# Radar Enhanced Multi-Armed Bandit for Rapid Beam Selection in Millimeter
Wave Communications

Akanksha Sneh1, Sumit Darak1, Shobha Sundar Ram1 and Manjesh Hanawal2
1Indraprastha Institute of Information Technology Delhi, New Delhi 110020
India
2Indian Institute of Technology Bombay, Mumbai 400076 India
E-mail: {akankshas, sumit, shobha<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Multi-arm bandit (MAB) algorithms have been used to learn optimal beams for
millimeter wave communication systems. Here, the complexity of learning the
optimal beam linearly scales with the number of beams, leading to high latency
when there are a large number of beams. In this work, we propose to integrate
radar with communication to enhance the MAB learning performance by searching
only those beams where the radar detects a scatterer. Further, we use radar to
distinguish the beams that show mobile targets from those which indicate the
presence of static clutter, thereby reducing the number of beams to scan.
Simulations show that our proposed radar-enhanced MAB reduces the exploration
time by searching only the beams with distinct radar mobile targets resulting
in improved throughput.

###### Index Terms:

multi-armed bandit, joint radar communication, upper confidence bound, analog
beamforming

##  I Introduction

Millimeter wave (mmW) unlicensed spectrum has been identified as a viable
solution for realizing high data rate communications between connected
vehicles [1, 2, 3, 4, 5]. The communication links are, however, characterized
by high atmospheric absorption and hence can be operational only in short-
range line-of-sight scenarios with highly directional beams realized through
analog or digital beamforming at the transmitter/receiver. Digital beamforming
allows for multiple simultaneous beams but is costly and complicated to
implement since multiple phase and time-synchronized RF/mmW chains are
required [6]. Analog beamforming is less costly since it involves a single
beam at a time. But there is considerable overhead expended by the
communication protocol and a long search/exploration time to scan the entire
field of view and select the best beams for each mobile user (MU). This
results in high latency and shorter service/exploitation time available for
communication causing low throughput.

There have been several recent works that have applied multi-armed bandit
(MAB) algorithms for reducing the exploration time of the best beams in order
to increase the exploitation time for subsequent mmW communications [7, 8, 9,
10]. MAB algorithms are a class of algorithms within the reinforcement
learning framework which provides a basis for making decisions under
uncertainty. MAB-based beam selection works, such as [8, 9], have relied on a
strategy where the base station (BS) waits for the feedback (reward) over the
uplink for the beam selection in subsequent time slots. In such time-slotted
communication, the transmitter can switch the beam only once in a slot, and
the duration of each slot depends on the time taken by the MU to process the
downlink signal and share the feedback over the uplink.

The use of radar signals for detecting the presence of targets can potentially
speed-up the beam search. However, there are certain challenges in the
integration of radar with communication physical layer. The use of an
auxiliary radar sensor for detecting MU, cannot be considered, as it would
increase the cost and complexity of the system. Further, the radar and
communication functionalities would have to be synchronized as well as managed
for interference. Instead, we propose that an integrated sensing and
communication system be utilized for mmW communication such as those proposed
by [11, 12]. Here, a common waveform on a common spectrum is used for joint
radar sensing and communications. Hence, no separate
hardware/spectrum/synchronization or interference management is required to
support both functionalities. Note that joint radar communication (JRC)
systems have been explored over the last several decades to tackle spectral
congestion issues [13]. While some works have studied how to manage mutual
interference that arises from the coexistence of both systems on a common
spectrum [14], others have exploited the communication signal as an
opportunistic illumination for passive radar receivers [15]. We identify our
work to belong to the third category of research that explores the
collaborative design of JRC systems to improve the performance of each
functionality [16, 17].

In this work, we propose incorporating a radar sensing mechanism into the MAB
framework at the BS to overcome the limitations listed above. In the proposed
framework, the radar at the BS is used to detect the presence of MU in the
candidate beams based on the strength of the scattered signal (_amplitude
gated radar enhanced MAB_) and the Doppler frequency shift (_Doppler gated
radar enhanced MAB_) introduced to the radar signal. Only those beams that
indicate the presence of a mobile radar target will be further scanned for the
presence of a MU. Radar detection-based decision-making will be much faster
than communication metric-based decision-making due to multiple reasons.
First, the feedback for a radar signal is nearly instantaneous since it is
based on the electromagnetic scattering of the signal by mobile targets.
Second, the exploration time is substantially reduced by restricting the
number of candidate beams that have to be scanned. Due to these factors, the
overall exploration time will be reduced resulting in rapid beam alignment and
improved overall communication throughput.
_Notation:_ In our paper, scalar variables, vectors, and matrices are denoted
with regular, and boldface lower and upper case characters respectively.
Vector superscript $T$ and symbol $\otimes$ denote transpose and convolution
operations.

##  II Rad-Com Signal Model

We first present the signal models for the JRC transmitter and receiver based
on the IEEE 802.11ad protocol where the Golay sequences in the communication
frame are exploited for radar sensing [11, 12, 17]. The digital waveform
$\operatorname{\mathbf{{x}}}_{q}[m]$ corresponds to the Golay sequence in the
$q^{th}$ packet transmitted at a pulse repetition interval of $T_{P}$ with
$m=1,2,\ldots,M$ samples. These digital packets are then converted into analog
signals $\operatorname{\mathbf{{x}}}_{q}(t)$ at the BS as follows:

$\displaystyle\operatorname{\mathbf{{x}}}_{q}(t)=\sum_{m=0}^{M}x_{q}[mT_{s}]\delta\left(t-mT_{s}-(q-1)T_{P}\right),$
(1)

where $T_{s}$ is the sampling time. The signal is then amplified with energy
$E_{s}$, convolved with a transmit shaping filter, $\mathbf{g}_{T}$, and then
passed through analog upconversion to the mmW carrier frequency $f_{c}$ as

$\displaystyle\operatorname{\mathbf{{x}}}_{{q_{uc}}}(t)=\sqrt{E_{s}}\left(\operatorname{\mathbf{{x}}}_{q}(t)\otimes\mathbf{g}_{T}(t)\right)e^{+j2\pi
f_{c}t}.$ (2)

The upconverted signal is then transmitted via analog beamforming through a
uniform linear array (ULA) of $P_{BS}$ elements after applying complex antenna
weight vector at BS transmitter (BS-TX),
$\mathbf{w}_{{BS}_{\theta}}\in\mathcal{C}^{P_{BS}\times 1}$ for a given angle
$\theta$, resulting in

$\displaystyle\mathbf{X}_{q_{uc}}(t)=\mathbf{w}_{{BS}_{\theta}}\mathbf{x_{q_{uc}}}^{T}(t).$
(3)

Here, $\mathbf{w}_{{BS}_{\theta}}=[1\;e^{-jk_{c}d_{BS}\sin\theta}\;\cdots\
e^{-jk_{c}d_{BS}(P_{BS}-1)\sin\theta}]$ where $k_{c}$ is the propagation
constant and $d_{BS}$ is the uniform element spacing. _It is important to note
that the problem of searching for a new beam only arises for a MU that has
changed its position and not a static user._

Radar received signal: Along a pre-determined beam angle $\theta$, we assume
that there are $B$ radar targets present in the channel including MU and other
discrete clutter scatterers. Then the received signal at the $P_{BS}$-element
ULA at the BS receiver (BS-RX) after being reflected from the targets, is

$\displaystyle\begin{split}\operatorname{\mathbf{\hat{\hat{x}}}}_{q}(t)=\sum_{b=1}^{B}\sigma_{b}\mathbf{w}_{{BS}_{\theta}}\mathbf{u}_{\theta}\mathbf{H_{r}}^{2}\mathbf{u}^{T}_{\theta}\left[\mathbf{X}_{q_{uc}}(t-2\tau_{b})\right]+\mathbf{\rho}(t),\end{split}$
(4)

where $\tau_{b}$ denotes the time delay caused by one-way propagation and
$\sigma_{b}$ is the strength of the reflection from each $b^{th}$ point target
obtained from Frii’s radar range equation. $\mathbf{H_{r}}^{2}$ is the
${P_{BS}\times{P_{BS}}}$ channel matrix that includes the direct path and
multipath due to static clutter scatterers (SCS) present in the environment
modeled in a manner described in [18] but for two-way propagation. The
steering vector from the ULA corresponding to $\theta$ is given by
$\mathbf{u}_{\theta}^{T}=[1\;e^{jk_{c}d_{BS}\sin\theta}\;\cdots\
e^{jk_{c}d_{BS}(P_{BS}-1)\sin\theta}]$ while $\mathbf{\rho}$ is the additive
circular symmetric white Gaussian noise at the BS-RX. We assume that each
radar target is moving with a constant radial velocity $v_{b}$, such that the
Doppler shift is $f_{b}=2v_{b}/\lambda$ where $\lambda$ is the wavelength.
After down-conversion and digitization, the received radar signal is

$\displaystyle\operatorname{\mathbf{\hat{\hat{x}}}}_{q}[m]=\sum_{b=1}^{B}\sigma_{b}\mathbf{w}_{BS}\mathbf{u}_{\theta}\mathbf{u}^{T}_{\theta}\mathbf{X}_{q_{uc}}\left[m-m_{b}\right]e^{-j2\pi
f_{b}qT_{P}}+\mathbf{\rho},$ (5)

where, $m_{b}$ is the sample index corresponding to $\tau_{b}$.

Radar signal processing: The radar received signal at BS-RX gathered over $Q$
packets is first converted to a radar rectangle of dimension $[M\times Q]$
which then goes to the radar signal processing block to obtain the range and
Doppler of the corresponding target. The range estimation output,
$\mathbf{\chi_{q}}$, is obtained through the matched filtering for each
$q^{th}$ packet,
$\mathbf{\chi_{q}}=\operatorname{\mathbf{\hat{\hat{x}}}}_{q}\otimes\operatorname{\mathbf{{x}}}_{q}.$
The output is processed through the ordered-statistics constant false alarm
(OS-CFAR) to estimate each $a^{th}$ peak with amplitude, $\hat{\sigma}_{a}$,
at the range $\hat{r}_{a}$. This $\hat{\sigma}_{a}$ information is used
subsequently for the _amplitude gated radar-enhanced MAB algorithm_ discussed
in the next section. Next, Doppler estimation is carried out through one-
dimensional multiple signal classification (MUSIC) for each $a^{th}$ peak
across the $Q$ packets to estimate the corresponding $\hat{f}_{a}$. This
$\hat{f}_{a}$ information is used for the _Doppler gated radar-enhanced MAB
algorithm_ discussed in the next section.

