# Interaction-Aware Decision-Making for Autonomous Vehicles in Forced Merging
Scenario Leveraging Social Psychology Factors
††thanks: ∗Xiao Li, Kaiwen Liu, Anouck Girard, and Ilya Kolmanovsky are with
the Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI
48109, USA. {hsiaoli, kwliu, anouck<EMAIL_ADDRESS>††thanks: †H. Eric Tseng
Retired Senior Technical Leader, Ford Research and Advanced Engineering, Chief
Technologist, Excelled Tracer LLC<EMAIL_ADDRESS>††thanks: This
research was supported by the University of Michigan / Ford Motor Company
Alliance program.

Xiao Li∗, Kaiwen Liu∗, H. Eric Tseng†, Anouck Girard∗, Ilya Kolmanovsky∗

###### Abstract

Understanding the intention of vehicles in the surrounding traffic is crucial
for an autonomous vehicle to successfully accomplish its driving tasks in
complex traffic scenarios such as highway forced merging. In this paper, we
consider a behavioral model that incorporates both social behaviors and
personal objectives of the interacting drivers. Leveraging this model, we
develop a receding-horizon control-based decision-making strategy, that
estimates online the other drivers’ intentions using Bayesian filtering and
incorporates predictions of nearby vehicles’ behaviors under uncertain
intentions. The effectiveness of the proposed decision-making strategy is
demonstrated and evaluated based on simulation studies in comparison with a
game theoretic controller and a real-world traffic dataset.

##  I Introduction

One of the major challenges in autonomous driving lies in ensuring safe
interaction with nearby traffic, particularly in highway merging scenarios.
Unlike the simple stop-and-go strategy used in urban intersections with stop
signs, the on-ramp ego vehicle must cooperate with high-speed vehicles and
transition itself into the highway traffic in a secure, but also timely
manner. Moreover, the varied social behaviors of different drivers can result
in diverse responses to the merging intent. In this dynamic interaction, the
ego vehicle must actively search for available space or create opportunities
for merging. A cooperative driver on the highway may decelerate or change
lanes to facilitate the merging process, while a self-interested driver may
maintain a constant speed and disregard the merging vehicle. Consequently,
understanding the driving intentions of the surrounding vehicles becomes
crucial for the ego vehicle to accomplish its task successfully.

Learning-based methods have been extensively explored for autonomous driving
applications. Without explicit interaction behavior modeling, Reinforcement-
Learning (RL) based methods have been utilized to learn end-to-end driving
policies [1, 2]. Additionally, researchers have employed imitation learning to
train decision-making modules [3, 4] that emulate expert behavior, such as a
Model Predictive Controller [3]. A comprehensive survey of RL methods in
autonomous driving research can be found in [5]. However, a significant
drawback of end-to-end RL-learned policies is the lack of interpretability;
furthermore, their ability to generalize may be limited by the interactive
behaviors observed in the training data.

To address this challenge, researchers have explored the integration of
learning-based methods with planning and control techniques. The Inverse
Reinforcement Learning (IRL) approach has been employed to learn the reward
function of human drivers for planning purposes [6, 7]. Additionally, neural
network models, such as the Social Generative Adversarial Network [8], have
been implemented in trajectory prediction modules for Model Predictive Control
(MPC) [9, 10]. Novel network architectures have also been designed to enhance
driving motion forecasting [11, 12]. However, a common issue in these learned
modules is their limited generalization capability beyond the training
dataset.

Game theoretic approaches have also been considered to represent interactions
between agents in traffic, such as the Level-k method [13], potential games
[14], and Stackleberg games [15, 16]. A Leader-Follower Game theoretic
Controller (LFGC) has been proposed specifically for modeling pairwise leader-
follower interactions in forced merging scenarios in [17, 18]. Inspired by the
concepts presented in [17, 18], we adopt a pairwise interaction formulation to
model vehicle cooperative behaviors, enabling better scalability for scenarios
involving multiple vehicles.

Differently from [17, 18], to create a more comprehensive model of human
driving, we incorporate a novel behavioral model that incorporates various
social psychology factors. Early studies in social psychology have revealed
that individuals don’t always act solely to maximize their own rewards in two-
person [19] or $n$-person experimental games [20]. Drawing inspiration from
the concept of Social Value Orientation (SVO) [20, 21, 22, 23], and its
application in the context of autonomous driving [24], we propose a novel
behavioral model that encompasses both the drivers’ inclination towards social
cooperation and their individual objectives as latent parameters. Leveraging
this proposed behavioral model, we can estimate the underlying driving
intentions of the interacting vehicles and make appropriate decisions for the
ego vehicle in forced merging scenarios. The algorithms we propose offer
several potential advantages:

  1. 1.

The proposed behavioral model incorporates aspects of both driver’s social
cooperativeness and personal objectives and captures a rich and realistic set
of behaviors.

  2. 2.

The algorithm uses a Bayesian filter to infer the latent driving intent
parameters, thereby handling uncertainties in the cooperation intent of
interacting vehicles.

  3. 3.

The derived decision-making module adopts a pairwise interaction formulation
and utilizes receding-horizon optimization-based control that leads to good
scalability while ensuring safety for forced merging applications.

This paper is organized as follows: In Sec. II, we introduce the problem
setting and the forced merging scenario. We also outline the assumptions made
regarding vehicle kinematics, action space, and driver’s action objectives. In
Sec. III, we present our behavioral model that incorporates the cooperation
intents and personal objectives of the interacting vehicles. In Sec. IV, we
present the decision-making module for the ego vehicle, that effectively
handles uncertainties in the driving intentions of the interacting vehicles.
In Sec. V, we demonstrate the ability of our behavioral model to reproduce
realistic driving behaviors. Furthermore, we validate the proposed controller
through simulations in comparison with the LFGC and real-world dataset
evaluations. Finally, the conclusions are given in Sec. VI.

##  II Problem Formulation

Figure 1: Schematic diagram of the highway forced merging problem: an ego
vehicle (red) interacts with highway vehicles (grey) to facilitate its
merging.

In this paper, we focus on the design of a decision-making module for
autonomous driving applications in forced merging scenarios. The decision-
making module plans high-level behaviors such as acceleration, deceleration,
or lane changing, and generates desired reference trajectories for the
autonomous vehicle. Subsequently, a lower-level controller is assumed to be
available that can control the steering and acceleration/braking of the
vehicle to track the reference trajectory. As illustrated in Fig. 1, the goal
is to design a behavior planner for the ego vehicle to merge into the target
highway lane while accounting for interactions with multiple highway vehicles
to ensure safe and effective merging.

###  II-A Vehicle Kinematics Model

We use the following discrete-time model to represent the vehicle kinematics,

$\left[\begin{array}[]{c}x(t+1)\\\ v_{x}(t+1)\\\
y(t+1)\end{array}\right]=\left[\begin{array}[]{c}x(t)+v_{x}(t)\Delta t\\\
v_{x}(t)+a(t)\Delta t\\\ y(t)+v_{y}(t)\Delta t\\\
\end{array}\right]+\tilde{w}(t),$ (1)

where $x$, $v_{x}$, and $a$ are the longitudinal position, velocity, and
acceleration, respectively; $y$ and $v_{y}$ are the lateral position and
velocity; $\Delta t>0$ is the sampling period between discrete time instances
$t$ and $t+1$; $\tilde{w}(t)\in\mathbb{R}^{3}$ is a disturbance representing
unmodeled dynamics.

We assume all the vehicles, including both ego and highway vehicles, follow
this dynamics model. For simplicity, Eq. (1) can be rewritten as,

$s_{i}(t+1)=f\big{(}s_{i}(t),u_{i}(t)\big{)}+\tilde{w}_{i}(t),\;i=0,1,2,\dots,$
(2)

where $s_{i}(t)=[x(t),y(t),v_{x}(t)]^{T}$ and $u_{i}(t)=[a(t),v_{y}(t)]^{T}$
represent the state and control of the $i$’th vehicle at time instance $t$,
respectively. In the following context, the subscript $i=0$ designates the ego
vehicle, while $i\in\left\\{1,2,\dots\right\\}$ represents another vehicle
with which the ego vehicle interacts.

###  II-B Action Space

We assume vehicles take actions from the action set $U$ that comprises:

  1. 1.

“Maintain”: keep the current lateral position and longitudinal speed;

  2. 2.

“Accelerate”: keep the current lateral position and accelerate at $a\;\rm
m/s^{2}$ without exceeding the upper speed limit $v_{\text{max}}\;\rm m/s$;

  3. 3.

“Decelerate”: keep the current lateral position and decelerate at $-a\;\rm
m/s^{2}$ without falling below the lower speed limit $v_{\text{min}}\;\rm
m/s$;

  4. 4.

“Steer to the left”: keep the current longitudinal speed and steer to the left
adjacent lane with a constant lateral velocity of
$\frac{w_{\text{lane}}}{T_{\text{lane}}}\;\rm m/s$;

  5. 5.

“Steer to the right”: keep the current longitudinal speed and steer to the
right adjacent lane with a constant lateral velocity of
$-\frac{w_{\text{lane}}}{T_{\text{lane}}}\;\rm m/s$.

Note that we assume a complete lane change takes $T_{\text{lane}}$ sec to move
into the adjacent lane with a lateral traveling distance of lane width
$w_{\text{lane}}$, which is reflected by the above actions. We also note that
more acceleration and deceleration levels $\left|a\right|$ can be introduced
to the action space, but we only consider one level here for simplicity.

###  II-C Driving Objectives

The driving objectives of each vehicle are reflected in the reward function
that depends on the following four variables:

  1. 1.

Traffic rules $c$: The binary variable $c\in\left\\{0,1\right\\}$ is an
indicator for either getting into a collision with other vehicles or getting
beyond the road boundaries. A safety bounding box is constructed that
overbounds each vehicle body in the $x-y$ plane with certain safety margins.
The value of $c=1$ indicates the overlap of two vehicles’ bounding boxes or
the overlap between the vehicle’s bounding box and the road boundary. The
value of $c=0$ indicates that the vehicle stays within the road and is not in
collision with other vehicles. The visualization of the highway road
boundaries is shown in Fig. 1 using solid black lines.

  2. 2.

Safety consciousness $h$: The variable $h\in[0,1]$ is derived from the Time-
to-Collision $(TTC)$ with a vehicle ahead in the same lane

$h=\frac{\operatorname*{sat}_{\left[T_{\text{min}},T_{\text{max}}\right]}(TTC)-T_{\text{min}}}{T_{\text{max}}-T_{\text{min}}},$

where $T_{\text{min}}=0.2$ sec is the minimum reaction time,
$T_{\text{max}}=3$ sec stands for an adequate time headway, and
$\operatorname*{sat}_{\left[a,b\right]}\left(\cdot\right)$ is a saturation
function between the minima $a$ and the maxima $b$. The reward function
depends on $h$ to encourage vehicles to keep an appropriate headway distance
and be conscious of potential collisions.

  3. 3.

Traveling time $\tau$: The variable $\tau\in[0,1]$ reflects the objective of
shortening the traveling time, and is a weighted summation of

$\tau_{x}=\frac{x-x_{0}}{x_{f}-x_{0}},\tau_{y}=1-\frac{1}{w_{\text{lane}}}\min\left(\left|y-y_{r}\right|,w_{\text{lane}}\right),$

where $x_{0}$ and $x_{f}$ (see Fig. 1) are the $x-$coordinates of the
beginning of the ramp and a goal placed a specified distance away from the end
of the ramp, respectively, while $y_{r}$ corresponds to the center of the
highway lane that is next to the ramp. The reward for $\tau_{x}$ promotes the
highway vehicle reaching the end of the highway in a shorter time. A higher
reward for $\tau_{y}$ is imposed for on-ramp vehicles to encourage merging
action.

  4. 4.

Control effort $e$: The reward for $e\in[0,1]$ promotes vehicles to drive at a
constant speed and to reduce acceleration/deceleration. The variable $e$
attains value of 1 under the action “maintain”; its value decreases if the
vehicle makes speed changes or lane changes.

##  III Social Behavior Modeling

In this section, we introduce our behavioral model that captures drivers’
interactive decision-making process during the forced merge scenario. Inspired
by social psychology studies [20, 21, 22, 23], and its application in the
context of autonomous driving [24], we define the SVO-based reward model in
Sec. III-A. In Sec. III-B, we integrate this reward model into the interacting
vehicle’s decision-making process.

