# NaRLE: Natural Language Models using Reinforcement Learning with Emotion
Feedback

###### Abstract

Current research in dialogue systems is focused on conversational assistants
working on short conversations in either task-oriented or open domain
settings. In this paper, we focus on improving task-based conversational
assistants online, primarily those working on document-type conversations
(e.g., emails) whose contents may or may not be completely related to the
assistant’s task. We propose “NARLE” a deep reinforcement learning (RL)
framework for improving the natural language understanding (NLU) component of
dialogue systems online without the need to collect human labels for customer
data. The proposed solution associates user emotion with the assistant’s
action and uses that to improve NLU models using policy gradients. For two
intent classification problems, we empirically show that using reinforcement
learning to fine tune the pre-trained supervised learning models improves
performance up to 43%. Furthermore, we demonstrate the robustness of the
method to partial and noisy implicit feedback.

Index Terms—  natural language understanding, deep reinforcement learning,
emotion recognition, intent detection

##  1 Introduction

In recent years, Intelligent Personal Digital Assistants (IPDA) have becoming
increasingly popular. Users interact with these assistants using natural
language (either spoken or through text) where the conversations are generally
short and chit-chat based. There is also emergence in assistants which
interact with document-type conversation, such as emails (figure 1(a)). These
assistants must work with a broader context scope and multiple sub-
conversations or intents occurring at each turn of the user input. The
increased scope and complexity that comes with multi-turn conversations
creates many challenges for assistants in this setting. Specifically,
extracting entities among other non-task entities, and ensuring dialogue state
updates from entities relevant to the task become challenging. Previous works
[1, 2] have shown that directly applying chit-chat methods to such document-
type conversations tasks leads to sub-optimal results.

In all conversation models, collecting and manually annotated training data is
a challenge and incurs significant cost. This problem is exacerbated by the
distributional shift of the input data in online decision-making, meaning a
supervised learning model must be retrained periodically to maintain its
accuracy. Moreover, for commercial assistants, most of the user data is not
available for eyes-on access, either for collecting or labeling, and hence
cannot be used for training or improving models via traditional supervised
learning. In this paper, we ask and answer the question ”Can we effectively
improve assistants in an online setting without explicit feedback and no eyes-
on access to the data?”

In this paper, we provide a simple yet effective method to improve the NLU
components of intelligent assistants in a privacy-preserving online setting.
The framework consists of two main parts. In the first part, we associate a
user’s emotion in his/her response to the assistant with the assistant’s
previous action. This is critical in document-type conversations which have
multiple intents present in each turn of a conversation where the emotion
might not be associated to the task being completed by the assistant. In the
second part, the relevant detected emotion is used as a weak reward signal for
the policy gradient algorithm to update the NLU models online. The signal is
associated with the previous intent/action the assistant took and is used to
improve its behavior for the previous step. It’s important to ensure users can
provide feedback with minimal effort. We accomplish this by detecting the
implicit emotion from the user’s natural conversation with the assistant
rather than requiring the user’s explicit feedback (such as ratings which are
difficult to collect and might not be reliable indicators on the assistant’s
performance).

##  2 Related work

(a) Example email to document-type assistant

(b) Architecture of proposed framework NARLE

Fig. 1: Improving natural language understanding using implicit feedback

Natural language understanding in conversations (including intent detection,
entity extraction, and dialogue state tracking) are well studied problems [3,
4, 5, 6]. Recently, there has been work in joint end-to-end models [7, 8].
Jointly modeling multiple tasks in an end-to-end manner solves the model
attribution problem and paves the way for directly optimizing the end-goal
objective of correctly completing a task. This led to works which used a
combination of supervised learning with RL [9, 10, 11]. Approaches which
preferred pre-training the dialogue model before interactive tuning using RL
suffered problems of dialogue state distribution mismatch. A solution to this
was proposed by [12], who proposed a hybrid imitation and RL approach. The
learning agent learned its policy from supervised training but used users’
explicit guidance or demonstrations to learn the correct action. The feedback
used in this case is either explicit feedback provided by user, or treating
task completion as positive feedback.

One key component in the proposed framework is extracting emotion from
document-type conversations by attributing the expressed emotion to the
correct cause, as emotions not emerging from the assistant’s actions should be
ignored. Previous works focusing on emotion cause extraction and its
variations evolve from rule-based methods [13] to modern learning-based models
[14, 15, 16, 17]. Instead of relying on the above methods, we adapt the
ScopeIt model [1] to extract not only task specific sentences but also
sentences expressing emotion towards the tasks of interest to the assistant.
Specifically, the ScopeIt is a neural model consisting of three parts: an
intra-sentence aggregrator which contextualizes information within sentence,
an inter-sentence aggregator which contextualizes information across
sentences, and a classifier.

##  3 Methodology

This section contains the details of the proposed deep RL framework, with
sections dedicated to the learning agent, the environment, and the learning
algorithm. The overall architecture is depicted in Figure 1(b).

###  3.1 The Learning Agent

The learning agent in our case is a sequence of a ScopeIt unit [1] and a NLU
model. The ScopeIt unit reduces the initial emails from the users to sentences
relevant to the assistant’s task, which will be supplied to the NLU model. In
our work, the ScopeIt module is modified to identify both task specific
sentences and sentences which provide surrounding information about the task.
The agent will learn a policy that maps the filtered email message to the
action based on the reward received from the environment. In this work, the
learning agent can either start from scratch or from a pre-trained model using
supervised learning.

###  3.2 The Environment

The environment models the dynamics of users’ interactions with the learning
agent. Upon receiving the actions from the agent, the workflow will
automatically generate a response based on the predicted action and the
existing data and knowledge base. Next, when users respond to the agent’s
action, their response may express implicit emotion towards the action of the
agents. Note that one key challenge in document-type conversations like emails
is associating the expressed emotion to the correct cause, as emotions not
emerging from the agent’s actions should be ignored. We modify ScopeIt module
to extract sentences expressing emotion towards the tasks of interest to the
assistant. Filtered sentences are embedded using BERT, generating an emotion
embedding. This emotion embedding is then used to classify the implicit
emotion into positive, negative, or neutral. The detected emotion is then
mapped to a numerical reward and fed back to the agent for updating the policy
network.

###  3.3 The Learning Algorithm

We optimize the NLU model by allowing the agent to interact with users and
learn from user feedback. We only use implicit emotions associated with the
task as the metric in designing the reward. A reward function $R(x)$ is
collected respectively for each positive (x = 1), negative (x = -1) and
neutral (x = 0) implicit feedback.

$R(x)=\begin{cases}+1&x=1\\\ -1&x=-1\\\ 0&x=0\end{cases}$

The agent always acts on-policy in order to ensure the agent is always acting
optimally for customers. This is accomplished through a softmax policy on the
actions/predictions from the policy network outputs. We apply the REINFORCE
algorithm [18] in optimizing the network parameters. Since the expectation of
the sample gradient is equal to the actual gradient, we measure returns from
real sample trajectories and use that to update our policy gradient.
Specifically,

$\displaystyle\nabla_{\theta}J(\theta)$
$\displaystyle=\mathbb{E}_{\pi}[Q^{\pi}(s,a)\nabla_{\theta}\ln(\pi_{\theta}(a|s))]$
$\displaystyle=\mathbb{E}_{\pi}[\nabla_{\theta}\ln(\pi_{\theta}(a|s))R]$

where $Q^{\pi}(s,a)$ is the value of state-action pair when we follow a policy
$\pi$.

##  4 Experiments

###  4.1 Dataset

The emotion detection model is trained offline on a private customer dataset
from our private preview service. The dataset for emotion recognition model is
built from the email messages that are either filtered or not filtered by the
ScopeIt module, and user feedback directed toward the actions of the agents.
The dataset is constructed through the following two steps.

Step 1: Identifying insertion positions We parse all email sentences based on
a set of punctuation (including comma, period, colon, question and exclamation
mark). Then, all positions after those punctuation are considered as candidate
locations for injecting emotional feedbacks.

Step 2: Inject users’ emotions We next inject the users’ emotions at the
candidate locations found in Step 1 randomly and label those samples. We also
randomly insert ”general positive or negative emotions” and label those
samples as neutral. This allows the agent to capture the nuance between
emotions directed toward the agent and general emotions.

###  4.2 Setup

We use the following transformer-based models for the emotion detection
problem: BERT, DistilBERT, ALBERT, and RoBERTa. For each model, a linear layer
is added on top of the pooled output to perform classification. We freeze some
transformer layers of all BERT-type models and the number of frozen layers are
learned in a trial-and-error manner. The model was end-to-end trained using
huggingface implementation [19] on 4 Nvidia K80 GPUs.