Communication received signal and processing: The one-way propagated
communication signal, $\operatorname{\mathbf{\hat{x}}}(t)$, is received at the
$P_{MU}$ element ULA at the MU receiver (MU-RX) as shown in

$\displaystyle\operatorname{\mathbf{\hat{x}}}(t)=\mathbf{w}_{MU_{\phi}}\mathbf{u}_{\phi}\mathbf{H_{c}}\mathbf{u}^{T}_{\theta}\left[\mathbf{X}_{q_{uc}}(t-\tau_{b})\right]+\mathbf{\delta}(t).$
(6)

Here, $\mathbf{w}_{MU_{\phi}}$ represents the weights applied at the MU-RX and
$\mathbf{u}_{\phi}=[1\;e^{jk_{c}d_{MU}\sin\phi}\;\cdots]$ is the steering
vector for the BS at $\phi$ for $d_{MU}$ antenna element spacing. $H_{c}$ is
the one-way propagation channel matrix ${P_{BS}\times{P_{MU}}}$ model [18] and
$\mathbf{\delta}$ is the additive circular-symmetric white Gaussian noise at
MU-RX. The signal $\hat{{\mathbf{x}}}(t)$ is received by MU and gets processed
and the corresponding signal-to-noise ratio (SNR) is sent back to the BS as
uplink feedback. Note that the processing time for the MU results in a greater
delay for the uplink signal to return to the BS compared to the nearly
instantaneous radar-scattered signal. Second, the uplink
$\operatorname{\mathbf{\hat{x}}}$ is distinguished from
$\operatorname{\mathbf{\hat{\hat{x}}}}$ at the BS-RX through cross-correlation
with $\operatorname{\mathbf{{x}}}$. Due to the nature of the Golay sequence,
the peak-to-sidelobe ratio after cross-correlation for
$\operatorname{\mathbf{\hat{\hat{x}}}}$ is very high compared to the
$\operatorname{\mathbf{\hat{x}}}$.

##  III Proposed MAB Framework for JRC

In this section, we set up the beam-selection problem between BS and MU as MAB
and develop the algorithms that speed up the beam selection. The standard
stochastic MAB consists of a set of $\mathcal{K}$ arms (predetermined beams)
and a single player (the BS-TX/RX) as shown in Fig.1a.

Figure 1: System model showing (a) Standard non-radar based MAB beam
selection, (b) Amplitude Gated Radar-Enhanced MAB, (c) Doppler Gated Radar-
Enhanced MAB. Beams with red and green boxes indicate zero and non-zero
Doppler targets respectively.

In each time slot, the BS-TX/RX, selects a single $k^{th}$ beam and receives
the reward - the SNR of the communication link, $S_{k}$, obtained from uplink
feedback along the beam. For each arm, the reward is assumed to be drawn
independently across time from distributions that are stationary and
independent across arms. The performance metric is equal to the difference
between the SNR of the optimal beam and the SNR over the selected beam. We
define this as regret which is given as

$R=TS_{k^{\prime}}-\mathop{\mathbb{E}}\Bigg{[}\sum_{k\in\mathcal{K}}S_{k}N_{k}\Bigg{]}$
(7)

where $T$ is the total number of time slots, $N_{k}$ is the number of times
the beam $k$ is selected by BS and $k^{\prime}$
=$\arg\\!\max\limits_{k\in\mathcal{K}}$ $S_{k}$. The expectation here is with
respect to the random number of pulls of the arms $(N_{k})$. Thus, the regret
can be minimized by selecting the optimal beam, $k^{\prime}$, i.e., the beam
with the highest SNR as many times as possible in a given horizon of size $T$.
In this paper, we limit our discussion to the upper confidence bound-based
(UCB) MAB algorithm [19] and provide regret bounds. The proposed idea can be
easily extended to other MAB algorithms such as UCB variants and Thompson
Sampling.

###  III-A SNR Based Beam Selection using UCB

We first present the conventional approach for beam selection using the UCB
algorithm given in Algorithm 1: $\mathbf{UCB}_{SNR}$. The communication is
assumed to be time-slotted. At each $t^{th}$ time slot, the BS-TX transmits
over the beam selected by the UCB algorithm. The algorithm selects each of the
$\mathcal{K}$ beams once at the beginning (lines 4-6), and thereafter, the
beam selection is based on the UCB index (lines 7-8). Here, $T_{RSP}$ denotes
the time slots required for RSP and it is set to 0 for $\mathbf{UCB}_{SNR}$.
We denote the index of the selected beam and corresponding reward, i.e.,
instantaneous normalized SNR, as $I_{t}$ and $W_{t}$, respectively. At the end
of each time slot, the parameters are updated (line 12). The UCB index is
calculated for each beam separately and it is given as

$UCB_{k}(t)=\frac{\hat{S}_{k}}{N_{k}}+\sqrt{\frac{2\log(t)}{N_{k}}},\vspace{-2mm}$
(8)

where $\hat{S}_{k}$ denotes the empirical mean of $k^{th}$ arm using samples
obtained till time $t$. The expected regret of $\mathbf{UCB}_{SNR}$ scales as
$\mathcal{O}(\sum_{k\in\mathcal{K}\backslash k^{\prime}}\frac{\log
T}{\Delta_{k}})$ [20] where $\Delta_{k}=S_{k^{\prime}}-S_{k}$ for all $k\neq
k^{\prime}$. Furthermore, it suffers from high exploration time especially
when $\mathcal{K}$ is large. Both these drawbacks limit the usefulness of
$\mathbf{UCB}_{SNR}$ for mmW communication with a large number of narrow
directional beams.

Algorithm 1 $\mathbf{UCB}_{SNR}$: SNR Based Beam Selection

1: Input: $\mathcal{K},T,T_{RSP}$

2: Initialize: $N_{k}\leftarrow 0$ and $S_{k}\leftarrow 0$ for all $k$

3: for $t=T_{RSP}+1,2\ldots T,$ do

4: if $t\leq\mathcal{K}$ then

5: Select beam, $I_{t}=t$.

6: else

7: $\forall k\in[\mathcal{K}]:$ compute $UCB_{k}(t)$ as given in Eq. (8)

8: Select beam, $I_{t}=\arg\\!\max\limits_{k\in[K]}UCB_{k}(t)$

9: end if

10: BS-TX transmits a data frame over $I_{t}$

11: MU observes instantaneous normalized SNR, $W_{t}$ and communicate to BS-RX
over the uplink.

12: $N_{I_{t}}\leftarrow N_{I_{t}}+1$ and $S_{I_{t}}\leftarrow
S_{I_{t}}+W_{t}$.

13: end for

###  III-B $\mathbf{UCB}_{SNR\\_AG}$: Amplitude Gated Radar-Enhanced MAB

In this section, we augment the $\mathbf{UCB}_{SNR}$ with the proposed radar-
based target detection as shown in Fig. 1b and described in Algorithm 2:
$\textbf{UCB}_{SNR\\_AG}$. Here, a radar target is detected in a beam when the
_amplitude/strength_ of the scattered signal in any one or more of the range
bins within the beam is above a pre-set threshold determined by CFAR. The
number of beams where potential targets are detected is $\tilde{\mathcal{K}}$
(dark-colored beams in the figure) where $\mathcal{K}\geq\tilde{\mathcal{K}}$.
Compared to Algorithm 1, the number of available beams is updated based on
radar target detection during the first time slot (line 3 of Algorithm.2).
Compared to $\mathbf{UCB}_{SNR}$, $\textbf{UCB}_{SNR\\_AG}$ potentially offers
lower regret due to the following reasons: 1) Faster target detection: The
identification of the presence of targets using radar is significantly faster
since returns of the scattered signals from the short-range targets are nearly
instantaneous with a short round-trip delay of the order of a few $ns$. For
5G, one slot is at least 4 $ms$ assuming downlink sub-frame (1 $ms$), uplink
sub-frame for reward feedback (1 $ms$), downlink (1 $ms$) and uplink data
processing (1 $ms$). On average, RSP time, $T_{RSP}$ for 10 radar packets is
36 $ms$, i.e., 9 slots are sufficient to find $\tilde{\mathcal{K}}$ [21];

Algorithm 2(or)3 $\textbf{UCB}_{SNR\\_AG}$ (or) $\mathbf{UCB}_{SNR\\_DG}$

1: Input: $\mathcal{K},T,T_{RSP}$

2: Initialize: $N_{k}\leftarrow 0$ and $S_{k}\leftarrow 0$ for all $k$

3: Find $\tilde{\mathcal{K}}$ using target detection (Subroutine 1)
$\vartriangleright\textbf{UCB}_{SNR\\_AG}$Find
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{K}}$}}$ using Doppler
detection (Subroutine 2) $\vartriangleright\mathbf{UCB}_{SNR\\_DG}$

4: Run $\mathbf{UCB}_{SNR}$ with $\tilde{\mathcal{K}}$ arms for rest of the
time horizon. $\vartriangleright\textbf{UCB}_{SNR\\_AG}$Run
$\mathbf{UCB}_{SNR}$ with
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{K}}$}}$ arms for rest of the
time horizon. $\vartriangleright\mathbf{UCB}_{SNR\\_DG}$

2) The proposed algorithm focuses on a subset of the total beams in which a
mobile target may be present which in turn reduces the exploration time. To
quantify this gain, let us fix a bandit instance. The set of beams is detected
by the radar, $\tilde{\mathcal{K}}$, is a random variable depending on the
distribution of scatterers. We can assume $\tilde{\mathcal{K}}$ includes the
optimal arm in each realization as it has the maximum signal strength and
radar is unlikely to miss the MU. Hence the optimal arm is the same in any
realized set $\tilde{\mathcal{K}}$. Expected regret over the set
$\tilde{\mathcal{K}}$ is $\mathcal{O}(\sum_{k\in\tilde{\mathcal{K}}\backslash
k^{\prime}}\frac{\log T}{\Delta_{k}})$. Clearly this bound is smaller than
$\mathcal{O}(\sum_{k\in{\mathcal{K}}\backslash k^{\prime}}\frac{\log
T}{\Delta_{k}})$ obtained for the previous case. Taking expectation over the
random realizations $\tilde{\mathcal{K}}$, we get expect regret of
$\textbf{UCB}_{SNR\\_AG}$ as
$\mathbb{E}\left[\mathcal{O}(\sum_{k\in{\tilde{\mathcal{K}}}\backslash
k^{\prime}}\frac{\log
T}{\Delta_{k}})\right]\leq\mathcal{O}(\sum_{k\in{\mathcal{K}}\backslash
k^{\prime}}\frac{\log T}{\Delta_{k}})$. Thus $\textbf{UCB}_{SNR\\_AG}$ is
better than that of $\mathbf{UCB}_{SNR}$ resulting in an improvement in the
performance.

