# Ghost Dark Energy in Tsallis and Barrow Cosmology

Esmaeil Ebrahimi1,2,3<EMAIL_ADDRESS>Ahmad Sheykhi1,2
<EMAIL_ADDRESS>1Department of Physics, College of Science, Shiraz
University, Shiraz 71454, Iran.  2Biruni Observatory, College of Science,
Shiraz University, Shiraz 71454, Iran.  3College of Physics, Shahid Bahonar
University, PO Box 76175, Kerman, Iran

###### Abstract

According to the thermodynamics-gravity conjecture, any modification to the
entropy expression leads to the modified cosmological field equations. Based
on this, we investigate the cosmological consequences of the modified
Friedmann equations when the entropy associated with the horizon is in the
form of Tsallis/Barrow entropy and the dark energy (DE) component is in the
form of ghost dark energy (GDE). We perform a dynamical system analysis and
see that the Tsallis GDE(TGDE) and Barrow GDE(BGDE) can exhibit a correct
phase space evolution for suitable range of the free parameters($0<\Delta<1$
for BGDE and $\beta<3/2$ for TGDE). We find that in BGDE and TGDE (with
$Q=3b^{2}H(\rho_{D}+\rho_{m})$), there exist an early radiation dominated
phase of expansion which is absent for GDE model in standard cosmology. It is
worth mentioning that seeking an unstable phase of matter dominated in TGDE
leads to a new constraint on $\beta$($<3/2$). Using the resulting range of
free parameters from latter step we find that both models are capable to
explain the cosmic evolution from deceleration to an accelerated phase. We
observe that increasing $\Delta$ ($\beta$) parameters leads to a delay in the
cosmic phase transition. We conclude that BGDE and TGDE are viable
cosmological models which predict a consistent phase transition of the cosmic
expansion for suitable ranges of the parameters. We also calculate the squared
sound speed for both models and find out that they are unstable against
perturbations. Next, we proceed the statefinder analysis and see that both
models are significantly distinct from $\Lambda CDM$ as well as with respect
to each other at past epochs, while both of them catch the standard cosmology
at far future. We also explore the impacts of the non-extensive parameters on
the density perturbations and find out that the pattern of density contrast
($\delta$) evolution in TGDE/BGDE is distinct from GDE in standard cosmology.

###### keywords:

Modified cosmology, Ghost dark energy, Tsallis entropy, Barrow entropy

##  1 Introduction

Modified gravity is an alternative explanation for the late time acceleration
of the cosmic expansion [1, 2, 3, 4]. Modified theories of gravity including
generalization of the Einstein gravity such as $f(R)$ [5, 6, 7, 8, 9, 10, 11,
12, 13], the Gauss-Bonet gravity [14, 15, 16] and Lovelock theory of
gravitation [17]. The main idea is to explain the early inflation epoch as
well as the late time acceleration of the cosmic expansion via modified
gravity.

On the other hand, cosmological models based on the modified entropy
expression have got a lot of attentions in recent years. According to
thermodynamics of black holes, the entropy and temperature associated with the
horizon are proportional to the horizon area and surface gravity,
respectively. Is there any direct connection between the gravitational field
equations and the laws of thermodynamics? Jacobsen was the first who answered
this question and derived the Einstein’s field equations of gravity from
Clausius relation $\delta Q=T\delta S$, where $\delta Q$ is the heat flux
crossing the horizon, $T$ and $S$ are, respectively, temperature and entropy
of the horizon [18]. The next great step put forwarded by Eric Verlinde [19]
who claimed that gravity is not a fundamental force and can be regarded as an
entropic force. In this approach gravitational interaction originates from
changes of the entropy associated with the information on the holographic
screen. Following [19] many efforts have been done to explore the entropic
nature of gravity [20, 21, 22, 23, 24, 25, 26, 27, 28].

In recent years a lot of attentions have arisen to the non-extensive entropy
formalism, where total entropy of a system cannot necessarily obtained via
summation of that of the subsystems [29, 30, 31]. The so called Tsallis
entropy is one of the most well-known case of the non-extensive entropies. In
2012, Citro and Tsallis discussed that entropy associated with the horizon of
a black hole does not obey the area law and instead may be written as [32],

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

where $A$ is the black hole horizon area, $\gamma$ is an unknown constant and
$\beta$ is known as Tsallis (or non-extensive) parameter. We assume $\beta$ is
a real parameter which shows the amount of the non-extensivity [32]. It can be
easily seen that the usual area law of black hole entropy is deduced in the
limiting case where $\beta\rightarrow 1$ and $\gamma\rightarrow
1/(4l_{p}^{2})$.