##  5 Results

###  5.1 Emotion recognition

We report both accuracy and the overall Macro F1 scores for all models in
Table 1. It is worth noting that the use of the ScopeIt model leads to
improved performance for all models. Both the BERT and the DistilBERT models
have similar accuracy. Hence, DistilBERT with two frozen layers is finally
selected due to its smaller size and lower inference latency.

Models | ScopeIt | F1 | Accuracy | Parameters
---|---|---|---|---
ALBERT |  | 89.93 | 90.36 | 11M
RoBERTa(-2) |  | 91.22 | 91.58 | 125M
BERT(-1) |  | 94.06 | 94.37 | 110M
DistilBERT(-1) |  | 94.15 | 94.45 | 66M
ALBERT | ✓ | 91.46 | 91.87 | 11M
RoBERTa(-2) | ✓ | 91.75 | 92.03 | 125M
BERT(-3) | ✓ | 94.96 | 95.42 | 110M
DistilBERT(-2) | ✓ | 94.92 | 95.39 | 66M
Table 1: Results for emotion recognition model. All scores are in percentage
and are reported at best accuracy. BERT(-1) represents the BERT classification
model with one frozen transformer layer.

(a) Multi-class training on random model

(b) Multi-class training on fine-tuned model

(c) Multi-label training on random model

(d) Multi-label training on fine-tuned model

Fig. 2: Learning curves for different model and settings

###  5.2 Multi-class Intent Detection

We first consider an intent detection model behind a conversational assistant
that assists users to schedule meetings. The assistant will use a DistilBERT
model to classify the intent of emails into three categories: modify a
meeting, cancel a meeting, and other (not relevant to task of meeting
scheduling). We first conduct experiments in learning the intent
classification model from scratch using only RL and summarize the results in
figure 2(a). We can see that all DistilBERT models start from random guesses,
but as the learning agent interacts more with users the task success rates
improve over time.

The type of feedback obtained is critical to NLU model training. Hence, three
different feedback mechanisms are studied in this work: full feedback, partial
feedback, and partial with noisy feedback. In the full feedback scenario,
every customer is assumed to leave implicit feedback. In partial feedback
scenario only 15% of requests are assumed to have implicit feedback. While in
partial with noisy feedback, out of the 15% partial feedback, one third of the
implicit feedback is incorrect. These feedback scenarios provide insight into
the NLU model performance and both quantity and quality of feedback required.

The orange curve shows the performance by assuming every customer leaves an
implicit feedback. Even under partial (i.e., 15%) feedback, the agent can
achieve comparable performance with the full feedback case after sufficient
number of turns. Finally, the agent learns more slowly and has lower accuracy
under the partial and noisy feedback case where one third of the 15% feedback
are wrongly labeled.

We next demonstrate the experimental results that use online RL training for a
limited data fine-tuned DistilBERT model. As shown in the blue line in figure
2(b), the supervised benchmark model only has about 63% accuracy. This is due
to limited eyes-on access to data and the domain mismatch of offline training
and online product scale-up. We study the same three feedback scenarios as
before. Note that a supervised pre-trained model leads to faster learning
rates for both full and partial feedback cases and a better accuracy for the
partial and noisy feedback.

###  5.3 Multi-label Intent Detection

We consider a multi-label intent classification model of the same
conversational assistant as before. In this case, there are a total of six
distinct actions and the action space is represented by a six-dimension vector
with 0s and 1s. Out of the total 64 ($2^{6}=64$) possible scenarios, only six
possible combinations are valid. The learning agent consists of an ensemble of
six separate DistilBERT binary classification models which work independently
to determine whether each single action should be taken or not.

We first train the learning agent from scratch where all six DistilBERT models
start from random guesses. The learning curves on accuracy for full, partial
as well as partial and noisy cases are shown in figure 2(c). We can see that
the learning accuracy improves for all three scenarios over time.
Specifically, the accuracy for both full and partial feedback can reach about
70%, which is nearly 10% higher than that of the partial noisy case.

We next use online RL training for the six separate fine-tuned DistilBERT
models from limited data. The learning curves are demonstrated in figure 2(d),
where the fine-tuned benchmark can be considerably improved over iterations
for all scenarios. Also, compared to learning from scratch, fine-tuned models
have faster learning rates and higher accuracies.

##  6 Conclusion

We propose a deep RL framework “NARLE” to improve the NLU models of a task-
oriented conversational assistant in an online manner for document-type
conversations. The proposed architecture scopes out emotion feedback relevant
to the assistant’s task and uses that as feedback to improve the performance
of the assistant in an adaptive way. The proposed framework is evaluated on
customer data, where the proposed method can improve a limited data fine-tuned
model up to 43%. We also show that the proposed method is robust to partial
and noisy feedback.

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#  Gravitational lens on de-Sitter background

Keita Takizawa  Hideki Asada  Graduate School of Science and Technology,
Hirosaki University, Aomori 036-8561, Japan

###### Abstract

Gravitational lenses are examined in de-Sitter (dS) background, for which the
existence of the dS horizon is taken into account and hyperbolic trigonometry
is used together with the hyperbolic angular diameter distance. Spherical
trigonometry is used to discuss a gravitational lens in anti-de Sitter (AdS)
background. The difference in the form among the dS/AdS lens equations and the
exact lens equation in Minkowski background begins at the third order, when a
small angle approximation is used in terms of lens and source planes. The
angular separation of lensed images is decreased by the third-order deviation
in the dS lens equation, while it is increased in AdS. In the present
framework on the dS/AdS backgrounds, we discuss also the deflection angle of
light, which does not include any term of purely the cosmological constant.
Despite the different geometry, the deflection angle of light rays in
hyperbolic and spherical geometry can take the same form. Through a coupling
of the cosmological constant with lens mass, the separation angle of multiple
images is larger (smaller) in dS (AdS) than in the flat case, for a given
mass, source direction, and angular diameter distances among the lens,
receiver and source.

###### pacs:

04.40.-b, 95.30.Sf, 98.62.Sb

##  I Introduction

Since the observation by Eddington and his collaborators Eddington , the
gravitational deflection of light has played an important role in astronomy
and gravitational physics. The Event Horizon Telescope (EHT) team has recently
succeeded a direct image of the immediate vicinity of the supermassive black
hole (BH) candidate of M87 galaxy EHT . In addition, the same team has
reported measurements of linear polarizations around the BH candidate EHT2021a
and has inferred the mass accretion rate and the strength of the magnetic
field EHT2021b . Such groundbreaking events have generated renewed interest in
the strong field gravitational lens.

Figure 1:  Intuitive illustration of how an orbit in a curved surface with a
boundary and its approximation on a flat background are different. The solid
red (in color) arrow denotes a curve on a bowl and the dashed blue (in color)
arrow denotes a line on a table. On the flat background, the dashed arrow
means an approximation of the curve on the bowl. The approximation is good at
small scale in the neighborhood of the bottom of the bowl, while the
difference of the two curves becomes significant at very large scale,
especially near the edge of the bowl. The edge has an analogy with the so-
called de-Sitter horizon in de-Sitter spacetime, though the latter is
traversable one way. The solid arrow terminates at the edge of the bowl. On
the other hand, one may ask if the dashed line can be more extended as
indicated by the dotted black (in color) arrow.

Gravitational lens equations, which relate source and image positions for a
given deflecting mass, are at the heart of the gravitational lens theory. For
most lens equations, the background spacetime is assumed to be Minkowskian. In
the flat background, Euclidean geometry holds for its spatial sector.
Therefore, Euclidean trigonometry is used to arrive at the standard form of
the lens equation SEF ; Petters ; Dodelson ; Keeton .

However, the current and future astronomy needs a more precise prediction from
a gravitational lensing theory especially relevant with the strong field
regime. The Schwarzschild de-Sitter spacetime (often called Kottler solution)
plays a theoretical model that allows us to examine the gravitational
deflection of light by a mass in the presence of the cosmological constant
$\Lambda$. Many attempts have been made on the simple model in the context of
the gravitational lensing e.g. Lake ; RI ; Finelli ; Park ; Sereno ; Bhadra ;
Simpson ; AK ; Ishihara2016 ; Piattella ; Butcher ; LW ; AB ; Arakida2021 ;
LLJ . Intuitively, the Minkowskian background can work as an approximation at
small scale, though there can be a significant departure of the Minkowskian
background from de-Sitter backgrounds especially at very large scale. See
Figure 1 for this intuition. Until recently, the definition for the deflection
angle of light has required the asymptotic flatness of a spacetime SEF ;
Petters ; Dodelson ; Keeton . However, recent works Ishihara2016 ;
Takizawa2020a  based on the Gauss-Bonnet theorem GW  have enabled us to well
define the deflection angle without assuming the asymptotic flatness. This new
method has been applied by several groups to various spacetime models
Ishihara2017 ; Jusufi2017a ; Jusufi2017b ; Jusufi2018 ; Ovgun2019 ; Ovgun2020
; Crisnejo2018 ; Crisnejo2019 ; Ono2017 ; Ono2018 ; Ono2019 . See Reference
Ono2019b  for a brief review on this subject.