###  III-C $\mathbf{UCB}_{SNR\\_DG}$: Doppler Gated Radar-Enhanced MAB

In the $\textbf{UCB}_{SNR\\_AG}$ algorithm, all the beams where radar targets
are present are selected. However, some of the beams correspond to SCS as
shown in Fig.1c. The SCS is distinguished to be of two types: some give rise
to direct scattering at the radar (termed $SCS_{1}$) with zero Doppler while
others give rise to Doppler-shifted returns at the radar through multipath
with respect to the MU (termed $SCS_{2}$). The proposed Doppler-enhanced MAB
algorithm is described in Algorithm 3: $\mathbf{UCB}_{SNR\\_DG}$. Since direct
path returns from SCS do not give any type of information regarding the MU,
they can be excluded from the list of candidate beams based on the Doppler
shift estimated from the radar signal processing described earlier.

Subroutine 1 $\mathbf{AG}:$ $\tilde{\mathcal{K}}$ Beams Selection Using Radar
Target Detection

1: Input: $\mathcal{K}$

2: Output: $\tilde{\mathcal{K}}$

3: for $\theta=1,2\ldots\mathcal{K},$ do

4: $\mathbf{\chi}_{\theta}$: Matched filtering across fast time samples

5: CFAR detection:

6: if $\mathbf{\chi}_{\theta}\geq\gamma$ then

7: Include beam, $\theta$ in subset $\tilde{\mathcal{K}}$

8: end if

9: end for

Hence the total number of beams where potential MU are detected is
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{K}}$}}$ and
$\mathcal{K}\geq\tilde{\mathcal{K}}\geq\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{\mathcal{K}}}$}}$.
Thus, fewer candidate beams (dark-colored beams in the figure) result in lower
exploration time. The theoretical explanation for the reduction in beams
follows the same logic provided for the previous algorithm and hence is not
repeated here. Note that static communication targets are not likely to have
first triggered the necessity for the selection of a new beam by the BS and
hence, static targets can be interpreted safely as SCS.

Subroutine 2 $\mathbf{DG}$:
$\operatorname{\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{K}}$}}}$ Beams
Selection Based on Doppler Estimation

1: Input: $\mathcal{K}$

2: Output: $\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{K}$}}$

3: for $\theta=1,2\ldots\mathcal{K},$ do

4: 1D-MUSIC for $\hat{r}_{a}$ across Q packets

5: if $\hat{f}_{a}\neq 0$ then

6: Include beam, $\theta$, in subset
$\tilde{\raisebox{0.0pt}[0.85pt]{$\tilde{\mathcal{K}}$}}$

7: end if

8: end for

##  IV Performance Analysis

We consider a three-dimensional (3D) Cartesian coordinate space with the
ground plane defined by the $x$ and $y$ axes and the height axis along $z$.
The BS is located at $[0,0,0]$ m with a $y$-aligned uniform linear array (ULA)
of 32 antennas with an antenna spacing of $\lambda/2$ where $\lambda$ is the
wavelength corresponding to the center frequency $f_{c}$ of 60 GHz. We adopt a
16QAM modulation and coding scheme with 512 OFDM subcarriers with a signal
bandwidth of 1.76 GHz. We assume that the channel consists of a single MU and
multiple SCS as shown in Fig. 1. The radar scattered returns of each of these
are confined to a single beam. We assume that the MU is also with a 32-element
ULA and is initially located at $[50,20,0]$m and subsequently moves with a
constant velocity of $v$ m/s along the $x$ axis. Both MU and SCS are modeled
as isotropic point scatterers and the SCS are distributed randomly across the
3D Cartesian space.

The throughput, $\Upsilon$, is calculated as
$\left(1-\frac{\sum_{i=1}^{N_{t}}BER_{i}}{N_{t}}\right)\frac{D}{T_{d}}$ where,
$BER_{i}$ corresponds to the bit error rate of $i^{th}$ time slot, $D$ and
$T_{d}$ correspond to the total number of bits and the time duration for each
slot respectively. We benchmark the $\Upsilon$ performance of the proposed
algorithms, $\textbf{UCB}_{SNR\\_AG}$ and $\mathbf{UCB}_{SNR\\_DG}$ with
conventional $\mathbf{UCB}_{SNR}$, lower upper confidence bound (LUCB)
described in [22], digital beamforming and trivial random beam selection
approach. We present the effects of the number of targets, number of beams,
the Doppler velocity resolution, and radar receiver SNR on $\Upsilon$. Each
result presented in this section is obtained after averaging over 15
independent experiments and each experiment’s duration/horizon is 2000 time
slots.
Effect of Number of Radar Scatterers: In Fig. 2(a), we compare $\Upsilon$ of
all the algorithms at different instants of the time horizon. Here, we assume
one MU and vary the number of SCS. The Doppler velocity of the MU is fixed to
3 m/s and the angular resolution is $4^{o}$ resulting in a total of 41
candidate beams spanning from $-80^{o}$ to $80^{o}$ and the Doppler velocity
resolution is 1 m/s.

Figure 2: (a) Throughput at different time slots for two targets, and (b)
Throughput at the end of the horizon for different numbers of targets.

It can be observed in the figure that the proposed algorithms offer higher
$\Upsilon$ than the benchmarked approaches - except for DBF - due to faster
identification of the optimal beam. DBF provides the best-case results since
all the beams are tested simultaneously. However, this approach is not pursued
since the implementation of multiple synchronized receiver chains is costly
and complex. In Fig. 2(b), we compare $\Upsilon$ of all algorithms at the end
of the horizon as the number of SCS increases. We observe that the performance
of all MAB-based approaches is significantly better than the random selection
approach validating the need for a learning algorithm. As expected, the
difference between $\mathbf{UCB}_{SNR}$ (non-radar based beam selection) and
$\textbf{UCB}_{SNR\\_AG}$ or $\mathbf{UCB}_{SNR\\_DG}$ (radar-based beam
selection) reduces as the number of targets increases. This is because when
there are a large number of SCS, there is a proportionate increase in the
number of candidate beams. Further, we can observe that there is a slight
improvement in $\Upsilon$ for $\mathbf{UCB}_{SNR\\_DG}$ than
$\textbf{UCB}_{SNR\\_AG}$ as beams with zero Doppler get eliminated.
Effect of Angular Resolution Next, we fix the number of targets to three,
which includes one MU and two SCS, and compare the performance for different
angular resolutions or the number of candidate beams. The rest of the
parameters are the same as Fig. 2.

Figure 3: (a) Throughput at different time slots for angular resolution of
$2^{o}$, and (b) Throughput at the end of the horizon for different angular
resolutions/number of candidate beams.

In Fig. 3(a), we compare the $\Upsilon$ for angular resolution of $2^{o}$ and
it can be observed that the proposed approaches offer higher $\Upsilon$ than
$\mathbf{UCB}_{SNR}$ and random approaches at all time slots. In Fig. 3(b), we
compare the $\Upsilon$ at the end of the horizon for different numbers of
candidate beams. As expected, $\Upsilon$ increases as the number of beams
reduces due to lower exploration time. However, proposed
$\textbf{UCB}_{SNR\\_AG}$ or $\mathbf{UCB}_{SNR\\_DG}$ algorithms
significantly outperform $\mathbf{UCB}_{SNR}$ and random approaches in all
cases. The $\mathbf{UCB}_{SNR\\_DG}$ offers higher $\Upsilon$ than
$\textbf{UCB}_{SNR\\_AG}$ for smaller angular resolution (i.e., large
$\mathcal{K}$) due to more opportunities of beam elimination. However, the
impact of Doppler processing reduces with the increase in angular resolution
due to the occurrence of fewer beams.
Effect of Velocity Resolution: Next, we study the effect of Doppler velocity
resolution on $\Upsilon$. The simulation is carried out for Doppler velocity
resolution of 1 m/s, 2 m/s, 3 m/s, 4 m/s and 5 m/s. We assume the MU with a
Doppler velocity of 3 m/s and three targets (one MU and two SCS).

Figure 4: (a) Throughput at different time slots for velocity resolution of
$15$m/s, and (b) Throughput at the end of the horizon for different velocity
resolutions.

In Fig. 4(a), we consider the velocity resolution of 5 m/s, which is not
sufficient to detect the MU and hence, the performance of
$\mathbf{UCB}_{SNR\\_DG}$ is poorer than that of $\textbf{UCB}_{SNR\\_AG}$. As
shown in Fig. 4(b), the velocity resolution of radar signal processing should
be carefully chosen. Thus, $\mathbf{UCB}_{SNR\\_DG}$ may not offer better
performance than $\textbf{UCB}_{SNR\\_AG}$ even though it incurs higher
computational cost due to Doppler estimation. As expected, $\Upsilon$ of
$\mathbf{UCB}_{SNR}$ and random approaches are independent of the velocity
resolution since they do not use this information for beam selection.
Effect of Radar Receiver SNR: Further, we analyze all the algorithms for four
different SNRs at the radar receiver ranging from -10 dB to 10 dB. Here, also
we simulate three targets (one MU and two SCS), and the velocity resolution is
set as 1 m/s.