Regarding this new form of entropy and its cosmological consequences, two
approaches has been followed in the literature. The first approach only
modifies the DE density due to the change in the entropy, while keeping the
cosmological field equations in its standard form. The second approach
modifies the cosmological field equations through modified entropy, beside to
plausible corrections to the DE density. Since the holographic dark energy
(HDE) is based on the definition of black hole entropy, thus redefinition of
the entropy will affect the resulting HDE. In this regard the modified HDE in
the context of standard cosmology has been extensively explored [33, 34, 35,
36]. It was shown that for a special choice of the parameter $\beta$ the
Tsallis HDE leads to an accelerated universe and can alleviate the age problem
[37]. On the other side, the thermodynamic-gravity conjecture reveals
modifications in the resulting Friedmann equations. Modified Friedmann
equations through Tsallis entropy were explored in [38]. Different aspects of
Tsallis cosmology were investigated and the model were also constrained with
observational data [39]. Growth of perturbations in Tsallis cosmology was
explored in [40]. In [41], the authors have argued that Tsallis cosmology can
alleviate $H_{0}$ and $\sigma_{8}$ tensions. Baryogenesis in Tsallis cosmology
is performed in [42] and the authors showed that induced effects of non-
extensivity to the Hubble function evolution in earlier epochs can lead a
mechanism to allow the baryogenesis.

Inspired by the fractal structure of the corona virus, Barrow proposed a
fractal structure for the horizon geometry of black holes [43]. This fractal
nature of the horizon, results to a finite volume and an infinite (or a
finite) area. Such a geometry for horizon leads to a non-extensive entropy
relation called Barrow entropy, which is given by

$S=\left(\frac{A}{A_{0}}\right)^{1+\frac{\Delta}{2}},$ (2)

where $A$ and $A_{0}$ are the black hole horizon and Planck area,
respectively. In the above relation $\Delta$ stands for the amount of the
quantum-gravitational deformation effects and ranges $\in[0,1]$. When
$\Delta=0$, the standard area law is restored and $A_{0}\longrightarrow 4G$,
while $\Delta=1$ represents the most intricate and fractal structure of the
horizon. Modified Friedmann equations from Barrow entropy were introduced in
[44, 45]. Applying the Barrow entropy, validity of the generalized second law
of thermodynamics has argued in [46]. It is worth mentioning that Barrow
cosmology cannot lead to a consistent picture of cosmic evolution in the
absence of the DE component. Thus exploring Barrow cosmology with DE component
is well motivated. For example, HDE in Barrow cosmology as well as
observational constraints on these models have been investigated [47, 48, 49,
50, 51, 52]. Structure formation in the context of Barrow cosmology was
addressed in [40].

It is important to note that although Barrow and Tsallis relations for the
entropy expressions are almost similar but they arise from completely
different perspective. Tsallis entropy originates from generalizing standard
thermodynamics, while Barrow entropy correction to the entropy comes from
quantum gravitational effects. It is also worth noting that $\Delta$ and
$\beta$ parameters, which denote the degree of non-extensivity, lie in
different ranges ($\Delta\epsilon$ [0,1] and $\beta\epsilon$[0,2]) for each
model.

On the other side, the GDE model was proposed to alleviate the QCD U(1)
problem [53, 54, 55, 56, 57]. Vacuum energy contribution of QCD field can act
as a DE component in the curved spacetime or in a dynamic background. Up to
first order the resulting vacuum energy in an expanding universe leads to
$\rho_{D}\propto\Lambda_{QCD}^{3}H$, where $H$ is the Hubble expansion rate
and $\Lambda_{\rm QCD}^{3}$ is QCD mass scale [58]. Taking $\Lambda_{QCD}\sim
100MeV$ and $H_{0}\sim 10^{-33}eV$, the resulting vacuum energy density equals
to $\rho_{GDE}\sim 10^{-3}eV$ in favor of observations.

It is important to note that in GDE model, we do not need to introduce any new
degree of freedom and thus this model is completely preferred with respect to
those add new degrees of freedom. First interesting out coming of this model
is its ’fine tuning’-free nature [59, 58]. Different aspects of GDE have been
investigated in standard cosmology. While dynamics of the universe in presence
of the GDE and DM was studied in [60], interacting GDE in a curved background
was discussed in [61]. Stability of a GDE dominated universe was addressed in
[59, 62]. Also a statefinder analysis of GDE model explored in [63].
Correspondence between the interacting GDE and a complex quintessence
cosmology was performed in [64].