Can the lens equation on de-Sitter backgrounds take the same form as that in
Minkowskian background? One may specifically ask if the de-Sitter lens
equation can be expressed in the same form when angular diameter distances are
properly defined. One possible answer is that there would exist a difference
between the two lens equations in the flat and de-Sitter backgrounds, even if
angular diameter distances are defined in a suitable manner. If this answer is
correct, what changes are made?

The main purpose of this paper is to examine gravitational lenses on de-Sitter
background. Henceforth, cases of $\Lambda>0$ and $\Lambda<0$ are referred to
as dS and AdS, respectively. A key tool in the present study is the optical
metric, which is known to describe light rays in stationary spacetimes AK2000
; GW ; Ishihara2016 ; Ono2017 . We shall show that the optical metric for dS
spacetime describes a hyperbolic space, while that for AdS spacetime
corresponds to spherical geometry. Therefore, we use hyperbolic trigonometry
in order to derive a gravitational lens equation on the dS background, where
we do not use a small angle approximation nor a thin lens one. For AdS case,
we use spherical trigonometry to derive a gravitational lens equation on the
AdS background. In both the derivations, we define angular diameter distances
on dS/AdS backgrounds.

This paper is organized as follows. In Section II, the optical metric is
examined on dS/AdS backgrounds. By using hyperbolic trigonometry, in Section
III, we discuss gravitational lens equations on dS background. In Section IV,
we make use of spherical trigonometry to study gravitational lens equations on
AdS background. In Section V, the Euclidean, hyperbolic and spherical lens
equations are unified into a single equation. The deflection angle of light on
dS/AdS backgrounds also is examined. Finally, this paper is summarized in
Section VI. Throughout this paper, we use the unit of $G=c=1$.

##  II Optical metric for dS/AdS spacetimes

###  II.1 Hyperbolic space: dS case

There are several known ways for slicing of dS spacetime, where the scalar
curvature of dS universe is positive. We work on the static coordinates of dS
spacetime, where the dS metric reads

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{\Lambda}{3}r^{2}\right)dt^{2}+\cfrac{dr^{2}}{1-\cfrac{\Lambda}{3}r^{2}}+r^{2}(d\Theta^{2}+\sin^{2}\Theta
d\phi^{2}),$ (1)

where $r_{H}\equiv\sqrt{3/\Lambda}$ is a radius of the dS horizon.

For later convenience, we normalize a radial coordinate in terms of the
cosmological constant as

$R\equiv\sqrt{\frac{\Lambda}{3}}r,$ (2)

such that dS metric can be reexpressed as

$\displaystyle ds^{2}=$ $\displaystyle-\left(1-R^{2}\right)dt^{2}$
$\displaystyle+\frac{3}{\Lambda}\left(\cfrac{dR^{2}}{1-R^{2}}+R^{2}(d\Theta^{2}+\sin^{2}\Theta
d\phi^{2})\right).$ (3)

In the present paper, we focus on a case that the lens, receiver and source
are located inside the dS horizon, namely $R<1$.

We consider the optical metric followed by the null condition $ds^{2}=0$,
which determines the light propagation AK2000 ; GW ; Ishihara2016

$\displaystyle d\ell^{2}$ $\displaystyle\equiv dt^{2}$
$\displaystyle=\frac{3}{\Lambda}\left(\cfrac{dR^{2}}{(1-R^{2})^{2}}+\frac{R^{2}}{1-R^{2}}(d\Theta^{2}+\sin^{2}\Theta
d\phi^{2})\right).$ (4)

Clearly, it is convenient to work on the conformally rescaled metric as

$\displaystyle d\hat{\ell}^{2}$
$\displaystyle\equiv\frac{\Lambda}{3}d\ell^{2}$
$\displaystyle=\cfrac{dR^{2}}{(1-R^{2})^{2}}+\frac{R^{2}}{1-R^{2}}(d\Theta^{2}+\sin^{2}\Theta
d\phi^{2}).$ (5)

We define a new radial coordinate $\rho$ by footnote-1

$\displaystyle R\equiv\tanh\rho,$ (6)

to reexpress the rescaled metric as

$\displaystyle d\hat{\ell}^{2}$
$\displaystyle=d\rho^{2}+\sinh^{2}\rho(d\Theta^{2}+\sin^{2}\Theta d\phi^{2}).$
(7)

Note that $\rho$ coordinate is different from $\chi$ coordinate that is
usually used in general relativistic cosmology. Eq. (7) means that the optical
metric in dS case describes a hyperbolic space, in which unlensed light rays
are geodesic curves.

The dS horizon is located at $\rho=+\infty$ in the metric Eq. (7). This means
that $\rho$ coordinate describes only the inside of the dS horizon. On the
other hand, it is apparent that the $r$ coordinate can go beyond the dS
horizon, if one treats perturbatively Eq. (1) around the Minkowskian
background even at very large scale.

###  II.2 Spherical space: AdS case

For AdS case ($\Lambda<0$), all we have to do is to replace $\sin\chi$ by
$\sinh\chi$. For this, we introduce footnote-2

$\displaystyle R$ $\displaystyle\equiv\sqrt{\frac{-\Lambda}{3}}r,$ (8)
$\displaystyle R$ $\displaystyle\equiv\tan\rho,$ (9)

to obtain the normalized optical metric as

$\displaystyle d\hat{\ell}^{2}$
$\displaystyle=d\rho^{2}+\sin^{2}\rho(d\Theta^{2}+\sin^{2}\Theta d\phi^{2}).$
(10)

This is a metric of spherical geometry.

##  III Gravitational lens in hyperbolic geometry

###  III.1 Hyperbolic trigonometry

We consider a photon orbit around a spherically symmetric lens at the origin
of the spatial coordinates $\rho=0$. Because of the symmetry, we can choose
the equatorial plane $\Theta=\pi/2$ as a photon orbit for its simplicity. This
choice can be done also for a photon orbit in the equatorial plane of an
axisymmetric lens with reflection symmetry.

Then, we can use the conformally normalized optical metric of a hyperbolic
plane as

$\displaystyle d\hat{\ell}^{2}$ $\displaystyle=d\rho^{2}+\sinh^{2}\rho
d\phi^{2}.$ (11)

It follows that the hyperbolic plane is homogeneous and isotropic. This allows
us to explicitly define angular diameter distance in a simple manner. See
Section III for more detail.

We consider hyperbolic triangles in order to discuss a gravitational lens
configuration on a hyperbolic plane. Figure 2 is a schematic illustration of
the gravitational lens configuration in the hyperbolic plane. For a
comparison, see also Figure 3 for a flat background.

Figure 2:  Schematic figure of a lens $L$, receiver $R$ and source $S$ in a
hyperbolic plane. For its simplicity, a line connecting L and R is drawn as a
straight one for reference. The red (in color) curve connecting R and S
denotes a lensed light ray, which is not a geodesic in the hyperbolic plane. A
dotted geodesic curve emanating from S is a tangent to the lensed light ray,
while the other dotted one from R is another tangent to the light ray. The
latter dotted line indicates the lensed image direction $\theta(=\Psi_{R})$
seen from the receiver. A geodesic curve between R and S is denoted by a long
dashed line, which indicates the unlensed source direction $\beta$. The lens
and source planes are vertical to the geodesic line RU. These curves are not
parallel to each other because of hyperbolicity. Note that the sum of the
inner angles for the hyperbolic quadrilateral LRVS does not equal to $2\pi$
according to the Gauss-Bonnet theorem for the curved surface (See e.g.
Ishihara2016 ; Takizawa2020b ). Therefore, the outer angle at the intersection
point V of the two tangents in the hyperbolic plane differs from
$\alpha(=\Psi_{R}-\Psi_{S}+\phi_{RS})$. In fact, the intersection V has
nothing to do with the derivation of the lens equations in this paper.  Figure
3:  A gravitational lens system in a Euclidean background Takizawa2020b . The
notations are the same as those in Figure 2. The lens and source planes are
parallel to each other, because they live in a Euclidean space.