Figure 5: (a) Throughput at different time slots for 10dB SNR, and (b)
Throughput at the end of the horizon for different SNRs.

Fig. 5(a) represents the viewgraph of $\Upsilon$ with respect to time slots
for a high SNR (10 dB). We observe in Fig. 5(b) that the regret improves for
the $\textbf{UCB}_{SNR\\_AG}$ and $\mathbf{UCB}_{SNR\\_DG}$ with increase in
SNR. At lower SNR, the poor prediction of the target presence amidst noise
results in higher regret. Note that since we consider the SNR with respect to
the radar receiver, change in SNR does not have any impact on the random beam
selection as well as $\mathbf{UCB}_{SNR}$ algorithms.
Impact of Doppler Processing: Further, we discuss the scenario where the
number of $SCS_{1}$ is greater than $SCS_{2}$ such that the candidate beams
selected by the $\mathbf{UCB}_{SNR\\_DG}$ are significantly lesser than those
selected by the $\textbf{UCB}_{SNR\\_AG}$. In Fig. 6(a), we consider one MU,
one $SCS_{2}$ and two $SCS_{1}$. The angular resolution is $4^{o}$ and the
velocity resolution is 1 m/s.

Figure 6: (a) Throughput at different time slots for four targets (including
two $SCS_{1}$, one $SCS_{2}$ and one MU, and (b) Throughput at the end of the
horizon for different number of $SCS_{1}$.

Since $SCS_{2}$ gives rise to Doppler shift at the BS through multipath from
the MU, the corresponding beam is retained for further scanning while the
beams corresponding to the two $SCS_{1}$ that give rise to zero-Doppler shift
are excluded in $\mathbf{UCB}_{SNR\\_DG}$. This results in better performance
of the algorithm compared to $\mathbf{UCB}_{SNR\\_AG}$. Further, the
improvement is greater as the number of $SCS_{1}$ increases for a fixed number
of $SCS_{2}$ as seen in Fig. 6(b).

##  V Conclusion

In this work, we demonstrate how radar-enhanced MAB within a JRC BS can
substantially reduce the exploration time by selecting only those candidate
beams that detect the presence of radar targets of which the MU may be one.
Further reduction in the exploration time is realized by distinguishing SCS
from MU through radar-based Doppler estimation. Simulation results demonstrate
an overall improvement in the communication link metrics with the reduction in
the exploration time through radar-enhanced MAB when compared with
conventional MAB algorithms.

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# Modified Friedmann Equations from Tsallis Entropy

Ahmad<EMAIL_ADDRESS>Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box
55134-441, Maragha, Iran

###### Abstract

It was shown by Tsallis and Cirto that thermodynamical entropy of a
gravitational system such as black hole must be generalized to the non-
additive entropy, which is given by $S_{h}=\gamma A^{\beta}$, where $A$ is the
horizon area and $\beta$ is the nonextensive parameter Tsa . In this paper, by
taking the entropy associated with the apparent horizon of the Friedmann-
Robertson-Walker (FRW) Universe in the form of Tsallis entropy, and assuming
the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, holds on the apparent
horizon, we are able to derive the corresponding Friedmann equations
describing the dynamics of the universe with any spatial curvature. We also
examine the time evolution of the total entropy and show that the generalized
second law of thermodynamics is fulfilled in a region enclosed by the apparent
horizon. Then, modifying the emergence proposal of gravity proposed by
Padmanabhan and calculating the difference between the surface degrees of
freedom and the bulk degrees of freedom in a region of space, we again arrive
at the modified Friedmann equation of the FRW Universe with any spatial
curvature which is the same as one obtained from the first law of
thermodynamics. We also study the cosmological consequences of Tsallis
cosmology. Interestingly enough, we find that this model can explain
simultaneously the late time acceleration in the universe filled with
pressureless matter without invoking dark energy, as well as the early
deceleration. Besides, the age problem can be circumvented automatically for
an accelerated universe and is estimated larger than $3/2$ age of the universe
in standard cosmology. Taking $\beta=2/5$, we find the age of the universe
ranges as $13.12$ Gyr $<t_{0}<16.32$ Gyr, which is consistent with recent
observations. Finally, using the Jeans’s analysis, we comment, in brief, on
the density perturbation in the context of Tsallis cosmology and found that
the growth of energy differs compared to the standard cosmology.

##  I Introduction

Although gravity is the most universal forces of nature, understanding its
origin has been a mystery for a long time. Einstein believed that gravity is
just the spacetime curvature and regarded it as an emergent phenomenon which
describes the dynamics of spacetime. In the past decades a lot of attempts
have been done to disclose the nature of gravity. A great step in this
direction put forwarded by Jacobson Jac  who studied thermodynamics of
spacetime and showed explicitly that Einstein’s equation of general relativity
is just an equation of state for the spacetime. Combining the Clausius
relation $\delta Q=T\delta S$, together with the entropy expression, he
derived Einstein field equations. This derivation is of great importance
because it confirms that the Einstein field equations is nothing but the first
law of thermodynamics for the spacetime. Following Jacobson, a lot of studies
have been carried out to disclose the deep connection between gravity and
thermodynamics Elin ; Cai1 ; Pad . The studies were also generalized to the
cosmological setups Cai2 ; Cai3 ; CaiKim ; Fro ; verlinde ; Cai4 ; CaiLM ;
Shey1 ; Shey2 , where it has been shown that the Friedmann equation of
Friedmann-Robertson-Walker (FRW) universe can be written in the form of the
first law of thermodynamics on the apparent horizon. Although Jacobson’s
derivation is logically clear and theoretically sound, the statistical
mechanical origin of the thermodynamic nature of general relativity remains
obscure.

In $2010$ Verlinde Ver  put forwarded the next great step toward understanding
the nature of gravity who claimed that gravity is not a fundamental force and
can be interpreted as an entropic force caused by changes of entropy
associated with the information on the holographic screen. Verlinde’s proposal
is based on two principles, namely the holographic principle and the
equipartition law of energy. Using these principles he derived the Newton’s
law of gravitation, the Poisson equation and in the relativistic regime the
Einstein field equations Ver  (see also Pad0 ). The investigation on the
entropic origin of gravity have been extended in different setups Cai5 ; Other
; newref ; sheyECFE ; Ling ; Modesto ; Yi ; Sheykh2  and references therein).
Although Verlinde’s proposal has changed our understanding on the origin and
nature of gravity, but it considers the gravitational field equations as the
equations of emergent phenomenon and leave the spacetime as a pre-existed
background geometric manifold. A new perspective towards emergence of
spacetime dynamics was suggested in $2012$ by Padmanabhan PadEm . He argued
that the spatial expansion of our Universe can be regarded as the consequence
of emergence of space and the cosmic space is emergent as the cosmic time
progresses. By calculating the difference between the number of degrees of
freedom in the bulk and on the boundary, Padmanabhan PadEm  derived the
Friedmann equation of the flat FRW Universe. This proposal was also applied
for deriving the Friedmann equations of a higher dimensional FRW Universe in
Einstein, Gauss-Bonnet and more general Lovelock cosmology CaiEm ; Yang . By
modification the Padmanabhan’s proposal, one may extract the Friedmann
equation of FRW Universe with any spatial curvature not only in Einstein
gravity, but also in Gauss-Bonnet and more general Lovelock gravity Sheyem .
The novel idea of Padmanabhan has got a lot of attentions in the literatures
FQ ; Ling2 ; Sheyem2 ; Sheyem3 ; FF ; Eu ; Wen .

It is important to note that in order to rewrite the Friedmann equations, in
any gravity theory, in the from of the first law of thermodynamics,
$dE=T_{h}dS_{h}+WdV$, on the apparent horizon and vice versa, one should
consider the entropy expression of the black hole in each gravity theory. The
only change one should done is replacing the black hole horizon radius $r_{+}$
by the apparent horizon radius ${\tilde{r}}_{A}$. However, the entropy
expression associated with the black hole horizon get modified from the
inclusion of quantum effects. Several type of quantum corrections to the area
law have been introduced in the literatures, among them are logarithmic and
power-law corrections. Logarithmic corrections, arises from the loop quantum
gravity due to thermal equilibrium fluctuations and quantum fluctuations Log ;
Rovelli ; Zhang . The logarithmic term also appears in a model of entropic
cosmology which unifies the inflation and late time acceleration YFCai .
Another form of correction to area law, namely the power-law correction,
appears in dealing with the entanglement of quantum fields inside and outside
the horizon sau1 ; sau2 ; Sau ; pavon1 . Another modification for the area law
of entropy comes from the Gibbs arguments who pointed out that in systems with
divergency in the partition function, like gravitational system, the
Boltzmann-Gibbs (BG) theory cannot be applied. As a result thermodynamical
entropy of such nonstandard systems is not described by an additive entropy
but must be generalized to the non-additive entropy Tsa0 . Based on this, and
using the statistical arguments, Tsallis and Cirto argued that the microscopic
mathematical expression of the thermodynamical entropy of a black hole does
not obey the area law and can be modified as Tsa ,

$\displaystyle S_{h}=\gamma A^{\beta},$ (1)

where $A$ is the black hole horizon area, $\gamma$ is an unknown constant and
$\beta$ known as Tsallis parameter or nonextensive parameter, which is a real
parameter which quantifies the degree of nonextensivity Tsa . It is obvious
that the area law of entropy is restored for $\beta=1$ and
$\gamma=1/(4L_{p}^{2})$. Through this paper we set $k_{B}=1=c=\hbar$ for
simplicity. In fact, at this limit, the power-law distribution of probability
becomes useless, and the system is describable by the ordinary distribution of
probability Tsa ; Barb .