###  III-A Social Value Orientation and Multi-modal Reward

We assume each vehicle $i$ interacts pairwise with each adjacent vehicle $j\in
A(i)$, where $A(i)$ contains indices of all the adjacent vehicles around $i$.
We assume each driver aims to achieve their personal objectives and, to a
certain extent, is cooperating with others. Hence, we model each driver’s
intention using a multi-modal reward function of the form

$\begin{array}[]{l}R_{i}\big{(}s(t),u(t)|\sigma_{i},w_{i}\big{)}=\frac{1}{\left|A(i)\right|}\sum_{j\in
A(i)}\cdot\\\ \Big{[}\theta_{1}(\sigma_{i})\cdot
r_{i}\big{(}s_{i}(t),u_{i}(t),s_{j}(t),u_{j}(t)|w_{i}\big{)}\\\
+\theta_{2}(\sigma_{i})\cdot
r_{j}\big{(}s_{j}(t),u_{j}(t),s_{i}(t),u_{i}(t)|w_{j}\big{)}\Big{]},\end{array}$
(3)

where $u(t)=[u_{i}^{T}(t),u_{A(i)}^{T}(t)]^{T}$ is the aggregated control
vector of all vehicles and $u_{A(i)}(t)$ is a column vector concatenating
$u_{j}(t)$ for all $j\in A(i)$; $s(t)=[s_{i}^{T}(t),s_{A(i)}^{T}(t)]^{T}$
reflects the state of the traffic at time $t$; $\left|A(i)\right|$ is the
number of interacting vehicles; $r_{i}(\cdot)$ and weights
$w_{i}\in\mathbb{R}^{3}$ model personal reward as a weighted summation of
personal objectives defined in Sec. II-C,

$r_{i}(s_{i},u_{i},s_{j},u_{j}|w_{i})=(\neg c)\cdot
w_{i}^{T}\cdot[h,\tau,e]^{T}.$ (4)

The symbol $\neg$ in Eq. (4) is the logical negative operator and $\sigma_{i}$
in Eq. (3) takes one of four values corresponding to four SVO categories and
specific values of $\theta_{1}(\sigma_{i})$ and $\theta_{2}(\sigma_{i})$:

$(\theta_{1},\theta_{2})=\left\\{\begin{array}[]{cl}(0,1)&\text{if
}\sigma_{i}=\text{ altruistic}\\\ (1/2,1/2)&\text{if }\sigma_{i}=\text{
prosocial}\\\ (1,0)&\text{if }\sigma_{i}=\text{ egoistic}\\\
(1/2,-1/2)&\text{if }\sigma_{i}=\text{ competitive}\\\ \end{array}\right..$
(5)

Note that in Eq. (3), $\theta_{1}$ and $\theta_{2}$ correspond to the weight
of the self-reward $r_{i}$ and the weight of the other drivers’ net reward,
respectively.

In this multi-modal reward given by Eq. (3), there are two latent parameters
$(\sigma_{i},w_{i})$ that represent different driving incentives: $w_{i}$
reflects different personal goals, and $\sigma_{i}$ represents different
social behaviors or levels of cooperativeness. For instance, a driver with
weights $w_{i}=[0,1,0]^{T}$ in Eq. (4) might consider driving at full speed
thereby minimizing the traveling time. As implied by Eq. (3), a “prosocial”
driver has equal weights between personal objectives and other drivers’
objectives; hence such drivers intend to cooperate with others in pursuing a
large net reward. Note that $w_{j}$ is the internal parameter of vehicle $j$
and is a latent variable affecting the decision of vehicle $i$ if
$\sigma_{i}\neq\text{``egoistic"}$ and $j\in A(i)$. Nonetheless, an altruistic
or prosocial (or competitive) driver of vehicle $i$ is likely to improve (or
diminish) other drivers’ rewards in all three variables if they do not know
other drivers’ objectives $w_{j}$ a priori. Therefore, we assume that during
the $i$’th vehicle’s decision-making, $w_{j}=[1/3,1/3,1/3]$ in Eq. (3) for
$j\in A(i)$.

###  III-B Driving Behavior Model

In our behavior model, we assume the driver of vehicle $i$ aims to maximize
the cumulative reward, defined as

$\begin{array}[]{l}Q^{\prime}_{i}\big{(}s(t),\gamma_{i}|\sigma_{i},w_{i}\big{)}=\\\
\operatorname*{\mathbb{E}}_{\gamma_{j},j\in
A(i)}\left[\sum\limits_{k=0}^{N-1}\lambda^{k}R_{i}\big{(}s(t+k),u(t+k)\big{|}\sigma_{i},w_{i}\big{)}\right],\end{array}$
(6)

where $\gamma_{i}=\left\\{u_{i}(t+k)\right\\}_{k=0}^{N-1}\in U^{N}$ is an
action sequence over a horizon of length $N$, and $\lambda\in[0,1]$ is a
discount factor. This cumulative reward is an averaged reward over all
possible action sequences $\gamma_{j}$ of vehicles $j\in A(i)$. Furthermore,
the driver of vehicle $i$ is assumed to adopt a receding horizon control
strategy, i.e.,

$u^{*}_{i}(t)=\operatorname*{argmax}\limits_{u\in
U}Q_{i}\big{(}s(t),u|\sigma_{i},w_{i}\big{)},$ (7)

where
$Q_{i}(s(t),u)=\mathbb{E}_{\gamma_{i}\in\Gamma^{1}(u)}\left[Q^{\prime}_{i}(s(t),\gamma_{i}|\sigma_{i},w_{i})\right]$
and
$\Gamma^{1}(u)=\left\\{\gamma_{i}=\left\\{u_{i}(t+k)\right\\}_{k=0}^{N-1}:u_{i}(t)=u\right\\}$
contains all the action sequences with the initial action $u$. Furthermore,
considering stochasticity in the decision-making process, a policy
distribution can be prescribed by adopting a softmax decision rule [25]:

$\mathbb{P}\big{(}u_{i}=u|\sigma_{i},w_{i},s(t)\big{)}\propto\exp\big{(}Q_{i}(s(t),u|\sigma_{i},w_{i})\big{)}.$
(8)

Based on the behavioral model defined above, the model parameters
$\sigma_{i},w_{i}$ can affect action policies to represent different driving
intentions. Since other drivers’ intentions are not known in a given traffic
scenario, the model parameters $\sigma_{i},w_{i}$ (i.e., drivers’ intentions)
need to be estimated and updated online so that the autonomous ego vehicle is
able to make optimal merging decisions.

##  IV Decision-Making Under Cooperation Intent Uncertainty

We now develop a decision-making algorithm to facilitate the forced merging
process. We first present a Bayesian filter for the ego vehicle that estimates
the latent variables $\sigma_{i},w_{i}$ of the interacting vehicles online.
Considering the uncertainties in our estimation and other drivers’ intentions,
we use a receding-horizon control formulation to simultaneously address the
safety and performance aspects of the forced merging.

###  IV-A Bayesian Inference of Latent Driving Intentions

At each time step, we assume that the ego vehicle can observe the traffic
nearby the $i$’th interacting vehicle where $i\in A(0)$, and the observed
traffic history is defined as

$\begin{array}[]{l}\xi(t)=\big{\\{}s(0),s(1),\dots,s(t),\\\
u_{A(i)}(0),u_{A(i)}(1),\dots,u_{A(i)}(t-1)\big{\\}},\end{array}$

where $s(t)=[s_{i}^{T}(t),s_{A(i)}^{T}(t)]^{T}$ and $u_{A(i)}(t)$ is a column
vector concatenating $u_{j}(t)$ for all $j\in A(i)$. The ego vehicle utilizes
the traffic history to estimate the posterior distribution
$\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t+1)\right)$ of the $i^{\prime}$th
interacting vehicle’s latent parameters using the following proposition:

######  Proposition 1.

Given a prior $\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right)$ and assuming
that the disturbance $\tilde{w}_{i}(t)\sim\mathcal{N}(0,Q)$ is zero-mean
Gaussian, the posterior distribution can be computed as

$\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t+1)\right)=\frac{\Lambda_{i}\left(\sigma_{i},w_{i},s(t),s_{i}\right)}{N_{i}(t)}\cdot\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right),$
(9)

where $N_{i}(t)$ is a normalization factor and
$\Lambda_{i}\left(\sigma_{i},w_{i},s(t),s_{i}\right)$ admits the following
form:

$\begin{array}[]{l}\Lambda_{i}\left(\sigma_{i},w_{i},s(t),s_{i}\right)=\sum\limits_{u\in
U}\;\mathbb{P}\left(u_{i}=u|\sigma_{i},w_{i},s(t)\right)\cdot\\\
\mathbb{P}\left(\tilde{w}_{i}(t)=s_{i}-f(s_{i}(t),u)\right),\end{array}$ (10)

where $\mathbb{P}\left(u_{i}=u|\sigma_{i},w_{i},s(t)\right)$ is defined in Eq.
(8).

Note that the above recursive Bayesian filter can be initialized using a
uniform distribution and the covariance matrix $Q$ is a tunable parameter.
Intuitively, if we consider the current traffic state $s(t)$ and the vehicle
$i$ is executing policy defined in Eq. (8) conditioned on parameters
$\sigma_{i}$ and $w_{i}$,
$\Lambda_{i}\left(\sigma_{i},w_{i},s(t),s_{i}\right)$ represents the
transition probability of the vehicle $i$ moving from $s_{i}(t)$ to
$s_{i}(t+1)=s_{i}$. The proof is presented as follows:

###### Proof.

Applying the Bayesian rule, the posterior admits the following form,

$\begin{array}[]{l}\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t+1)\right)=\mathbb{P}\left(\sigma_{i},w_{i}|s(t+1),u_{A(i)}(t),\xi(t)\right)\\\
=\frac{\mathbb{P}\left(s(t+1)|\sigma_{i},w_{i},u_{A(i)}(t),\xi(t)\right)}{\mathbb{P}\left(s(t+1)|u_{A(i)}(t),\xi(t)\right)}\cdot\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right).\end{array}$

Moreover, the likelihood term can be rewritten as follows,

$\begin{array}[]{l}\mathbb{P}\left(s(t+1)|\sigma_{i},w_{i},u_{A(i)}(t),\xi(t)\right)\\\
=\mathbb{P}\left(s_{A(i)}(t+1)|u_{A(i)}(t),s_{A(i)}(t)\right)\cdot\\\
\mathbb{P}\left(s_{i}(t+1)|\sigma_{i},w_{i},s(t)\right)\end{array},$

whereby the posterior reduces to

$\begin{array}[]{l}\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t+1)\right)\propto\\\
\mathbb{P}\left(s_{i}(t+1)|\sigma_{i},w_{i},s(t)\right)\cdot\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right),\end{array}$

and
$\Lambda_{i}\left(\sigma_{i},w_{i},s(t),s_{i}\right)=\mathbb{P}\left(s_{i}(t+1)|\sigma_{i},w_{i},s(t)\right)$
is the transition probability conditioned on the model parameters
$\sigma_{i}$, $w_{i}$ and the current traffic state $s(t)$. ∎

###  IV-B Receding-horizon Optimization-based Control

We leverage the receding horizon control to achieve safe merging. The
objective of successful merging without collisions can be translated into
maximizing predictive cumulative reward,

$\begin{array}[]{l}Q^{\prime}_{0}(s(t),\gamma_{0})=\frac{1}{\left|A(0)\right|}\sum\limits_{i\in
A(0)}\operatorname*{\mathbb{E}}\limits_{\sigma_{i},w_{i}\sim\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right)}\\\
\left\\{\operatorname*{\mathbb{E}}\limits_{\gamma_{i}\sim\mathbb{P}\left(\gamma_{i}|\sigma_{i},w_{i},s(t)\right)}\left[\overline{Q}^{\prime}_{0}\big{(}s_{0}(t),s_{i}(t),\gamma_{0},\gamma_{i}\big{)}\right]\right\\},\\\
\end{array}$ (11)

where $A(0)$ contains indices of all the vehicles near the ego vehicle;
$\gamma_{0}$, $\gamma_{i}$ are action sequences of length $N$ corresponding to
the ego vehicle and the vehicle $i$;
$\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right)$ is the latent parameter
estimated distribution and is updated by the Bayesian filter; Similar to Eq.
(8), $\mathbb{P}\left(\gamma_{i}|\sigma_{i},w_{i},s(t)\right)$ is an augmented
policy distribution derived from Eq. (6) such that
$\mathbb{P}\left(\gamma_{i}|\sigma_{i},w_{i},s(t)\right)\propto\exp\left(Q^{\prime}_{i}(s(t),\gamma_{i}|\sigma_{i},w_{i})\right)$;
$\overline{Q}^{\prime}_{0}\left(\cdot\right)$ computes the discounted
cumulative reward of the ego vehicle according to

$\begin{array}[]{l}\overline{Q}^{\prime}_{0}(s_{0}(t),s_{i}(t),\gamma_{0},\gamma_{i})\\\
=\sum\limits_{k=0}^{N-1}\lambda^{k}r_{0}(s_{0}(t+k),u_{0}(t+k),s_{i}(t+k),u_{i}(t+k)),\end{array}$

and the one-step reward function $r_{0}(s_{0},u_{0},s_{i},u_{i})=(\neg
c)\cdot\tau$ encourages collision avoidance and faster merging. Note that we
can also customize the reward function $r_{0}$ for other implementation
requirements, e.g., add variable $\tau$ of other drivers to $r_{0}$ to
mitigate traffic congestion introduced by the ego vehicle’s merging action.