Although taking a non-extensive form of the entropy in Barrow and Tsallis
cases modifies the Friedmann equations, however the resulting Barrow and
Tsallis cosmology are not well satisfying specially in facing the accelerated
expansion of the universe. It is worth mentioning that in [38] one of the
present authors argued that for $\beta<1/2$, Tsallis cosmology can explain the
late time accelerated expansion without invoking any kind of dark energy.
Nevertheless, in [39], the authors found an observational constraint on
Tsallis parameter($\beta$) and the best fit value for Tsallis parameter is
($\beta\approx 0.999$) which denotes a discrepancy between Tsallis cosmology
(in the absence of any DE component) and observational evidences. Thus
considering an additional component of energy (such as GDE) in these modified
cosmological models, is still well motivated. From a closer look one should
note that two important feature in DE models are still of main motivations in
exploring the GDE in Barrow and Tsallis cosmology. First is that the Barrow
(Tsallis) HDE as well as Barrow ADE suffers instability issues regarding the
squared sound speed analysis [34, 65, 66]. Second reason comes from the
behavior of the models in facing the phantom line. For example the Barrow ADE
can not cross this border[34, 65, 66]. Hence, there exist enough motivations
to describe the GDE in Barrow and Tsallis cosmology. Thus, we would like to
explore the influences of the entropy based modified cosmological models when
the DE component of the universe is taken as GDE. We shall explore the
cosmological consequences of this model in the context of modified
Barrow/Tsallis modified cosmology and study the evolution of the cosmological
parameters.

The outlook of the article is as follows. In the next section, using the
thermodynamics-gravity correspondence, we review derivation of the modified
Friedmann equations based on Tsallis/Barrow entropy. In section 3, we
investigate GDE in the context of the modified Barrow cosmology. In section 4,
we explore GDE in the context of the modified Tsallis cosmology. The last
section is devoted to closing remarks.

##  2 Tsallis and Barrow cosmology

In this section, we derive the modified Friedmann equations using the first
law of thermodynamics at apparent horizon of a FRW universe. To this goal we
assume that the entropy associated with the apparent horizon is in the form of
Barrow/Tsallis entropy. We take the spatial subspace of the universe as a
maximally symmetric space. In this case the form of the line element reads

$ds^{2}=h_{\mu\nu}dx^{\mu}dx^{\nu}+\tilde{r}^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2}),$ (3)

where $\tilde{r}=a(t)r,x^{0}=t,x^{1}=r$ and
$h_{\mu\nu}=diag\left(-1,\frac{a^{2}(t)}{1-kr^{2}}\right)$ is two dimensional
metric. $k=-1,0,+1$ corresponds to open, flat and closed universe
respectively. In order to choose a boundary at which first and second laws of
thermodynamics hold on, the suitable choice is apparent horizon. The
definition of the apparent horizon is [67]

$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+{k}/{a^{2}}}}.$ (4)

where $H=\dot{a}/a$ is the Hubble parameter. The resulting temperature with
the apparent horizon is [67]

$T_{h}=-\frac{1}{2\pi\tilde{r}_{A}}\left(1-\frac{\dot{\tilde{r}}_{A}}{2H\tilde{r}_{A}}\right).$
(5)

Further, we assume the cosmic fluid could be described through

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (6)

where $\rho$ and $p$ are the energy density and pressure of the cosmic fluid
respectively. Conservation equation $(\nabla_{\mu}T^{\mu\nu}=0)$ leads to the
continuity equation as

$\dot{\rho}+3H(\rho+p)=0.$ (7)

In an expanding background any change in the volume element is associated with
the following amount of the work density [68]

$W=-\frac{1}{2}T^{\mu\nu}h_{\mu\nu}.$ (8)

Above definition of work density in terms of the energy momentum tensor
elements, (6), becomes

$W=\frac{1}{2}(\rho-p).$ (9)

Now we are ready to extract Friedman equations through first law of
thermodynamics. On this way, we start with the following equation

$dE=T_{h}dS_{h}+WdV,$ (10)

and assume that above equation is satisfied on the apparent horizon(4). Here
$V=4\pi\tilde{r}^{3}/3$ is the volume of a sphere with radius $\tilde{r}$,
$S_{h}$ is the associated entropy with apparent horizon and $E=\rho V$ is the
total energy of the universe. Differentiating $E=\rho V$ and mixing the result
with the continuity equation ($d\rho=-3H(\rho+p)dt$), on gets

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

Given this equation at hand, one can proceed to get the evolution equation of
any expanding universe with an arbitrary entropy definition. In this paper we
would like to obtain the resulting equation for Barrow/Tsallis entropy
definitions.

