# PeleNet: A Reservoir Computing Framework for Loihi

Carlo Michaelis

(24. November 2020)

###### Abstract

High-level frameworks for spiking neural networks are a key factor for fast
prototyping and efficient development of complex algorithms. Such frameworks
have emerged in the last years for traditional computers, but programming
neuromorphic hardware is still a challenge. Often low level programming with
knowledge about the hardware of the neuromorphic chip is required. The PeleNet
framework aims to simplify reservoir computing for the neuromorphic hardware
Loihi. It is build on top of the NxSDK from Intel and is written in Python.
The framework manages weight matrices, parameters and probes. In particular,
it provides an automatic and efficient distribution of networks over several
cores and chips. With this, the user is not confronted with technical details
and can concentrate on experiments.

Note: This is a draft

## Introduction

Several different neuromorphic hardware chips have been developed in recent
years (reviewed by Schuman et al., 2017; Young et al., 2019; Rajendran et al.,
2019). All of them promise to be a key factor in future neuroscientific
research as well as technological developments in artificial intelligence. The
main benefit of neuromorphic systems is their low power consumption and speed
(Rajendran et al., 2019). Shown for Loihi for example from Tang et al. (2019).
This advantage of brain inspired hardware comes with a solution to the von
Neumann bottleneck (Backus, 1978). While novel neuromorphic hardware becomes
more and more powerful, algorithms for such hardware systems are still in an
early stage. A specific field of spiking neural network algorithms, which can
be used for neuromorphic hardware, is reservoir computing. For details about
reservoir computing I refer the reader to the literature (Jaeger, 2001; Maass
et al., 2002; Jaeger, 2007; Schrauwen et al., 2007; Lukoševičius et al., 2012;
Goodfellow et al., 2016).

Here, I focus on the neuromorphic hardware chip Loihi (Davies et al., 2018).
The chip is digital and includes a current-based (CUBA) leaky integrate-and-
fire (LIF) neuron. A chip contains $128$ cores and each core time-multiplexes
$1024$ compartments. In addition, every chip contains three conventional x86
CPUs. Several chips can be used in parallel on one board. The parameters of
the synapses and the compartments can be adapted, but the neuron model itself
is fixed. A single compartment neuron can be extended to a multi compartment
neuron which comes with the cost of less available neurons. Intel provides a
software development kit, the so-called NxSDK which is written in Python and
already allows higher-level programming of the chip (Lin et al., 2018). In
addition, C scripts can be used to run code on the x86 CPUs of the chip. The
NxSDK allows to define compartment prototypes and compartment groups which can
be combined to neuron prototypes and neurons. Using connection prototypes and
connections these compartments and neurons can be interconnected using a
connection matrix. Spikes can be injected using spike generators and several
different types of probes can be defined. Finally, learning rules can be used
to make connections in the reservoir plastic.

However, the NxSDK defines compartments for each core separately. If bigger
networks are used, it is necessary to split the connections manually to the
Loihi cores. Compartments between cores need then to be interconnected
manually, which results in $n_{\text{conn}}=n_{\text{core}}^{2}$ connection
matrices, if all potential connections should be possible. For example, if we
create $3$ compartment groups, distributed to $n_{\text{core}}=3$ cores, and
we want to interconnect all of them, we need to define $n_{\text{conn}}=9$
connection weight matrices, which is illustrated in Figure 1B. Note that this
amount of matrices is necessary even for sparse reservoirs networks, since we
do not want to exclude any possibility a priori. In addition to handling these
connection matrices, also probes can only be taken for each core individually.
Probing the whole network from the example above requires defining and
handling $3$ probes for the compartment groups and $9$ probes for the
connection weight matrices. Note that this problem is only difficult in
recurrent structures, especially in a reservoir where all neurons can
potentially connect to each other. In feed-forward structures, even in deep
spiking neural networks (already applied to several neuromorphic hardware
systems, see e.g. Diehl et al., 2016; Schmitt et al., 2017; Patino-Saucedo et
al., 2020; Massa et al., 2020), this problem is less predominant since the
layers are connected in series. Multiple probes still need to be defined, but
the connection matrices scale linear with the number of neuron groups and not
quadratic as in reservoirs.

Figure 1: (A) The reservoir network consists of a pool of excitatory neurons
(red) and a pool of inhibitory neurons (blue). Those pools are connected
within and in-between. The excitatory neurons can be stimulated by an input
(orange) and/or noise. Optionally a pool of output neurons (purple) can be
defined. The spiking data can be read out from the reservoir itself and/or
from the output neurons. PeleNet supports configuring a learning rule for the
connections of the excitatory neurons (dotted arrow). (B) The example shows
$12$ neurons which are distributed to three cores, where each core contains
four neurons. Each core is color coded (grey, green, purple). Every triangle
symbolises a potential connection between two neurons. The color of the
triangle indicates which cores need to be interconnected to interconnect the
related neurons. In this case, neurons spread over three cores require $9$
connection matrices.

The PeleNet framework was developed to solve the distribution of connections
efficiently and to make it easy for the user. In the framework, the
experimenter only needs to define one connection matrix for every part of the
network (e.g. for the reservoir or for the output layer). After the
simulation, the user gets usable probes for every part of the network. In
addition, PeleNet provides different distributions for initializing the
connection weights, defining learning rules, creating standard plots, logging
relevant computation steps and a collection of utils for calculating
statistics and handling data. Moreover, the framework is a whole new
abstraction layer on top of the NxSDK. Compartments, connections and probes
are defined implicitly and are controlled via parameters. Due to its modular
and object oriented architecture, the framework can easily be extended with
additional functionality. Here, I give a brief overview of the code structure
and the main features. The code is available on Github 111PeleNet on Github:
https://github.com/sagacitysite/pelenet under the MIT license.

## Design and implementation

Pele is the goddess of volcanoes and fire in the Hawaiian religion (Nimmo,
1986; Emerson, 2013). She has the control over lava and volcanoes and is inter
alias in control of the volcano _Loihi_. The name of the PeleNet framework is
an eponym of the goddess Pele.

The framework is build to allow experiments with reservoir networks on Loihi.
As shown in Figure 1A, PeleNet currently supports reservoir networks that
follow Dale’s law. The reservoir contains a pool of excitatory and a pool of
inhibitory neurons. Those neuron pools are connected within and in-between.
Additionally an input, noise and an output is available. The spiking data can
be read out from the excitatory, inhibitory and output neurons. Every
experiment can contain one or multiple trials. Figure 3 shows an example with
$10$ trials. The spiking activity can optionally be reset after every trial.
With this, it is possible to simulate much faster, since is is not necessary
to initialize the network again after every trial.

Programmatically, PeleNet consists of two main parts. One part contains some
helper functions and external libraries which are not available as a package.
This part is imported from the PeleNet framework internally, the user does not
need to import modules from this part. The other part consists of the PeleNet
code itself which is imported and used by the user.

### Libraries

The lib folder currently contains code to generate an anisotropic connectivity
matrix and some helper functions and classes. The code for initializing an
anisotropic connectivity matrix is public on Github 222Code for generating
anisotropic connection weights on Github: https://github.com/babsey/spatio-
temporal-activity-sequence/tree/6d4ab597c98c01a2a9aa037834a0115faee62587. The
underlying principle was introduced in (Spreizer et al., 2019) and used in
(Michaelis et al., 2020) for generating robust robotic trajectories, using the
PeleNet framework. The helper folder contains custom exception functions for
invalid parameters or invalid function arguments and a Singleton class to
decorate classes in the PeleNet framework to make use of the singleton design
pattern.