First, we briefly summarize hyperbolic trigonometry Anderson-book ; Ratcliffe-
book , with which most physicists may not be familiar. Let us consider a
triangle $ABC$, where $A$, $B$ and $C$ denote the vertices of the triangle and
they mean also the inner angles. The sides of the triangle follow geodesics in
the hyperbolic plane. See Figure 4 for the hyperbolic triangle $ABC$.

Figure 4:  Schematic figure of a hyperbolic triangle ABC. The vertices are
denoted by A, B and C. The sides of the triangle are geodesics indicated by a
thin solid line. The arc length of them is $\rho_{AB}$, $\rho_{BC}$ and
$\rho_{CA}$.

In all the formulae stated here, arc length of the sides $a$, $b$ and $c$ are
measured in absolute unit, in which the Gaussian curvature $K$ of the plane is
$-1$. For instance, in this paper, the absolute length of the side $AB$ (which
is in the normalized optical metric) is denoted as $\rho_{AB}$, which is
different from the proper length measured by the original optical metric. In
the case that $A$ is the receiver, we shall omit it simply as $\rho_{B}$,
because the receiver is often chosen as the coordinate origin in the
gravitational lens study.

The hyperbolic law of sines is

$\displaystyle\frac{\sinh\rho_{BC}}{\sin A}=\frac{\sinh\rho_{CA}}{\sin
B}=\frac{\sinh\rho_{AB}}{\sin C}.$ (12)

The hyperbolic law of cosines is

$\displaystyle\cosh\rho_{AB}=\cosh\rho_{BC}\cosh\rho_{CA}-\sinh\rho_{BC}\sinh\rho_{CA}\cos
C.$ (13)

The trigonometry of right angles is as follows. If $C=\pi/2$, then

$\displaystyle\sin A=$ $\displaystyle\frac{\sinh\rho_{BC}}{\sinh\rho_{AB}},$
(14) $\displaystyle\cos A=$
$\displaystyle\frac{\cosh\rho_{BC}\sinh\rho_{CA}}{\sinh\rho_{AB}}$
$\displaystyle=$ $\displaystyle\frac{\tanh\rho_{CA}}{\tanh\rho_{AB}},$ (15)
$\displaystyle\cosh\rho_{AB}=$ $\displaystyle\cosh\rho_{BC}\cosh\rho_{CA}.$
(16)

Eq. (16) is a hyperbolic generalization of Pythagorean theorem. It recovers
$\rho_{AB}^{2}=\rho_{BC}^{2}+\rho_{CA}^{2}$ at very small scale of
$\rho_{AB}\ll 1$.

###  III.2 Lens, receiver and source in a hyperbolic plane

For the right triangle PRL, we obtain

$\displaystyle\sin\theta=$
$\displaystyle\frac{\sinh\rho_{PL}}{\sinh\rho_{L}},$ (17)

where Eq. (14) is used. Similarly, for the right triangle LQS,

$\displaystyle\sin(\pi-\Psi_{S})=$
$\displaystyle\frac{\sinh\rho_{QL}}{\sinh\rho_{LS}}.$ (18)

Both of $\rho_{PL}$ and $\rho_{QL}$ mean the arc length $\rho_{b}$ of the
impact parameter of a single light ray in the normalized optical metric. See
Figure 2. For the single light ray, they must be equal to each other.
Therefore, $\rho_{PL}=\rho_{QL}$. By eliminating $\rho_{PL}$ and $\rho_{QL}$
from Eqs. (17) and (18), we thus find a relation between $\Psi_{S}$ and
$\theta$ as

$\displaystyle\Psi_{S}=\pi-\arcsin\left(\frac{\sinh\rho_{L}}{\sinh\rho_{LS}}\sin\theta\right).$
(19)

Next, we consider the right triangle LSU, where the point U is chosen such
that the side SU is perpendicular to LU. In the hyperbolic plane, the inner
angle between LR and LS is the same as the longitude $\phi_{RS}$ in dS
spacetime. Using Eq. (14) at the vertex L of LSU, we find

$\displaystyle\sin(\pi-\phi_{RS})=\frac{\sinh\rho_{SU}}{\sinh\rho_{LS}}.$ (20)

Similarly, from the triangle RSU, we obtain

$\displaystyle\sin\beta=\frac{\sinh\rho_{SU}}{\sinh\rho_{S}}.$ (21)

Here, $\beta$ denotes the angle $R$, which is the same as the directional
angle of the unlensed source in dS spacetime. From Eqs. (20) and (21), we
obtain a relation between the longitude $\phi_{RS}$ and the source direction
$\beta$ as

$\displaystyle\phi_{RS}=\pi-\arcsin\left(\frac{\sinh\rho_{S}}{\sinh\rho_{LS}}\sin\beta\right).$
(22)

We work on the angle of the gravitational deflection of light that has been
defined by Ishihara et al. as Ishihara2016

$\displaystyle\alpha\equiv\theta-\Psi_{S}+\phi_{RS}.$ (23)

It has been recently proven that the definition holds even in a non-
asymptotically flat spacetime Takizawa2020b .

By substituting Eqs. (19) and (22) into Eq. (23), we obtain

$\displaystyle\alpha-\theta=\arcsin\left(\frac{\sinh\rho_{L}}{\sinh\rho_{LS}}\sin\theta\right)-\arcsin\left(\frac{\sinh\rho_{S}}{\sinh\rho_{LS}}\sin\beta\right).$
(24)

###  III.3 Hyperbolic angular diameter distance

In the normalized hyperbolic plane, $\hat{d}$ denotes the angular diameter
distance between two points, while we use the conventional notation $d$ for
the angular diameter distance in the original metric. In dS case, the physical
angular diameter distance is $d=\hat{d}\sqrt{3/\Lambda}$. From Eq. (11), the
angular diameter distance is defined as

$\displaystyle\hat{d}_{L}$ $\displaystyle\equiv\sinh\rho_{L},$ (25)
$\displaystyle\hat{d}_{S}$ $\displaystyle\equiv\sinh\rho_{S},$ (26)
$\displaystyle\hat{d}_{LS}$ $\displaystyle\equiv\sinh\rho_{LS}.$ (27)

Thereby, Eq. (24) can be rewritten explicitly in terms of the angular diameter
distance as

$\displaystyle\alpha-\theta=\arcsin\left(\frac{\hat{d}_{L}}{\hat{d}_{LS}}\sin\theta\right)-\arcsin\left(\frac{\hat{d}_{S}}{\hat{d}_{LS}}\sin\beta\right).$
(28)

Eq. (28) takes the same form as Eq. (17) of Reference Takizawa2020b  for the
flat background footnote-3 . It is not a surprising coincidence, because the
both derivations do not rely on whether the quadrilateral LRQS lives in a flat
space. The sum of the internal angles is not necessarily $L+R+V+S=2\pi$.
Indeed, the formulation by Takizawa et al. Takizawa2020b  stands on fully
curved backgrounds. However, this point has not been stressed Takizawa2020b .

We should note also that the addition law holds for the flat angular diameter
distance in a Euclidean space, but it does not in a curved space. Indeed,
$\hat{d}_{L}+\hat{d}_{LS}\neq\hat{d}_{S}$ in hyperbolic geometry.

###  III.4 Lens and source planes in hyperbolic geometry

In the above, we have considered the angular diameter distance between points.
On the other hand, most studies on the gravitational lens employ angular
diameter distances between a point (usually chosen as the receiver position)
and the so-called lens plane (or the source plane). In a Euclidean space, the
lens and source planes are parallel to each other. In a non-Euclidean space,
however, the two planes are not parallel to each other.

In the hyperbolic space, we define a lens plane as a surface which consists of
a family of geodesics, every of which is vertical to the geodetic line LR at
the point L. We consider a point in the hyperbolic space, at which the
geodetic line connecting L and R is vertical to another geodetic emanating
from the source point S. We denote the point as U. By the same way, we define
a source plane that is vertical to the geodetic line connecting L and R at the
point U. See Figure 2 for the lens and source planes.

In terms of the lens and source planes, we can define the angular diameter
distances. The normalized angular diameter distance from the receiver to the
lens plane is defined as that from the receiver to the point L on the lens
plane. That is,

$\displaystyle\hat{D}_{L}$ $\displaystyle\equiv\hat{d}_{L}$
$\displaystyle=\sinh\rho_{L},$ (29)

where we use Eq. (25).