Here, we model the ego vehicle interactions pairwise with the nearby vehicles
$i\in A(0)$. For each interaction vehicle $i$, the outer expectation in Eq.
(11) considers all possible combinations of parameters $\sigma_{i},w_{i}$ that
illustrate the cooperation intent and personal objectives of the interacting
vehicle $i$. The inner expectation then takes an average return of all
plausible decisions by vehicle $i$ conditioned on the parameters
$\sigma_{i},w_{i}$ under our behavior model. The reward function
$\overline{Q}^{\prime}_{0}\left(\cdot\right)$ represents the cumulative gain
of the pairwise interaction between the ego vehicle and the vehicle $i$.

Based on the cumulative reward, the ego vehicle is controlled using a receding
horizon control strategy, i.e.,

$u^{*}_{0}(t)=\operatorname*{argmax}\limits_{u\in U}Q_{0}(s(t),u),$ (12)

where
$Q_{0}(s(t),u)=\mathbb{E}_{\gamma_{0}\in\Gamma^{1}(u)}\left[Q^{\prime}_{0}(s(t),\gamma_{0})\right]$
and $\Gamma^{1}(u)$ contains all action sequences of length $N$ with $u$ being
their initial action.

######  Remark.

Note that the reward computation in Eq. (11) is not reactive. Our algorithm
does not predict other vehicles’ reactions to the ego actions in the
prediction. To address this concern, we can also adopt a game-theoretic
formulation similar to [17, 18, 24]. However, such a formulation is
computationally demanding in practice. Due to the formulation of pairwise
interaction in Eq. (11), our algorithm can solve Eq. (12) effectively via an
exhaustive search that scales linearly with the number of interacting
vehicles.

##  V Simulation and Experimental Results

Here, we demonstrate the effectiveness of the proposed behavioral model and
the forced merging control algorithm. The behavioral model is first
demonstrated by reproducing real-world driving behaviors. Then, the
effectiveness of the proposed forced merging control algorithm is illustrated
through simulation studies in comparison with the LFGC [17, 18] against
interacting vehicles controlled by our behavior model and through a
naturalistic driving dataset present in High-D [26].

Figure 2: Examples of reproducing real-world highway merging and overtaking
behaviors: In each example, the target vehicle is represented by a green box,
and interacts with the traffic. All traffic vehicles are visualized as boxes
of different edge colors filled with grey. The trajectory of vehicles is shown
in dashed lines with vehicles’ positions every second marked as boxes in green
(for the target vehicle) and circles filled with grey (for other traffic
vehicles). The virtual vehicles’ trajectories are visualized using red dashed
lines and red boxes and match the actual ones closely. a) A traffic of 12
seconds is sampled from frames 1-300 in scene 59 from the High-D dataset. The
trajectory of vehicle 14 is reproduced using our behavioral model with
$\sigma_{14}=\text{egoistic},$ $w_{14}=[0,2/3,1/3]^{T}$, i.e., vehicle 14 is
an egoistic driver, and minimizes the traveling time by merging to the
highway. b) A traffic of 14 seconds is sampled from frames 1-350 in scene 59
from the High-D dataset. The trajectory of vehicle 13 is reproduced using our
behavioral model with $\sigma_{13}=\text{competitive},$ $w_{13}=[0,1,0]^{T}$,
i.e., vehicle 13 is a competitive driver, and minimizes the traveling time by
overtaking the leading vehicles.

###  V-A Reproducing Real-world Traffic

We leverage a drone-recorded naturalistic traffic dataset called the High-D
dataset [26]. It contains 60 traffic recordings among which three
recordings/scenes (58-60) have merging ramps. We first calibrate the values of
the model parameters $a$ and $T_{\text{lane}}$ from the High-D dataset (see
Fig. 3). The lane width $w_{\text{lane}}=3.5\;\rm m$ in High-D. We choose
$\left|a\right|=6\;\rm m/s^{2}$ because the longitudinal accelerations and
decelerations of High-D vehicles are within an interval of $[-6,6]\;\rm
m/s^{2}$. And we select $T_{\text{lane}}=4\;\rm sec$ since the majority of the
vehicles in the High-D dataset take $4\sim 6\;\rm sec$ to change lane.

Figure 3: Histogram of vehicle driving statistics in High-D dataset: (a)
Longitudinal acceleration/deceleration (y-axis in log scale). (b) Time
duration for a complete lane change.

We aim to reproduce the real-world driving behavior (see Fig. 2) in the
recording segments using our behavioral model with certain parameters
$\sigma_{i},w_{i}$. Specifically, to reproduce the behavior of a target
vehicle $i$, we initialize our behavioral model with its actual initial state
$s_{i}(t=0)$ from the recording segment. At each time step of 2 seconds, we
assume a virtual vehicle $i$ is controlled by our behavior model, and can
observe the surrounding traffic $s(t)=[s_{i}(t)^{T},s_{A(i)}(t)^{T}]^{T}$.
With prescribed parameters $\sigma_{i},w_{i}$, the virtual vehicle $i$ updates
its control decision every time step using the behavioral model defined in Eq.
(7) with MPC prediction horizon $N=3$ spanning 6 seconds. The kinematics of
the virtual vehicle obeys Eq. (1) with a sampling rate of 25 $\rm Hz$.
Furthermore, to introduce realistic lateral behaviors, the underlining lane
change trajectories are modeled by 5th-order polynomials as proposed in [27].

As shown in Fig. 2a), the virtual truck $14$ in green is controlled by our
behavioral model with parameters $\sigma_{14}=\text{egoistic},$
$w_{14}=[0,2/3,1/3]^{T}$. This combination of parameters implies the virtual
vehicle cares about the self-reward solely by the formulation in Eq. (3) and
Eq. (5) and tries to minimize the traveling time because it has the largest
weight $2/3$ for $\tau$ in Eq. (4). As a result, the virtual truck merges into
the highway, and the trajectory in the green boxes matches the actual target
vehicle in the red boxes. As shown in Fig. 2b), in front of the virtual
vehicle $13$, there are two vehicles $12$ and $15$ driving slowly. With
parameters $\sigma_{13}=\text{competitive},$ $w_{14}=[0,1,0]^{T}$, the
behavioral model controls the virtual vehicle to compete with the two vehicles
while minimizing its traveling time. This interaction results in an overtaking
behavior for the virtual vehicle which qualitatively matches the actual
traffic recording. These examples provide evidence of our behavioral model
being able to capture realistic driving behaviors. Note that there are
quantitative mismatches in positions between the reproduced and the actual
results in Fig. 2. Such mismatches partially result from the model assumption
in Eq. (1); improvements to the kinematic model are left as future work.

###  V-B Forced Merging in Simulation Compared with LFGC

We build up a highway with five vehicles (see Fig. 4) to simulate a forced
merging scenario. The ego vehicle interacts with the surrounding traffic using
the proposed controller in Eq. (12). In this example, each time step
corresponds to one second and the lane change takes two time steps. Other
vehicles are controlled using our behavioral model with different parameters
$\sigma_{i},w_{i}$. All the traffic vehicles are modeled as “egoistic”
drivers, i.e., $\sigma_{i}=$ “egoistic” for all $i=1,\dots,4$. We assign
vehicle $i=1,3,4$ with weights $w_{i}=[0,0,1]$ to encourage minimization of
control effort $e$ such that the three vehicles will keep a constant speed.
Vehicle 2 with weights $w_{2}=[1,0,0]$ tries to search for a larger headway
space $h$, therefore, changes to the inner lane from $t=0$ to $t=2$ seconds.

As shown in Fig. 4, we also deploy the LFGC to control the ego vehicle in the
same highway traffic setting, and the results are plotted overlaid with ours
in comparison. Note that the parameters $\sigma_{i}$ and $w_{i}$ of all
vehicles 1-4 are not available to ego vehicle 0. As a result, the ego vehicle
needs to interact with other vehicles and estimate their intentions online to
facilitate its merging. During the simulation, our ego vehicle successfully
infers the cooperative intents of vehicle $1,2,3$ from their behaviors, and it
decides to first accelerate to surpass vehicle $1$ from $t=0$ to $t=1$, and
merges into the gap created by the lane changing of vehicle $2$.

Figure 4: Forced merging comparison in a simulation environment: Ego vehicle
controlled by our algorithm in the red box is interacting with the adjacent
vehicles in colored boxes. The LFGC is tested using the same highway traffic
setting, and the results are plotted overlaid using grey boxes. Four sub-
figures demonstrate the interactions at $t=0,1,2,3$ seconds, respectively.
After accelerating to surpass vehicle 1, our ego vehicle properly merges into
the highway at $t=3$ seconds.

We also provide our Bayesian filter estimation results (see Fig. 5) at time
step $t=1$. Here, we confine the domain of the weights $w_{i}$ into a finite
subset $W\subset[0,1]^{3}$ as follows

$W=\left\\{\begin{aligned} ,\;&[0,1,1]/2,\;[0,1,0],\;[1,1,1]/3,\\\
[1,0,1]/2,\;&[1,1,0]/2,\;[1,0,0]\end{aligned}\right\\},$ (13)

where each weight case is a normalized combination of zeros and ones.
Moreover, as shown in Fig. 5, the $\sigma_{i}=$“altruistic” category stands
alone and is not correlated with the reward weights $w_{i}$ because the
“altruistic” driver does not care about the self-reward as modeled in Eqs.
(3), (4) and (5). Notably, the actual parameters $\sigma_{i},w_{i}$ are among
the ones of the highest probability. Meanwhile, for vehicle $2$, the cases
with the highest probability mostly emphasize traveling time $\tau$ and
headway distance $h$ while those of vehicles $1,3$ emphasize control effort
$e$. Using these probability distributions, the ego vehicle can predict the
driving and cooperation intent of vehicles $1,2,3$, and plan its task
accordingly, as demonstrated in Fig. 4.

Figure 5: Bayesian filter estimations of posterior
$\mathbb{P}\left(\sigma_{i},w_{i}|\xi(t)\right)$: Three sub-figures visualize
the parameter belief of the interacting vehicles $i=1,\;2,\;3$, respectively,
with $t=1$ second. The $x$ axis labels 7 different weights $w_{i}\in W$ with
order listed in Eq. (13). The $y$ axis labels the SVO categories $\sigma_{i}$,
with the “altruistic” one standing alone to the right. The heatmap presents a
higher probability with a deeper blue and the actual model parameters are
circled in red.

In comparison, the ego vehicle controlled by the LFGC estimates the
probability of vehicles being a leader or being a follower, i.e.,

$\mathbb{P}\left(i=\text{``leader"}\right)=1-\mathbb{P}\left(i=\text{``follower"}\right),i=1,2,3,4.$

However, due to the lane-changing behavior, the LFGC cannot distinguish
between vehicle $2$ being a “leader” and it being a “follower”, namely,
$\mathbb{P}\left(i=\text{``leader"}\right)=\mathbb{P}\left(i=\text{``follower"}\right)=0.5$.
This results in a deceleration decision of the LFGC from $t=0$ to $t=1$.
Seeing the constant-speed vehicle $1$ as a “leader”, the LFGC further keeps a
low speed from $t=1$ to $t=2$ and decides to merge after vehicle $1$ at $t=2$
while the ego vehicle controlled by our algorithm is in the middle of a lane
change. This provides evidence that, compared to the LFGC, our behavioral
model captures a richer and more realistic set of behaviors and the controller
integrated with our behavioral model can achieve faster merging in certain
cases.

###  V-C Validation on Real-world Dataset

To validate our controller in real-world traffic, we consider traffic segments
(see Fig. 6) that contain merging vehicles from the High-D dataset. Similar to
Sec. V-A, we use $w_{\text{lane}}=3.5\;\rm m$, $\left|a\right|=6\;\rm
m/s^{2}$, $T_{\text{lane}}=4\;\rm sec$, and an MPC prediction horizon $N=3$
spanning 6 seconds. We initialize our virtual ego vehicle using the initial
state of the merging vehicle on the ramp. Afterward, we control the virtual
ego vehicle with the proposed controller. We use the finite weight set $W$ in
Eq. (13) for parameter estimation in the Bayesian filter.

As shown in Fig. 6, the virtual ego vehicle interacts with two trucks $1$ and
$2$ that drive approximately at constant speeds. Thus, the ego vehicle first
accelerates to create adequate merge space and speed advantage, then
successfully merges between the two trucks. Another example is provided in
Fig. 7, where the ego vehicle keeps a constant speed for 2 seconds to create
merge space after vehicle 2. Notably, in both examples, we can observe that
the ego vehicle’s trajectories are similar to the ones of the actual target
vehicles, shown in green boxes.