The Barrow entropy formula reads [43]

$S=\left(\frac{A}{A_{0}}\right)^{1+\frac{\Delta}{2}}.$ (12)

Taking differential form of this equation and using Eqs. (5), (10) and (11),
after a little algebra one arrives at

$-\frac{2+\Delta}{2\pi
A_{0}}\left(\frac{4\pi}{A_{0}}\right)^{\frac{\Delta}{2}}\frac{d\tilde{r}}{\tilde{r}^{3-\Delta}}=\frac{d\rho}{3}.$
(13)

Integrating the above equation and using (4), we reach [44]

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{2\pi
A_{0}}{3}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{3}\right)^{\Delta/2}\rho,$
(14)

which can be rewritten as [44]

$\left(H^{2}+\frac{k}{a^{2}}\right)^{1-\Delta/2}=\frac{8\pi G_{\rm
eff}}{3}\rho,$ (15)

where the effective gravitational constant $G_{\rm eff}$ is defined as [44]

$G_{\rm
eff}=\frac{A_{0}}{4}\left(\frac{2-\Delta}{2+\Delta}\right)\left(\frac{A_{0}}{3}\right)^{\Delta/2}.$
(16)

It could be easily seen that in the limit $\Delta\rightarrow 0$ and
$A_{0}\rightarrow 4G$, we have $G_{\rm eff}\rightarrow G$, and the standard
Friedmann equation is retrieved.

Starting with definition of the Tsallis entropy relation, $S=\gamma
A^{\beta}$, and performing an almost same set of derivations the modified
Friedmann equation in Tsallis case can be obtained. The resulting first
Friedmann equation in Tsallis case reads [38]

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

where $l^{2}_{p}=\frac{2-\beta}{4\beta\gamma}(4\pi)^{1-\beta}$. One should
note that in the limiting case $\beta=1$ and $\gamma=1/(4l^{2}_{p})$ the
standard Friedmann equation is recovered. Since in present work we do not use
the second Friedmann equation we do not mention about that. However a detailed
derivation of the Tsallis and Barrow cosmology equations can be seen in [38,
44, 45].

It is worthy to note that despite the non-extensive statistics modifications
in the resulting cosmology, the new frameworks are still in need of an
additional DE component to overcome the late time acceleration issue
conveniently [44]. Thus it is still well motivated to consider the GDE in the
context of Tsallis/Barrow modified cosmology.

##  3 Ghost dark energy in Barrow cosmology

Here, we would like to consider a flat universe filled with a pressureless
matter and the GDE components. In this case the first Friedmann equation in
the modified Barrow cosmology reads

$\displaystyle H^{2-\Delta}=\frac{8\pi G_{\rm eff}}{3}(\rho_{m}+\rho_{D}),$
(18)

where $\rho_{m}$ and $\rho_{D}$ are energy density of matter and GDE,
respectively. In an expanding background the continuity equations are

$\displaystyle\dot{\rho}_{D}+3H\rho_{D}(1+w_{D})$ $\displaystyle=-Q,$ (19)
$\displaystyle\dot{\rho}_{m}+3H\rho_{m}$ $\displaystyle=Q,$ (20)

where $w_{D}=p_{D}/\rho_{D}$ is equation of state parameter (EoS) of dark
component and $Q$ denotes the chance of interaction between matter and DE
component. One should note that the DE and DM components are introduced to
literature for curing different issues and these component are both separately
conserved. However, in the absence of a symmetry regarding separate
conservation, people give a chance to mutual interaction between DE and DM.
Further, presence of interaction between DE and DM makes a closer agreement to
observational evidences from the galaxy cluster Abell A586[69]. Thus according
to these theoretical and observational reasons it is well motivated to
consider an interaction between dark sector components. Interacting models
were first presented by Wetterich in[70]. Simplest form of interaction term
has the form $Q\propto H\rho$, where $\rho$ could be $\rho_{D}$, $\rho_{m}$ or
$\rho_{m}+\rho_{D}$. Various models of dark energy are explored in presence of
such interaction terms[71, 72, 73, 74, 75, 76, 77]. Here, we will consider the
choice $Q=3b^{2}H(\rho_{D}+\rho_{m})$ in Eq.(19). Beside it worth to mention
that some non-linear choices of $Q$ has been also examined in the
literature[78, 79, 80].