It is worth mentioning that in deriving Friedmann equations from the first law
of thermodynamics, the entropy expression associated with the horizon plays a
crucial role CaiLM . Thus, it is interesting to see how the Friedmann equation
get modified if the entropy-area relation gets corrections by some reasons.
Starting from the first law of thermodynamics at apparent horizon of a FRW
universe, and assuming that the associated entropy with apparent horizon has a
logarithmic quantum corrected relation, the modified Friedmann equations were
derived in SheyLog . Also, taking the associated entropy with apparent horizon
as the power-law-corrected relation, one is able to obtain the corrected
Friedmann equation by using the first law of thermodynamics at the apparent
horizon SheyPL . Besides, if thermodynamical interpretation of gravity near
apparent horizon is generic feature, one should also ont only check the first
law of thermodynamics but also the generalized second law of thermodynamics.
The latter is a universal principle governing the evolution of the total
entropy of the Universe. In the context of the accelerating Universe, the
generalized second law of thermodynamics has been explored in wang1 ; wang2 ;
Shey3 ; akbar . It should be noted that dark energy and a modified Friedmann
equations in the context of Tsallis entropy and from different perspective,
were first investigated in Barb ; Nunes . It was argued that Tsallis entropy
parameter change the strength of the gravitational constant and consequently
the energy density of the dark components of the universe, requiring more
(less) dark energy to provide the observed late time universe acceleration
Nunes . They also explored some phenomenological aspects as well as some
observational constraints from a modified Friedmann equations induced by
Tsallis entropy Barb ; Nunes . In the context of the nonextensive Kaniadakis
statistics Kan , the Jeans length was investigated and the results were
compared with the Jeans length obtained in the non-extensive Tsallis
statistics Abero . Recently, modified cosmology through nonextensive Tsallis
entropy have been investigated in Em . It was shown that the universe exhibits
the usual thermal history, with the sequence of matter and dark energy eras,
and depending on the value of nonextensive parameter $\beta$ the equation of
state of dark energy can even cross the phantom-line Em . In this work, we
shall derive the modified Friedmann equation in a universe which its entropy
is given by the nonextensive Tsallis entropy. Then, we investigate the
cosmological implications of the obtained modified Friedmann equations in the
matter and radiation dominated era. In our study, we do not need to invoke the
dark energy component and the early deceleration as well as the late time
acceleration of the universe expansion can be achieved in the presence of
radiation and pressureless matter. Besides, we shall show that our model can
solve the age problem of the universe. Our approach and the obtained Friedmann
equations, from the first law of thermodynamics completely differ from those
investigated in Barb ; Nunes ; Em .

This paper is organized as follows. In the next section, we derive the
modified Friedmann equations by applying the first law of thermodynamics,
$dE=T_{h}dS_{h}+WdV$, at apparent horizon of a FRW universe and assuming the
entropy associated with apparent horizon is in the form of Tsallis entropy
(1). In section III, we examine the generalized second law of thermodynamics
for the total entropy including the corrected entropy-area relation together
with the matter field entropy inside the apparent horizon. In section IV, we
adopt the modified version of Padmanabhan’s proposal PadEm  for deriving the
Friedmann equation corresponding to Tsallis entropy (1). By assuming the
difference between the number of degrees of freedom in the bulk and on the
boundary is proportional to the volume change of the spacetime, we find the
modified Friedmann equation which coincides with the one obtained from the
first law of thermodynamics. In section V, we investigate the cosmological
consequences of the modified Friedmann equations derived from the non-
extensive Tsallis entropy. The last section is devoted to conclusion and
discussion.

##  II Modified Friedman Equation from the First law of thermodynamics

We assume the background spacetime is spatially homogeneous and isotropic
which is described by the line element

$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (2)

where $\tilde{r}=a(t)r$, $x^{0}=t,x^{1}=r$, and $h_{\mu\nu}$=diag
$(-1,a^{2}/(1-kr^{2}))$ represents the two dimensional metric. The open, flat,
and closed universes corresponds to $k=0,1,-1$, respectively. We also assume
the physical boundary of the Universe, which is consistent with laws of
thermodynamics, is the apparent horizon with radius

$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (3)

The associated temperature with the apparent horizon can be defined as Cai2

$T_{h}=\frac{\kappa}{2\pi}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(4)

where $\kappa$ is the surface gravity. For $\dot{\tilde{r}}_{A}\leq
2H\tilde{r}_{A}$, the temperature becomes $T\leq 0$. To avoid the negative
temperature one may define $T=|\kappa|/2\pi$. Also, within an infinitesimal
internal of time $dt$ one may assume $\dot{\tilde{r}}_{A}\ll 2H\tilde{r}_{A}$,
which physically means that the apparent horizon radius is kept fixed. Thus
there is no volume change in it and one may define $T=1/(2\pi\tilde{r}_{A})$
CaiKim . The profound connection between temperature on the apparent horizon
and the Hawking radiation has been considered in cao , which further confirms
the existence of the temperature associated with the apparent horizon.

We assume the matter and energy content of the Universe is in the form of
perfect fluid with stress-energy tensor

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (5)

where $\rho$ and $p$ are the energy density and pressure, respectively. The
conservation of the energy-stress tensor in the FRW background,
$\nabla_{\mu}T^{\mu\nu}=0$, leads to the continuity equation as

$\dot{\rho}+3H(\rho+p)=0,$ (6)

where $H=\dot{a}/a$ is the Hubble parameter. Following Hay2 , we define the
work density as

$W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu}.$ (7)

A simple calculation gives

$W=\frac{1}{2}(\rho-p).$ (8)

The work density term is indeed the work done by the volume change of the
Universe, which is due to the change in the apparent horizon radius. We assume
the first law of thermodynamics on the apparent horizon is satisfied and has
the form

$dE=T_{h}dS_{h}+WdV.$ (9)

It is clear that this equation is similar to the standard first law of
thermodynamics, unless the work term $-pdV$ is replaced by $WdV$. For a pure
de Sitter space where $\rho=-p$, the work term reduces to the standard $-pdV$
and one arrives at the standard first law of thermodynamics.

We suppose the total energy content of the universe inside a $3$-sphere of
radius $\tilde{r}_{A}$, is $E=\rho V$ where
$V=\frac{4\pi}{3}\tilde{r}_{A}^{3}$ is the volume enveloped by 3-dimensional
sphere with the area of apparent horizon $A=4\pi\tilde{r}_{A}^{2}$. Taking
differential form of the total matter and energy inside the apparent horizon,
we find

$dE=4\pi\tilde{r}_{A}^{2}\rho
d\tilde{r}_{A}+\frac{4\pi}{3}\tilde{r}_{A}^{3}\dot{\rho}dt.$ (10)

Using the continuity equation (6), we obtain

$dE=4\pi\tilde{r}_{A}^{2}\rho d\tilde{r}_{A}-4\pi
H\tilde{r}_{A}^{3}(\rho+p)dt.$ (11)

We further assume the entropy associated with the apparent horizon is in the
form of Tsallis entropy (1), where now $A$ is the apparent horizon area of the
Universe. Differentiating the modified entropy-area relation (1), we get

$dS_{h}=\gamma\beta
A^{\beta-1}dA=8\pi\gamma\beta(4\pi{r}_{A}^{2})^{\beta-1}\tilde{r}_{A}d\tilde{r}_{A}.$
(12)

Substituting Eqs. (8), (11) and (12) in the first law (9) and using relation
(5) we get the differential form of the Friedmann equation as

$\frac{\gamma\beta}{\pi\tilde{r}_{A}^{3}}\left(4\pi\tilde{r}_{A}^{2}\right)^{\beta-1}d\tilde{r}_{A}=H(\rho+p)dt.$
(13)

Using the continuity equation (6), we can rewrite it as

$-\frac{2}{\tilde{r}_{A}^{3}}\left(4\pi\tilde{r}_{A}^{2}\right)^{\beta-1}d\tilde{r}_{A}=\frac{2\pi}{3\gamma\beta}d\rho.$
(14)

Next, we integrate Eq. (14),

$-2\left(4\pi\right)^{\beta-1}\int{\tilde{r}_{A}^{2\beta-5}d\tilde{r}_{A}}=\frac{2\pi}{3\gamma\beta}\rho,$
(15)

which can be written as

$\frac{1}{\tilde{r}_{A}^{4-2\beta}}=\frac{2\pi(2-\beta)}{3\gamma\beta}\left(4\pi\right)^{1-\beta}\rho,$
(16)

where we have set the integration constant equal to zero. Substituting
$\tilde{r}_{A}$ from Eq.(3) we obtain

$\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=\frac{8\pi L_{p}^{2}}{3}\rho,$
(17)

provided we define

$\gamma\equiv\frac{2-\beta}{4\beta L_{p}^{2}}\left(4\pi\right)^{1-\beta}.$
(18)

Equation (17) is nothing but the modified Friedmann equation obtained through
the non-additive Tsallis entropy. In this way we have not only derived the
modified Friedmann equation by starting from the first law of thermodynamics
at apparent horizon of a FRW universe, and assuming that the associated
entropy with apparent horizon has a corrected relation (1), but also find a
general definition for the unknown constant $\gamma$ in the entropy expression
(1). In the limiting case where $\beta=1$, we recover the standard Friedmann
equation in Einstein gravity. Besides, from Eq. (18), we have always $\beta<2$
which puts an upper bound on the non-additive parameter of the Tsallis
entropy.

##  III Generalized Second law of thermodynamics

Now we want to examine the validity of the generalized second law of
thermodynamics in a region enclosed by the apparent horizon. Combining Eq.
(14) with Eq. (6) and using relation (18), we arrive at

$\frac{2}{\tilde{r}_{A}^{3}}(2-\beta)\dot{\tilde{r}}_{A}\tilde{r}_{A}^{2\beta-2}=8\pi
L_{p}^{2}H(\rho+p).$ (19)

Solving for $\dot{\tilde{r}}_{A}$ we find

$\dot{\tilde{r}}_{A}=\frac{4\pi
L_{p}^{2}H}{2-\beta}\tilde{r}_{A}^{5-2\beta}(\rho+p).$ (20)

Since $\beta<2$, thus the sign of $\dot{\tilde{r}}_{A}$ depends on the sign of
$\rho+p$. For $\rho+p>0$, which physically means the dominant energy condition
holds, we have $\dot{\tilde{r}}_{A}>0$. Let us now turn to find out
$T_{h}\dot{S_{h}}$:

$T_{h}\dot{S_{h}}=\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right)\frac{d}{dt}\left[\gamma(4\pi\tilde{r}_{A}^{2})^{\beta}\right].$
(21)

After some simplifications and using Eq. (20) we obtain

$T_{h}\dot{S_{h}}=4\pi
H{\tilde{r}_{A}^{3}}(\rho+p)\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(22)

At present, our Universe is undergoing an acceleration phase and thus its
equation of state parameter can cross the phantom line ($w_{D}=p/\rho<-1$).
This implies that in an accelerating universe the dominant energy condition
may violate, $\rho+p<0$, implying that the second law of thermodynamics,
$\dot{S_{h}}\geq 0$, does not hold. Therefore, we should examine the validity
of the generalized second law of thermodynamics, namely
$\dot{S_{h}}+\dot{S_{m}}\geq 0$.