Figure 6: An example of forced merging validation on the High-D dataset: the
virtual ego vehicle in the red box is initialized using the target vehicle’s
initial state. Its adjacent vehicles are in colored boxes while other traffic
vehicles are visualized in grey ones. Four sub-figures demonstrate the
interactions at $t=0,2,4,6$ seconds, respectively. After accelerating, the
virtual ego vehicle properly merges onto the highway.

Figure 7: Another example of forced merging validation on the High-D dataset:
the virtual ego vehicle first keeps a constant speed, then merges onto the
highway.

Meanwhile, in the High-D dataset, there are in total of 75 merging vehicles in
scenes 58-60. For each merging vehicle, we repeat the aforementioned
procedures to set up the test environment and control the virtual ego vehicle
to merge onto the highway. The test results are presented in Table I. We
consider a test case a success if the ego vehicle successfully merges without
collisions. A failure case implies that the ego vehicle either collides with
other vehicles or fails to merge by the end of the ramp. Our algorithm can
achieve a $100\%$ success rate among the $75$ test cases, and properly merges
the virtual vehicles into the naturalistic traffic.

TABLE I: High-D forced merging test results on scenes 58, 59, and 60: the traffic recordings in three scenes are recorded during different time intervals on Wednesday 07/2018 in the same highway location. Scene Number | 58 | 59 | 60 | Total
---|---|---|---|---
Time Interval (AM) | 9:15-9:22 | 9:23-9:31 | 9:37-9:53 |
Number of Merges | 18 | 21 | 36 | 75
Success | 18 | 21 | 36 | 75
Failure | 0 | 0 | 0 | 0
Success Rate (%) | 100 | 100 | 100 | 100

##  VI Conclusion

In this paper, we proposed a driving behavioral model that takes different
social value orientations and personal objectives into consideration. Based on
the proposed behavioral model, we developed a Bayesian filter that estimates
online the latent cooperation intent of the interacting vehicles, and proposed
a control algorithm that simultaneously achieves the merging objective and
ensures driving safety. Finally, we demonstrate the effectiveness of the
proposed behavioral model and the forced merge control algorithm by
reproducing real-world trajectories and evaluating the merging performance in
simulations in comparison with the LFGC and a naturalistic traffic dataset.

## References

  * [1] M. Hügle, G. Kalweit, M. Werling, and J. Boedecker, “Dynamic interaction-aware scene understanding for reinforcement learning in autonomous driving,” in _International Conference on Robotics and Automation (ICRA)_. IEEE, 2020, pp. 4329–4335.
  * [2] A. Mavrogiannis, R. Chandra, and D. Manocha, “B-gap: Behavior-rich simulation and navigation for autonomous driving,” _Robotics and Automation Letters_ , vol. 7, no. 2, pp. 4718–4725, 2022.
  * [3] Y. Pan, C.-A. Cheng, K. Saigol, K. Lee, X. Yan, E. Theodorou, and B. Boots, “Agile autonomous driving using end-to-end deep imitation learning,” _arXiv preprint arXiv:1709.07174_ , 2017.
  * [4] X. Mei, Y. Sun, Y. Chen, C. Liu, and M. Liu, “Autonomous navigation through intersections with graph convolutionalnetworks and conditional imitation learning for self-driving cars,” _arXiv preprint arXiv:2102.00675_ , 2021\.
  * [5] B. R. Kiran, I. Sobh, V. Talpaert, P. Mannion, A. A. Al Sallab, S. Yogamani, and P. Pérez, “Deep reinforcement learning for autonomous driving: A survey,” _Transactions on Intelligent Transportation Systems_ , vol. 23, no. 6, pp. 4909–4926, 2021.
  * [6] D. Sadigh, S. Sastry, S. A. Seshia, and A. D. Dragan, “Planning for autonomous cars that leverage effects on human actions.” in _Robotics: Science and systems_ , vol. 2. Ann Arbor, MI, USA, 2016, pp. 1–9.
  * [7] C. You, J. Lu, D. Filev, and P. Tsiotras, “Advanced planning for autonomous vehicles using reinforcement learning and deep inverse reinforcement learning,” _Robotics and Autonomous Systems_ , vol. 114, pp. 1–18, 2019\.
  * [8] A. Gupta, J. Johnson, L. Fei-Fei, S. Savarese, and A. Alahi, “Social gan: Socially acceptable trajectories with generative adversarial networks,” in _Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2018, pp. 2255–2264.
  * [9] J. L. V. Espinoza, A. Liniger, W. Schwarting, D. Rus, and L. Van Gool, “Deep interactive motion prediction and planning: Playing games with motion prediction models,” in _Learning for Dynamics and Control Conference (L4DC)_. PMLR, 2022, pp. 1006–1019.
  * [10] S. Bae, D. Isele, A. Nakhaei, P. Xu, A. M. Añon, C. Choi, K. Fujimura, and S. Moura, “Lane-change in dense traffic with model predictive control and neural networks,” _Transactions on Control Systems Technology_ , 2022.
  * [11] M. Liang, B. Yang, R. Hu, Y. Chen, R. Liao, S. Feng, and R. Urtasun, “Learning lane graph representations for motion forecasting,” in _European Conference on Computer Vision (ECCV)_. Springer, 2020, pp. 541–556.
  * [12] B. Ivanovic, K. Leung, E. Schmerling, and M. Pavone, “Multimodal deep generative models for trajectory prediction: A conditional variational autoencoder approach,” _Robotics and Automation Letters_ , vol. 6, no. 2, pp. 295–302, 2020.
  * [13] N. Li, D. Oyler, M. Zhang, Y. Yildiz, A. Girard, and I. Kolmanovsky, “Hierarchical reasoning game theory based approach for evaluation and testing of autonomous vehicle control systems,” in _Conference on Decision and Control (CDC)_. IEEE, 2016, pp. 727–733.
  * [14] M. Liu, I. Kolmanovsky, H. E. Tseng, S. Huang, D. Filev, and A. Girard, “Potential game-based decision-making for autonomous driving,” _Transactions on Intelligent Transportation Systems_ , pp. 1–14, 2023.
  * [15] P. Hang, C. Lv, Y. Xing, C. Huang, and Z. Hu, “Human-like decision making for autonomous driving: A noncooperative game theoretic approach,” _Transactions on Intelligent Transportation Systems_ , vol. 22, no. 4, pp. 2076–2087, 2020.
  * [16] J. F. Fisac, E. Bronstein, E. Stefansson, D. Sadigh, S. S. Sastry, and A. D. Dragan, “Hierarchical game-theoretic planning for autonomous vehicles,” in _International Conference on Robotics and Automation (ICRA)_. IEEE, 2019, pp. 9590–9596.
  * [17] K. Liu, N. Li, H. E. Tseng, I. Kolmanovsky, and A. Girard, “Interaction-aware trajectory prediction and planning for autonomous vehicles in forced merge scenarios,” _Transactions on Intelligent Transportation Systems_ , 2022.
  * [18] K. Liu, N. Li, H. E. Tseng, I. Kolmanovsky, A. Girard, and D. Filev, “Cooperation-aware decision making for autonomous vehicles in merge scenarios,” in _Conference on Decision and Control (CDC)_. IEEE, 2021, pp. 5006–5012.
  * [19] D. M. Kuhlman and A. F. Marshello, “Individual differences in game motivation as moderators of preprogrammed strategy effects in prisoner’s dilemma.” _Journal of Personality and Social Psychology_ , vol. 32, no. 5, p. 922, 1975\.
  * [20] W. B. Liebrand, “The effect of social motives, communication and group size on behaviour in an n-person multi-stage mixed-motive game,” _European Journal of Social Psychology_ , vol. 14, no. 3, pp. 239–264, 1984.
  * [21] D. M. Messick and C. G. McClintock, “Motivational bases of choice in experimental games,” _Journal of Experimental Social Psychology_ , vol. 4, no. 1, pp. 1–25, 1968.
  * [22] W. B. Liebrand, R. W. Jansen, V. M. Rijken, and C. J. Suhre, “Might over morality: Social values and the perception of other players in experimental games,” _Journal of Experimental Social Psychology_ , vol. 22, no. 3, pp. 203–215, 1986.
  * [23] D. W. Griesinger and J. W. Livingston Jr, “Toward a model of interpersonal motivation in experimental games,” _Behavioral science_ , vol. 18, no. 3, pp. 173–188, 1973.
  * [24] W. Schwarting, A. Pierson, J. Alonso-Mora, S. Karaman, and D. Rus, “Social behavior for autonomous vehicles,” _Proceedings of the National Academy of Sciences_ , vol. 116, no. 50, pp. 24 972–24 978, 2019.
  * [25] R. S. Sutton and A. G. Barto, _Reinforcement learning: An introduction_. MIT press, 2018.
  * [26] R. Krajewski, J. Bock, L. Kloeker, and L. Eckstein, “The highd dataset: A drone dataset of naturalistic vehicle trajectories on german highways for validation of highly automated driving systems,” in _International Conference on Intelligent Transportation Systems_. IEEE, 2018, pp. 2118–2125.
  * [27] I. Papadimitriou and M. Tomizuka, “Fast lane changing computations using polynomials,” in _Proceedings of the 2003 American Control Conference, 2003._ , vol. 1. IEEE, 2003, pp. 48–53.


# Brane-vector dark matter and its connection to inflation and primordial
gravitational waves

Cao H. Nam<EMAIL_ADDRESS>(corresponding author) Phenikaa
Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa
University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam  Tran N. Hung
<EMAIL_ADDRESS>Phenikaa Institute for Advanced Study and
Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong,
Hanoi 12116, Vietnam

###### Abstract

The scalar mode describing the fluctuation of the 3-brane (the observable
universe) in a five-dimensional bulk spacetime compactified on a circle is
absorbed by the Kaluza-Klein U(1) gauge field, leading to a massive brane-
vector living on the 3-brane. The brane-vector can be responsible for dark
matter because it is odd under a $\mathrm{Z}_{2}$ symmetry, neutral under the
Standard Model (SM) symmetries, and couples extremely weak to the SM particles
due to its gravitational origin. Interestingly, the brane-vector dark matter
could leave particular imprints on the cosmic microwave background (CMB) and
the primordial gravitational waves. Hence, the precise measurements of the CMB
and the observations of the primordial gravitational waves generated during
the inflation can provide a potential way to probe the extra-dimensions and
branes which are the main ingredients of string/M theory.

##  I Introduction

Extra-dimensions and branes (hypersurfaces embedded in a higher-dimensional
spacetime or the bulk) play a fundamental role in string/M theory which has
been considered as a promising candidate for quantum gravity. String/M theory
predicts that the spacetime has spatial extra-dimensions wrapped upon a
compact internal space and open strings describing the non-gravitational
sector with their endpoints are attached to branes. Whereas, closed strings
describing the gravitational sector can propagate freely in the bulk. The low
energy regime of string/M theory hence can imply a scenario that our
observable universe is actually a 3-brane on which the Standard Model (SM)
fields are confined, the directions transverse to the 3-brane are compact, and
only the gravitational degrees of freedom and other exotic ones can move
freely in the bulk [1, 2]. In such a scenario, the constraints on the size of
the extra-dimensions could be obtained from deviations from the usual Newton’s
law [3, 4, 5, 6] and the experiments of gravitational waves (GW) [7, 8, 9].
Therefore, the experimental bounds of the size of the extra-dimensions can be
relaxed, where the size of the extra-dimensions is as large as the micron
scale. Interestingly, the compactified extra-dimensions can provide a possible
solution to the hierarchy problem where the fundamental Planck scale can be
much lower than the observed Planck scale [10, 11].

The brane tension is an important parameter that characterizes the typical
size of the brane fluctuations along the extra-dimensions. If the brane
tension is large enough, corresponding to the branes being too heavy, the
branes can be considered as rigid objects. On the contrary, when the brane
tension is much lower than the fundamental scale, the branes can fluctuate
along the extra-dimensions. The fluctuations corresponding to the oscillating
brane are realized as the Nambu-Goldstone bosons, also well-known as the
branons, which arise as a result of the spontaneously broken isometry of the
internal space due to the presence of the brane [12]. The branons are the
physical scalar excitations in the low energy spectrum of the effective field
theory generically only in the global case where the physical degrees of
freedom describing the higher-dimensional gravitation are negligible.

From the Kaluza-Klein (KK) theory, we know that the isometry of the internal
space leads to the gauge transformations where the gauge fields are the off-
diagonal components of the bulk metric [13, 14] or the connection on the
spacetime fiber bundle with the typical fiber being the internal manifold [15,
16]. This implies when the isometry of the internal space is broken
spontaneously by the presence of the brane, the KK gauge fields would absorb
the branons as their longitudinal modes, corresponding to a gravitational
Brout-Englert-Higgs mechanism. This leads to massive vector fields appearing
on the brane, as well-known as brane-vector [17, 18, 19, 20, 21]. The mass of
the brane-vector is determined by the brane tension, and hence it is a free
parameter. Because the brane tension is much lower than the fundamental Planck
scale for the case of the flexible branes, we expect that the mass of the
brane-vector can range from the eV regime to the TeV regime. The brane-vector
interacts with the SM fields via their energy-momentum tensor with the
coupling determined by the observed Planck scale as a result of the fact that
the brane-vector has a gravitational origin as new dynamic degrees of freedom
describing the behavior of the spacetime at the short-distances [22].

