TS AF6
+Exosorre rrediated
+secretion
+MDM2Cop-1PIRH2CyelinGSiah-1WiplANp73
+p53 regative feedback
+041156/4/20
+(c) Kanehisa LaboratoriesFor every pathway, member genes that were missing in the
+given dataset were removed from the gene set. Consider a
+pathway G that consist of n genes {gk}n
+k=1 whose mea-
+surements were available in the dataset. In the context of
+binary classification, we assume that the expression level of
+gene gk (k = 1, 2, . . . , n) has a phenotype-dependent dis-
+tribution. Let us denote the conditional probability density
+function (PDF) of gene gk expression level under phenotype
+1 as f 1
+k(x) and the conditional PDF under phenotype 2
+as f 2
+k(x) with x representing the expression level of gene
+gk. In our case, we classify radiation exposures into three
+categories: zero-dose, low-dose, and high-dose. We compare
+low-dose and high-dose samples separately to zero-dose
+samples, which means that if zero-dose samples are treated
+as phenotype 1, either low-dose or high-dose samples will
+be treated as phenotype 2.
+After examining different probability distribution mod-
+els, we assumed that both f 1
+k(x) and f 2
+k(x) are Guassian in
+this study. Having these conditional PDFs, we can calculate
+the log-likelihood ratio (LLR) between the two phenotypes
+at a given expression level x of gene gk as follows
+Lk(x) = log[f 1
+k(x)/f 2
+k(x)]
+(1)
+For any given gene gk in the pathway G, the associated log-
+likelihood ratio Lk(x) in (1) indicates which phenotype is
+more likely based on the expression level x of gene gk. By
+combining the evidence–in the form of LLR–from all the
+member genes in the pathway, we can assess the overall
+activity level of the pathway at hand to infer which of the
+two phenotypes the collective expression pattern of its mem-
+ber genes points to and how significantly so, as discussed
+in [16]. More specifically, provided with a set {xj,k}m
+j=1 of
+m samples (i.e., gene expression measurements) for each
+gene gk, we first calculated activity levels {Sj}m
+j=1 defined
+as
+Sj =
+n
+�
+k=1
+Lk(xj,k)
+(2)
+The activity level Sj in (2) incorporates information from
+every gene in the pathway of interest and can be used to
+predict the phenotype (class label) based on the overall
+activation level of the given pathway in sample j.
+Note that to calculate the log-likelihood ratio Lk(x) in
+(1), we must first estimate the conditional PDF f c
+k(x) for
+each phenotype c ∈ {1, 2}. We assume that the expression
+of gene gk under the phenotype c follows a Gaussian dis-
+tribution with a mean of µc
+k and a standard deviation of σc
+k.
+These parameters were calculated using all of the available
+samples that correspond to the phenotype c. After that, the
+estimated conditional PDFs can be utilized to compute the
+log-likelihood ratios. In practice, we often have insufficient
+training data to estimate the PDFs of f 1
+k(x) and f 2
+k(x)
+with confidence. As a result, the computation of the log-
+likelihood ratio may be sensitive to relatively small changes
+in the gene expression levels. To alleviate this issue, we
+normalized the data as recommended in [16]. Namely, Lk(x)
+was normalized to obtain �Lk(x) as follows
+�Lk(x) =
+Lk(x) − E[Lk(x)]
+�
+E[(Lk(x) − E[Lk(x)])2]
+.
+(3)
+While the use of (1) and (2) without normalization for infer-
+ring the pathway activity level would be equivalent to using
+a Naive Bayes model (NBM) for classifying the phenotype
+(class label) given the expression profile of the member
+genes that belong to a given pathway, this normalization step
+in (3) makes the pathway activity scoring scheme diverge
+from the traditional NBM.
+3.2. Pathways as potential markers for discriminat-
+ing low-dose response from high-dose response
+To examine the ability of a pathway to discriminate
+between two phenotypes, we computed the t-test statistics
+scores using the activity levels Sj for all member genes (as
+defined in (2)) and averaged the absolute value of the t-test
+scores to compute an aggregated differential activity score.