In the similar manner, the normalized angular distance from the receiver to
the hyperbolic source plane is defined as that from the receiver to the point
U on the source plane

$\displaystyle\hat{D}_{S}\equiv\sinh\rho_{U}.$ (30)

The normalized angular distance between the lens and source planes is defined
as that from the point L to the point U

$\displaystyle\hat{D}_{LS}$ $\displaystyle\equiv\sinh\rho_{LU}.$ (31)

Note that $\hat{D}_{L}+\hat{D}_{LS}\neq\hat{D}_{S}$ in hyperbolic geometry.
This is because $\sinh\rho_{L}+\sinh\rho_{LU}\neq\sinh\rho_{U}$, though
$\rho_{L}+\rho_{LU}=\rho_{U}$. Again, we should note that the physical angular
diameter distance needs the factor as $D=\hat{D}\sqrt{3/\Lambda}$.

By using the cosine formula Eq. (15) for the right triangle RSU, we obtain

$\displaystyle\cos\beta=\frac{\cosh\rho_{SU}\sinh\rho_{U}}{\sinh\rho_{S}}.$
(32)

From Eqs. (21) and (32), we find

$\displaystyle\tan\beta$ $\displaystyle=\frac{\sin\beta}{\cos\beta}$
$\displaystyle=\frac{\tanh\rho_{SU}}{\sinh\rho_{U}}.$ (33)

By using this for Eq. (30), we obtain

$\displaystyle\hat{D}_{S}=\frac{\tanh\rho_{SU}}{\tan\beta}.$ (34)

From this, we can see

$\displaystyle\sinh^{2}\rho_{SU}=\frac{\hat{D}_{S}^{2}\tan^{2}\beta}{1-\hat{D}_{S}^{2}\tan^{2}\beta}.$
(35)

By using Eq. (16) for the right triangle LSU, we obtain

$\displaystyle\cosh\rho_{LS}=\cosh\rho_{SU}\cosh\rho_{LU}.$ (36)

This leads to

$\displaystyle\sinh^{2}\rho_{LS}$ $\displaystyle=\cosh^{2}\rho_{LS}-1$
$\displaystyle=\left(\cosh\rho_{LU}\cosh\rho_{SU}\right)^{2}-1$
$\displaystyle=\frac{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}{1-\hat{D}_{S}^{2}\tan^{2}\beta},$
(37)

where Eqs. (31) and (35) are used in the last line.

Next, we consider the angle L in the right triangle LSU. We find

$\displaystyle\sin L=\frac{\hat{d}_{S}}{\hat{d}_{LS}}\sin\beta,$ (38)

where Eq. (14) is used. We thus obtain

$\displaystyle\tan L$ $\displaystyle=\frac{\sin L}{\cos L}$
$\displaystyle=\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta,$ (39)

where Eqs. (15), (16) and (37) are used. By combining Eqs. (38) and (39), we
obtain

$\displaystyle\arcsin\left(\frac{\hat{d}_{S}}{\hat{d}_{LS}}\sin\beta\right)=\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right).$
(40)

In terms of the angular diameter distance $\hat{D}$, Eq. (28) can be
reexpressed as

$\displaystyle\alpha-\theta=$
$\displaystyle\arcsin\left(\sqrt{\frac{1-\hat{D}_{S}^{2}\tan^{2}\beta}{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}}\hat{D}_{L}\sin\theta\right)$
$\displaystyle-\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right),$
(41)

where Eqs. (29), (30), (31), (33) (37) and (40) are used. The inside of the
square root in Eq. (41) must be nonnegative. Therefore,

$\displaystyle|\tan\beta|\leq\frac{1}{\hat{D}_{S}},$ (42)

which gives the allowed region of the source direction. This is due to the
existence of the dS horizon.

The hyperbolic lens equation Eq. (41) is slightly different from the Takizawa
lens equation in the flat background as Takizawa2020b

$\displaystyle\alpha-\theta=$
$\displaystyle\arcsin\left(\frac{D_{L}}{\sqrt{D_{LS}^{2}+D_{S}^{2}\tan^{2}\beta}}\sin\theta\right)$
$\displaystyle-\arctan\left(\frac{D_{S}}{D_{LS}}\tan\beta\right).$ (43)

The only difference is caused because Euclidean Pythagorean theorem does not
stand in hyperbolic space, especially for the right triangle such as RSU and
LSU. In other words, the methods of deriving Eqs. (41) and (43) do depend on
whether the triangle LSU lives in a flat space.

Eqs. (41) and (43) bear a striking resemblance to each other, though they are
based on two completely different geometry. Does it mean that the cosmological
constant makes almost no effects on gravitational lens observations? No. Eq.
(41) is written by using the angular diameter distances in hyperbolic geometry
due to the presence of the cosmological constant. This means that the
cosmological constant significantly affects the gravitational lens.

##  IV Gravitational lens in spherical geometry

###  IV.1 Spherical trigonometry

In the similar manner to the previous section, we focus on photon orbits on
the equatorial plane $\theta=\pi/2$ in spherical geometry, which corresponds
to AdS case. Then, the normalized optical metric in the plane is

$\displaystyle d\hat{\ell}^{2}$ $\displaystyle=d\rho^{2}+\sin^{2}\rho
d\phi^{2}.$ (44)

Let us briefly summarize the spherical trigonometry Ratcliffe-book . See
Figure 5 for a spherical triangle $ABC$.

Figure 5:  Schematic figure of a spherical triangle. The notations are the
same as those in Figure 4

The laws of sines and cosines are

$\displaystyle\frac{\sin\rho_{BC}}{\sin A}=\frac{\sin\rho_{CA}}{\sin
B}=\frac{\sin\rho_{AB}}{\sin C},$ (45)
$\displaystyle\cos\rho_{AB}=\cos\rho_{BC}\cos\rho_{CA}-\sin\rho_{BC}\sin\rho_{CA}\cos
C.$ (46)

The spherical trigonometry of right angles is as follows. If $C=\pi/2$,

$\displaystyle\sin A=$ $\displaystyle\frac{\sin\rho_{BC}}{\sin\rho_{AB}},$
(47) $\displaystyle\cos A=$
$\displaystyle\frac{\cos\rho_{BC}\sin\rho_{CA}}{\sin\rho_{AB}}$
$\displaystyle=$ $\displaystyle\frac{\tan\rho_{CA}}{\tan\rho_{AB}},$ (48)
$\displaystyle\cos\rho_{AB}=$ $\displaystyle\cos\rho_{BC}\cos\rho_{CA},$ (49)

where the last equation is a generalization of Pythagorean theorem in
spherical geometry. Eq. (49) approaches
$\rho_{AB}^{2}=\rho_{BC}^{2}+\rho_{CA}^{2}$ at very small scale of
$\rho_{AB}\ll 1$.

###  IV.2 Gravitational lens configuration in spherical geometry

It is usually convenient to use a correspondence between hyperbolic functions
and spherical ones, when we wish to obtain expressions in spherical geometry
from known hyperbolic ones, and vice versa. If this correspondence were
applied to Eq. (41), this equation would remain the same, because it does not
include any hyperbolic function. However, this is not the case as shown below.

Figure 6 shows a gravitational lens system in spherical geometry. For the
right triangle PRL, we obtain

$\displaystyle\sin\theta=$ $\displaystyle\frac{\sin\rho_{PL}}{\sin\rho_{L}},$
(50)

where Eq. (47) is used. From the right triangle LQS,

$\displaystyle\sin(\pi-\Psi_{S})=$
$\displaystyle\frac{\sin\rho_{QL}}{\sin\rho_{LS}}.$ (51)

Figure 6:  A gravitational lens system in AdS background. The notations are
the same as those in Figure 2. The lens and source planes are not parallel to
each other, because they live in a curved space.

For the single light ray, $\rho_{PL}$ and $\rho_{QL}$ must be equal to each
other, because it means the impact parameter. Hence, $\rho_{PL}=\rho_{QL}$. By
using this for Eqs. (50) and (51), we obtain

$\displaystyle\Psi_{S}=\pi-\arcsin\left(\frac{\sin\rho_{L}}{\sin\rho_{LS}}\sin\theta\right).$
(52)

Next, we consider the right triangle LSU. Using Eq. (47) at the vertex L of
LSU, we find

$\displaystyle\sin(\pi-\phi_{RS})=\frac{\sin\rho_{SU}}{\sin\rho_{LS}}.$ (53)

For the triangle RSU, we obtain

$\displaystyle\sin\beta=\frac{\sin\rho_{SU}}{\sin\rho_{S}},$ (54)

where $\beta$ is the angle $R$, namely the (unlensed) source angle in AdS
spacetime.