In this paper, we explore the brane-vector as a candidate for dark matter (DM)
whose stability is guaranteed by a $\mathrm{Z}_{2}$ parity as a result of the
4D general coordinate invariance on the brane. The coupling of the brane-
vector to the SM particles is strongly suppressed by the observed Planck
scale, hence the relic abundance of the brane-vector DM could not generated by
the thermal processes that appeared after the inflation. On the other hand,
the brane-vector DM which couples extremely weak to the SM particles could be
produced by the gravitational mechanism due to the quantum fluctuations in the
inflationary stage [23, 24, 25, 26, 27]. The transverse polarizations of a
massive vector field that couples minimally to 4D gravity are conformally
invariant in analogy to the massless vector field, since its production is
suppressed and could be negligible during the inflation [26, 27]. In this
sense, the observed relic abundance of the brane-vector DM should result from
the production of its longitudinal polarization.

The Friedmann equation describing the evolution of the brane universe consists
of linear and quadratic terms in terms of the energy density. The quadratic
term is inversely proportional to the mass of the brane-vector or the brane
tension since it plays an important role in the early universe. (Whereas, the
linear term is dominant when the energy density is much smaller than the brane
tension corresponding to the late time of the universe evolution.) This novel
feature of the modified Friedmann equation combined with the production of the
brane-vector DM during the inflation leads to a prediction of the tensor-to-
scalar ratio and the reheating temperature in terms of the brane-vector mass.
Note that, the reheating temperature can be estimated by the primordial GW
spectrum [28]. In this way, the brane-vector DM could leave specific imprints
on the cosmic microwave background (CMB) and the primordial GWs. Therefore,
the precise measurements of the CMB as well as the future detections of the
primordial GWs and the B-mode polarization in the CMB can allow us to probe
the extra-dimensions and branes.

This paper is organized as follows. In Sec. II, we study the four-dimensional
effective field theory derived from a flexible $3$-brane which propagates in a
bulk spacetime compactified on a circle $S^{1}$ and on which the SM and
inflaton fields are confined. We show that the low-energy spectrum of the
effective theory includes a brane-vector that possesses a discrete
$\mathrm{Z}_{2}$ parity and hence could be responsible for the DM. Then, we
compute the relic abundance of the brane-vector DM based on the production of
its longitudinal mode during the inflation. In Sec. III, we study the imprints
of the brane-vector on the spectrum of the CMB and the primordial GWs carrying
the information of the inflation era for two typical potentials of the
inflation: i) $\alpha$-attractor potential and ii) natural potential. Finally,
we present a brief conclusion in Sec. IV.

##  II  Brane-vector DM

For simplicity without loss of generality, we consider the dynamics of a
$3$-brane propagating in a five-dimensional bulk spacetime compactified on a
circle $S^{1}$. The action of the system is given by a sum of the bulk action
and the brane action as follows

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{\text{bulk}}+S_{\text{brane}},$ (1) $\displaystyle S_{\text{bulk}}$
$\displaystyle=$ $\displaystyle\frac{M^{3}_{5}}{2}\int
d^{5}X\sqrt{-G}\left(\mathcal{R}^{(5)}-2\Lambda_{5}\right),$ (2)
$\displaystyle S_{\text{brane}}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-\widetilde{g}}\left[-f^{4}+\mathcal{L}_{\text{SM}}+\frac{1}{2}\widetilde{g}^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right].$
(3)

In the bulk action (2), $M_{5}$ is the five-dimensional Planck scale,
$\mathcal{R}^{(5)}$ is the Ricci scalar of the bulk spacetime, $G$ is the
determinant of the bulk metric, $X^{M}$ with $M=0,1,2,3,4$ are the coordinates
of the points on the bulk spacetime, and $\Lambda_{5}$ refers to the bulk
cosmological constant which is necessarily negative to restore the
conventionally Hubble expansion law in the low-energy regime [29]. In the
brane action (3), $f$ is the tension of the $3$-brane, the coordinates of the
points on the $3$-brane are parameterized by $x^{\mu}$ with $\mu=0,1,2,3$, and
$\widetilde{g}$ is the determinant of the induced metric
$\widetilde{g}_{\mu\nu}$ on the $3$-brane. The SM fields that are described by
the Lagrangian $\mathcal{L}_{\text{SM}}$ and the inflaton field $\phi$ are
confined to the $3$-brane.

The presence of the $3$-brane could be created spontaneously by a proper
physical process, such as the tachyon condensation when turning on the
magnetic fluxes [30]. This configuration should be realized as the ground
state. As a result, the formation of the $3$-brane would break spontaneously
the local isometry of the bulk spacetime in the compactified extra dimension.
As we see below, the Nambu-Goldstone (NG) mode describing the excitation of
the $3$-brane would be eaten by the KK gauge field, corresponding to the local
$U(1)$ isometry of the circle $S^{1}$, hence it becomes massive.

###  II.1 Dimensional reduction of the bulk action

Let us study the compactification of the five-dimensional bulk spacetime on
the circle $S^{1}$ that is parametrized by an angle $\theta$. The bulk metric
can be decomposed into the four-dimensional fields as follows

$\displaystyle
G_{MN}=\left(\begin{array}[]{cc}g_{\mu\nu}+\rho^{2}A_{\mu}A_{\nu}&\rho^{2}A_{\mu}\\\
\rho^{2}A_{\nu}&\rho^{2}\\\ \end{array}\right),$ (6)

where $g_{\mu\nu}$ is the four-dimensional metric tensor, $A_{\mu}$ is the KK
gauge field corresponding to the local isometry in the fifth compactified
dimension, and $\rho$ is a scalar field whose vacuum expectation value (vev)
determines physically the radius of the circle $S^{1}$. With this
decomposition, we can write the bulk action (2) in terms of the four-
dimensional fields as follows.

$\displaystyle S_{\text{bulk}}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{P}}}{2\langle\rho\rangle}\int
d^{4}x\sqrt{-g}\rho\left(\mathcal{R}^{(4)}-2\Lambda_{5}-\frac{\rho^{2}}{4}F_{\mu\nu}F^{\mu\nu}+\frac{2}{3}\frac{\partial^{\mu}\rho\partial_{\mu}\rho}{\rho^{2}}\right),$
(7)

where $g$ is the determinant of the four-dimensional metric $g_{\mu\nu}$,
$\mathcal{R}^{(4)}$ is the four-dimensional Ricci scalar,
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength tensor of the KK gauge field, and $M_{\text{P}}$ is the observed
Planck scale determined as $M^{2}_{\text{P}}=2\pi\langle\rho\rangle M^{3}_{5}$
with $\langle\rho\rangle$ denoting the vev of the scalar field $\rho$ or the
radius of the circle $S^{1}$. We can check that the gravitational action (7)
is invariant under the following transformation

$\displaystyle\theta$ $\displaystyle\longrightarrow$
$\displaystyle\theta+\alpha(x),$ (8) $\displaystyle A_{\mu}$
$\displaystyle\longrightarrow$ $\displaystyle
A_{\mu}-\partial_{\mu}\alpha(x).$ (9)

A set of such transformations forms the local isometry in the compactified
dimension associated with a particular class of general coordinate
transformations. In particular, the off-diagonal element of the bulk metric
transforms as a gauge field of the U(1) local symmetry.

Reducing the bulk action (2) on the internal manifold $S^{1}$, we obtain an
effective action that consists of the four-dimensional Einstein gravity plus a
cosmological constant coupled to the massless U(1) gauge field and the scalar
field as follows

$\displaystyle S_{4D}$ $\displaystyle\supset$ $\displaystyle\int
d^{4}x\sqrt{g}\left[\frac{M^{2}_{\text{P}}}{2}\left(\mathcal{R}^{(4)}-2\Lambda_{5}\right)-\frac{1}{4\kappa^{2}}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\partial^{\mu}r\partial_{\mu}r+\cdots\right],$
(10)

where we have expanded the scalar field $\rho$ around its vev as
$\rho=\langle\rho\rangle(1+\sqrt{3}r/2)$ with $r$ to be the radion field,
$\kappa$ is the KK gauge coupling given as

$\displaystyle\kappa\equiv\sqrt{2}\frac{\langle\rho\rangle^{-1}}{M_{\text{P}}},$
(11)

and the ellipse refers to the terms related to the self-couplings of the
radion field and its couplings to the four-dimensional metric and the KK gauge
field.

###  II.2 Effective action on the $3$-brane

The induced metric on the $3$-brane can be determined by the embedding
geometry as $ds^{2}_{\text{brane}}=\widetilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
where $\widetilde{g}_{\mu\nu}=G_{MN}\partial_{\mu}Y^{M}\partial_{\nu}Y^{N}$
with $Y^{M}(x)$ being the embedding function which describes the position of
the $3$-brane in the bulk spacetime. In the static gauge, the embedding
function is given by $Y^{M}(x)=(x^{\mu},Y^{\theta}(x))$. With the ansatz of
the bulk metric as given by Eq. (6), the induced metric
$\widetilde{g}_{\mu\nu}$ is found as follows

$\displaystyle\widetilde{g}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}-\rho^{2}\left(A_{\mu}+\partial_{\mu}Y^{\theta}\right)\left(A_{\nu}+\partial_{\nu}Y^{\theta}\right),$
(12) $\displaystyle=$ $\displaystyle
g_{\mu\nu}-\langle\rho\rangle^{2}\left(A_{\mu}+\partial_{\mu}Y^{\theta}\right)\left(A_{\nu}+\partial_{\nu}Y^{\theta}\right)+\mathcal{O}(r).$

When the $3$-brane is created at a certain point $Y^{\theta}_{0}$ in the fifth
compactified dimension which minimizes the brane action (3) and corresponds to
the ground state. In this situation, the presence of the $3$-brane breaks
spontaneously the U(1) local isometry in the fifth compactified dimension. The
fluctuations of the $3$-brane around the ground state in the fifth
compactified dimension can be parametrized as
$Y^{\theta}(x)=Y^{\theta}_{0}+\pi(x)$ where $\pi(x)$ is the NG mode associated
with the excitation of the $3$-brane along the direction of the U(1) local
isometry breaking. As a result, the KK gauge field would absorb this NG mode
as its longitudinal mode, leading to the brane-vector. This brane-vector is
expressed by the definition $X_{\mu}\equiv A_{\mu}+\partial_{\mu}\pi$. The
field strength tensor of the brane-vector is given by

$\displaystyle
X_{\mu\nu}\equiv\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu}=F_{\mu\nu}.$ (13)

It should be noted that the NG mode describing the fluctuation of the
$3$-brane in the fifth compactified dimension would be the physical excitation
in the decoupling limit of the KK gauge field corresponding to take the limit
$\kappa\rightarrow 0$. This physical excitation is well known as the branon
which is massive in the situation that the U(1) global isometry in the fifth
compactified dimension is realized as an approximate symmetry. Interestingly,
it was indicated that the branon can be responsible for the DM candidate [31].
However, as indicated in Ref. [22], this limit leads to the violation of the
Sublattice Weak Gravity Conjecture [32, 33], the Swampland Distance Conjecture
[34], and the Festina Lente bound [35, 36] which are recognized as the
constraints of quantum gravity on the low-energy effective theories. On the
other hand, this limit would belong to the Swampland which consists of the
effective field theories that cannot be ultraviolet completed in quantum
gravity.

In this way, we can rewrite the induced metric on the $3$-brane in terms of
the four-dimensional metric and the brane-vector as follows

$\displaystyle\widetilde{g}_{\mu\nu}=g_{\mu\nu}-\langle\rho\rangle^{2}X_{\mu}X_{\nu}+\mathcal{O}(r).$
(14)

Using this result, we can expand the square root of the determinant of the
induced metric on the $3$-brane as follows

$\displaystyle\sqrt{-\widetilde{g}}=\sqrt{-g}\left(1-\frac{\langle\rho\rangle^{2}}{2}X^{\mu}X_{\mu}+\cdots\right),$
(15)

where $g$ refers to the determinant of the four-dimensional metric
$g_{\mu\nu}$. With Eqs. (14) and (15), we can find an effective action on the
$3$-brane as follows

$\displaystyle S_{4D}$ $\displaystyle\supset$ $\displaystyle\int
d^{4}x\sqrt{-g}\left(-f^{4}+\frac{m^{2}_{X}}{2\kappa^{2}}X_{\mu}X^{\mu}+\frac{\lambda}{\kappa^{2}}X^{\mu}X^{\nu}T^{\text{mt}}_{\mu\nu}+\mathcal{L}_{\text{SM}}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)+\cdots\right),$
(16)

where $m_{X}=\sqrt{2}f^{2}/M_{\text{P}}$ is the mass of the brane-vector which
is acquired due to the spontaneous breaking of the U(1) local isometry in the
fifth compactified dimension, $T^{\text{mt}}_{\mu\nu}$ is the energy-momentum
tensor of the matte fields (the SM and inflation fields) which is given by

$\displaystyle
T^{\text{mt}}_{\mu\nu}=2\frac{\delta\mathcal{L}_{\text{mt}}}{\delta
g^{\mu\nu}}-g_{\mu\nu}\mathcal{L}_{\text{mt}},$ (17)

with $\mathcal{L}_{\text{mt}}$ denote the Lagrangian of the matter fields
living on the $3$-brane, and $\lambda=1/M^{2}_{\text{P}}$ is the coupling
constant of the brane-vector to the matter fields.