### PeleNet structure

The central entity of PeleNet is the experiment, which inherits from an
abstract experiment. In an experiment one or several networks can be defined
and used. In the experiment, a parameter set is defined and overwrites values
of the default parameter object. The parameters are passed to every network,
when it is initialized by the experiment. In addition, every network contains
a plot object, which has access to the data sets and probes of the simulation.
Passing data to one of the plot methods is therefore in most cases not
necessary. Hence, the arguments in den plot method shape the plots
appropriately in size, limits, labels, colors, etc. Two singleton objects are
globally available to all other objects. The system singleton contains a
datalog object which logs all important steps in a log file. It also logs the
parameter set which was used for a particular experiment and basic plots and
optionally data sets from this experiment. The utils singleton provides a
bunch of methods to handle and evaluate data, like dimensionality reduction,
smoothing, calculating the spectral radius of the weight matrix and different
kind of statistics. Figure 2 gives an overview of the dependencies between the
classes in PeleNet.

Figure 2: The code structure of the PeleNet framework. Classes have a green
background, singletons are in purple and collections of methods for a class
are yellow.

### Network

The major component of the PeleNet framework is the network. It contains a
collection of methods, distributed over several files. Since this is the core
of the framework, I will give some more details about the behavior in the
following listing.

#### weights

Methods in the weights file initialize the weight matrices for all parts of
the network. The basic weight matrices are for connecting the excitatory and
inhibitory parts of the reservoir. Weights can be initialized using constant
values or a log-normal or normal distribution. In addition, a 2D topological
anisotropic weight matrix can be initialized. All weight matrices are stored
sparsely in a compressed sparse row (CSR) format.

#### connect

Takes a weight matrix for every part of the network (e.g. one weight matrix
for the whole reservoir), splits it in parts (called chunks in the framework),
distributes these parts to the cores and interconnects them to each other. It
is currently still necessary to define the number of neurons that should be
used per core as a parameter. For an efficient distribution we need to
consider two aspects. First, every core time-multiplexes up to $1024$ neurons,
the less neurons we simulate on every core, the faster the simulation will be.
Second, the more cores are used, the more connection matrices are required,
which slows down the initialization of the network. For an optimal run-time
performance we should therefore reduce the number of neurons per core as much
as possible and for an optimal initialization performance we should use as
much neurons per core as possible. It is currently up to the user to choose
best fitting values for this trade-off, but in most cases it is probably the
fast simulation time which is desired. Later versions of the framework may
allow the user to choose a preferred method and handles the distribution of
neurons automatically.

#### input

Adds different types of inputs to the network. All of them base on the spike
generators from the NxSDK. Currently topological inputs (in case of a
2D-network), noisy inputs (leave-n-neurons out in every trial), sequences of
inputs and varying input positions per trial are supported. Topological inputs
define a square of stimulated neurons in the excitatory layer of the reservoir
at a defined position. Noisy inputs stimulate a specific number of reservoir
neurons, but in every trial some few neurons are left out such that the input
differs slightly in every trial. A sequence of inputs are multiple input
regions which are stimulated in a row within one trial such that relations
between them can be learned (e.g. with spike-timing-dependent plasticity). An
example for a input sequence is shown in Figure 3. Finally, it is possible to
define separate input regions, where one input region is randomly chosen in
every trial.

#### noise

Noise is currently generated by random inputs from spike generators which are
connected to randomly chosen neurons in the network. Future implementations
will probably make use of random changes in the current of a synapse or the
membrane voltage of the neuron.

#### output

Adds output neurons to the reservoir. Currently the only available output is a
pooling layer, which was used in Michaelis et al. (2020). The pooling layer
was used for a faster read out and performed regularization for the
anisotropic network.

#### probes

Contains several methods to define and process probes. First, probes are
defined for every core and every connection matrix chunk. Second, after a
successful simulation, probe data are post-processed and stacked together to
complete and useful data sets. Note that the output of a connection weight
probes is in CSR format again such that this matrix can directly be used as an
initial connection matrix for another simulation.

#### snips

Handles small C-scripts that run on the x86 cores of the Loihi chips (so
called SNIPs). These scripts are located in the pelenet/snips folder.
Currently a reset SNIP is available that resets the membrane voltages after a
trial. It is important to note that the plasticity is currently not stopped
while resetting the membrane voltages, which can cause problems when the $uk$
variables ares used. But this feature is under development and will probably
be added soon.

⬇

1# Import abstract experiment

2from ._abstract import Experiment

3

4”””

5@desc: An experiment with a sequential input, trained over several trials

6”””

7class SequenceExperiment(Experiment):

8

9 ”””

10 @desc: Define parameters for this experiment

11 ”””

12 def defineParameters(self):

13 return {

14 # Experiment

15 ’seed’: 1, # Set fixed random seed

16 ’trials’: 10, # Number of trials

17 ’stepsPerTrial’: 60, # Number of simulation steps for every trial

18 # Neurons

19 ’refractoryDelay’: 2, # Refactory period

20 ’voltageTau’: 100, # Voltage time constant

21 ’currentTau’: 5, # Current time constant

22 ’thresholdMant’: 1200, # Spiking threshold for membrane potential

23 # Network

24 ’reservoirExSize’: 400, # Number of excitatory neurons

25 ’reservoirConnPerNeuron’: 35, # Number of connections per neuron

26 ’isLearningRule’: True, # Apply a learning rule

27 ’learningRule’: ’2^-2*x1*y0 - 2^-2*y1*x0 + 2^-4*x1*y1*y0 - 2^-3*y0*w*w’,

28 # Input

29 ’inputIsSequence’: True, # Activates sequence input

30 ’inputSequenceSize’: 3, # Number of input clusters in sequence

31 ’inputSteps’: 20, # Number of steps a trace input is active

32 ’inputGenSpikeProb’: 0.8, # Probability of spike for the input generator

33 ’inputNumTargetNeurons’: 40, # Number of neurons activated by the input

34 # Probes

35 ’isExSpikeProbe’: True, # Probe excitatory spikes

36 ’isInSpikeProbe’: True, # Probe inhibitory spikes

37 ’isWeightProbe’: True # Probe weight matrix at the end of every trial

38 }

Listing 1: Defining an experiment in the pelenet/experiments folder. The
defineParameters method is required. Lifecycle methods are optional (not
shown). In addition, the experiment can be extended by custom methods for e.g.
data evaluation or visualization.

### Experiments

An experiment inherits from an abstract experiment and is created in the
pelenet/experiments folder. The abstract experiment inherits again from the
ABC package, which allows defining abstract methods. The defineParameters
method in the abstract experiment class is implemented as an abstract method
that is then necessary in every experiment, otherwise an exception is thrown.
The abstract experiment also provides some default functionality, which can
optionally be overwritten. It initializes all necessary objects for the
experiment and contains a default build process. Also the execution of the
simulation follows a default behavior. In both cases, it is preferred to
control the behavior of the experiment via parameters instead of overwriting
methods. If the parameters do not cover the wanted behavior, it is suggested
to make use of the available lifecycle methods. Available lifecycle methods
are:

  * •

onInit: Called after the experiment was initialized.

  * •

afterBuild: Called after all network parts are connected (i.e. weight matrix,
inputs, outputs, noise, probes).

  * •

afterRun: Called after the simulation has finished and all data are post-
processed.

Only if the parameters and the lifecycle methods are not sufficient to solve
the intended behavior it is suggest to overwrite the build and run methods of
the abstract experiment.

In practice, Jupyter notebooks are used to quickly evaluate the results of an
experiment. It is suggested to use Jupyter notebooks only for visualization of
the results and for prototyping. Code for the experiment should be included in
the defined experiment. An example of a simple experiment is shown in Code
Listing 1. In Code Listing 2 the experiment is used. The plotted spike train
from the Code Listing 2 (line $25$) is shown in Figure 3.