The fractional energy densities could be defined as

$\Omega_{m}=\frac{\rho_{m}}{\rho_{cr}},\ \ \
\Omega_{D}=\frac{\rho_{D}}{\rho_{cr}},$ (21)

where the critical energy density is $\rho_{cr}=3H^{2-\Delta}/(8\pi G)$. Thus,
the first Friedmann equation in terms of the fractional energy densities reads

$\Omega_{m}+\Omega_{D}=1.$ (22)

Our main mission in this paper is to disclose the impacts of the GDE in
entropy corrected cosmology frameworks. As we mentioned in the introduction,
the energy density of the GDE in an expanding background is written as [67]

$\rho_{D}=\alpha H,$ (23)

where $\alpha$ denotes a constant of order $\Lambda_{QCD}^{3}$. At first step
we consider the EoS parameter for the GDE in the modified Barrow cosmology.
Solving the continuity equation for $w_{D}$, one finds

$w_{D}=-1-\frac{\dot{\rho}_{D}}{3H\rho_{D}}-\frac{Q}{3H\rho_{D}}.$ (24)

Taking the time derivative from Eqs. (23) and (18) and using the fractional
density definitions, one gets

$\frac{\dot{\rho}_{D}}{\rho_{D}}=\frac{\dot{H}}{H}=-\frac{3H(\Omega_{D}w_{D}+1)}{2-\Delta}.$
(25)

Taking Eq. (24) into account and using relation
$Q=3b^{2}H(\rho_{D}+\rho_{m})$, one finds

$w_{D}=-\frac{1}{2-\Omega_{D}-\Delta}\left[1-\Delta+\frac{b^{2}}{\Omega_{D}}(2-\Delta)\right].$
(26)

One can easily check that for $\Delta=0$, the above relation for $w_{D}$
reduces to the corresponding relation in GDE [60]. The deceleration parameter
is another important quantity in our setup. Using definition of the
deceleration parameter as

$q=-1-\frac{\dot{H}}{H^{2}}=-1+\frac{3}{2-\Delta}\left(1+\Omega_{D}w_{D}\right),$
(27)

and substitute Eq. (26), we get

$q=-1+\frac{3(b^{2}+\Omega_{D}-1)}{\Omega_{D}-2+\Delta}$ (28)

Evolution of the DE fractional density can complete our equation set. To this
end we use $\dot{\Omega}_{D}=H\Omega^{{}^{\prime}}_{D}$ which prime denotes a
differentiation with respect to $x=\ln{a}(a_{0}=1)$. Taking a time derivative
of $\Omega_{D}$ one can get

$\Omega^{{}^{\prime}}_{D}=(1-\Delta)\frac{3\Omega_{D}(b^{2}+\Omega_{D}-1)}{\Omega_{D}-2+\Delta}$
(29)

Once again one can check that above equations approach a correct limit for
$\Delta=0$ in both interacting and non-interacting cases in agreement with
[60].

###  3.1 Dynamical system analysis of Barrow cosmology

A primary description in disclosing the BGDE cosmology is the phase-space
analysis. This approach is widely discussed in cosmological and astrophysical
frameworks[81, 82, 83, 84]. The dynamical system analysis allows us to obtain
a qualitative description of dynamics of cosmological models. This analysis
could be performed in ample details but our aim here is to see overall
features of the model. Specifically, we would like to see if the BGDE is
capable to result in radiation, DM and DE dominated phases with correct
timeline order. Beside, we can obtain a prior range of free parameters through
this technique. On this way, we take the universe filled by radiation, CDM and
the GDE. Next, we introduce {$\Omega_{D},\Omega_{m}$} as the dynamical
variables and obtain the corresponding evolution equations
{$\Omega^{\prime}_{D}=0,\Omega^{\prime}_{m}=0$}. Then algebraic solution of
the autonomous set of the first order equations reveals evolutionary behavior
of each model[81, 82, 83, 84, 85].

We already introduced the evolution equation of $\Omega_{D}$ in Eq.(29).
Performing a same calculations for $\Omega_{m}$, we will obtain the resulting
dynamical autonomous system of the BGDE as

$\Omega^{\prime}_{D}=\frac{\Omega_{\mathrm{D}}\left(3b^{2}\Omega_{\mathrm{D}}+3b^{2}\Omega_{m}+4\Omega_{\mathrm{D}}+\Omega_{m}-4\right)\left(\Delta-1\right)}{\Omega_{\mathrm{D}}-2+\Delta},$
(30)