From the actions (10) and (16), we observe that first, the brane-vector is odd
under a discrete symmetry $\mathrm{Z}_{2}$. This means that the effective
action (16) is invariant under the transformation $X_{\mu}\rightarrow-
X_{\mu}$, which means that the brane-vector always appears in the pairs.
Hence, this discrete symmetry would guarantee the stability of the brane
vector. Second, in analogy to the graviton, the brane-vector couples to the
energy-momentum tensor of the matter fields with the coupling strength
suppressed strongly by the observed Planck scale. This is due to the fact that
the brane vector has a geometric or gravitational origin associated with the
off-diagonal component of the bulk metric. Because the brane-vector is stable,
neutral, and couples very weakly to the matter fields, it behaves as the DM.

###  II.3 Production of brane-vector particles during the inflation

Because the brane-vector is a massive vector field, it has three physical
polarizations corresponding to two transverse components and one longitudinal
component which is the NB mode describing the fluctuation of the $3$-brane in
the fifth compactified dimension. However, the production of the transverse
polarizations of the brane-vector by the quantum fluctuations is suppressed
during the inflation because the transverse polarizations are conformally
invariant. This implies that the contribution of the transverse polarizations
to the production of the energy density of the brane-vector can be negligible.
Whereas, unlike transverse polarizations, the production of the longitudinal
polarization is unsuppressed because it is not conformally invariant.
Therefore, the relic abundance of the brane-vector DM which is observed today
is essentially due to the production of its longitudinal polarization during
the inflation.

The relic abundance of the brane-vector observed today is

$\displaystyle\Omega_{X}=\frac{\rho_{X}}{\rho_{c}},$ (18)

where $\rho_{c}$ is the critical density and $\rho_{X}$ is the present energy
density of the brane-vector. As discussed above, the contribution to the
present energy density of the brane-vector is approximately that of its
longitudinal mode and it is calculated as follows [27]

$\displaystyle\rho_{X}\simeq\rho_{X,L}=\frac{H^{2}_{i}}{8\pi^{2}}\frac{a_{*}k^{2}_{*}}{a^{3}_{0}}\int
d\ln
k\frac{k^{2}}{k^{2}_{*}}\left[\frac{a_{0}}{a_{*}}\left|\frac{X_{L}(k,t_{0})}{X_{0}(k)}\right|^{2}+\frac{a^{3}_{0}}{a_{*}\left(k^{2}+a^{2}_{0}m^{2}_{X}\right)}\left|\frac{\partial_{t}X_{L}(k,t_{0})}{X_{0}(k)}\right|^{2}\right],$
(19)

where $H_{i}$ refers to the Hubble parameter during the inflation, $a_{0}$ and
$a_{*}$ are the values of the scale factor at the present time and the moment
of $H=m_{X}$, respectively, the wavenumber $k_{*}$ is defined as
$k_{*}=a_{*}m_{X}$. $X_{L}(k,t_{0})$ is the solution of the equation of motion
for the longitudinal polarization of the brane-vector at the present time,
given by

$\displaystyle\left(\partial^{2}_{t}+\frac{3k^{2}+a^{2}m^{2}_{X}}{k^{2}+a^{2}m^{2}_{X}}H\partial_{t}+\frac{k^{2}}{a^{2}}+m^{2}_{X}\right)X_{L}(k,t)=0,$
(20)

with an initial condition at the horizon exit given as

$\displaystyle
X_{L}(k,t)\Big{|}_{\text{exit}}=X_{0}(k)\left(1-\frac{ik}{aH_{i}}\right)\exp\left\\{\frac{ik}{aH_{i}}\right\\}.$
(21)

By solving numerically Eq. (20), we can find $X_{L}(k,t)$ and then substitute
it into the integral given in Eq. (19) to obtain an approximate result which
is equal to $1.2$ [27]. As a result, we derive the energy density of the
brane-vector observed at the present time as follows

$\displaystyle\rho_{X}\simeq\frac{3m^{2}_{X}H^{2}_{i}}{20\pi^{2}}\left(\frac{a_{*}}{a_{0}}\right)^{3}.$
(22)

From the expansion law along the $3$-brane as [37, 38]

$\displaystyle
H^{2}+\frac{k^{2}}{a^{2}}=\left(\frac{\rho_{\text{m}}+M^{2}_{\text{P}}m^{2}_{X}/2}{6M^{3}_{5}}\right)^{2}+\frac{\Lambda_{5}}{6},$
(23)

and the relation $a_{*}\simeq a_{\text{mre}}\sqrt{H_{\text{mre}}/m_{X}}$ with
$a_{\text{mre}}$ and $H_{\text{mre}}$ denoting the scale factor and the Hubble
parameter at the time of the matter-radiation equality, we can determine the
ratio $a_{*}/a_{0}$ in Eq. (22) as

$\displaystyle\frac{a_{*}}{a_{0}}=\left(\frac{\rho_{\text{r},0}}{(\sqrt{7}-1)M^{2}_{\text{P}}}\right)^{1/4}\frac{1}{\sqrt{m_{X}}},$
(24)

where $\rho_{\text{r},0}$ is the energy density of the radiation at the
present time. In addition, using the tensor power spectrum amplitude in the
brane-world scenario as $A_{t}\simeq 12x_{0}\left[H_{i}/(2\pi
M_{\text{P}})\right]^{2}$ with $x_{0}\equiv
2\sqrt{3H^{2}_{i}M^{2}_{\text{P}}/(2f^{4})}$ [39] and the relation between the
tensor power spectrum amplitude $A_{t}$ and the scalar power spectrum
amplitude $A_{s}$ given by the tensor-to-scalar ratio $r$ as $r=A_{t}/A_{s}$,
we can determine the Hubble parameter $H_{i}$ in terms of the inflationary
quantities that can be observed in the CMB spectrum as

$\displaystyle
H^{3}_{i}=\frac{\pi^{2}A_{s}r}{6\sqrt{3}}m_{X}M^{2}_{\text{P}}.$ (25)

Finally, we find an expression for the relic abundance of the brane-vector DM
in terms of its mass, the tensor-to-scalar ratio and the scalar power spectrum
amplitude as

$\displaystyle\Omega_{X}h^{2}$ $\displaystyle\simeq$ $\displaystyle
0.16\times\left(\frac{rA_{s}M^{2}_{\text{P}}}{10^{26}\text{GeV}^{2}}\right)^{2/3}\left(\frac{m_{X}}{10^{3}\text{GeV}}\right)^{7/6}.$
(26)

Using the constraints from both the Planck data on the observed DM relic
abundance $\Omega_{\text{CDM}}h^{2}\simeq 0.12$ [40] and the $95\%$ CL upper
limit on the tensor-to-scalar ratio as $r<0.056$ [41], we depict the allowed
parameter space for the tensor-to-scalar ratio $r$ and the brane-vector mass
$m_{X}$ in Fig. 1.

---
Figure 1: The allowed parameter region (the white region) is given in the
$r-m_{X}$ plane where the light gray and light purple regions are excluded by
the measurement of the DM relic density by the Planck collaboration [40] and
no detection of the CMB B-mode polarization leading to the $95\%$ CL upper
limit on $r$ [41], respectively.

We observe that these constraints impose a lower bound on the brane-vector
mass as $m_{X}\gtrsim 262.45$ GeV. This bound can be transferred to an upper
bound on the radius of the compactified extra-dimension. First, we need to
determine the brane-vector mass in terms of the radius of the compactified
extra-dimension. We observe that the expansion law on the $3$-brane as given
by Eq. (23) would lead to the standard expansion law in the low energy limit
of $\rho_{\text{m}}\ll M^{2}_{\text{P}}m^{2}_{X}/2$ and the bulk cosmological
constant satisfying

$\displaystyle\Lambda_{5}=-\frac{6M^{6}_{5}}{M^{4}_{\text{P}}}.$ (27)

Using this relation, we find an expression for the brane-vector mass in terms
of $\langle\rho\rangle$ as follows

$\displaystyle m_{X}=\frac{\sqrt{3}}{\pi\langle\rho\rangle}.$ (28)

Finally, we determine the constraint on the radius of the compactified extra-
dimension as $\langle\rho\rangle\lesssim 4.14\times 10^{-19}$ m in order for
the brane-vector responsible for the DM candidate. Obviously, this upper bound
satisfies the constraint coming from the fifth-force experiments [5, 6] which
set an upper bound $\langle\rho\rangle\lesssim\mathcal{O}(40)$ $\mu$m. In
addition, we find that the brane-vector mass is about a few hundred GeV for
the region $r\in[\mathcal{O}(10^{-3}),\mathcal{O}(10^{-2})]$ which is
predicted in some potential inflation models and is experimentally accessible
in the near future/present experiments.

##  III  Connection of brane-vector DM to inflation and primordial
gravitational wave

As discussed above, the brane-vector DM has the geometric origin by which its
coupling to the SM particles would be strongly suppressed by the observed
Planck scale. Hence, it is extremely hard to detect the signatures of the
brane-vector DM at the colliders. However, the inflation and the primordial
GWs generated during the inflation are very sensitive to the presence of the
brane-vector DM. This can be seen from the non-standard expansion law given by
Eq. (23) with the allowed mass region of $m_{X}\gtrsim 262.45$ GeV. The very
different behavior of the expansion rate compared to the usual one due to the
presence of the brane-vector DM becomes significant in the energy density
region around/above $5.32\times 10^{39}$ GeV4. Therefore, the production of
the brane-vector DM due to the quantum fluctuations during the inflation would
leave imprints in the spectrum of the CMB and the primordial GWs whose
accurate measurements allow us to probe the brane-vector DM and hence the
compactified extra dimension and the brane universe.

In the following, we assume that the brane-vector particles constitute almost
the observed relic abundance by which the tensor-to-scalar ratio can be
expressed in terms of the brane-vector mass as follows

$\displaystyle r\simeq\left(\frac{50.55\ \text{GeV}}{m_{X}}\right)^{7/4}.$
(29)

From this relation, we observe that if the tensor-to-scalar ratio is not too
small, the energy density during the inflation is much larger than
$M^{2}_{\text{P}}m^{2}_{X}/2$. As a result, the expansion law during the
inflation is given by

$\displaystyle H_{i}\simeq\frac{V(\phi)}{\sqrt{3}M^{2}_{\text{P}}m_{X}}.$ (30)

The equation of motion which plays the behavior of the inflaton during the
inflation in the slow-roll approximation reads

$\displaystyle 3H_{i}\dot{\phi}\simeq-V^{\prime}(\phi).$ (31)

Also, we can determine the slow-roll parameters in terms of the inflation
potential, its derivatives, and the brane-vector mass as follows

$\displaystyle\epsilon\simeq\frac{M^{4}_{\text{P}}m^{2}_{X}}{V(\phi)}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2},\
\
\eta\simeq\frac{M^{4}_{\text{P}}m_{X}^{2}}{V(\phi)}\frac{V^{\prime\prime}(\phi)}{V(\phi)}.$
(32)

These slow-roll parameters are related to the scalar spectral index $n_{s}$
and the tensor-to-scalar ratio $r$ which can be measured in the CMB spectrum
as

$\displaystyle n_{s}-1=-6\epsilon+2\eta,\ \ r=16\epsilon.$ (33)

The number of e-folds between the moment that the comoving mode with the wave
number $k$ exits the horizon and the end of the inflation is computed as

$\displaystyle N_{k}$ $\displaystyle=$
$\displaystyle\int^{\phi_{\text{e}}}_{\phi_{k}}\frac{H_{i}}{\dot{\phi}}d\phi$
(34) $\displaystyle=$
$\displaystyle-\frac{1}{M^{4}_{\text{P}}m^{2}_{X}}\int^{\phi_{\text{e}}}_{\phi_{k}}\frac{V^{2}(\phi)}{V^{\prime}(\phi)}d\phi,$

where $\phi_{k}$ and $\phi_{\text{e}}$ refer to the values of the inflaton at
the time of the mode $k$ exiting the horizon and the end of the inflation,
respectively. We can determine $\phi_{\text{e}}$ from $\epsilon\approx 1$
which occurs at the end of the inflation.