### Parameters

⬇

1# Load pelenet modules

2from pelenet.experiments.sequence import SequenceExperiment

3

4# Overwrite default parameters from pelenet/experiments/sequence.py

5parameters = {

6 # Experiment

7 ’seed’: 2, # Change random seed

8}

9

10# Initialize experiment

11# The name is optional, it is extended to the folder in the log directory

12# The parameters defined above are handed over to the experiment object

13exp = SequenceExperiment(

14 name=’random-network-sequence-learning’,

15 parameters=parameters

16)

17

18# Build the network (weight matrix, inputs, probes, etc)

19exp.build()

20

21# Run the network simulation, afterwards the probes are post-processed to
nice arrays

22exp.run()

23

24# Plot spike trains of the excitatory and inhibitory neurons

25exp.net.plot.reservoirSpikeTrain(figsize=(12,6))

Listing 2: An example for running the experiment. For this Jupyter notebooks
can be used. Parameters of the experiment can be overwritten to allow fast
experimentation. At the end of the script an included plotting method is
called to show spike trains. They are directly plotted in Jupyter and stored
in the log folder related to this simulation.

The parameter system is a powerful tool of PeleNet for defining an experiment.
All NxSDK functionalities, which are included in PeleNet, are covered by the
parameter set. It is not required to know anything about the NxSDK at all, if
the functionality provided by PeleNet is sufficient for the user. The default
parameters are split in three parts, parameters for the experiment, the system
and derived parameters. The system parameters cover information about e.g. the
used Loihi board, settings for the matplotlib library, logging and paths. The
experiment parameters contain e.g. neurons, connections, inputs, outputs,
noise, probes and also a learning rule for the reservoir. Finally, the derived
parameters are calculated from the system and experiment parameters and are
useful for the framework or for the user. Some parameters are sanity checked
to avoid serious issues with false parameters. These checks are constantly
extended. Note that the parameters are well documented in the
pelenet/parameters file to understand their meaning, but it is not suggested
to overwrite the parameters there. For overwriting parameters the
defineParameters method is available in the experiment. All parameter values
which are defined in this method overwrite the default parameter set. Derived
parameters are calculated after the parameters are defined in an experiment.

## Discussion

Currently the neuromorphic hardware community grows fast and it is probably
only a matter of time until new or updated hardware systems will emerge. The
field of applications is very broad for such hardware, including the
estimation of linear models, like LASSO (Shapero et al., 2014; Davies et al.,
2018), non-parametric classification with k-nearest neighbor (Frady et al.,
2020), deep spiking neural network (Massa et al., 2020) or reservoir networks
(Michaelis et al., 2020). High-level libraries and frameworks are needed to
cover these different specialized application areas. While for deep spiking
neural networks the SNN Toolbox (Rueckauer et al., 2017; Rueckauer and Liu,
2018) is available for ANN to SNN conversions (also for Loihi) or SLAYER is
available for training SNNs via backpropagation, reservoir network frameworks
are still rare for neuromorphic computing.

Figure 3: Example for a spike raster plot that shows inputs and trials. The
experiment was performed with $10$ trials and a input sequence with three
inputs.

The here presented PeleNet framework simplifies the implementation of
reservoir networks on the neuromorphic hardware Loihi. The framework is an
abstraction layer on top of the NxSDK from Intel. PeleNet allows an efficient
distribution of the network over an arbitrary number of Loihi cores and chips.
Probes are combined to data sets, which can directly be used for further
evaluations. Parameters define the experiments. The “parameter approach”
allows an easy initialization process and keeps the experiments clear.
Finally, the plot systems already includes a bunch of default plots for
reservoir computing.

Beside its already existing features, the framework is still in an early
stage. Some functionalities of the Loihi chip are not provided yet. This
includes inter alia current and voltage noise, usage of the tag for learning
rules and synaptic delays. The learning rule covers the excitatory neurons in
the reservoir, future version will also be able to apply a learning rule to
output neurons. In addition, the framework is not yet available as a pip
python package and still requires a manual installation of dependencies.
Therefore one of the next steps will be to provide a package release of
PeleNet. It is also intended to add an optimization functionality which runs
several experiments in order to find optimal parameters, according to a
criterion. The parameter system is already designed to support future
optimization scripts. Finally, it is planned to add unit and integration tests
to make sure that simulations are performed correctly. Currently four Jupyter
notebooks are available as functional tests.

The aim of PeleNet is to speed up the implementation of reservoir computing
experiments on Loihi. In a more broad perspective it has even the potential to
push the field of reservoir computing implementations on neuromorphic hardware
in general. Despite its early development stage, I am confident that the
framework is already useful for computational studies.

## License and code availability

The framework PeleNet is published under the MIT License and is therefore
freely available without warranty. The code is available on Github333PeleNet
on Github: https://github.com/sagacitysite/pelenet. Contributions are highly
appreciated.

## Acknowledgement

I sincerely thank Dr. Christian Tetzlaff for providing me with funding for my
doctoral thesis and his supervision. In addition, I want to thank Intel for
providing the Kapoho Bay to our lab and ongoing support. Finally I want to
thank Andrew B. Lehr, since he is always willing to discuss problems
constructively and influenced the implementation of the anisotropic network
significantly.

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# Probing flavored regimes of leptogenesis with gravitational waves from
cosmic strings

Marco Chianese<EMAIL_ADDRESS>Dipartimento di Fisica “Ettore
Pancini”, Universit‘a degli studi di Napoli “Federico II”, Complesso Univ.
Monte S. Angelo, I-80126 Napoli, Italy  INFN - Sezione di Napoli, Complesso
Univ. Monte S. Angelo, I-80126 Napoli, Italy 3Scuola Superiore Meridionale,
Universit‘a degli studi di Napoli “Federico II”, Largo San Marcellino 10,
80138 Napoli, Italy  Satyabrata Datta<EMAIL_ADDRESS>Department
of Physics and Institute of Theoretical Physics, Nanjing Normal University,
Nanjing, 210023, China  Gennaro Miele<EMAIL_ADDRESS>Dipartimento di Fisica
“Ettore Pancini”, Universit‘a degli studi di Napoli “Federico II”, Complesso
Univ. Monte S. Angelo, I-80126 Napoli, Italy  INFN - Sezione di Napoli,
Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy 3Scuola Superiore
Meridionale, Universit‘a degli studi di Napoli “Federico II”, Largo San
Marcellino 10, 80138 Napoli, Italy  Scuola Superiore Meridionale, Largo S.
Marcellino 10, I-80138 Napoli, Italy  Rome Samanta<EMAIL_ADDRESS>Scuola
Superiore Meridionale, Largo S. Marcellino 10, I-80138 Napoli, Italy  INFN -
Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
3Scuola Superiore Meridionale, Universit‘a degli studi di Napoli “Federico
II”, Largo San Marcellino 10, 80138 Napoli, Italy  Ninetta Saviano
<EMAIL_ADDRESS>INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo,
I-80126 Napoli, Italy 3Scuola Superiore Meridionale, Universit‘a degli studi
di Napoli “Federico II”, Largo San Marcellino 10, 80138 Napoli, Italy  Scuola
Superiore Meridionale, Largo S. Marcellino 10, I-80138 Napoli, Italy

###### Abstract

Cosmic strings radiate detectable gravitational waves in models featuring
high-scale symmetry breaking, e.g., high-scale leptogenesis. In this Letter,
for the first time, we show that different flavored regimes of high-scale
leptogenesis can be tested distinctly with the spectral features in cosmic
string-radiated gravitational waves. This is possible if the scalar field that
makes right-handed neutrinos massive is feebly coupled to the Standard Model
Higgs. Each flavored regime, sensitive to low-energy neutrino experiments,
leaves a marked imprint on the gravitational waves spectrum. A three-flavor
and a two-flavor regime could be probed by a characteristic fall-off of the
gravitational wave spectrum at the LISA-DECIGO-ET frequency bands with
preceding scale-invariant amplitudes bounded from above and below. We present
Gravitational Waves windows for Flavored Regimes of Leptogenesis (GWFRL)
falsifiable in the upcoming experiments.