From Eqs. (53) and (54), a relation between $\phi_{RS}$ and $\beta$ is found
as

$\displaystyle\phi_{RS}=\pi-\arcsin\left(\frac{\sin\rho_{S}}{\sin\rho_{LS}}\sin\beta\right).$
(55)

By substituting Eqs. (52) and (55) into Eq. (23), we obtain

$\displaystyle\alpha-\theta=\arcsin\left(\frac{\sin\rho_{L}}{\sin\rho_{LS}}\sin\theta\right)-\arcsin\left(\frac{\sin\rho_{S}}{\sin\rho_{LS}}\sin\beta\right).$
(56)

###  IV.3 Angular diameter distance in spherical geometry

In the normalized spherical surface, we consider the angular diameter distance
between two points as $\hat{d}$, while $d$ denotes the angular diameter
distance in the original spherical space. Namely,
$d=\hat{d}\sqrt{3/(-\Lambda)}$. The normalized angular diameter distances are
defined as

$\displaystyle\hat{d}_{L}$ $\displaystyle\equiv\sin\rho_{L},$ (57)
$\displaystyle\hat{d}_{S}$ $\displaystyle\equiv\sin\rho_{S},$ (58)
$\displaystyle\hat{d}_{LS}$ $\displaystyle\equiv\sin\rho_{LS},$ (59)

such that Eq. (56) can be rewritten simply as

$\displaystyle\alpha-\theta=\arcsin\left(\frac{\hat{d}_{L}}{\hat{d}_{LS}}\sin\theta\right)-\arcsin\left(\frac{\hat{d}_{S}}{\hat{d}_{LS}}\sin\beta\right).$
(60)

Eq. (60) takes the same form as Eq. (17) of Reference Takizawa2020b  for the
flat background. Note that $\hat{d}_{L}+\hat{d}_{LS}\neq\hat{d}_{S}$ also in
spherical geometry.

###  IV.4 Lens and source planes in spherical geometry

In spherical geometry, we consider lens and source planes as shown by Figure
6.

For the lens and source planes, we define the normalized angular diameter
distance from the receiver to the lens plane as

$\displaystyle\hat{D}_{L}$ $\displaystyle\equiv\hat{d}_{L}$
$\displaystyle=\sin\rho_{L},$ (61)

where we use Eq. (57). By the same way, the normalized angular distance from
the receiver to the source plane is defined as

$\displaystyle\hat{D}_{S}\equiv\sin\rho_{RU}.$ (62)

The normalized angular distance between the lens and source planes is the same
as that between the points L and U,

$\displaystyle\hat{D}_{LS}$ $\displaystyle\equiv\sin\rho_{LU}.$ (63)

Note that $\rho_{L}+\rho_{LU}=\rho_{U}$ but
$\hat{D}_{L}+\hat{D}_{LS}\neq\hat{D}_{S}$ in spherical geometry, because
$\sin\rho_{L}+\sin\rho_{LU}\neq\sin\rho_{U}$. We should remember also that the
physical angular diameter distance can be obtained from the normalized one by
$D=\hat{D}\sqrt{3/(-\Lambda)}$.

By using the cosine formula Eq. (48) for the right triangle RSU, we obtain

$\displaystyle\cos\beta=\frac{\cos\rho_{SU}\sin\rho_{U}}{\sin\rho_{S}}.$ (64)

From Eqs. (54) and (64), we find

$\displaystyle\tan\beta$ $\displaystyle=\frac{\sin\beta}{\cos\beta}$
$\displaystyle=\frac{\tan\rho_{SU}}{\sin\rho_{U}}.$ (65)

We use this for Eq. (62) to obtain

$\displaystyle\hat{D}_{S}=\frac{\tan\rho_{SU}}{\tan\beta}.$ (66)

From this, we can see

$\displaystyle\sin^{2}\rho_{SU}=\frac{\hat{D}_{S}^{2}\tan^{2}\beta}{1-\hat{D}_{S}^{2}\tan^{2}\beta}.$
(67)

Using Eq. (49) for the right triangle LSU leads to

$\displaystyle\cos\rho_{LS}=\cos\rho_{SU}\cos\rho_{LU}.$ (68)

This leads to

$\displaystyle\sin^{2}\rho_{LS}$ $\displaystyle=1-\cos^{2}\rho_{LS}$
$\displaystyle=1-\left(\cos\rho_{LU}\cos\rho_{SU}\right)^{2}$
$\displaystyle=\frac{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}{1+\hat{D}_{S}^{2}\tan^{2}\beta},$
(69)

where Eqs. (63) and (67) are used in the last line.

By the same way to Eq. (40), we consider the angle L in the right triangle LSU
to obtain also in spherical geometry

$\displaystyle\arcsin\left(\frac{\hat{d}_{S}}{\hat{d}_{LS}}\sin\beta\right)=\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right).$
(70)

In terms of the angular diameter distance $\hat{D}$, Eq. (60) is rearranged as

$\displaystyle\alpha-\theta=$
$\displaystyle\arcsin\left(\sqrt{\frac{1+\hat{D}_{S}^{2}\tan^{2}\beta}{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}}\hat{D}_{L}\sin\theta\right)$
$\displaystyle-\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right),$
(71)

where Eqs. (61), (62), (63), (65) (69) and (70) are used.

There exists the only difference between Eqs. (41) and (71). The sign of one
term in the square root is opposite. For this reason, the present section
avoids a conventional method of only replacing hyperbolic functions by
spherical ones. The sign difference comes from Eqs. (37) and (69) for
$\sinh\rho_{LS}^{2}$ and $\sin\rho_{LS}^{2}$, respectively.

We should note that Eq. (71) is based on the angular diameter distances in
spherical geometry due to the presence of the negative cosmological constant.

##  V Discussions

###  V.1 Unified form in Euclidean, hyperbolic and spherical geometry

Eqs. (28) and (60) are in the same form as the flat lens equation. However,
they are not in practical use, because the receiver cannot directly measure
the distance from the lens to the source though the distance from the receiver
to the source is in principle a direct observable. Hence, the lens equations
in terms of the lens and source planes have much more practical use SEF ;
Petters ; Dodelson ; Keeton .

First, we unify Eqs. (41), (43) and (71) in a single form as

$\displaystyle\alpha-\theta=$
$\displaystyle\arcsin\left(\sqrt{\frac{1+K\hat{D}_{S}^{2}\tan^{2}\beta}{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}}\hat{D}_{L}\sin\theta\right)$
$\displaystyle-\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right),$
(72)

where $K$ denotes $1$, $0$ and $-1$ for spherical, flat and hyperbolic
geometry, respectively. $K$ is corresponding to the Gaussian curvature of the
normalized background surface.

How does the only difference among Eqs. (41), (43) and (71) affect the lensed
image position? What is the physical effect of the $K$ term in Eq. (72)? To
investigate this issue below, we shall use small angle approximations to
follow the method by Takizawa et al. (2020b) Takizawa2020b .

###  V.2 Iterative behaviors

Let us introduce a book-keeping parameter $\varepsilon$. For a given source
position angle as $\beta=\varepsilon\beta_{(1)}$, the image position angle and
the deflection angle are expanded in a Taylor series as

$\displaystyle\theta$
$\displaystyle=\sum_{k=1}^{\infty}\varepsilon^{k}\theta_{(k)},$ (73)
$\displaystyle\alpha$
$\displaystyle=\sum_{k=1}^{\infty}\varepsilon^{k}\alpha_{(k)}.$ (74)

For the third-order solution in a flat background, please refer to Eqs.
(56)-(58) of Reference Takizawa2020b  for instance. In the similar
calculations to Ref. Takizawa2020b , we find the third-order solution for the
lens equation in hyperbolic/spherical geometry. The new term appears owing to
the background curvature. From Eq. (72), we obtain the new term as

$\displaystyle\theta_{(3)}^{New}=K\frac{\hat{D}_{L}\hat{D}_{S}^{2}}{2\hat{D}_{LS}^{2}(\hat{D}_{L}+\hat{D}_{LS})}\beta_{(1)}^{2}\theta_{(1)}.$
(75)

At the third order level, $\Lambda$ ($K=-1$) decreases $\theta$ compared with
in the flat background when the angular diameter distances are the same as
each other, while $\theta$ is increased in AdS ($K=1$). However, there is
subtlety in this statement. Eqs. (56)-(58) of Reference Takizawa2020b  uses
the addition law $D_{S}=D_{L}+D_{LS}$ because of the flat background. In a
curved background, however, they have to be modified because
$\hat{D}_{L}+\hat{D}_{LS}\neq\hat{D}_{S}$.