$\Omega^{\prime}_{m}=\frac{\left(f_{1b}\Omega_{m}^{2}+f_{2b}\Omega_{m}+3b^{2}\Omega_{\mathrm{D}}\left(\Omega_{\mathrm{D}}-2+\Delta\right)\right)}{\Omega_{\mathrm{D}}-2+\Delta},$
(31)

where

$\displaystyle f_{1mb}$ $\displaystyle=$
$\displaystyle\left(-3\Delta+6\right)b^{2}+2-\Delta,$ $\displaystyle f_{2mb}$
$\displaystyle=$
$\displaystyle\left(\left(-3\Delta+9\right)\Omega_{\mathrm{D}}+3\Delta-6\right)b^{2}$
(32) $\displaystyle+$
$\displaystyle\left(-4\Delta+5\right)\Omega_{\mathrm{D}}+\Delta-2.$

fixed point | $\Omega_{D}$ | $\Omega_{m}$ | $q$ | Nature
---|---|---|---|---
$P_{1}$ | 0 | 0 | $>0$ | unstable for $0<\Delta<1$
$P_{2}$ | 0 | 1 | $>0$ | saddle $0<\Delta<1$
$P_{3}$ | $-b^{2}+1$ | $b^{2}$ | $<0$ | stable
Table 1: The admitted fixed points of BGDE in presence of an interaction
term($Q=3b^{2}H(\rho_{D}+\rho_{m})$).

The resulting fixed points are listed in Tab.(1). The physical phase space is
compact(i.e. $0\leq\Omega_{i}\leq 1$). The first point($P_{1}$) denotes a
radiation dominated phase at the beginning of the cosmic evolution. It is
worth to mention that in [86] the interacting GDE(with
$Q=3b^{2}H(\rho_{D}+\rho_{m})$) in standard cosmology does not admit a
radiation dominated epoch while in Barrow cosmology this point is present. For
this fixed point the corresponding eigenvalues(whose their signs determine the
stability) are $\lambda_{1}=3b^{2}+1$ and
$\lambda_{2}=\frac{4\left(\Delta-1\right)}{\Delta-2}$. It is evident that this
fixed point represents an unstable character for all acceptable(It is accepted
that $b^{2}<1$ and $0<\Delta<1$). Using the deceleration parameter
$q=-1+\frac{3(b^{2}-1)}{\Delta-2}$, one can easily find that at this point the
universe shows a decelerative behavior as we expect.

Second fixed point($P_{2}:\Omega_{m}=1,\Omega_{D}=0$), corresponds to a matter
dominated universe. The eigenvalues of the stability matrix are
($\lambda_{1}=-3b^{2}-1,\lambda_{2}=-3\frac{\Delta-1}{\Delta-2}(b^{2}-1)$).
Taking $\Delta<1$ and $b^{2}<1$(which both are favored by literature), $P_{2}$
is a saddle fixed point. Thus the universe will have a transition to an other
available fixed point. In this era the deceleration parameter is still
positive and has a same form$q=-1+\frac{3(b^{2}-1)}{\Delta-2}$ as the $P_{1}$.
At this point $w_{eff}=\Omega_{D}w_{D}\approx 0^{-}$ which corresponds to a
matter dominated epoch.

Last fixed point($P_{3}:\Omega_{m}=b^{2},\Omega_{D}=-b^{2}+1$) is evidently a
late time DE dominated attractor. According to eigenvalues,$\lambda_{1}=-4$
and $\lambda_{2}=3\frac{\Delta-1}{\Delta-b^{2}-1}(b^{2}-1)$, this epoch is
stable for favored ranges of free parameters $0<\Delta<1$ and $b^{2}<1$. In
this era the universe undergoes an accelerated expansion($q=-1(w_{eff}=-1)$)
irrespective of the free parameters. For completeness we plotted the phase
space diagram of ($\Omega_{m},\Omega_{D}$) in fig.(1).

Figure 1: Phase space evolution of the BGDE model. We set $b^{2}=0.01$ and
take $\Delta=0.1$. The solid red line identifies the universe with
$\Omega_{D0}=0.69$ and $\Omega_{m0}=0.30996$

###  3.2 Numerical demonstration of the BGDE

According to the new range of free parameters$(\Delta,b)$, obtained from
previous task, we depict $\Omega_{D}$, $w_{D}$ and $q$ in Figs. 2-4. In these
figures we set $\Omega_{D0}=0.69$. According to Fig. 2 we observe that for
BGDE at early universe $\Omega_{D}\rightarrow 0$ while the model predicts a
phase of totally DE dominated ($\Omega_{D}\rightarrow 1$) in the future. This
figure also reveals that larger choices of $\Delta$ leaves a larger
contribution of GDE at early epochs. Lower panel of Fig. 2 shows the impact of
adding an interaction to evolution of $\Omega_{D}$. Fig.3 displays evolution
of the $w_{D}$. This figure implies that increasing $\Delta$ significantly
affects the behavior of the GDE, specially at higher redshift. Lower panel of
this figure states that introducing an interaction term in the Barrow
cosmology leads the GDE model to cross the phantom line($w=-1$), favored by
observations, at late epochs while this behavior is absent in non-interacting
GDE case in Barrow cosmology signaling that the fate of universe will end with
a big rip singularity. This behavior is similar to what already presented in
the literature for GDE in standard cosmology[61, 60]. It could be seen that
$w_{D}$ will almost be independent of $\Delta$ at future approaches a same
limit $w_{D}<-1$ depending on $b^{2}$.