Introduction. Within the framework of the seesaw mechanism of light neutrino
masses, leptogenesis [1] is a simple process for obtaining the observed baryon
asymmetry of the universe [2]. In this process, CP asymmetric and out-of-
equilibrium decays of heavy right-handed neutrinos (RHNs) to lepton and the
Standard Model (SM) Higgs doublets, create a lepton asymmetry, which is then
processed to baryon asymmetry by sphalerons [3]. In the thermal leptogenesis
scenario, the lepton asymmetry gets produced at a temperature $T_{\rm
lepto}\sim M_{N}$, where $M_{N}$ is the mass of the decaying RHN – also
referred to as the scale of thermal leptogenesis [4, 5]. Despite being a
simple mechanism, leptogenesis operating on scales $M_{N}>\mathcal{O}\left(\rm
TeV\right)$ are not directly testable, e.g., with collider experiments.
Accurate incorporation of flavor effects [6, 7, 8] nonetheless opens up the
opportunity to test high-scale leptogenesis with low-energy neutrino
observables, specifically with the leptonic CP violation [9] – a measurement
of which is one of the primary goals of the ongoing neutrino experiments such
as T2K [10] and NO$\nu$A [11]. Depending on the interaction strengths of the
charged lepton flavors, there exist three distinct regimes of high-scale
leptogenesis: $M_{N}\gtrsim 10^{12}~{}{\rm GeV}$ (one-flavor/vanilla, 1FL),
$10^{9}~{}{\rm GeV}\lesssim M_{N}\lesssim 10^{12}~{}{\rm GeV}$ (two-flavor,
2FL), and $M_{N}\lesssim 10^{9}~{}{\rm GeV}$ (three-flavor, 3FL). While the
1FL regime is not sensitive to neutrino mixing parameters, the 2FL and 3FL
regimes can offer low-energy neutrino phenomenology, which however may not be
unequivocal, because of plenty of free parameters in the seesaw Lagrangian.
Lately, there have been efforts to obtain a direct possible signature of high-
scale leptogenesis with primordial stochastic gravitational waves background
(SGWB) by focusing on the RHN sector, specifically, on the leptogenesis scale.
This is interesting and wholesome because, unlike electromagnetic radiation,
GWs traverse the early universe unimpeded with unerring information about
their origin. Therefore, no matter how high the leptogenesis scale is, it can
be probed with GWs. The idea originates from contemplating the origin of RHN
mass at a more fundamental level [12, 13, 14, 15]. In Ref. [12] it has been
argued that the RHNs become massive because of the spontaneous breaking of a
$U(1)_{B-L}$ gauge invariance, which many Grand Unified Theories (GUT)
naturally embed [16, 17, 18, 19]. Besides generating RHN mass, a gauged
$U(1)_{B-L}$ breaking inevitably produces cosmic strings that radiate
gravitational waves, which may be a signature of leptogenesis. Let us consider
the $U(1)_{B-L}$ invariant RHN mass term $f_{N}\overline{N_{R}}\Phi
N^{C}_{R}$, where $f_{N}$ is the Yukawa coupling, $N_{R}$ is the RHN field and
$\Phi$ being the $B-L$ scalar with vacuum expectation value $v_{\Phi}$. After
the $U(1)_{B-L}$ breaking, the RHNs obtain mass given by
$M_{N}=f_{N}v_{\Phi}$, cosmic strings are formed [20, 21, 22] and radiate GWs
with a scale-invariant spectrum at higher frequencies with an overall
amplitude $\Omega_{GW}\propto v_{\Phi}$ [23, 24]. While the scenario is
interesting, note however, that the GW spectrum being dependent only on
$v_{\Phi}$ is not so sensitive to $M_{N}$ because of the additional free
parameter $f_{N}$. As such, a GW spectrum for $v_{\Phi}=10^{14}$ GeV may
correspond to 3FL for $f_{N}=10^{-6}$ as well as 2FL for $f_{N}=10^{-4}$.
Therefore, a more realistic GW signature of leptogenesis requires dependence
on both parameters $v_{\Phi}$ and $f_{N}$, hence on $M_{N}$.

In this Letter, we show that the $M_{N}$-dependent GW spectrum and a concrete
link between GWs and high-scale leptogenesis can be achieved in the context of
$U(1)_{B-L}$ model with a peculiar scalar field dynamics [25]. After the phase
transition, $\Phi$ rolls down to its true vacuum and oscillates coherently to
provide an early matter-domination when it is long-lived. If the lifetime of
the scalar field gets determined by $v_{\Phi}$ and $f_{N}$, then $M_{N}$
regulates the beginning and the end of the matter-domination. Differently from
Ref. [12], the amplitude and the spectral features of the GWs radiated from
cosmic strings are imprinted by the $M_{N}$-dependent matter domination, thus
carrying information on $M_{N}$, and therefore, different regimes of flavored
leptogenesis. In particular, the scale invariance in the GW spectrum at higher
frequencies breaks at a frequency $f_{\rm dec}$ determined by the end of the
matter domination at the temperature $T_{\rm dec}(M_{N})$ (see Fig. 1). We
point out that such a scenario allows us to probe the RHN mass scale markedly,
in agreement with all the relevant constraints on the parameter space (see
Fig. 2): i) the bound from the recent observations of nHz GWs by Pulsar Timing
Arrays (PTA) [26, 27, 28, 29, 30] which disfavor GWs from cosmic strings [31];
ii) the condition that the cosmic strings radiate predominantly GWs than
particles [32, 33]; iii) the requirement that the $\Phi$ decays before the
Big-Bang-Nucleosynthesis (BBN) [34, 35]. We highlight that if the break of the
GW spectrum occurs in the LISA (ET) frequency band, most likely it is a
signature of a 3FL (2FL) regime. Remarkably, the amplitudes and the spectral
break frequencies of the GW spectra predicted by our model are very well
delimited and they are all within reach of planned gravitational wave
experiments such as LISA, DECIGO and ET [36, 37, 38]. This ensures unique and
highly testable GW signatures of flavor regimes of high-scale leptogenesis.

The parameter space for $M_{N}$-dependent matter domination. The seesaw
Lagrangian for neutrino masses and leptogenesis is given by

$\displaystyle-\Delta\mathcal{L}\subset
f_{Di\alpha}\overline{N_{Ri}}\tilde{H}^{\dagger}L_{\alpha}+\frac{1}{2}f_{Ni}\overline{N_{Ri}}\Phi
N_{Ri}^{C}+{\rm h.c.}\,,$ (1)

where $f_{D}$ is the neutrino Dirac Yukawa coupling and $H(L)$ is the SM Higgs
(lepton doublet). The fields $N_{R}$ and $L$ have $B-L$ charge $-1$, whereas
the same charges for $\Phi$ and $H$ are $2$ and $0$, respectively. The scalar
field dynamics is dictated by the finite temperature potential restoring
symmetry at higher temperatures given by [39, 40]