The totally modified form is thus

$\displaystyle\theta_{(3)}=$
$\displaystyle\frac{\hat{D}_{LS}}{\hat{D}_{L}+\hat{D}_{LS}}\alpha_{(3)}-\frac{\hat{D}_{S}}{3(\hat{D}_{L}+\hat{D}_{LS})}\left(1-\frac{\hat{D}_{S}^{2}}{\hat{D}_{LS}^{2}}\right)\beta_{(1)}^{3}$
$\displaystyle-\frac{(1-K)\hat{D}_{L}\hat{D}_{S}^{2}}{2\hat{D}_{LS}^{2}(\hat{D}_{L}+\hat{D}_{LS})}\beta_{(1)}^{2}\theta_{(1)}$
$\displaystyle-\frac{\hat{D}_{L}}{6(\hat{D}_{L}+\hat{D}_{LS})}\left(1-\frac{\hat{D}_{L}^{2}}{\hat{D}_{LS}^{2}}\right)\theta_{(1)}^{3},$
(76)

where the new term makes a correction in the second line. Instead of assuming
a specific model of the lens object, here, we consider a general one, for
which the deflection angle of light at $O(\varepsilon^{3})$ is denoted as
$\alpha_{(3)}$. $\alpha_{(3)}$ can be calculated by using the lower order
solutions $\theta_{(1)}$ and $\theta_{(2)}$ for a given lens model. In some
model, $\theta_{(2)}$ vanishes Takizawa2020b .

To be rigorous, $\theta$ is not always a direct observable, because it is the
angle measured from the lens direction but the direction is unknown in several
cases. On the other hand, the separation angle between two lensed images, each
of which is located on the opposite sides of the lens, is a direct observable.
The separation angle is decreased (increased) by not $\Lambda>0$ ($\Lambda<0$)
alone but its coupling with the lens mass.

The above correction due to the new $K$ term is a mathematical consequence of
introducing the lens and source planes into curved backgrounds.

###  V.3 Deflection angle of light on dS/AdS backgrounds

Up to this point, we have not mentioned details of $\alpha$. Takizawa et al.
have shown Eq. (23) for the deflection angle of light can be justified even in
a non-asymptotically flat spacetime Takizawa2020a . They have calculated
$\alpha$ in the Schwarzschild de-Sitter spacetime. For their $\alpha$, the
background spacetime is implicitly Minkowskian, because $\alpha$ vanishes only
when $M=0$ and $\Lambda=0$. Hence, their $\alpha$ should be used in Eq. (43).

On the other hand, Eqs. (28) and (41) are valid in hyperbolic geometry due to
the positive cosmological constant, while Eqs. (60) and (71) hold in spherical
geometry due to the negative cosmological constant.

In the present method, the effects of purely the cosmological constant are
included in the angular diameter distance of the dS/AdS backgrounds.
Therefore, the deflection angle of light $\alpha^{dS}$ is subtracted by the
effects of purely $\Lambda$. For the clarity, we denote the deflection angle
of light explicitly as $\alpha^{dS}(p_{i},\Lambda)$, where the lens object is
parameterized by $p_{i}$ ($i=1,2,\cdots$) in addition to $\Lambda$. In the
Kerr de-Sitter spacetime for instance, $p_{i}$ corresponds to the mass or spin
parameter. The deflection angle of light on dS/AdS backgrounds is thus

$\displaystyle\alpha^{dS}\equiv\alpha(p_{i},\Lambda)-\alpha(p_{i}=0,\Lambda).$
(77)

Let us explain why Eq. (77) is justified. See Figure 7 for two triangles LRS
in a hyperbolic plane. One triangle LRS has a side RS that is a hyperbolic
geodesic indicated by a dashed blue (in color) line, while the other LRS has
another side RS that means a true light ray denoted by a solid red (in color)
line. From Figure 7, we find
$\alpha(p_{i},\Lambda)=\Psi_{R}-\Psi_{S}+\phi_{RS}$, and
$\alpha(p_{i},=0,\Lambda)=\Psi_{R}^{dS}-\Psi_{S}^{dS}+\phi_{RS}$. By using Eq.
(77), therefore, we obtain

$\displaystyle\alpha^{dS}=(\Psi_{R}-\Psi_{R}^{dS})+(\Psi_{S}^{dS}-\Psi_{S}).$
(78)

This allows us to interpret $\alpha^{dS}$ as the deflection angle of the light
ray (the solid red (in color) line) with respect to the reference line (the
dashed blue (in color) line). This point is explained also in the caption of
Figure 7. As a result, Eq. (77) has the meaning of the deflection angle of
light on the dS background footnote-4 . By the same way, one can see that Eq.
(77) gives the deflection angle of light on the AdS background as the
spherical surface.

Figure 7:  Schematic figure of the deflection angle of light on a hyperbolic
plane as the dS background. The solid lines LR and LS denote the radial
geodesics emanating from the lens to the receiver and the source,
respectively. The inner angle at L is $\phi_{RS}$ which is a given parameter
characterizing the angle separation between R and S. The dashed blue (in
color) curve denotes the geodesic connecting S and R. The solid red (in color)
curve denotes a true light ray from S to R. The light ray by a gravitating
mass in the presence of $\Lambda$ does not follow a geodesic in hyperbolic
geometry. The angle between the radial line LR and the geodesic RS is denoted
as $\Psi_{R}^{dS}$, while that between the radial line LR and the light ray is
denoted as $\Psi_{R}$. The dotted arrow denotes the tangent of the radial line
LS. The angle between the radial line LS and the geodesic RS is an outer
angle, which is denoted as $\Psi_{S}^{dS}$. The angle between the radial line
LS and the light ray is an outer angle, which is denoted as $\Psi_{S}$. We
consider the tangents of the solid red (in color) line and those of the dashed
blue (in color) one at the two points R and S. The angle difference between
the two tangents at the source position S is $\Psi_{S}-\Psi_{S}^{dS}$, which
is denoted as $\delta_{S}$. The angle difference between the two tangents at
the receiver point R is $\Psi_{R}^{dS}-\Psi_{R}$, which is denoted as
$\delta_{R}$. The right-had side of Eq. (78) is the sum of the two angle
differences, namely $\delta_{R}+\delta_{S}$. This can be thus interpreted as
the deflection angle of the solid red (in color) line to the reference line as
the dashed blue (in color) one.

As an example, we assume the Schwarzschild de-Sitter spacetime, for which
$\alpha$ is calculated in References Ishihara2016 ; Takizawa2020a . Their
$\alpha$ is invariant under transformations of spatial coordinates, because it
can be expressed as the areal integral of the Gaussian curvature of the plane.
By using Eq. (77) for their expression of $\alpha$, we obtain the deflection
angle on the dS/AdS backgrounds as

$\displaystyle\alpha^{dS}=$
$\displaystyle\frac{r_{g}}{b}\left(\sqrt{1-b^{2}u^{2}_{S}}+\sqrt{1-b^{2}u^{2}_{R}}\right)$
$\displaystyle+\frac{r_{g}b\Lambda}{12}\left(\frac{1}{\sqrt{1-b^{2}u^{2}_{S}}}+\frac{1}{\sqrt{1-b^{2}u^{2}_{R}}}\right)$
$\displaystyle+O\left(r^{2}_{g},\Lambda^{2}\right),$ (79)

where $r_{g}\equiv 2m$ for the mass $m$.

Note that, in the present formulation, $\alpha^{dS}$ does not include any term
of purely the cosmological constant. This is because we work on the dS
background and hence the effects of purely the cosmological constant are fully
included in the well-defined angular diameter distance.

On the dS background, the impact parameter of light $b$ is
$b=\sqrt{3/\Lambda}R_{b}=\sqrt{3/\Lambda}\tanh\rho_{b}$, where $R_{b}$ denotes
the normalized impact parameter and $\rho_{b}$ is the arc length corresponding
to the impact parameter of light.

$\displaystyle\frac{r_{g}}{b}=$
$\displaystyle\sqrt{\frac{\Lambda}{3}}\frac{r_{g}}{\tanh\rho_{b}}$
$\displaystyle=$ $\displaystyle
r_{g}\sqrt{\frac{\Lambda}{3}}\left(\frac{1}{\hat{D}_{b}}+\frac{\hat{D}_{b}}{2}+O(\hat{D}^{2}_{b})\right)$
$\displaystyle=$ $\displaystyle\frac{r_{g}}{D_{b}}+\frac{r_{g}\Lambda
D_{b}}{6}+O\left(r_{g}\Lambda^{2}D_{b}^{3}\right),$ (80)

where we use in the second line

$\displaystyle\frac{1}{\tanh\rho_{b}}=\frac{1}{\sinh\rho_{b}}+\frac{\sinh\rho_{b}}{2}+O\left(\sinh^{3}\rho_{b}\right),$
(81)

and the angular diameter distance and the normalized one corresponding to
$\rho_{b}$ are denote denoted as $\hat{D}_{b}\equiv\sinh\rho_{b}$ and
$D_{b}\equiv\sqrt{3/\Lambda}\hat{D}_{b}$, respectively.