Figure 2: Evolution of $\Omega_{D}$ parameter for GDE in Barrow cosmology. In
upper panel we set $b^{2}=0.02$ while in lower panel we take $\Delta=0.1$.

Figure 3: Evolution of $w_{D}$ parameter for GDE in Barrow cosmology. In upper
panel we set $b^{2}=0.02$ while in lower panel we take $\Delta=0.1$.

Figure 4: Evolution of $q$ parameter for GDE in Barrow cosmology. In upper
panel we set $b^{2}=0.02$ while in lower panel we take $\Delta=0.1$.

Figure 4, shows that introducing non-extensive behavior to the GDE leads a
delay in the time of transition from deceleration to acceleration($z_{t}$).
According to this figure the GDE in barrow cosmology with $\Delta=0.1$ and
$b^{2}=0$ has $z_{t}=0.24$ while taking an interaction term with $b^{2}=0.03$
leads $z_{t}=0.50$. Thus one can conclude that an interacting Barrow GDE is
more favored by observations[87].

In facing any model of DE we expect a stable and clustered universe at present
epoch. Squared sound speed($v^{2}_{s}$), is a key quantity which reveals
signals of instability against perturbations in a semi-Newtonian approach. In
a short wavelength limit the perturbations in the cosmic background could be
as

$\delta\propto e^{\pm i\omega t},$ (33)

where $\omega\propto v_{s}$. The negative $v_{s}^{2}$, can lead to a growing
mode of perturbations and make the cosmic fluid unstable. However, the
positive sign of the $v_{s}^{2}$ has an oscillating behavior which implies a
stable behavior against perturbations. So the sign of the squared sound speed
is of special importance in the stability issue. The definition of the
adiabatic squared sound speed reads

$v_{s}^{2}=\frac{dP}{d\rho}.$ (34)

In order to obtain the squared sound speed in a background filled with matter
and DE components the definition could be written as [67]

$v_{s}^{2}=\frac{dP}{d\rho}=\frac{\dot{P}}{\dot{\rho}}=\frac{\dot{w}_{D}\rho_{D}+w_{D}\dot{\rho}_{D}}{\dot{\rho}_{D}+\dot{\rho}_{m}}.$
(35)

After a little algebra and using Eqs.(29), (24) we reach

$v^{2}_{s}=\frac{(1-\Delta)\Omega^{2}_{D}+(b^{2}(3-2\Delta)+\Delta-1)\Omega_{D}-b^{2}(2-\Delta)^{2}}{(\Omega_{D}-2+\Delta)^{2}}.$
(36)

Figure 5: The chance of stability against perturbations is discussed through
plots of $v^{2}_{s}$ versus $z$. In upper panel we set $b^{2}=0.02$ while in
lower panel we take $\Delta=0.1$.

It is worthy to note that above equation reduces to the corresponding relation
in the standard GR cosmology for $\Delta=0$[60]. we plotted the squared sound
speed for different choices of $\Delta$ and $b^{2}$ in Fig.5. It is clear that
Barrow ghost dark energy(BGDE) is unstable against perturbations for all
choices of $\Delta$ and $b^{2}$. However it can be seen from lower panel of
Fig.5 that for $\Delta=0.1$ and $b^{2}=0$, the model meets the borderline at
future. In next step we consider the so-called statefinder analysis in the
case of BGDE to find the evolutionary status of the model with respect to the
$\Lambda$CDM. The statefinder analysis introduced for first time in [88],
where a pair of dimensionless, geometrical parameters,{r,s} presented to
discriminate different DE models. These pair of parameters read

$\displaystyle r$ $\displaystyle=$
$\displaystyle\frac{\dddot{a}}{aH^{3}}=2q^{2}+q-q^{{}^{\prime}},$ (37)
$\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{r-1}{3(q-\frac{1}{2})}.$

labelrs where these parameters are background parameters since they are
constructed from scale factor ($a$) and its temporal derivatives. For
$\Lambda$CDM, in a flat background the statefinder pair take a fixed value {r,
s} = {1, 0}. Using above definitions we obtain these parameters in the case of
our study. The results are