$\displaystyle
V(\Phi,T)=\frac{\lambda}{4}\Phi^{4}+D(T^{2}-T_{0}^{2})\Phi^{2}-ET\Phi^{3}\,,$
(2)

where $D,E$ and $T_{0}$ are functions of $B-L$ gauge coupling $g^{\prime}$,
$\lambda$ is the self-interaction coupling, and the zero temperature potential
$V(\Phi,0)=-\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4}\Phi^{4}$ determines
the vacuum expectation value $v_{\Phi}=\mu/\sqrt{\lambda}$. The last term in
Eq. (2) generates a potential barrier causing a secondary minimum at $\Phi\neq
0$, which becomes degenerate with the $\Phi=0$ one at $T=T_{c}$. The potential
barrier vanishes at $T_{0}~{}(\lesssim T_{c})$, and the minimum at $\Phi=0$
becomes a maximum. The transition from $\Phi=0$ to $\Phi=v_{\Phi}$ can be
treated as a second-order transition with an extremely quick disappearance of
the potential barrier if the order parameter $\Phi_{c}/T_{c}\ll 1$ (weakly
first-order transition). Such a smooth transition can be obtained
approximately within the ballpark of $\lambda\simeq{g^{\prime}}^{3}$ and for
${g^{\prime}}\lesssim 10^{-2}$ [13, 41]. Once $\Phi$ rolls down, it oscillates
around $v_{\Phi}$ behaving as matter [42, 43, 25]. Three important
observations are in order. First, the $U(1)_{B-L}$ breaking scale can be as
high as the GUT scale, implying $v_{\phi}\lesssim 5\times 10^{15}~{}{\rm
GeV}$. Second, the Hubble friction at $T_{c}$ should also be negligible for
the rolling of the field down to $v_{\Phi}$; this can be achieved by
considering $m_{\Phi}$ larger than the Hubble parameter $\mathcal{H}(T_{c})$.
Third, RHNs become massive after the phase transition at $T=T_{c}$. Therefore,
the scale of leptogenesis $T_{\rm lepto}\sim M_{N}$ is bounded from above as
$T_{\rm lepto}\sim M_{N}\lesssim T_{c}$.

We aim to make $\Phi$ long-lived to induce a matter-domination epoch with a
lifetime determined by $v_{\phi}$ and $f_{N}$. In high-scale leptogenesis
scenario, the coupling $f_{N}$ should be generically large to have $M_{N}$
comparable with $v_{\Phi}$. In such a case, if $m_{\Phi}>2M_{N}$, tree-level
decay of $\Phi$ to RHN pairs ($\Phi\rightarrow NN$) would be too quick to make
$\Phi$ long-lived. This restricts ourselves to the case
$M_{N}>m_{\Phi}=\sqrt{2\lambda}v_{\Phi}$. Therefore, along with $M_{N}\lesssim
T_{c}$, the $\Phi$ scalar is long-lived within the folowing range

$\displaystyle\sqrt{2g^{\prime}}g^{\prime}v_{\Phi}<M_{N}\lesssim
2\sqrt{g^{\prime}}v_{\Phi}\,,$ (3)

where we have assumed $\lambda={g^{\prime}}^{3}$. In this case, the mass of
$\Phi$ is suppressed by a factor $\sqrt{g^{\prime}}$ compared to the
$M_{Z^{\prime}}=\sqrt{2}g^{\prime}v_{\Phi}$, thus kinematically forbidding the
tree-level decays of $\Phi$ to $Z^{\prime}$ pairs. Note interestingly that the
seesaw Lagrangian in Eq. (1) can provide an effective coupling $\lambda_{\Phi
H,{\rm 1-loop}}\sim f_{D}^{2}f_{N}^{2}/(2\pi^{2})$ at one-loop to trigger
$\Phi\rightarrow HH$ decay. Therefore, despite $f_{N}$ being large, we can
have a smaller decay rate as it is proportional to $\lambda_{\Phi H,{\rm
1-loop}}^{2}$. This decay involves both: the RHN mass and the active neutrino
mass $m_{\nu}$ via the Dirac Yukawa coupling
$f_{D}=\sqrt{m_{\nu}f_{N}v_{\Phi}/v_{H}^{2}}$. In our discussion, we shall
consider this radiative $\Phi\rightarrow HH$ decay with the assumption that
$\lambda_{\Phi H,{\rm 1-loop}}\gg\lambda_{\Phi H,{\rm tree}}$. The decay rate
up to a logarithmic factor is given by [44, 45, 25]

$\displaystyle\Gamma_{\Phi}^{HH}\simeq\Gamma_{0}\left(\frac{g^{\prime}}{10^{-5}}\right)^{3}\left(\frac{v_{\Phi}}{\rm
10^{13}~{}GeV}\right)^{2}\left(\frac{m_{\Phi}}{\rm 10^{8}~{}GeV}\right)\,,$
(4)

with $\Gamma_{0}=1.3\times 10^{-17}~{}{\rm GeV}$. We have assumed
$m_{\nu}\simeq 0.01$ eV, the SM Higgs vacuum expectation value $v_{H}=174$
GeV, and $f_{N}\simeq{g^{\prime}}\ll 1$, which implies $M_{N}\simeq
M_{Z^{\prime}}\ll v_{\Phi}$. The latter assumption should not drastically
change the phenomenological implication of the scenario because of the
constraint in Eq. (3).

The evolution of the energy density of $\Phi$ and the duration of matter
domination epoch, which ends when $\Phi$ decay [25], can be tracked by solving
the Friedman equations (see Supplemental Material I for further details). The
decay temperature of the scalar field can be obtained analytically considering
$\Gamma_{\Phi}^{HH}\simeq\mathcal{H}$, and is given by

$\displaystyle T_{\rm dec}\simeq
6.35\left(\frac{g^{\prime}}{10^{-5}}\right)^{3/4}\left(\frac{M_{N}}{10^{9}~{}{\rm
GeV}}\right)^{3/2}{\rm GeV}\,,$ (5)

where we consider $M_{N}=g^{\prime}v_{\Phi}$. The scalar decays must occur
before BBN, i.e., $T_{\rm dec}\gtrsim T_{\rm BBN}\sim 10~{}{\rm MeV}$.
Moreover, because the scalar field is long-lived, it produces entropy which
dilutes any pre-existing relic, including the baryon asymmetry. The amount of
entropy production in our scenario can be computed as $\kappa\simeq
140({g^{\prime}}/{10^{-3}})^{-1/4}({M_{N}}/{10^{9}~{}\rm GeV})^{-1/2}$. The
entropy dilution by a factor of $\kappa^{-1}$ must be considered in the
computation of baryon asymmetry. Even in the low $M_{N}$ region, the scenario
does not allow large entropy dilution [25], consequently the baryon asymmetry
can be comfortably generated. Finally, we mention that we can safely neglect
the additional one-loop decay $\Phi\rightarrow f\bar{f}V$ (the two-body decay
$\Phi\rightarrow f\bar{f}$ is suppressed due to chirality flip [46]), where
$f$ and $V$ are SM fermions and vector bosons. Indeed, this decay dominates
over the $\Phi\rightarrow HH$ only for $g^{\prime}\gtrsim 10^{-2}$ and smaller
values of $v_{\Phi}$, which is outside our parameter space.

Imprints of flavored leptogenesis on GWs from cosmic string. Gravitational
waves are radiated from cosmic string loops chopped off from the long strings
resulting from the spontaneous breaking of the gauged $U(1)_{B-L}$ [23, 24].
Long strings are described by a correlation length
$L=\sqrt{\mu/\rho_{\infty}}$, with $\rho_{\infty}$ is the long string energy
density and $\mu$ is the string tension defined as [47]

$\displaystyle\mu=\pi
v_{\Phi}^{2}~{}h\left(\lambda,g^{\prime}\right)~{}~{}{\rm
with}~{}~{}h\simeq\frac{1}{\ln({2{g^{\prime}}^{2}}/{\lambda})}\,,$ (6)

for $\lambda\ll 2{g^{\prime}}^{2}$. In the present analysis, we take
$\lambda={g^{\prime}}^{3}$ therefore $h\simeq 1/\ln(2/g^{\prime})$. The time
evolution of a radiating loop of initial size $l_{i}=\alpha t_{i}$ is given by
$l(t)=l_{i}-\Gamma G\mu(t-t_{i})$, where $\Gamma\simeq 50$ [23, 24],
$\alpha\simeq 0.1$ [48, 49], $G=M_{\rm Pl}^{-2}$, and $t_{i}$ being the
initial time of loop creation. The total energy loss from a loop is decomposed
into a set of normal-mode oscillations with frequencies
$f_{k}=2k/l_{k}=a(t_{0})/a(t)f$, where $k=1,2,3...k_{\rm max}$, $f$ is the
present-day frequency at $t_{0}$, and $a$ is the scale factor. The total GW
energy density is computed by summing all the $k$ modes giving [48, 49]