With the equations related to the inflation and the brane-vector DM production
during the inflation, we can predict the scalar spectral index $n_{s}$ and the
tensor-to-scalar ratio $r$ in terms of the number of e-folds $N_{k}$. The
relation between $r$ and $N_{k}$ is given by

$\frac{r^{15/7}}{1078.47}=\frac{M^{4}_{\text{P}}\text{GeV}^{2}}{V(\phi_{k})}\left(\frac{V^{\prime}(\phi_{k})}{V(\phi_{k})}\right)^{2},$
(35)

where $\phi_{k}=\phi_{k}(N_{k},r)$ which is a function of the number of
e-folds and the tensor-to-scalar ratio and can be determined from Eqs. (29)
and (34). Whereas, the relation between $n_{s}$ and $N_{k}$ is determined by

$\displaystyle n_{s}=1-\frac{3}{8}r(N_{k})+2\eta(N_{k}).$ (36)

###  III.1 $\alpha$-attractor potential

The inflation with the $\alpha$-attractor potential is given by [42, 43]

$\displaystyle
V(\phi)=\Lambda^{4}\left[1-\exp\left(-\frac{\phi}{\bar{M}}\right)\right]^{2n},$
(37)

where $\phi$ is the inflaton field coupled minimally to gravity on the
3-brane, $\Lambda$ is related to the inflationary energy scale,
$\bar{M}\equiv\sqrt{3\alpha/2}M_{\text{P}}$ with $\alpha$ to be a parameter
determining the inverse curvature of the Kähler manifold, and $n$ is a
parameter. Note that, in the case of $\alpha=1$ and $n=1$, the
$\alpha$-attractor potential leads to the Starobinsky model [44].

By substituting the potential (37) into Eq. (32), we obtain the slow-roll
parameters as follows

$\displaystyle\epsilon$ $\displaystyle=$ $\displaystyle\left[\frac{2n\beta
e^{-\phi/\bar{M}}}{\left(1-e^{-\phi/\bar{M}}\right)^{n+1}}\right]^{2},$ (38)
$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\left(1-\frac{e^{\phi/\bar{M}}}{2n}\right)\epsilon,$ (39)

where

$\displaystyle\beta\equiv\frac{M_{\text{P}}^{2}m_{X}}{\bar{M}\Lambda^{2}}.$
(40)

At the moment that the mode $k$ crosses the comoving Hubble horizon
corresponding to $X_{k}=\exp\left(-\phi_{k}/\bar{M}\right)\ll 1$, we obtain
the following approximation

$\displaystyle X_{k}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{\epsilon}}{2n\beta},$ (41) $\displaystyle\eta$
$\displaystyle\simeq$ $\displaystyle\epsilon-\beta\sqrt{\epsilon}.$ (42)

As a result, we can find approximate expressions for the scalar spectral index
$n_{s}$ and the tensor-to-scalar ratio $r$ as follows

$\displaystyle\epsilon$ $\displaystyle\simeq$
$\displaystyle\frac{1}{16}\left[\sqrt{4(1-n_{s})+\beta^{2}}-\beta\right]^{2},$
(43) $\displaystyle r$ $\displaystyle\simeq$
$\displaystyle\left[\sqrt{4(1-n_{s})+\beta^{2}}-\beta\right]^{2}.$ (44)

Note that, from Eqs. (29) and (44), we can express the parameter $\beta$ in
terms of the scalar spectral index $n_{s}$ and the brane-vector mass $m_{X}$
as follows

$\displaystyle\beta\simeq
0.02\left[4(1-n_{s})-958.48\left(\frac{\text{GeV}}{m_{X}}\right)^{7/4}\right]\left(\frac{m_{X}}{\text{GeV}}\right)^{7/8}.$
(45)

The value of the inflaton at the end of the inflation corresponding to
$\epsilon\simeq 1$ reads

$\displaystyle
X_{\text{e}}\simeq\frac{1+n+2n\beta-\sqrt{1-n^{2}+4n(1+n)\beta+4n^{2}\beta^{2}}}{n(1+n)},$
(46)

where $X_{\text{e}}\equiv\exp\left(-\phi_{\text{e}}/\bar{M}\right)$. Then, the
number of e-folds is calculated by

$\displaystyle
N_{k}=\frac{1}{2n\beta^{2}}\left[B(1-X_{k},2+2n,-1)-B(1-X_{\text{e}},2+2n,-1)\right],$
(47)

where $B(z,p,q)=\int^{z}_{0}t^{p-1}(1-t)^{q-1}dt$ is the incomplete beta
function.

In order to find the relation between $N_{k}$ and $n_{s}$, we solve
numerically Eq. (47) for various values of the number of e-folds. The result
is shown in Fig. 2 where $n$ is taken the values from 1 to 10 and the mass of
the brane-vector DM is considered as $350$ GeV and $1000$ GeV.

Figure 2: The number of e-folds $N_{k}$ versus the scalar spectral index
$n_{s}$ for the $\alpha$-attractor potential inflation. The light red and blue
bands correspond to the brane-vector mass $m_{X}=350$ GeV and $1000$ GeV,
respectively. The parameter $n$ takes the value from $1$ to $10$ in every band
where the dashed and solid boundary curves correspond to $n=1$ and $n=10$,
respectively. The gray band refers to the observation constraint on the scalar
spectral index as $n_{s}=0.965\pm 0.004$ (68$\%$ CL) [40].

We observe that the brane-vector DM would be consistent with the inflation
which requires 50 - 60 e-folds for its high mass region. The low mass region
of the brane-vector DM may be favorable to the inflation if the parameter $n$
in the inflation potential is large enough.

By solving numerically Eq. (47) and using Eq. (29), we obtain the relation
between the number of e-folds $N_{k}$ and the mass of the brane-vector DM, as
depicted in Fig. 3, with $n=1$ and $n=10$. Our prediction is shown by the
purple bands with the scalar spectral index taking the value from $0.961$ to
$0.969$. In adition, we show the $68\%$ and $95\%$ CL experimental constraints
for $n_{s}$ and $r$ [41] corresponding to the (dashed) blue and (dashed) red
contours.

Figure 3: The number of e-folds $N_{k}$ versus the mass of the brane-vector DM
with $n=1$ (the left panel) and $n=10$ (the right panel). Every purple band
represents the prediction with the scalar spectral index taking the value from
$0.965-0.004$ to $0.965+0.004$. The solid and dashed red contours correspond
to the $68\%$ and $95\%$ CL Planck18 constraints for $n_{s}$ and $r$,
respectively [41]. Whereas, the solid and dashed blue contours correspond to
the $68\%$ and $95\%$ CL Planck18+BK15 constraints, respectively [41].

We see that the number of e-folds required by the inflation is satisfied in
the high mass region of the brane-vector DM. Whereas, in order for the low
mass region to meet this criterion, the parameter $n$ in the inflation
potential needs to be large enough. These are completely consistent with what
is observed in Fig. 2.

We arrive to investigate the reheating constraint for the brane-vector DM
produced during the inflation. The connection between the parameters of the
inflation and those of the reheating can be obtained by the following relation

$\displaystyle\frac{\rho_{\text{re}}}{\rho_{\text{e}}}=e^{-3N_{\text{re}}(1+\omega_{\text{re}})},$
(48)

where $\rho_{\text{re}}$ and $\rho_{\text{e}}$ are the energy density at the
end of reheating (re) and the end of inflation, respectively, and
$N_{\text{re}}$ is the number of e-folds during the reheating corresponding to
the equation-of-state parameter denoted by $\omega_{\text{re}}$. The energy
density at the end of reheating is determined in terms of the reheating
temperature $T_{\text{re}}$ and the effective number of relativistic degrees
of freedom $g_{\text{re}}$ at the reheating epoch as
$\rho_{\text{re}}=\pi^{2}g_{\text{re}}T^{4}_{\text{re}}/30$. With the energy
density at the end of inflation given by the inflation potential at the end of
the inflation $V_{\text{e}}$ as $\rho_{\text{e}}\simeq 3V_{\text{e}}/2$, we
express the reheating temperature as follows

$\displaystyle T_{\text{re}}$ $\displaystyle=$
$\displaystyle\left[\left(\frac{45}{\pi^{2}g_{\text{re}}}\right)^{3}\frac{\pi^{2}A_{s}}{2}\right]^{1/12}\left(M^{2}_{\text{P}}m_{X}\right)^{1/3}\left(\frac{50.55\
\text{GeV}}{m_{X}}\right)^{7/48}e^{-\frac{3}{4}N_{\text{re}}(1+\omega_{\text{re}})}$
(49)
$\displaystyle\times\left[\frac{n(n-2\beta)-1+\sqrt{1-n^{2}+4n(1+n)\beta+4n^{2}\beta^{2}}}{n(n+1)}\right]^{n/2}.$

In order to determine $N_{\text{re}}$, we use the relation
$k/(a_{0}H_{0})=a_{k}H_{k}/(a_{0}H_{0})$ with $k=0.002$ Mpc-1 being the pivot
scale and $a_{k}$ ($H_{k}$) is the scale factor (the Hubble parameter) at the
moment which the mode $k$ crosses the horizon, which leads to [45]

$\displaystyle\frac{k}{a_{0}H_{0}}=e^{-\left(N_{k}+N_{\text{re}}\right)}\frac{a_{\text{re}}}{a_{\text{mre}}}\frac{a_{\text{mre}}H_{\text{mre}}}{a_{0}H_{0}}\frac{H_{k}}{H_{\text{mre}}}.$
(50)

From the conservation of entropy, we can find the ratio of $a_{\text{re}}$ to
$a_{0}$ as follows

$\displaystyle\frac{a_{\text{re}}}{a_{0}}=\left(\frac{43}{11g_{s,\text{re}}}\right)^{1/3}\frac{T_{0}}{T_{\text{re}}},$
(51)

where $g_{s,\text{re}}$ is the effective number of relativistic degrees of
freedom that contribute to the entropy and $T_{0}$ is the present temperature
of the universe. Then, the number of e-folds at the end of reheating is given
by

$\displaystyle N_{\text{re}}$ $\displaystyle=$
$\displaystyle\frac{4}{1-3\omega_{\text{re}}}\left[67.38-N_{k}-\ln\left(\frac{k}{a_{0}H_{0}}\right)-\frac{1}{12}\ln
g_{\text{re}}+\frac{1}{3}\ln\frac{g_{\text{re}}}{g_{s,\text{re}}}+\frac{1}{4}\ln\left(\frac{\pi^{2}A_{s}}{3}\right)\right.$
(52) $\displaystyle\left.+\frac{7}{16}\ln\left(\frac{50.55\
\text{GeV}}{m_{X}}\right)+\frac{n}{2}\ln\left(\frac{n(n+1)}{n(n-2\beta)-1+\sqrt{1-n^{2}+4n(1+n)\beta+4n^{2}\beta^{2}}}\right)\right].$

From Eqs. (49) and (52), we can find the relation between the reheating
temperature $T_{\text{re}}$ and the mass of the brane-vector DM. In Fig. 4, we
depict this relation for $n=1$ (the left panel) and $n=10$ (the right panel).
The region is bounded by the solid and dashed purple curve represents our
prediction with the $68\%$ CL value of the scalar spectral index as
$n_{s}=0.965\pm 0.004$.

Figure 4: The reheating temperature versus the mass of the brane-vector DM for
various values of the parameter $n$ with $\omega_{\text{re}}=0$. The left and
right panels correspond to $n=1$ and $n=10$, respectively. The solid and
dashed purple curves correspond to $n_{s}=0.969$ and $n_{s}=0.961$,
respectively. The red and blue contours correspond to the $68\%$ CL Planck18
and Planck18+BK15 constraints, respectively. The grey region is excluded by
the constraint of Big Bang Nucleosynthesis (BBN).

We see that the present observation of $n_{s}$ leads to the large uncertainty
for the prediction of the reheating temperature in terms of the mass of the
brane-vector DM. The CMB experiments in the future which will be able to
decrease the uncertainty of the $n_{s}$ measurement will provide a more
accurate prediction for this relation.