From the Gibbs equation we have Pavon2

$T_{m}dS_{m}=d(\rho V)+pdV=Vd\rho+(\rho+p)dV,$ (23)

where $T_{m}$ and $S_{m}$ are, respectively, the temperature and the entropy
of the matter fields inside the apparent horizon. We assume the local
equilibrium hypothesis holds. Therefore, the thermal system bounded by the
apparent horizon remains in equilibrium so that the temperature of the system
must be uniform and the same as the temperature of its boundary, $T_{m}=T_{h}$
Pavon2 . Note that if the temperature of the fluid differs much from that of
the horizon, there will be spontaneous heat flow between the horizon and the
bulk fluid and the local equilibrium hypothesis will no longer hold. Thus,
from the Gibbs equation (23) we can obtain

$T_{h}\dot{S_{m}}=4\pi{\tilde{r}_{A}^{2}}\dot{\tilde{r}}_{A}(\rho+p)-4\pi{\tilde{r}_{A}^{3}}H(\rho+p).$
(24)

In order to examine the generalized second law of thermodynamics, we should
study the evolution of the total entropy $S_{h}+S_{m}$. Adding equations (22)
and (24), we get

$T_{h}(\dot{S_{h}}+\dot{S_{m}})=2\pi{\tilde{r}_{A}}^{2}(\rho+p)\dot{\tilde{r}}_{A}=\frac{A}{2}(\rho+p)\dot{\tilde{r}}_{A}.$
(25)

where $A$ is the apparent horizon area. Inserting $\dot{\tilde{r}}_{A}$ from
Eq. (20) into (25) we find

$T_{h}(\dot{S_{h}}+\dot{S_{m}})=\frac{8\pi^{2}}{2-\beta}L_{p}^{2}H{\tilde{r}_{A}}^{7-2\beta}(\rho+p)^{2}.$
(26)

Since $\beta<2$, the right hand side of the above equation is always a non-
negative function during the universe history, which means that
$\dot{S_{h}}+\dot{S_{m}}\geq 0$. This implies that for a universe with Tsallis
entropy the total entropy namely the entropy of the boundary together with the
matter entropy inside the bulk is a non decreasing function of time and hence
the generalized second law of thermodynamics is fulfilled.

##  IV Modified Friedmann equation from emergence of cosmic space

According to the Padmanabhan’s proposal PadEm , the basic equation governing
the evolution of the Universe can be derived by relating the emergence of
space to the difference between the number of degrees of freedom in the
holographic surface and the one in the emerged bulk. In this viewpoint, the
spatial expansion of our Universe can be regarded as the consequence of
emergence of space and the cosmic space is emergent, following the progressing
in the cosmic time. Thus, he argued that in an infinitesimal interval $dt$ of
cosmic time, the increase $dV$ of the cosmic volume, is given by PadEm

$\frac{dV}{dt}=L_{p}^{2}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}),$ (27)

where $N_{\mathrm{sur}}$ is the number of degrees of freedom on the boundary
and $N_{\mathrm{bulk}}$ is the number of degrees of freedom in the bulk. Using
this new idea, Padmanabhan derived the Friedmann equation of a flat FRW
Universe PadEm . It was argued that this proposal failed to derive the
Firedmann equations of a nonflat FRW universe in other gravity theories such
as Gauss-Bonnet and Lovelock gravity CaiEm . The modification of Padmanabhan’s
proposal was done by the present author Sheyem , who argued that in a nonflat
Universe the proposal should be generalized as

$\frac{dV}{dt}=L_{p}^{2}\frac{\tilde{r}_{A}}{H^{-1}}\left(N_{\mathrm{sur}}-N_{\mathrm{bulk}}\right).$
(28)

This implies that the volume increase, in a nonflat universe, is still
proportional to the difference between the number of degrees of freedom on the
apparent horizon and in the bulk, but the function of proportionality is not
just a constant, and is equal to the ratio of the apparent horizon and Hubble
radius. Clearly, for spatially flat universe, $\tilde{r}_{A}=H^{-1}$, and one
recovers the proposal (27).

Now, we would like to see whether one can derive the modified Friedmann
equation from the relation (28), when the entropy associated with the apparent
horizon get modified to Tsallis entropy. First of all, we define the effective
area of the holographic surface corresponding to the entropy (1) as

$\displaystyle\widetilde{A}=A^{\beta}=\left(4\pi{\tilde{r}_{A}^{2}}\right)^{\beta}.$
(29)

Next, we calculate the increasing in the effective volume as

$\displaystyle\frac{d\widetilde{V}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\tilde{r}_{A}}{2}\frac{d\widetilde{A}}{dt}=\beta(4\pi\tilde{r}_{A}^{2})^{\beta}\dot{\tilde{r}}_{A}$
(30) $\displaystyle=$
$\displaystyle\beta(4\pi)^{\beta}\frac{\tilde{r}_{A}^{5}}{(2\beta-4)}\
\frac{d}{dt}\left(\tilde{r}_{A}^{2\beta-4}\right).$

Inspired by (30), we propose that the number of degrees of freedom on the
apparent horizon with Tsallis entropy, is given by

$N_{\mathrm{sur}}=\frac{4\gamma\beta}{2-\beta}(4\pi\tilde{r}_{A}^{2})^{\beta}.$
(31)

We also assume the temperature associated with the apparent horizon is the
Hawking temperature, which is given by CaiKim

$T=\frac{1}{2\pi\tilde{r}_{A}},$ (32)

and the energy contained inside the sphere with volume
$V=4\pi\tilde{r}^{3}_{A}/3$ is the Komar energy

$E_{\mathrm{Komar}}=|(\rho+3p)|V.$ (33)

According to the equipartition law of energy, the bulk degrees of freedom obey

$N_{\mathrm{bulk}}=\frac{2|E_{\mathrm{Komar}}|}{T}.$ (34)

In order to have $N_{\rm bulk}>0$, we take $\rho+3p<0$ PadEm . Thus the number
of degrees of freedom in the bulk is given by

$N_{\rm bulk}=-\frac{16\pi^{2}}{3}\tilde{r}_{A}^{4}(\rho+3p),$ (35)

We also replace $L_{p}^{2}$ with $1/(4\gamma)$ and $V$ with $\widetilde{V}$ in
proposal (28) and write it down as

$4\gamma\frac{d\widetilde{V}}{dt}=\frac{\tilde{r}_{A}}{H^{-1}}(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).$
(36)

Substituting relations (30), (31) and (35) in Eq. (36), after simplifying, we
arrive at

$\displaystyle(2-\beta)\tilde{r}_{A}^{2\beta-5}\
\frac{\dot{\tilde{r}}_{A}}{H}-\tilde{r}_{A}^{2\beta-4}$ $\displaystyle=$
$\displaystyle\frac{4\pi L_{p}^{2}}{3}(\rho+3p),$ (37)

where we have also used definition (18). Multiplying the both hand side of Eq.
(37) by factor $2\dot{a}a$, after using the continuity equation (6), we reach

$\frac{d}{dt}\left(a^{2}\tilde{r}_{A}^{2\beta-4}\right)=\frac{8\pi
L_{p}^{2}}{3}\frac{d}{dt}(\rho a^{2}).$ (38)

Integrating, yields

$\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=\frac{8\pi L_{p}^{2}}{3}\rho,$
(39)

where we have set the integration constant equal to zero. This is nothing but
the modified Friedmann equation motivated from Tsallis entropy-area relation
(1). We see that our result from the emergence approach coincides with the
modified Friedmann equation derived from the first law of thermodynamics in
section II. Our study indicates that the approach presented here is enough
powerful and further supports the viability of the Padmanabhan’s perspective
of emergence gravity and its modification given by Eq. (36).