+The aggregated score–which we refer to as the pathway
+activity score–was then used as an indicator of the pathway’s
+discriminative power [21]. It should be noted that low-dose
+and high-dose samples were analyzed separately to detect
+most strongly differentially activated pathways under each
+radiation exposure level. We had three types of samples:
+zero radiation, low-dose radiation (0.005 Gy to 0.1 Gy),
+and high-dose radiation (0.5 Gy). Despite the fact that
+different low-dose levels of ionizing radiation have been
+tested, we treated all dose levels between 0.005 Gy and 0.1
+Gy as the same type (i.e., low-dose radiation). Based on this
+categorization, we ranked all relevant KEGG pathways to
+based on the strongest differential pathway activity between
+zero-dose against low-dose radiations, and separately, based
+on zero-dose against high-dose radiations. This is illustrated
+in Fig. 1(C).
+4. Results
+4.1. Pathway analysis results
+To begin, we evaluated all relevant pathways in the
+KEGG database and ranked the pathways based on their
+discriminative power following the procedures elaborated
+in Sec. 3 and illustrated in Fig. 1. In particular, we ranked
+the pathways based on their discriminative power, assessed
+based on the aggregated differential activity score obtained
+by averaging the absolute value of the t-test scores of the
+member genes in a given pathway [21] and estimating the
+p-value.
+Fig. 2(a) shows the top five pathways that have been
+identified as being the most deferentially activated in the
+presence of low-dose radiation.
+The top pathway was associated with Natural killer cell
+mediated cytotoxicity, focusing on natural killer cells, which
+are innate immune system lymphocytes involved in early
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+Figure 2. Ranking of most differentially activated pathways and their
+discriminative power in terms of the pathway activity score. (a) Top
+differentially activated pathways under low-dose radiation exposure. The
+aggregated t-test scores reflect the discriminative power of the pathways
+for discriminating between zero-dose and low-dose samples. (b) Top differ-
+entially activated pathways for high-dose radiation exposure (zero-dose vs
+high-dose). Comparison between (a) and (b) show a significant difference
+between the list of top pathways that are differentially activated under low-
+dose radiation and those under high-dose radiation.
+defenses against both allogeneic and autologous cells. Many
+studies have been conducted to investigate the direct effects
+of low-dose ionizing radiation (LDIR) on natural killer cells
+and the potential mechanism [22], [23]. The results of the
+experiments showed that a simplified strategy based on
+LDIR leads to effective expansion and increased activity of
+natural killer cells, providing a novel approach for adoptive
+cellular immunotherapy.
+The second pathway is related to Adherens junction (AJ),
+which is the most common type of intercellular adhesion. AJ
+initiates and maintains cell adhesion while also controlling
+the actin cytoskeleton. In [24], three types of junctional
+proteins were chosen for immunohistochemical labeling,
+and experimental results showed that not only high, but
+also low and moderate doses of cranial irradiation increase
+cerebral vessel permeability in mice. In-vitro studies showed
+that irradiation alters junctional morphology, reduces cell
+number, and causes senescence in brain endothelial cells.
+Another study [25] discovered that gamma-radiation, even
+at low doses, rapidly disrupts tight junctions, adherens
+junctions, and the actin cytoskeleton, resulting in barrier
+dysfunction in the mouse colon in vivo. Radiation-induced
+epithelial junction disruption and barrier dysfunction are
+mediated by oxidative stress, which can be mitigated by
+NAC supplementation prior to IR.
+Another pathway linked to Sphingolipid metabolism was
+also highly ranked. Sphingolipids, a type of membrane
+lipid, are bioactive molecules that play a variety of roles
+in fundamental cellular processes such as cell division,
+differentiation, and cell death. Many studies on the effect
+of sphingolipids on cancer treatment have been conducted.
+Microbeam radiation can induce radiosensitivity in elements
+within the cytoplasm, according to [26]. The effect could be
+inhibited by agents that disrupt the formation of lipid rafts
+(filipin), demonstrating once again that membranes could be
+a target of ionizing radiation. The authors of [27] concluded
+that, while other pathways are activated to induce radiation
+or chemoresistance, sphingolipids play a significant role.
+The JAK-STAT signaling pathway and Glycosphin-
+golipid biosynthesis have also been revealed to be very
+important in the study of radiation effects. For example,
+erythropoietin (EPO), which was originally identified as an
+erythrocyte growth factor, is now used to treat anemia and
+fatigue in cancer patients receiving radiation therapy and
+chemotherapy. The study in [28] demonstrated previously
+unknown EPO-mediated HNSCC cell invasion via the Janus
+kinase (JAK)-signal transducer and activator of transcription
+(STAT) signaling pathway. On the other hand, the findings in
+[29] suggest that glycosphingolipid biosynthesis on the cell
+surface contributed to the activation of ionizing radiation-
+induced apoptosis via ceramide production. The functional
+importance of this pathway to eradicating cancer cells with
+ionizing radiation has been proven, with sphingolipid break-
+down activated as a mechanism of ceramide formation after
+cell irradiation.