The spectrum of inflationary GWs [46], which is a function of the frequency
$f$, is connected to the brane-vector DM as

$\displaystyle\Omega_{\text{GW}}h^{2}\left(f\right)$ $\displaystyle\simeq$
$\displaystyle\frac{A_{s}}{3}\left(\frac{\pi\Omega_{\text{m}}hf}{a_{0}H_{0}}\right)^{2}\left(\frac{50.55\
\text{GeV}}{m_{X}}\right)^{7/4}\exp\left\\{-\frac{1}{8}\left(\frac{50.55\
\text{GeV}}{m_{X}}\right)^{7/4}\log\left(\frac{f}{f_{0}}\right)\right\\}$ (53)
$\displaystyle\times\frac{g_{\text{in}}}{3.36}\left[\frac{3.9}{g_{s,{\text{in}}}}\right]^{4/3}\left[\frac{3j_{1}(z_{k})}{z_{k}}\right]^{2}T^{2}_{1}\left(f/f_{\text{eq}}\right)T^{2}_{2}(f/f_{\text{re}}),$

where “in” denotes the moment that the mode $k$ crosses the horizon,
$j_{1}(z)$ is the spherical Bessel function of the first kind, $z_{k}\equiv
4\pi f/(a_{0}H_{0})$, the functions $T_{1,2}(x)$ describe the change of the
primordial GW spectrum at the frequency $f_{\text{re}}\simeq
0.26\text{Hz}\times T_{\text{re}}/(10^{7}\text{GeV})$ [47] due to the presence
of the reheating and are given as follows

$\displaystyle T_{1}(x)$ $\displaystyle=$
$\displaystyle\left(1+1.57x+3.42x^{2}\right)^{1/2},$ $\displaystyle T_{2}(x)$
$\displaystyle=$ $\displaystyle\left(1-0.32x+0.99x^{2}\right)^{-1/2}.$ (54)

In Fig. 5, we show the prediction of the primordial GW spectrum for various
values of the brane-vector mass and the parameter $n$. In addition, we include
the projected sensitivities of the different GW observations, represented by
the colored regions, which are the future SKA [48], the space-based LISA [49],
BBO [50], DECIGO [51], and Ultimate-DECIGO [52].

Figure 5: The spectrum of inflationary GWs as a function of the frequency for
various values of the brane-vector mass and the parameter $n$. The scalar
spectral index $n_{s}$ runs from $0.965$ (represented by the dashed blue, red,
and purple curves) to $0.967$ (represented by the solid blue, red, and purple
curves). The left and right panels correspond to $n=1$ and $n=10$,
respectively. The bands bounded by the blue, red, and purple curves correspond
to $m_{X}=1186$ GeV, $798$ GeV, and $702$ GeV, respectively.

We observe that Ultimate-DECIGO can probe the reheating temperature and the
brane-vector DM in its allowed mass region if the parameter $n$ in the
inflation potential is large enough. On the contrary, if $n$ is low, Ultimate-
DECIGO only allows us to probe the low mass region of the brane-vector DM. In
this sense, the observation of the primordial GWs produced during the
inflation provides a potential way to probe the extra-dimensions and branes:
the reheating temperature and the mass of the brane-vector DM which are
determined by the spectrum of the primordial GWs are compared to the
prediction which is given in Eqs. (49) and (52).

###  III.2 Natural potential

We study the connection of the brane-vector DM to the inflation with the
natural potential or the axion potential [53] which appears naturally in the
compactifications of M/string theory [54] and provides a plausible solution to
the strong CP problem [55]. The inflation potential in the natural inflation
is given as follows

$\displaystyle
V(\phi)=\Lambda^{4}\left[1-\cos\left(\frac{\phi}{f}\right)\right],$ (55)

where $f$ is a parameter. Using Eq. (32) with the inflation potential (55), we
find the slow-roll parameters as follows

$\displaystyle\epsilon=\gamma^{2}\frac{1+\cos(\phi/f)}{[1-\cos(\phi/f)]^{2}}$
(56) $\displaystyle\eta=\gamma^{2}\frac{\cos(\phi/f)}{[1-\cos(\phi/f)]^{2}}$
(57)

where

$\displaystyle\gamma\equiv\frac{M_{\text{P}}^{2}m_{X}}{f\Lambda^{2}}.$ (58)

At the horizon crossing of the mode $k$, corresponding to $\epsilon\ll 1$, we
find the value of the inflaton field in terms of the spectral index $n_{s}$
and the brane-vector mass as follows

$\displaystyle\cos\left(\frac{\phi_{k}}{f}\right)\simeq-1+\frac{4}{25\gamma^{2}}\left[5(1-n_{s})-\gamma\left(\gamma+\sqrt{(5-5n_{s})/2+\gamma^{2}}\right)\right]\equiv
X_{k},$ (59)

where the parameter $\gamma$ can be expressed as

$\displaystyle\frac{\gamma}{\sqrt{2}}\simeq\frac{1-n_{s}-300\left(\text{GeV}/m_{X}\right)^{7/4}}{\sqrt{1-n_{s}-240\left(\text{GeV}/m_{X}\right)^{7/4}}}.$
(60)

The number of e-folds is

$\displaystyle
N_{k}=\frac{1}{\gamma^{2}}\left[X_{k}-X_{f}-2\log\left(\frac{1+X_{k}}{1+X_{f}}\right)\right],$
(61)

where $X_{f}\equiv\cos(\phi_{f}/f)$ with $\phi_{f}$ being the value of the
inflaton field at the end of inflation which is determined by the condition
$\epsilon\simeq 1$ leading to

$\displaystyle X_{f}\simeq
1+\frac{\gamma}{2}\left(\gamma-\sqrt{\gamma^{2}+8}\right).$ (62)

In Fig. 6, we show the dependence of the scalar spectral index $n_{s}$ on the
number of e-folds $N_{k}$.

Figure 6: The number of e-folds $N_{k}$ versus the scalar spectral index
$n_{s}$ in the natural inflation for various values of the brane-vector mass
from $m_{X}=280$ GeV (dashed red curve) to $m_{X}=1000$ GeV (solid red curve).
The dark and light gray bands correspond to the value of $n_{s}$ at 68$\%$ CL
and $95\%$ CL, respectively.

We can see that the brane-vector DM is only consistent with the natural
inflation in the $2\sigma$ region for the value of $n_{s}$ and its low mass
region around 280 GeV. However, as we see above, the low mass region of the
brane-vector DM is hard to detect in the spectrum of the primordial GWs in the
near future.

##  IV Conclusion

In the present paper, we have explored dark matter and its connection to the
inflationary observations and the spectrum of the primordial gravitational
waves in a scenario in which our observable universe is realized as a flexible
3-brane that could oscillate in a five-dimensional bulk spacetime compactified
on a circle. (Such a scenario can be considered as a low-energy limit of ten-
dimensional string theory which is reduced on a five-dimensional internal
manifold.) The off-diagonal components of the bulk metric that is the gauge
field corresponding to the U(1) isometry of the internal space would absorb
the scalar mode describing the fluctuation of the 3-brane in the bulk. As a
result, this leads to a massive vector field living on the 3-brane, so-called
the brane-vector. Because the coupling of the brane-vector is strongly
suppressed by the observed Planck scale, the observed relic abundance of the
brane-vector dark matter is essentially due to the production of its
longitudinal mode by the quantum fluctuations in the inflationary stage. In
this way, the tensor-to-scalar ratio is determined in terms of the brane-
vector mass or the brane tension. In addition, the mass of the brane-vector
enters into the Friedmann equation describing the evolution of the brane
universe, which is very sensitive with respect to the region of the high
energy density such as the inflation scale. Therefore, we have studied the
imprints of the brane-vector dark matter in the inflationary observables and
the spectrum of the primordial gravitational waves. This can open a promising
observation window to detect the extra-dimensions and branes which play a
fundamental role in string/M theory.

## References

  * [1] P. Horava and E. Witten, Nucl. Phys. B 475 (1996) 94.
  * [2] A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram, Phys. Rev. D 59, 086001 (1999).
  * [3] E. G. Adelberger, B. R. Heckel, and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 53 (2003) 77.
  * [4] E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl, and S. Schlamminger, Prog. Part. Nucl. Phys. 62 (2009) 102.
  * [5] D. J. Kapner, T. S. Cook, E. G. Adelberger, J. H. Gundlach, B. R. Heckel, C. D. Hoyle, and H. E. Swanson, Phys. Rev. Lett. 98, 021101 (2007).
  * [6] J. Murata and S. Tanaka, Class. Quant. Grav. 32, 033001 (2015).
  * [7] J. García-Bellido, S. Nesseris, and M. Trashorras, JCAP 07 (2016) 021.
  * [8] L. Visinelli, N. Bolis, and S. Vagnozzi, Phys. Rev. D 97, 064039 (2018).
  * [9] S. Chakraborty, K. Chakravarti, S. Bose, and S. SenGupta, Phys. Rev. D 97, 104053 (2018).
  * [10] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B 429, 263 (1998).
  * [11] C. H. Nam, Eur. Phys. J. C 81, 1102 (2021).
  * [12] R. Sundrum, Phys. Rev. D 59, 085009 (1999).
  * [13] D. Bailin and A. Love, Rep. Prog. Phys. 50, 1087 (1987).
  * [14] J. M. Overduin and P.S. Wesson, Phys. Rep. 283, 303 (1997).
  * [15] R. Coquereaux and A. Jadczyk, World Sci. Lect. Notes Phys. 16, 1 (1988).
  * [16] C. H. Nam, Eur. Phys. J. C 79, 384 (2019).
  * [17] A. Dobado and A. L. Maroto, Nucl. Phys. B 592, 203 (2001).
  * [18] G. Kang and Y. S. Myung, Phys. Rev. D 63, 064036 (2001).
  * [19] T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Rev. D 75, 065028 (2007).
  * [20] T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Rev. D 78, 115004 (2008).
  * [21] T. E. Clark, S. T. Love, M. Nitta, T. ter Veldhuis, and C. Xiong, Phys. Lett. B 671, 383 (2009).
  * [22] C. H. Nam, Phys. Lett. B 841, 137930 (2023).
  * [23] L. Ford, Phys. Rev. D 35, 2955 (1987).
  * [24] D. H. Lyth and D. Roberts, Phys. Rev. D 57, 7120 (1998).
  * [25] K. Dimopoulos, Phys. Rev. D 74, 083502 (2006).
  * [26] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, JCAP 1206, 013 (2012).
  * [27] P. W. Graham, J. Mardon, and S. Rajendran, Phys. Rev. D 93, 103520 (2016).
  * [28] K. Nakayama, S. Saito, Y. Suwa, and J. Yokoyama, JCAP 0806, 020 (2008).
  * [29] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, Phys. Rept. 513 (2012) 1.
  * [30] M. Majumdar and A.-C. Davis, JHEP 0203 (2002) 056.
  * [31] J. A. R. Cembranos, A. Dobado, and A. L. Maroto, Phys. Rev. Lett. 90, 241301 (2003).
  * [32] B. Heidenreich, M. Reece, and T. Rudelius, JHEP 08 (2017) 025.
  * [33] B. Heidenreich, M. Reece, and T. Rudelius, Eur. Phys. J. C 78, 337 (2018).
  * [34] H. Ooguri and C. Vafa, Nucl. Phys. B 766, 21 (2007).
  * [35] M. Montero, T. Van Riet, and G. Venken, JHEP 01 (2020) 039.
  * [36] M. Montero, C. Vafa, T. Van Riet and G. Venken, JHEP 10 (2021) 009.
  * [37] P. Binetruy, C. Deffayet, and D. Langlois, Nucl. Phys. B 565, 269 (2000).
  * [38] P. Binetruy, C. Deffayet, U. Ellwanger, and D. Langlois, Phys. Lett. B 477, 285 (2000).
  * [39] D. Langlois, R. Maartens, and D. Wands, Phys. Lett. B489 (2000) 259.
  * [40] Y. Akrami _et al._ (Planck collaboration), Astron. Astrophys. 641, A6 (2020).
  * [41] Y. Akrami _et al._ (Planck collaboration), Astron. Astrophys. 641, A10 (2020).
  * [42] J. J. M. Carrasco, R. Kallosh, and A. Linde, JHEP 10 (2015) 147.
  * [43] J. J. M. Carrasco, R. Kallosh, and A. Linde, Phys. Rev. D 92, 063519 (2015).
  * [44] A. A. Starobinsky, Phys. Lett. B 91, 99 (1980).
  * [45] A. R. Liddle and S. M. Leach, Phys. Rev. D 68, 103503 (2003).
  * [46] K. Nakayama and J. Yokoyama, JCAP 1001, 010 (2010).
  * [47] S. Kuroyanagi, K. Nakayama, and S. Saito, Phys. Rev. D 84, 123513 (2011).
  * [48] G. Janssen _et al._ , PoS AASKA14, 037 (2015).
  * [49] LISA collaboration, _Laser Interferometer Space Antenna_ , arXiv:1702.00786.
  * [50] J. Crowder and N. J. Cornish, Phys. Rev. D 72, 083005 (2005).
  * [51] N. Seto, S. Kawamura, and T. Nakamura, Phys. Rev. Lett. 87, 221103 (2001).
  * [52] H. Kudoh, A. Taruya, T. Hiramatsu, and Y. Himemoto, Phys. Rev. D 73, 064006 (2006).
  * [53] K. Freese, J. A. Frieman, and A. V. Olinto, Phys. Rev. Lett. 65, 3233 (1990).
  * [54] P. Svrcek and E. Witten, JHEP 06 (2006) 051.
  * [55] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).