$\displaystyle\Omega_{GW}(f)=\sum_{k=1}^{k_{\rm
max}}\frac{2k\mathcal{F}_{\alpha}G\mu^{2}\Gamma_{k}}{f\rho_{c}}\int_{t_{i}}^{t_{0}}\left[\frac{a(t)}{a(t_{0})}\right]^{5}n_{\omega}\left(t,l_{k}\right){\rm
d}t\,,$ (7)

where $\rho_{c}$ is the critical energy density of the universe,
$\mathcal{F}_{\alpha}\simeq 0.1$ is an efficiency factor [48] and
$n_{\omega}\left(t,l_{k}\right)$ is the loop number density. The latter can be
computed from the velocity-dependent-one-scale model as [50, 51, 52, 53]

$\displaystyle
n_{\omega}(t,l_{k})=\frac{A_{\beta}}{\alpha}\frac{(\alpha+\Gamma
G\mu)^{3(1-\beta)}}{\left[l_{k}(t)+\Gamma G\mu
t\right]^{4-3\beta}t^{3\beta}}\,,$ (8)

where $\beta=2/3(1+\omega)$ with $\omega$ being the equation of state
parameter of the universe, and $A_{\beta}=5.4$ ($A_{\beta}=0.39$) for
radiation-dominated (matter-dominated) universe [53]. The quantity
$\Gamma_{k}={\Gamma k^{-\delta}}/{\zeta(\delta)}$ quantifies the emitted power
in $k$-th mode, with $\delta=4/3$ ($\delta=5/3$) for loops containing cusps
(kinks) [54]. We shall present the results for the dominant $k=1$ mode which
captures the qualitative features of our analysis.

The integral in Eq. (7) is subjected to two Heaviside functions
$\Theta(t_{i}-t_{\rm fric})\Theta(t_{i}-l_{c}/\alpha)$ which set cut-offs on
the GW spectrum at very high frequencies $f_{*}$ above which it falls as
$f^{-1}$. The quantity $t_{\rm fric}$ represents the time until which the
motion of the string network is damped by friction [55], and $l_{c}$ is a
critical length above which GW emission is dominant over particle production
as shown by high-resolution numerical simulations [32, 33]. The critical
length is $l_{c}\simeq\delta_{w}(\Gamma G\mu)^{-\gamma}$ where
$\delta_{w}=(\sqrt{\lambda}v_{\Phi})^{-1}$ is the width of the string and
$\gamma=2$ ($\gamma=1$) for loops containing cusps (kinks). Note that we deal
with small values of $\lambda$ leading to “fat” strings, which are not
conventional while studying GWs signatures from cosmic strings in BSM models
[12]. In particular, in this case, the particle production cut-off in the GW
spectrum is much stronger than the one usually considered for “thin strings”
($\lambda\simeq 1$). Thus, our scenario is also the first concrete BSM model
that discusses the results of [32, 33] for small $\lambda$.

Figure 1: The GW spectrum from cosmic string as predicted by our model for
three benchmark choices for the parameter $g^{\prime}$ and $T_{\rm dec}$, with
$M_{N}$ uniquely fixed by Eq. (5) (see text for details). Blue and green
spectra correspond to 3FL, while the red one is for 2FL. The symbols show the
analytical calculations of the GW plateau amplitude and the break frequency
according to Eq.s (9) and (10), respectively.

In Fig. 1, we show the GW spectrum from Eq. (7) (for $k=1$) predicted by our
model for three benchmark scenarios: $g^{\prime}=10^{-6}$ and $T_{\rm
dec}=0.05~{}{\rm GeV}$ (blue line), $g^{\prime}=10^{-5}$ and $T_{\rm
dec}=1~{}{\rm GeV}$ (green line), $g^{\prime}=10^{-4}$ and $T_{\rm
dec}=10^{3}~{}{\rm GeV}$ (red line), with the scale $M_{N}$ of leptogenesis
being uniquely defined by Eq. (5) to be $1.3\times 10^{8}$, $2.9\times 10^{8}$
and $9.2\times 10^{9}$ in units of GeV, respectively. Without any intermediate
matter-dominated epoch, the GW spectrum has two main contributions: a low-
frequency peak owing to the GW radiation from the loops which are created in
the radiation epoch and decay in the standard matter epoch, and an almost
scale-invariant plateau at high frequencies as [48, 49, 56]

$\displaystyle\Omega_{GW}^{\rm
plt}(f)=\frac{128\pi\mathcal{F}_{\alpha}G\mu}{9\zeta(\delta)}\frac{A_{R}}{\epsilon_{R}}\Omega_{R}\left[(1+\epsilon_{R})^{3/2}-1\right]\,,$
(9)

that arises from loop creation and decay in the radiation-dominated epoch
only. In Eq. (9), $\epsilon_{R}=\alpha/\Gamma G\mu\gg 1$, $A_{R}\equiv
A_{\beta}\simeq 5.4$ and $\Omega_{R}\sim 9\times 10^{-5}$. Note that
$\Omega_{GW}^{\rm plt}(f)\propto\sqrt{\mu}\propto v_{\Phi}$, implying, the
larger the symmetry breaking scale, the stronger the amplitude of GW. In our
scenario, which features instead an intermediate matter-dominated epoch, the
plateau breaks at a high frequency $f_{\rm dec}$, beyond which the spectrum
falls as $\Omega_{GW}(f>f_{\rm dec})\sim f^{-1}$ (see Fig. 1). The analytical
expression for this spectral break frequency $f_{\rm dec}$ can be computed as
[57]

$\displaystyle f_{\rm dec}\simeq\sqrt{\frac{8z_{\rm eq}}{\alpha\Gamma
G\mu}}\left(\frac{t_{\rm eq}}{t_{\rm dec}}\right)^{1/2}t_{0}^{-1}\,,$ (10)

where $z_{\rm eq}\simeq 3400$ is the red-shift at the standard matter-
radiation equality taking place at time $t_{\rm eq}$, while $t_{\rm dec}$ is
the time at which the scalar field $\Phi$ decays (matter domination ends). For
the three benchmark cases in Fig. 1, the break of the GW spectrum occurs in
the LISA [36], DECIGO [37] and ET [38] frequency bands, respectively.

In order to robustly claim the spectral fall as a signature of
$M_{N}$-dependent matter domination, we must require both $t_{\rm
dec}>l_{c}/\alpha$ and $t_{\rm dec}>t_{\rm fric}$. The first condition leads
to the constraint

$\displaystyle M_{N}\gtrsim M^{c}_{N}\left(\frac{T_{\rm dec}}{\rm
GeV}\right)^{2/5}\left(\frac{g^{\prime}}{10^{-5}}\right)^{7/10}h(g^{\prime})^{-2/5}\,,$
(11)

with $M^{c}_{N}=3.8\times 10^{7}$ GeV. Similar constraints can be derived for
loops containing kinks, which however are less stringent. The second condition
$t_{\rm dec}>t_{\rm fric}$ implies

$\displaystyle M_{N}\gtrsim M^{\rm fric}_{N}\left(\frac{T_{\rm dec}}{\rm
GeV}\right)^{1/2}\left(\frac{g^{\prime}}{10^{-5}}\right)h(g^{\prime})^{-1/2}\,,$
(12)

with $M^{\rm fric}_{N}=2.0\times 10^{4}$ GeV.