##  V Tsallis Cosmology

In this section, we would like to investigate cosmological consequences of the
modified Friedmann equation derived in Eqs. (17) and (39). Let us begin by
deriving the second modified Friedmann equation in Tsallis cosmology. Taking
the time derivative of the first Friedmann equation (39), we get

$2H(2-\beta)\left(\dot{H}-\frac{k}{a^{2}}\right)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\beta}=\frac{8\pi
L_{p}^{2}}{3}\dot{\rho}.$ (40)

Using the continuity equation (6), we arrive at

$\displaystyle(2-\beta)\left(\dot{H}-\frac{k}{a^{2}}\right)\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\beta}=-4\pi
L_{p}^{2}(\rho+p)$
$\displaystyle-\frac{3}{2}\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}-4\pi
L_{p}^{2}p,$ (41)

where we have also used Eq. (39) in the last step. Now using the fact that
$\dot{H}=\ddot{a}/a-H^{2}$, after some calculations, we can rewrite the above
equation as

$\displaystyle(4-2\beta)\frac{\ddot{a}}{a}+(2\beta-1)\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=-8\pi
L_{p}^{2}p.$ (42)

This is indeed the second modified Friedmann equation governing the evolution
of the Universe in Tsallis cosmology which is based on the nonextensive
Tsallis entropy. When $\beta=1$, the above equation reduces to the second
Friedmann equation in standard cosmology,

$\displaystyle 2\frac{\ddot{a}}{a}+\left(H^{2}+\frac{k}{a^{2}}\right)=-8\pi
L_{p}^{2}p.$ (43)

We can also obtain the equation for the second time derivative of the scale
factor. To this aim, we combine the first and second modified Friedmann
equations (39) and (42). We find

$\displaystyle\frac{\ddot{a}}{a}=-\frac{4\pi
L_{p}^{2}\rho}{3(2-\beta)}\left[(2\beta-1)\rho+3p\right].$ (44)

Since our Universe is currently undergoing an acceleration phase
($\ddot{a}>0$), thus from the above equation we should have

$\displaystyle(2\beta-1)\rho+3p<0\ \ \longrightarrow\ \
\omega<\frac{1-2\beta}{3},$ (45)

where $\omega=p/\rho$ is the equation of state parameter. Again, for $\beta=1$
the above condition for the accelerated universe leads to the well-known
inequality $\omega<-1/3$ for the equation of state in standard cosmology. A
close look on the inequality (45) show that for $\beta\geq 1/2$, we should
always have $\omega<0$. However, for $\beta<1/2$, we can have $\omega\geq 0$
in an accelerated universe. For example, for $\beta=1/3$ the above inequality
implies $\omega<1/9$. This is an interesting result, which indicates that in
Tsallis cosmology, the late time accelerated universe can be achieved even in
the presence of the ordinary matter with positive equation of state parameter.
Thus, if we assume our Universe is now dominated with pressureless matter with
($\omega=0$), then it can undergo an accelerated expansion provided we choose
the nonextensive parameter $\beta$ less than $1/2$, without needing to an
additional component of energy such as dark energy.

Next, we are going to find the scalae factor as a function of time in Tsallis
cosmology. For simplicity we only consider the flat universe with $k=0$,
although the calculations can be easily extended to the nonflat universe
($k\neq 0$). We shall consider the radiation and matter dominated era,
separately.

###  V.1 Matter-dominated era

It is well-known that the pressure caused by random motions of galaxies and
clusters of galaxies is negligible. This is due to the fact that the random
velocities of the galaxies are of the order of $10^{8}$ $\rm cm\rm s^{-1}$ or
less GRbook . Besides, the average energy density of the universe is of the
order of $10^{-30}\rm g\rm cm^{-3}$. Thus, we can write

$\displaystyle p\sim\rho<v^{2}>\sim 10^{-5}\rho c^{2}\ll\rho c^{2}.$ (46)

This implies that the pressure is negligible compared with $\rho c^{2}$. Thus
if the energy density of the universe is dominated by the nonrelativistic
matter with negligible pressure, we can write $p=0$. In this case, the
continuity equation (6) can be written $\dot{\rho}(t)+3H\rho(t)=0$, which can
be easily integrated to yield

$\displaystyle\frac{\rho(t)}{\rho(t_{0})}=\left(\frac{a(t)}{a(t_{0})}\right)^{-3},$
(47)

where $t_{0}$ is the present time of the universe. Therefore, $\rho a^{3}=\rm
constant.$, which is exactly the same as in standard cosmology in the matter
dominated era. Substituting $\rho$ from (47) in the first Friedmann equation
(39), we arrive at

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{4-2\beta}=\frac{8\pi
L_{p}^{2}\rho_{0}a_{0}^{3}}{3a^{3}},$ (48)

which can be rewritten as

$\displaystyle\left(\frac{d{a}}{dt}\right)^{4-2\beta}=C_{1}a^{1-2\beta},$ (49)

where we have defined

$\displaystyle C_{1}\equiv\frac{8\pi L_{p}^{2}\rho_{0}a_{0}^{3}}{3}.$ (50)

Eq. (49) has the following solution,

$\displaystyle a(t)=(C_{2}t)^{(4-2\beta)/3},$ (51)

where

$\displaystyle
C_{2}\equiv\frac{3}{2}\frac{C_{1}^{1/(4-2\beta)}}{(2-\beta)}>0.$ (52)

In the limit of standard cosmology where $\beta=1$, the above solution
restores

$\displaystyle a(t)=\left[\frac{3}{2}\sqrt{C_{1}}\right]^{2/3}t^{2/3},$ (53)

Let us now calculate the second time derivative of the scale factor. We find

$\displaystyle\ddot{a}(t)=\frac{C_{2}^{(4-2\beta)/3}}{9}(4-2\beta)(1-2\beta)\
t^{-(2+2\beta)/3},$ (54)

Again, we see that in the matter dominated universe, the accelerated expansion
($\ddot{a}(t)>0$) can be achieved provided we take $\beta<1/2$, which is
consistent with our previous discussion. Thus, in the framework of Tsallis
cosmology, without invoking any kind of dark energy and only with pressureless
matter, the late time acceleration of the universe expansion can be
understood.

We can also calculate the evolution of the energy density, the Hubble and the
deceleration parameters as

$\displaystyle\rho(t)$ $\displaystyle\propto$
$\displaystyle\frac{1}{t^{4-2\beta}},$ (55) $\displaystyle H(t)$
$\displaystyle=$ $\displaystyle\frac{\dot{a}}{a}=\frac{4-2\beta}{3t},$ (56)
$\displaystyle q(t)$ $\displaystyle=$
$\displaystyle-\frac{a\ddot{a}}{\dot{a}^{2}}=\frac{2\beta-1}{4-2\beta}.$ (57)

Again, all above parameters are reduced to the one in standard cosmology for
$\beta=1$. From the deceleration parameter, we see that for $\beta<1/2$, we
have $q<0$ and $\ddot{a}>0$, a result which confirm the accelerated universe.
Now, we look at the Hubble parameter which can be employed for estimating the
age of the universe at the present time ($t=t_{0}$). We have

$t_{0}=\frac{4-2\beta}{3H_{0}},$ (58)

where $H_{0}=H(t_{0})$ is the Hubble constant. In an accelerated universe we
have $\beta<1/2$, which implies $4-2\beta>3$, and thus

$t_{0}>\frac{1}{H_{0}}=\frac{3}{2}\left(\frac{2}{3H_{0}}\right),$ (59)

Note that $2/(3H_{0})$ is the age of the universe in standard cosmology. Let
us see how the above relation can alleviate the problem of age in standard
cosmology. The Hubble constant is usually written as

${H_{0}}=100h\ \rm km\ \rm sec^{-1}\rm Mpc=2.1332\ h\times 10^{-42}\rm GeV.$
(60)

where $h$ describes the uncertainty on the value $H_{0}$. The observations of
the Hubble Key Project constrain this value to be $h=0.72\pm 0.08$ age1 . Thus
the Hubble time is $t_{H}=1/H_{0}=9.78\times 10^{9}h^{-1}\rm years$. Using
this value for $h$, the age of the Universe in standard cosmology is in the
range $8.2$ Gyr $<t_{0}<10.2$ Gyr DEbook . Carretta et al. age2  estimated the
age of globular clusters in the Milky Way to be $12.9\pm 2.9$ Gyr, whereas
Jimenez et al. age3  obtained the value $13.5\pm 2$ Gyr. Hansen et al. age4
constrained the age of the globular cluster $M4$ to be $12.7\pm 0.7$ Gyr by
using the method of the white dwarf cooling sequence. In most cases the ages
of globular clusters are larger than $11$ Gyr which indicates that the cosmic
age estimated in standard cosmology is inconsistent with the ages of the
oldest globular clusters. It was argued that, in standard model of cosmology,
this problem cannot be circumvented unless the cosmological constant (dark
energy) is taken into account DEbook .

However, in the framework of Tsallis cosmology, the problem of age can be
circumvented automatically for an accelerated universe. More precisely, from
relation (59) the age of the universe in Tsallis cosmology is larger than
$3/2$ age of the universe in standard cosmology, namely

$t_{0}|_{T}>\frac{3}{2}t_{0}|_{S},$ (61)

provided we choose $\beta<1/2$. Here subscript “T” and “S” stand for Tsallis
and Standard cosmology, respectively. As an example, taking $\beta=2/5$, from
relation (58) we find $t_{0}|_{T}=1.6\ t_{0}|_{S}$. Consequently, the age of
the accelerated universe in Tsallis cosmology is in the range $13.12$ Gyr
$<t_{0}|_{T}<16.32$ Gyr, which is larger than the age of the oldest globular
clusters. Therefore, the cosmic age problem can be properly alleviated in an
accelerated universe in the framework of Tsallis cosmology without invoking
additional component of energy.

###  V.2 Radiation-dominated era

It is well-known that, in an expanding universe, the proper momenta of freely
moving particles decreases in term of scale factor as $1/a(t)$. Thus, small
random velocities of particles seen today should have been large in the past
when the scale factor was much smaller than its present value GRbook . As a
result, the pressureless approximation is break down in the early universe. In
this subsection, we are going to find the solution for the modified Friedmann
equation in the framework of Tsallis cosmology by assuming that the universe
is filled with a highly relativistic gas (radiation) with equation of state
$p=\rho/3$. In this case from the continuity equation,
$\dot{\rho}(t)+4H\rho(t)=0$, we can get

$\displaystyle\frac{\rho(t)}{\rho(t_{1})}=\left(\frac{a(t)}{a(t_{1})}\right)^{-4},$
(62)

where $t_{1}$ is an arbitrary reference time in the radiation dominated era.
Thus, in this case $\rho a^{4}=\rm constant.$, which is similar to the result
obtained in the radiation dominated era in standard cosmology. Substituting
$\rho$ from (62) in the first Friedmann equation (39), we get

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{4-2\beta}=\frac{8\pi
L_{p}^{2}\rho_{1}a_{1}^{4}}{3a^{4}},$ (63)

where $\rho_{1}=\rho(t_{1})$, $a_{1}=a(t_{1})$ and we have again considered
the flat universe ($k=0$), although the $k=\pm 1$ cases can also be easily
solved. The above equation can be rewritten as

$\displaystyle\left(\frac{d{a}}{dt}\right)^{4-2\beta}=B_{1}a^{-2\beta},$ (64)

where

$\displaystyle B_{1}\equiv\frac{8\pi L_{p}^{2}\rho_{1}a_{1}^{4}}{3}.$ (65)