+In a similar manner, Fig. 2(b) shows the top five path-
+ways that have been identified as being most differentially
+activated in the presence of high-dose radiation. The genes
+found in the identified pathways are closely related to the
+radiotherapy regimen. Graft-versus-host disease (GVHD),
+for example, is a fatal complication of allogeneic hematopoi-
+etic stem cell transplantation in which immunocompetent
+donor T cells attack genetically diverse host cells. Many
+clinical studies have found a link between GVHD severity
+and radiation dose, with more severe GVHD after condi-
+tioning regimens that included radiation therapy compared
+to those that only included chemotherapy [30], [31]. Another
+example is allograft rejection. By definition, the recipient’s
+alloimmune response to nonself antigens expressed by donor
+tissues causes allograft rejection. According to research,
+the complex pathophysiology involves host tissue damage
+caused by the conditioning regimen (chemotherapy and/or
+irradiation) [32]. After nonmyeloablative conditioning with
+low-dose irradiation, the use of recombinant fusion protein
+promotes mixed lymphoid chimerism.
+Interestingly, we can see that there is relatively small
+overlap between the set of pathways there were most re-
+sponsive to low-dose radiation exposure and those that were
+responsive to high-dose radiation exposure. For example, as
+shown in Fig. 2, only one pathway (i.e., Natural killer cell
+mediated cytotoxicity) was among the top 5 differentially
+activate pathways under both low-dose and high-dose ra-
+diation. However, we can see more pathways in common
+as we go down the list further. For example, when we
+compare the top ten pathways that are the most responsive
+
+Natural killer cell mediated cytotoxicity
+Pathway Name
+Adherens junction
+Sphingolipid metabolism
+JAK-STAT signaling pathway
+Glycosphingolipid biosynthesis
+2.5
+12.5
+0.0
+5.0
+7.5
+10.0
+Aggregated t-test scoreNatural killer cell mediated cytotoxicity
+Graft-versus-host disease
+Viral myocarditis
+Allograft rejection
+Autoimmune thyroid disease
+2.5
+12.5
+0.0
+5.0
+7.5
+10.0
+Aggregated t-test scoreto low-dose and high-dose radiation exposure, we find four
+common pathways: Natural killer cell mediated cytotoxic-
+ity, Adherens junction, Glycosphingolipid biosynthesis, and
+Antigen processing and presentation.
+4.2. Differential dose effect on radiation responsive
+pathways
+Next, we investigated the differential dose effects on the
+top pathways that were most responsive to either low-dose
+or high-dose radiation exposure. As noted earlier in Sec. 3.1,
+the probabilistic pathway activity inference scheme [16],
+which we adopted in this current study, is equivalent to using
+a simple probabilistic graphical model (PGM)–namely, a
+NBM–when we use (2) for calculating the pathway activity
+score based on the LLRs of the member genes belonging
+to the pathway. We wanted to find out whether this PGM
+constructed to detect the presence of low-dose (or high-dose)
+radiation exposure yields consistent activity inference results
+as the radiation dose level changes.
+Figure 3 shows the inference result based on the PGM
+trained to discriminate between zero-dose and low-dose
+samples. The y-axis shows the aggregated LLRs and the
+x-axis corresponds to the radiation dose level. For each
+dose level, the dots show the distribution of the pathway
+activity scores for all samples radiated at the given dose
+level. The results are shown for the top five pathways that
+were found to be most responsive to low-dose radiation.
+As we can see in Fig. 3, all low-dose responsive pathways
+yielded similar trends, where the inferred differential activity
+levels generally decreased as the radiation exposure level
+increased. These results imply that these pathways, and the
+gene expression profiles of the members therein, may reflect
+potential molecular signatures underlying the biological re-
+sponse to low-dose radiation exposure.
+We carried out a similar analysis based on the top five
+high-dose radiation response pathways that were identified
+in our study. The analysis results are summarized in Fig. 4.