The parameter space of our model is also constrained by the recent PTA
observations of stochastic GW which disfavor cosmic strings due to a different
spectral slope [31]. Thus, PTA data require $\Omega_{GW}<\Omega_{GW}^{\rm
PTA}$ at $f=30$ nHz giving $G\mu\lesssim 7.0\times 10^{-11}$. This translates
to the constraint on $M_{N}$ as

$\displaystyle M_{N}\lesssim M^{\rm
PTA}_{N}\left(\frac{g^{\prime}}{10^{-5}}\right)h(g^{\prime})^{-1/2}\,,$ (13)

whit $M^{\rm PTA}_{N}\simeq 5.6\times 10^{8}$ GeV. All these constrained are
satisfied by the benchmark cases shown in Fig. 1.

We note that there could be additional contribution to the GW spectrum at very
high frequencies due to the cosmic string loops in the first radiation epoch
before $\Phi$ starts to dominate. However, the amplitude at higher frequencies
is suppressed not only because of the entropy production, but also because the
hard particle production cut-off makes the spectrum red.

Figure 2: The white (shaded) regions correspond to the allowed (excluded)
model parameter space in the $T_{\rm dec}$-$M_{N}$ plane for
$g^{\prime}=10^{-6},10^{-5},10^{-4}$ (left to right). The sensitivity of LISA
[36], DECIGO [37] and ET [38] are shown with the black curves. The diagonal
colored lines represent the relation Eq. (5) predicted by our model. The
symbols correspond to the benchmark cases whose GW spectra are reported in
Fig. 1. The black horizontal thin lines mark the separation between the two-
flavor (2FL) and the three-flavor (3FL) regimes of thermal leptogenesis.

Results and discussion. In Fig. 2, we show the sensitivities of LISA, DECIGO
and ET in the plane $T_{\rm dec}$-$M_{N}$, assuming $g^{\prime}=10^{-6}$ (left
panel), $g^{\prime}=10^{-5}$ (middle panel) and $g^{\prime}=10^{-4}$ (right
panel), together with all the conditions and constraints discussed above. We
underline that the dependence of the GW sensitivities on $g^{\prime}$ comes
from the relation $M_{N}=(G\mu{g^{\prime}}^{2}M_{\rm
Pl}^{2}/\pi/h(g^{\prime}))^{1/2}$. The white regions in each plot can be
generally regarded as allowed parameter space when taking $T_{\rm dec}$ and
$M_{N}$ as independent parameters. However, in our model, they are related by
Eq. (5) which restricts the allowed parameter space along the solid colored
lines in the plots. Overall, the requirement of negligible particle production
from cosmic string, the recent PTA data, and the BBN constraint limit the
parameter space to be $g^{\prime}\sim[10^{-6},10^{-4}]$ and $M_{N}/{\rm
GeV}\sim[10^{7},10^{10}]$.

We find that, irrespective of the values of $g^{\prime}$, going from light to
heavy $M_{N}$ region along the colored lines, the spectral break frequency
$f_{\rm dec}$ increases according to Eq.s (5) and (10). This trend allows us
to point out that if the spectral break occurs in the LISA band, it most
likely corresponds to $M_{N}\lesssim 10^{9}$ (3FL regime), as proven by the
allowed segment of the blue line in the left plot. With a similar argument, a
spectral break in the ET band would correspond to a 2FL regime of
leptogenesis, as depicted by the red line in the right plot. An intermediate
situation with $g^{\prime}=10^{-5}$, both the 2FL and 3FL regions would
provide a spectral break in the DECIGO frequency band. Therefore, in this
frequency band, the spectral break signature of 2FL and 3FL becomes ambiguous.

The allowed parameter space of our model can be translated into allowed
regions for the GW plateau amplitude $\Omega_{\rm GW}^{\rm plt}h^{2}$ and the
break frequency $f_{\rm dec}$ (see Supplemental Material II for details).
Usually, they are limited from above by the PTA data. However, in our model,
they are also bounded from below by the BBN constraint due to the matter-
dominated epoch and by the requirement that the spectral break is not due to
the particle production from cosmic strings. We find that our scenario implies
$10^{-12}\lesssim\Omega_{\rm GW}^{\rm plt}h^{2}\lesssim 10^{-10}$ and
$10^{-3}\lesssim f_{\rm dec}/{\rm Hz}\lesssim 10^{3}$ which we define as the
Gravitational Waves windows for Flavored Regimes of Leptogenesis (GWFRL).
These ranges, within which a robust GW signature of high-scale leptogenesis is
achieved, will be fully explored by planned GW experiments such as LISA,
DECIGO and ET. These GWFRL windows also trivially satisfy two important
constraints. First, $G\mu<6.3\times 10^{-3}g^{\prime}h(g^{\prime})$ due to the
Hubble friction being always less than $m_{\Phi}$ at $T\simeq T_{c}$. Second,
the ratio of the $\Phi$ vacuum energy to the radiation at $T_{c}$, which in
our case is $\rho_{\Phi}(T_{c})/\rho_{R}(T_{c})=4.7\times 10^{-4}g^{\prime}$,
is always smaller than one, thus avoiding a second period of inflation.

There could be possible corrections to the GWFRL windows relaxing some of our
assumptions. First, we have considered $\lambda={g^{\prime}}^{3}$ but a more
rigorous parameter space scan could be performed to find a smooth transition
of the field and at the same time being consistent with other gauge coupling
dependent constraints and a negligible $\Phi\rightarrow Z^{\prime}Z^{\prime}$
decays. Second, we have presented our results on GWs for $k=1$ mode. However,
when summing over a large number of modes, the overall amplitude and the
spectral break frequency would change slightly. Third, we have assumed
$f_{N}=g^{\prime}$. Weakening this assumption but being still consistent with
Eq. (3) would anyway not modify the overall trend of the parameter space.
Indeed, smaller $f_{N}$ ($M_{N}$) would correspond to smaller $T_{\rm dec}$
and therefore smaller $f_{\rm dec}$, meaning a 3FL regime likely to show up
the spectral break at lower frequencies than the 2FL regime. Summarizing, we
point out that the GWFRL windows we have derived in the present paper are
quite robust against our assumptions. In conclusions, we demonstrate that
different flavor regimes of high-scale leptogenesis can be realistically
probed with GW signatures from cosmic strings, thus establishing synergy
between low-energy neutrino physics and gravitational waves.

Acknowledgements:  We thank Tanmay Vachaspati for useful insight regarding the
particle production cut-off. The work of MC, GM, RS, and NS is supported by
the research project TAsP (Theoretical Astroparticle Physics) funded by the
Istituto Nazionale di Fisica Nucleare (INFN). The work of NS is further
supported by the research grant number 2022E2J4RK “PANTHEON: Perspectives in
Astroparticle and Neutrino THEory with Old and New messengers” under the
program PRIN 2022 funded by the Italian Ministero dell’Università e della
Ricerca (MUR). The work of SD is supported by the National Natural Science
Foundation of China (NNSFC) under grant No. 12150610460.

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Supplemental Material for
Probing flavored regimes of leptogenesis with gravitational waves from cosmic
strings

Marco Chianese, Satyabrata Datta, Gennaro Miele, Rome Samanta, and Ninetta
Saviano

The Supplemental Material is organized as follows. In Sec. I, we discuss the
evolution of the radiation and the scalar field energy densities, and the
produced entropy with the inverse of temperature. We derive the analytical
expressions for the decay temperature $T_{\rm dec}$ and the amount $\kappa$ of
the produced entropy which match the numerical results well. In Sec. II, we
detail the constraints leading to the Gravitational Waves window for Flavored
Regimes of Leptogenesis (GWFRL).