Eq. (64) admits a solution of the form,

$\displaystyle a(t)=B_{1}^{1/4}\left(\frac{2t}{2-\beta}\right)^{1-\beta/2}.$
(66)

When $\beta=1$, we arrive at

$\displaystyle a(t)=\sqrt{2}B_{1}^{1/4}t^{1/2},$ (67)

which is the corresponding solution in standard cosmology. The second time
derivative of the scale factor is obtained as

$\displaystyle\ddot{a}(t)=-\frac{\beta}{2}B_{1}^{1/4}\left(\frac{2}{2-\beta}\right)^{-\beta/2}t^{-(1+\beta/2)}<0.$
(68)

This shows that in the radiation dominated era, the universe has been in an
decelerated phase ($\ddot{a}(t)<0$). Now, we calculate the energy density, the
Hubble and the deceleration parameters in the radiation dominated era. We find

$\displaystyle\rho(t)$ $\displaystyle\propto$
$\displaystyle\frac{1}{t^{4-2\beta}},$ (69) $\displaystyle H(t)$
$\displaystyle=$ $\displaystyle\frac{\dot{a}}{a}=\frac{2-\beta}{2t},$ (70)
$\displaystyle q(t)$ $\displaystyle=$
$\displaystyle-\frac{a\ddot{a}}{\dot{a}^{2}}=\frac{\beta}{2-\beta}.$ (71)

Therefore, for $\beta<2$, we have always $q>0$ ($\ddot{a}(t)<0$), which
confirms that we have a decelerated universe in the radiation dominated era.
This implies that in the early stage of the universe where the relativistic
particles have been dominated, our Universe has been in a decelerated phase
and at the late time it undergoes an accelerated expansion. In summary,
Tsallis cosmology can explain the evolution of the universe from the early
deceleration to the late time acceleration, without invoking dark energy,
which is consistent with recent observations.

###  V.3 Density perturbation

In order to study the density perturbation, we follow the method which was
first introduced by James Jeans GRbook . Assuming a flat FRW universe in the
matter-dominated era, the fundamental differential equation governing the
fractional density perturbation $\delta=\rho_{1}/\rho$ in an expanding
universe becomes GRbook

$\displaystyle\ddot{\delta}+\frac{2\dot{a}}{a}\dot{\delta}+\left(\frac{v_{s}^{2}\kappa^{2}}{a^{2}}-4\pi
G\rho\right)\delta=0,$ (72)

where $v_{s}^{2}=p_{1}/\rho_{1}$ is the speed of sound, $\rho_{1}$ and $p_{1}$
stand for the perturbed part of the energy density and pressure, respectively,
and $\kappa$ is the co-moving wavevector. The solution of the above equation
depends on the sign of the function

$\displaystyle f(t)=\frac{v_{s}^{2}\kappa^{2}}{a^{2}(t)}-4\pi G\rho(t).$ (73)

For $f(t)>0$, the solution for $\delta(t)$ is periodic and the perturbation
propagate in the medium as sound waves. For $f(t)<0$, however, lead to
imaginary frequency, indicating a growth of $\delta(t)$ with time. In what
follows we are interested in the second case where $f(t)<0$ and further assume
$\frac{v_{s}^{2}\kappa^{2}}{a^{2}}\ll 4\pi G\rho$. In this case Eq. (72) can
be written

$\displaystyle\ddot{\delta}+\frac{2\dot{a}}{a}\dot{\delta}-4\pi
G\rho\delta=0.$ (74)

In the framework of standard model of cosmology where $a(t)\propto t^{2/3}$,
Eq. (74) can be further rewritten as

$\displaystyle\ddot{\delta}+\frac{4}{3t}\dot{\delta}-\frac{2}{3t^{2}}\delta=0.$
(75)

which has a solution in the form $\delta(t)=\delta_{0}t^{2/3}$, and thus the
perturbation grows with time. On the other hand, when the Friedmann equations
are governed by the Tsallis cosmology, the scale factor in the matter
dominated era is, $a(t)\propto t^{(4-2\beta)/3}$. On substituting in Eq. (74),
we arrive at

$\displaystyle\ddot{\delta}+\frac{4}{3t}(2-\beta)\dot{\delta}-\frac{3}{2}\left(\frac{4-2\beta}{3t}\right)^{4-2\beta}\delta=0.$
(76)

The solution of the above equation is given by

$\displaystyle\delta(t)=\delta_{0}\times$ $\displaystyle
BesselJ\left(\frac{5-4\beta}{6\beta-6},\frac{2(\beta-2)^{2}}{9\beta-9}\sqrt{-6\left(\frac{4-2\beta}{3}\right)^{-2\beta}}t^{\beta-1}\right),$

To have an insight on the form of $\delta(t)$, let us consider a special case
where $\beta=5/4$ at the early stage of the universe. In this case the
expansion of solution (V.3) is given by

$\displaystyle\delta(t)=\delta_{0}\left[1-\alpha_{1}t^{1/2}+\alpha_{2}t+...\right]$
(78)

where $\alpha_{i}$ are constants. Thus, the growth of energy differs in
Tsallis cosmology compared to the standard cosmology. It is important to note
that this study is very brief and we leave the details of investigations on
the density perturbation in Tsallis cosmology for future studies.

##  VI conclusion and discussion

It was already pointed out by Gibbs in 1902 that in systems whose partition
function diverges, like gravitational system, the Boltzmann-Gibbs (BG) theory
cannot be applied. As a result, the thermodynamical entropy of such
nonstandard systems is not described by an additive entropy but with
appropriately generalized nonadditive entropies. Based on this, and using the
statistical arguments, Tsallis and Cirto Tsa  argued that the microscopic
mathematical expression of the thermodynamical entropy of a black hole does
not obey the area law and can be modified as in Eq. (1). Besides, it is well-
known that the entropy of the whole universe, considered as a system with
apparent horizon radius has a similar expression to the entropy associated
with black hole horizon. The only change is needed replacing the black hole
horizon radius $r_{+}$ by the apparent horizon radius $\tilde{r}_{A}$ of the
Universe.

In this paper, we have assumed the entropy associated with the apparent
horizon of FRW universe is in the form of the Tsallis entropy and examined its
consistency with the laws of thermodynamics. For this purpose, we first
assumed the the first law of thermodynamics, $dE=T_{h}dS_{h}+WdV$, holds on
the apparent horizon and the Tsallis entropy has the form (1). Then, we showed
that the first law of thermodynamics on the apparent horizon can be rewritten
in the form of the modified Friedmann equations of a FRW universe with any
spatial curvature. We have also examined the generalized second law of
thermodynamics, by studying the time evolution of the total entropy including
the entropy of the matter and energy inside the Universe together with the
Tsallis entropy associated with the apparent horizon. Assuming the local
equilibrium hypothesis, we confirmed that the generalized second law of
thermodynamics is fulfilled in a region enclosed by the apparent horizon.

We have also considered the idea of emergence of cosmic space proposed by
Padmanabhan PadEm  and its modification given by Eq. (36) Sheyem . Assuming
the entropy associated with the apparent horizon in the form of Tsallis
entropy, we determine the number of degrees of freedom on the boundary. Then,
by calculating the difference between the degrees of freedom on the boundary,
$N_{\rm sur}$, and the degrees of freedom in the bulk, $N_{\rm bulk}$, we
showed that one can extract the Friedmann equations corresponding to the
Tsallis entropy-area relation (1) which coincides with the one obtained from
the first law of thermodynamics. This study strongly supports the viability of
the novel idea proposed in PadEm ; Sheyem . Note that, we have only modified
the number of degrees of freedom on the boundary due to the change in the
entropy expression, while we assumed the bulk degrees of freedom, $N_{\rm
bulk}$, has the same expression as in standard cosmology. The reason comes
from the fact that we have assumed $N_{\rm bulk}$ depends only on the matter
degrees of freedom, while $N_{\rm sur}$ crucially depends on the entropy
expression or/and underlying theory of gravity.

We have also investigated some cosmological consequences of the modified
Friedmann equations. For this purpose, we first derived the second modified
Friedmann equation as well as the equation for the second time derivative of
the scale factor. Then, we considered the matter-dominated era and the
radiation-dominated era, separately. We found that, in the matter dominated
era which is filled with pressureless matter, this model admits an accelerated
universe ($\ddot{a}(t)>0$) provided $\beta<1/2$. Besides, the problem of age
of universe can be circumvented automatically for an accelerated universe. By
calculating the Hubble parameter in the matter dominated era, we estimated the
age of the universe larger than $3/2$ age of the universe in standard
cosmology. For example, taking $\beta=2/5$, we estimated the age of the
accelerated universe in Tsallis cosmology in the range $13.12$ Gyr
$<t_{0}|_{T}<16.32$ Gyr, which is larger than the age of the oldest globular
clusters. Therefore, the cosmic age problem can be alleviated in an
accelerated universe in the framework of Tsallis cosmology without invoking
additional component of energy. In the radiation dominated era we have always
$q>0$ ($\ddot{a}(t)<0$), which confirms the deceleration phase for the early
stage of the universe. Therefore, Tsallis cosmology can explain the evolution
of the universe from the early deceleration to the late time acceleration
which is consistent with recent observations. Finally, employing the Jeans’s
analysis, we explored in brief, density perturbation in the matter dominated
era in the context of Tsallis cosmology. We showed that due to the
modification of the Friedmann equations and hence the scale factor, the growth
of energy is changed compared to the standard cosmology.

It is important to note that many issues remain to be studied. First, the
details of the effects of the modified Friedmann equations on the growth of
energy density in the early stage of the universe as well as its influences on
the structure formation deserve further investigation. It is also interesting
to consider other forms of entropies e.g., the generalized Kaniadakis entropy
Kan  to investigate the modified Friedmann equations as well as its
cosmological consequences. These issues are under investigations and the
results will be appeared in the future works.

###### Acknowledgements.

I am grateful to the referee for very constructive comments which helped me
improve the paper significantly. I also thank Shiraz University Research
Council. This work has been supported financially by Research Institute for
Astronomy and Astrophysics of Maragha (RIAAM), Iran.

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