+As before, the y-axis shows the pathway activity score
+obtained by aggregating the LLRs of the member genes in
+the pathway at hand. It should however be noted that, in this
+case, the LLR is obtained by comparing the likelihood ratios
+between zero-dose response and high-dose response. The
+resulting PGM is therefore trained to discriminate between
+zero-dose samples and high-dose samples. Interestingly, ex-
+cept for the first pathway (i.e., Natural killer cell mediated
+cytotoxicity), which was the top-ranked pathway in both
+low-dose as well as high-dose differential activity analysis
+(see Fig. 2), the pathway activity levels did not change
+significantly as the dose level increased. Considering that
+the pathway activity scores reflect the presence of potential
+molecular signatures of high-dose radiation response, this
+may imply that these top pathways that were responsive
+to high-dose radiation exposure might not be substantially
+perturbed when the radiation dose level is relatively low.
+4.3. Reproducibility of the identified pathways
+We conducted cross-validation experiments to assess the
+reproducibility of pathway analysis results and the signifi-
+cance of the identified pathways. To begin the experiment,
+we randomly selected 70% of zero-dose, low-dose, and
+high-dose samples, and we repeated this process ten times,
+taking into account the total size of our dataset. The top-
+ranked pathways identified by the algorithm are depicted
+in Fig. 5. Because the different sample selection introduces
+randomness, we first counted the show-up cases of pathways
+from the top ten most activated pathways. Then, we ranked
+our cross-validation results based on the total number of
+counts (shown in blue color). We also computed the mean
+and standard deviation of the aggregated t-test scores for
+each pathway (shown in red color). The cross-validation
+experiments for low-dose radiation responsive pathways are
+shown in see Fig. 5(a). As we can see, Fig. 5(a) demonstrates
+the consistency of the identified pathways when compared
+to the results originally obtained using the whole dataset
+(see Fig. 2 for comparison). Pathways Natural killer cell
+mediated cytotoxicity and JAK-STAT signaling pathway, for
+example, have been identified as being highly related to low-
+dose radiation response. We suspect that the difference is
+due to the radiation dose level. As previously discussed,
+we discovered a direct relationship between dose level and
+activation. Such differences are expected in a mixed and
+random combination of different dose levels.
+Noticeably, such consistency was not observed in the
+high-dose experiments shown in Fig. 5(b). In many top-
+ranked pathways, as shown in Fig. 4, there is a weak dis-
+tinction between high-dose samples. The last column, which
+represents the distribution of the calculated aggregated t-test
+scores of high-dose samples, in particular, shows a narrow-
+band distribution (See Fig. 4(b), (d), and (e)). Because the
+calculated statistical scores are so close, when randomness
+is introduced into data sampling, the cross-validation results
+in Fig. 5(b) appear more random. To validate this, we
+expanded our ranked pathway list to the top 30 pathways
+and found a larger number of overlapping pathways between
+the experiments using full dataset and the cross-validation
+experiments using only 70% of the dataset. In this case, the
+average ranking of the pathways Natural killer cell mediated
+cytotoxicity and Allograft rejection, for example, were 17th
+and 22nd, respectively. It should be noted that the radiation
+dose level that we categorized as “high-dose” in this study is
+still relatively low. We expect that gene expression analysis
+of samples that underwent higher-dose radiation exposure
+may result in more consistent pathway identification results
+with clear molecular signatures.
+Finally, we also investigated the assumption regarding
+the conditional distribution of the gene expression values.
+We used the one-sample Kolmogorov-Smirnov (KS) test to
+determine the goodness of fit. The test compares the under-
+lying distribution F(x) of a sample to a given distribution
+G(x), which in our case is a Gaussian distribution. The
+null hypothesis holds that the two distributions are identical,
+with F(x) = G(x) for all x; the alternative holds that
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+Figure 3. The pathway activity level measured in terms of the aggregated log-likelihood ratios (LLRs) in response to different levels of radiation exposure.
+Dose-dependent activity level is shown for the top five pathways that were most differentially activated under low-dose radiation exposure. (a) Natural
+killer cell mediated cytotoxicity (b) Adherens junction (c) Sphingolipid metabolism (d) JAK-STAT signaling pathway (e) Glycosphingolipid biosynthesis.
+All plots in (a)–(e) for the top low-dose response pathways display similar trends, where the differential activity levels reflecting the presence of potential
+molecular signatures of low-dose radiation response decrease as the radiation dose level increases.