##  I scalar field evolution and entropy production

The scalar field $\Phi$ with initial vacuum energy
$\rho_{\Phi}\left(T_{c}\right)\simeq\lambda v_{\Phi}^{4}/4$ dominates the
energy density for a period, and injects entropy as it decays. We account for
this effect by solving the following Friedmann equations:

$\displaystyle\frac{{\rm d}\rho_{R}}{{\rm
d}t}+4\mathcal{H}\rho_{R}=\Gamma_{\Phi}^{HH}\rho_{\Phi}\,,~{}~{}\frac{{\rm
d}\rho_{\Phi}}{{\rm
d}t}+3\mathcal{H}\rho_{\Phi}=-\Gamma_{\Phi}^{HH}\rho_{\Phi}\,,~{}~{}\frac{{\rm
d}s}{{\rm d}t}+3\mathcal{H}s=\Gamma_{\Phi}^{hh}\frac{\rho_{\Phi}}{T}\,,$ (S1)

and upon recasting them as

$\displaystyle\frac{{\rm d}\rho_{R}}{{\rm
d}z}+\frac{4}{z}\rho_{R}=0\,,~{}~{}\frac{{\rm d}\rho_{\Phi}}{{\rm
d}z}+\frac{3}{z}\frac{\mathcal{H}}{\tilde{\mathcal{H}}}\rho_{\Phi}+\Gamma_{\Phi}^{HH}\frac{1}{z\tilde{\mathcal{H}}}\rho_{\Phi}=0\,,$
(S2)

where $\rho_{R}$ and $\rho_{\Phi}$ are the energy densities of radiona and the
scalar field as a function of$z=T_{c}/T$, $\mathcal{H}$ is the Hubble
parameter, $s$ is the entropy density of the thermal bath, and the
temperature-time relation has been derived from the third of Eq. (S1) as

$\displaystyle\frac{1}{T}\frac{{\rm d}T}{{\rm
d}t}=-\left(\mathcal{H}+\frac{1}{3g_{*s}(T)}\frac{{\rm d}g_{*s}(T)}{{\rm
d}t}-\Gamma_{\Phi}^{HH}\frac{\rho_{\Phi}}{4\rho_{R}}\right)=-\tilde{\mathcal{H}}\,,$
(S3)

with $a$ and $g_{*s}$ being the scale factor and the number of degree of
freedom, respectively. The production of entropy from the $\Phi$ decays is
computed by solving

$\displaystyle\frac{{\rm d}a}{{\rm
d}z}=\left(1+\Gamma_{\Phi}^{hh}\frac{\rho_{\Phi}}{4\rho_{R}\tilde{\mathcal{H}}}\right)\frac{a}{z}\,,$
(S4)

and computing the ratio of $\tilde{S}\sim a^{3}/z^{3}$ after and before the
scalar field decay. The amount of entropy production $\kappa=\tilde{S}_{\rm
after}/\tilde{S}_{\rm before}$ can be analytically quantified as

$\displaystyle\mathbb{\kappa}\simeq\left[\frac{\rho_{R}\left(T_{c}\right)}{\rho_{\Phi}\left(T_{c}\right)}\frac{T_{\rm
dec}}{T_{c}}\right]^{-1}\simeq\frac{\rho_{\Phi}\left(T_{c}\right)}{3^{1/4}\left(\frac{30}{\pi^{2}g_{*}}\right)^{-3/4}T_{c}^{3}\sqrt{\Gamma_{\Phi}^{HH}\tilde{M}_{\rm
Pl}}}\simeq
140\left(\frac{g^{\prime}}{10^{-3}}\right)^{-1/4}\left(\frac{M_{N}}{10^{9}~{}{\rm
GeV}}\right)^{-1/2}\,,$ (S5)

by considering $\Gamma_{\Phi}^{HH}\simeq\mathcal{H}$ which gives the decay
temperature $T_{\rm dec}$

$\displaystyle T_{\rm
dec}\simeq\left(\frac{90}{\pi^{2}g_{*}}\right)^{1/4}\sqrt{\Gamma_{\Phi}^{HH}\tilde{M}_{\rm
Pl}}\simeq
6.35\left(\frac{g^{\prime}}{10^{-5}}\right)^{3/4}\left(\frac{M_{N}}{10^{9}~{}{\rm
GeV}}\right)^{3/2}{\rm GeV}\,,$ (S6)

with $\tilde{M}_{\rm Pl}=2.4\times 10^{18}~{}{\rm GeV}$ being the reduced
Planck constant, and $g_{*}\simeq 106.75$ is the effective degrees of freedom
that contribute to the radiation. The evolution of the radiation, the scalar
field normalized energy densities $\Omega_{i}=\rho_{i}/\rho_{\rm tot}$, and
the entropy production factor $\kappa$ corresponding to the benchmarks
discussed in the main text are shown in Fig. S1.

##  II Gravitational Waves window for Flavored Regimes of Leptogenesis (GWFRL)

Figure S1: Evolution of the radiation and the scalar field normalized energy
densities $\Omega_{i}=\rho_{i}/\rho_{\rm tot}$, and the entropy production
factor $\kappa$ corresponding to the benchmarks discussed in the main text.
Because only the ratio of entropies before and after the $\Phi$ decay matters,
we set $\kappa(T_{C})=1$. The vertical lines represent the cut-off
temperatures $T_{\rm fric}\gg T_{\rm fric}\gg T_{\rm dec}$, implying that the
fall in the GW spectra occurs due to the $M_{N}$-dependent matter domination.

Figure S2: The white regions are allowed windows for the GW plateau amplitude
(left) and the spectral break frequencies (right), referred to as
Gravitational Waves Window for Flavored Regimes of Leptogenesis (GWFRL) in the
main text. These windows are derived from the spectral features of GWs
radiated from the fundamental mode oscillation of cosmic string loops. The
symbols correspond to the benchmark cases discussed in the main text.

We here detail the upper and lower bounds on the the GW plateau amplitude
$\Omega^{\rm plt}_{GW}h^{2}$ and the break frequency $f_{\rm dec}$ which allow
us to define the GWFRL windows reported in Fig. S2. These regions are bounded
from below by either the BBN or the particle production constraint, and from
above by the PTA observations. The lower bounds on the GW amplitude (from Eq.
(9)) come from the lower bound on $G\mu$ imposed by BBN and particle
production from cosmic strings, which read

$\displaystyle(G\mu)^{\rm BBN}$ $\displaystyle\gtrsim$ $\displaystyle
3.8\times
10^{-14}\left(\frac{g^{\prime}}{10^{-5}}\right)^{-3}h(g^{\prime})\,,$ (S7)
$\displaystyle(G\mu)^{c}$ $\displaystyle\gtrsim$ $\displaystyle 6.4\times
10^{-16}\left(\frac{g^{\prime}}{10^{-5}}\right)^{3}h(g^{\prime})^{-1}\,.$ (S8)

Similarly, the lower bounds on the spectral break frequency are derived as

$\displaystyle f^{\rm BBN}_{\rm dec}$ $\displaystyle\gtrsim$ $\displaystyle
3.2\times
10^{-2}\left(\frac{g^{\prime}}{10^{-5}}\right)^{3/2}h(g^{\prime})^{-1/2}~{}{\rm
Hz}\,,$ (S9) $\displaystyle f^{c}_{\rm dec}$ $\displaystyle\gtrsim$
$\displaystyle 1.2\times
10^{-2}\left(\frac{g^{\prime}}{10^{-5}}\right)^{3}h(g^{\prime})^{-1}~{}{\rm
Hz}\,.$ (S10)

The upper bound on the plateau amplitude is given by

$\displaystyle(G\mu)^{\rm PTA}\lesssim 7.0\times 10^{-11}\,.$ (S11)

while for $f^{\rm PTA}_{\rm dec}$ we have

$\displaystyle f^{\rm PTA}_{\rm dec}\lesssim 2.0\times
10^{-1}\left(\frac{g^{\prime}}{10^{-5}}\right)^{9/4}h(g^{\prime})^{-3/4}\rm
Hz\,.$ (S12)