By using Eq. (14) for the right triangle LPR in Figure 2, we obtain

$\displaystyle\sin\theta=\frac{D_{b}}{D_{L}}.$ (82)

By using this for $D_{b}$ in Eq. (80), we find

$\displaystyle\frac{r_{g}}{b}=$
$\displaystyle\frac{r_{g}}{D_{L}\sin\theta}+\frac{r_{g}\Lambda
D_{L}\sin\theta}{6}+O\left(r_{g}\Lambda^{2}D_{b}^{3}\right).$ (83)

For more simplicity, we employ small angle approximations. Then, we find

$\displaystyle b^{2}u^{2}_{R}$ $\displaystyle=\theta^{2}+\frac{1}{3}\Lambda
D_{L}^{2}\theta^{2}+O(\theta^{4},\theta^{4}D^{2}\Lambda,\theta
D^{4}\Lambda^{2}),$ (84) $\displaystyle b^{2}u^{2}_{S}$
$\displaystyle=\left(\frac{D_{L}}{D_{LS}}\right)^{2}\theta^{2}+\frac{1}{3}\Lambda
D_{L}^{2}\theta^{2}+O(\theta^{4},\theta^{4}D^{2}\Lambda,\theta
D^{4}\Lambda^{2}),$ (85)

where the latter equation can be obtained by noting Eq. (19). By using Eqs.
(83), (84) and (85), Eq. (79) is simplified as

$\displaystyle\alpha^{dS}=$
$\displaystyle\frac{2r_{g}}{D_{L}\theta}-\frac{r_{g}\theta}{2D_{L}}\left[1+\left(\frac{D_{L}}{D_{LS}}\right)^{2}\right]$
$\displaystyle+\frac{r_{g}\Lambda
D_{L}\theta}{6}+O\left(r_{g}^{2},r_{g}\theta^{3},r_{g}\Lambda
D\theta^{3},r_{g}\Lambda^{2}D^{3}\right).$ (86)

In the AdS case, $\tanh$ functions should be replaced by $\tan$ ones. This
leads to

$\displaystyle\frac{1}{\tan\rho_{b}}=\frac{1}{\sin\rho_{b}}-\frac{\sin\rho_{b}}{2}+O\left(\sin^{3}\rho_{b}\right),$
(87)

which has a difference in the sign of the second term of the right-hand side,
compared with Eq. (81). However, we obtain

$\displaystyle\frac{r_{g}}{b}=$
$\displaystyle\frac{r_{g}}{D_{L}\sin\theta}+\frac{r_{g}\Lambda
D_{L}\sin\theta}{6}+O\left(r_{g}\Lambda^{2}D_{b}^{3}\right),$ (88)

where the second term of the right-hand side has the same sign as the dS case
because of the factor $\sqrt{3/(-\Lambda)}$ in the angular diameter distance,
though the second term in the right-hand side of Eq. (87) has the minus sign.
As a consequence, we obtain for the AdS

$\displaystyle\alpha^{AdS}=$
$\displaystyle\frac{2r_{g}}{D_{L}\theta}-\frac{r_{g}\theta}{2D_{L}}\left[1+\left(\frac{D_{L}}{D_{LS}}\right)^{2}\right]$
$\displaystyle+\frac{r_{g}\Lambda
D_{L}\theta}{6}+O\left(r_{g}^{2},r_{g}\theta^{3},r_{g}\Lambda
D\theta^{3},r_{g}\Lambda^{2}D^{3}\right).$ (89)

Eq. (89) is in the same form as Eq. (86), though the two background geometries
are very different.

In order to discuss effects of $\Lambda$ on lensing observations, we consider
Eqs. (28) and (60). In terms of the angular diameter distances among the three
points L, R and S, the dS/AdS lens equations take exactly the same form as the
flat one. We assume an ideal situation that the angular diameter distances
$d_{L}$, $d_{S}$ and $d_{LS}$ take the same values for each of the flat, dS
and AdS cases. Then, the difference in the three lens equations can come only
from the deflection angle of light. According to Eqs. (86) and (89), the
deflection angle of light for a given lens mass is larger (smaller) in dS
(AdS) than in the flat case, where $d_{L}=D_{L}$ is used. This means that,
through a coupling of $\Lambda$ with $m$, the separation angle of multiple
images is increased (decreased) by $\Lambda>0$ ($\Lambda<0$), for given $m$,
$\beta$, $d_{L}$, $d_{S}$ and $d_{LS}$.

It is worthwhile to mention also that $\alpha$ in References Ishihara2016 ;
Takizawa2020a  apparently diverges as $b\to\infty$, while Eq. (79) is not
divergent because the present formulation takes full account of the curvature
of dS backgrounds. This means that the present formulation can describe the
lensing behavior on much larger scale than the conventional method based on
the Minkowskian background. See Figure 8 for a schematic illustration of light
rays on hyperbolic, flat and spherical surfaces.

Figure 8:  Lensed light rays on three distinct manifolds that are
characterized by $\Lambda$. Each light ray is denoted by a solid red (in
color) arrow.

###  V.4 Strong deflection

Finally, we mention the strong deflection case, for which light rays have the
winding number $N$, where $N$ is a positive integer. We thus extend Eq. (72)
as

$\displaystyle\alpha-\theta=$
$\displaystyle\arcsin\left(\sqrt{\frac{1+K\hat{D}_{S}^{2}\tan^{2}\beta}{\hat{D}_{LS}^{2}+\hat{D}_{S}^{2}\tan^{2}\beta}}\hat{D}_{L}\sin\theta\right)$
$\displaystyle-\arctan\left(\frac{\hat{D}_{S}}{\hat{D}_{LS}}\tan\beta\right)+2\pi
N.$ (90)

Fully nonlinear investigations of the strong deflection with Eq. (90) are left
for future.

##  VI Summary

Gravitational lens equations have been discussed in dS background, for which
the existence of the dS horizon has been taken into account and hyperbolic
trigonometry has been used together with the hyperbolic angular diameter
distance. We have used spherical trigonometry in order to discuss
gravitational lens equations in AdS background.

In terms of the angular diameter distances between two points, the lens
equations on the dS/AdS backgrounds Eqs. (28) and (60) take exactly the same
form as that in the flat background Takizawa2020b . On the other hand, there
exists a difference among those in terms of the angular diameter distances
using the lens and source planes. See Eqs. (41), (43) and (71). The only
difference in the form is indicated as the $K$ term in Eq. (72).

In small angle approximations, the difference in the form among the dS/AdS
lens equations and the exact lens equation in Minkowski background begins at
the third order. The angular separation of lensed images is decreased by the
third-order deviation in the dS lens equation, while it is increased in AdS.

We have discussed also the deflection angle of light to match with the lens
equations on the dS/AdS backgrounds. This new form of the deflection angle of
light does not include any term of purely the cosmological constant. In
addition, we wish to stress that the deflection angle of light rays in both
hyperbolic and spherical geometry can take the same form within the present
framework. It would be worthwhile to fully understand a reason for the
coincidence.

Through a coupling of $\Lambda$ with $m$, the separation angle of multiple
images is increased (decreased) by $\Lambda>0$ ($\Lambda<0$), for a given
mass, source direction and angular diameter distances among the lens, receiver
and source.

The above results imply that a similar behavior in the dS/AdS lensing may
occur at the third order level of the general relativistic cosmological
perturbations. Along this direction, a study on realistic cosmological
backgrounds with $\Lambda$ is left for future.

###### Acknowledgements.

We thank Marcus Werner and Toshiaki Ono for the useful discussions on the
optical metric and the background subtraction. We wish to thank Mareki Honma
for the conversations on the EHT method and technology. We thank Hideyoshi
Arakida, Yuuiti Sendouda, Ryuichi Takahashi, Masumi Kasai, Ryuya Kudo,
Ryunosuke Kotaki, Masashi Shinoda, and Hideaki Suzuki for the useful
conversations. This work was supported in part by Japan Society for the
Promotion of Science (JSPS) Grant-in-Aid for Scientific Research, No. 20K03963
(H.A.), and in part by Ministry of Education, Culture, Sports, Science, and
Technology, Grant No. 17H06359 (H. A.)

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