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+Figure 4. The pathway activity level measured in terms of the aggregated log-likelihood ratios (LLRs) in response to different levels of radiation exposure.
+As before, dose-dependent activity level is shown for the top five pathways that were most differentially activated under high-dose radiation exposure. (a)
+Natural killer cell mediated cytotoxicity (b) Graft-versus-host disease (c) Viral myocarditis (d) Allograft rejection (e) Autoimmune thyroid disease. Except
+for the top pathway in (a), the differential activity levels reflecting the presence of potential molecular signatures of high-dose radiation response do not
+significantly change as the radiation dose level increases. This implies that the pathways that are responsive to high-dose radiation exposure may not be
+substantially perturbed under relatively lower-dose radiation exposure.
+they are not. We classify the samples as having a Gaussian
+distribution if the P-value is greater than 0.05; otherwise,
+they have a non-Gaussian distribution. Figure 6 depicts
+the computed results, which show that 70.45 percent of
+the low-dose samples and 89.63 percent of the high-dose
+samples adhere to the Gaussian assumption. This indicates
+that during the pathway analysis, it is appropriate to assume
+that the conditional distribution of the gene expression data
+is Gaussian.
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+Figure 5. Cross validation results of the top ranked pathways. (a) Cross-
+validation results for pathways most responsive to low-dose radiation.
+(b) Cross-validation results for pathways most responsive to high-dose
+radiation.
+Figure 6. Kolmogorov-Smirnov (KS) test results. We checked the normality
+of the gene expression values in low-dose and high-dose samples using the
+KS test. Results indicate that the Gaussian assumption holds in most cases.
+5. Conclusion
+The current study aimed to unveil molecular signatures
+of biological responses exposed to low or very low doses
+of ionizing radiation through pathway-based analysis of
+genome-wide expression profiles. Gene expression patterns
+under the radiation exposure at six different dose levels
+ranging from 5 mGy to 500 mGy were investigated, where
+the measurements in the original study [18] were made using
+blood samples obtained from five different donors during
+five independent irradiation sessions. Our investigation was
+conducted at the pathway level, as pathway-based gene
+expression analysis is known to yield more robust and repro-
+ducible results and as it may shed light on potential molecu-
+lar mechanisms underlying low-dose radiation response. To
+determine the differential activity level of a given pathway
+under different levels of radiation exposure, a probabilistic
+pathway activity inference scheme was adopted that aggre-
+gates the log-likelihood ratios (LLRs) of the member genes
+in a given pathway to infer its differential activity. This
+allows robust detection of pathways, whose member genes
+display possibly subtle yet consistent coordinated expression
+patterns in response to low-dose radiation exposure. We
+searched through the KEGG database to prioritize pathways
+based on their differential activity levels modulated by low-
+dose radiation exposure. Our analysis identified the top
+pathways that may be associated with low-dose radiation re-
+sponse. Findings in this study reflect the complicated nature
+of the biological response to low-dose ionizing radiation,
+as well as the fact that low-dose exposures affect many
+different gene pathways that are not significantly altered
+after higher doses of radiotherapy.
+One limitation of the current study is the small sample
+size of the analyzed dataset (GSE43151). While it has been
+challenging to find large-scale human gene expression data
+under low-dose radiation exposure, should such data be
+available in the future, their analysis would shed further light
+onto the unique molecular signatures of low-dose radiation
+response. Furthermore, the pathway activity level inference
+scheme in (2) makes specific modeling assumptions, upon
+which the derived results depend. In fact, the adopted
+scheme [16] assumes that the gene expression levels of
+the member genes in a given pathway are conditionally
+independent given the class label (e.g., presence/absence of
+radiation exposure as was considered in the current study)
+and follow Gaussian distributions. Although we carried out
+some preliminary validation of this modeling assumption
+(e.g., see Fig. 6), it would be also worth validating the
+pathway analysis results using other methods [33], [34],
+which may be potentially pursued in our future studies.
+Acknowledgements
+This work is supported by the U.S. Department of
+Energy, Office of Science, RadBio program under Award
+KP1601011/FWP CC121.
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+0
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+High doseNatural killer cell mediated cytotoxicity
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+Pathway
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+Aggregated t-test scoreGlycosphingolipid biosynthesis - lacto and neolacto series
+T cell receptor signaling pathway
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+JAK-STAT signaling pathway
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