$\displaystyle r$ $\displaystyle=$
$\displaystyle-\frac{1}{(\Omega_{D}-2+\Delta)^{3}}\times$
$\displaystyle\big{[}$ $\displaystyle
9\big{(}-\frac{\Delta^{3}}{9}+\big{(}(1-\Omega_{D})b^{2}-\Omega^{2}_{D}+\frac{1}{3}(5\Omega_{D}-1)\big{)}\Delta^{2}$
$\displaystyle+$
$\displaystyle\big{(}\big{(}b^{2}+\frac{5}{3}\big{)}\Omega^{2}_{D}+\big{(}b^{4}-b^{2}-\frac{8}{3}\big{)}\Omega_{D}-2b^{4}+\frac{2}{3}\big{)}\Delta$
$\displaystyle-$
$\displaystyle\frac{10\Omega^{3}_{D}}{9}+\big{(}-4b^{2}+\frac{8}{3}\big{)}\Omega^{2}_{D}+\big{(}-3b^{4}+8b^{2}-\frac{7}{3}\big{)}\Omega_{D}$
$\displaystyle+$ $\displaystyle 4b^{2}(b^{2}-1)+\frac{8}{9}\big{)}\big{]}$

$\displaystyle
s=\frac{-2\left(b^{2}+\Omega_{D}-1\right)}{\left(2b^{2}-\Delta+\Omega_{D}\right)\left(\Omega_{D}-2+\Delta\right)^{2}}\times$
(39) $\displaystyle\left(-\Omega^{2}_{D}+\left(\Delta
b^{2}-\Delta^{2}-3b^{2}+2\Delta+1\right)\Omega_{D}-2\left(-2+\Delta\right)\left(b^{2}-\frac{\Delta}{2}\right)\right)$

Figure 6: Evolution of $r$ parameter for different choices of $\Delta$ and
$b^{2}$.

Figure 7: Evolution of $s$ parameter for different choices of $\Delta$ and
$b^{2}$.

Figure 8: The pattern of $\\{s,r\\}$ for different choices of $\Delta$ and
$b^{2}$.

Evolution of the statefinder parameters for the BGDE is plotted against $z$ in
Figs.6-8. We find from these figures that in all choices of $b^{2}$ and
$\Delta$ the BGDE behaves evidently distinct from the $\Lambda-CDM$. However
in future BGDE catches $\\{r=1,s=0\\}$ in all cases of interest. This analysis
reveals that BGDE starts from a matter dominated region while enters the realm
of standard cosmology in the future.

##  4 Ghost dark energy in Tsallis cosmology

In this section we would like to consider cosmological implications of TGDE.
In this case the modified Friedmann equation reads

$\left(H^{2}+\frac{k}{a^{2}}\right)^{2-\beta}=\frac{8\pi l^{2}_{P}}{3}\rho,$
(40)

where $\beta$ is the non-extensive Tsallis parameter, which is a real
parameter responsible of quantifying the degree of non-extensivity. Necessity
of $\gamma>0$ implies that $\beta<2$ [38]. Here, we work in a flat spacetime
and thus the Friedmann equation could be written as

$H^{4-2\beta}=\frac{8\pi l^{2}_{P}}{3}\rho.$ (41)

Our universe is assumed to be filled with pressureless matter and the GDE
components ($\rho=\rho_{m}+\rho_{D}$). Once again we use a same set of
continuity equations (19). Using Eqs. (19) and (40) and following same steps
as the previous section, one can obtain the EoS for the TGDE as

$w_{D}=\frac{\left(3-2\beta\right)\Omega_{D}-2b^{2}\left(-2+\beta\right)}{\Omega_{D}\left(\Omega_{D}-4+2\beta\right)},$
(42)

where the interaction term in continuity equation is taken of the form
$Q=3b^{2}H(\rho_{D}+\rho_{m})$. It is also a matter of calculation to derive
the evolutionary form of $\Omega_{D}$ as well as the deceleration parameter
$q$. The results are

$\Omega^{\prime}_{D}=-\frac{3\Omega_{D}\left(b^{2}+\Omega_{D}-1\right)\left(-3+2\beta\right)}{\Omega_{D}-4+2\beta},$
(43) $q=\frac{3b^{2}+2\Omega_{D}-2\beta+1}{\Omega_{D}-4+2\beta}.$ (44)

It is worth mentioning that when $\beta\rightarrow 1$, in both interacting
($b\neq 0$) and non-interacting ($b=0$) cases, the above equations reduce to
the corresponding equations in standard cosmology [60].

###  4.1 Dynamical system analysis in Tsallis cosmology

Here, as in the BGDE, we at first perform a same dynamical analysis in the
case of Tsallis cosmology. The corresponding autonomous set of equations in
this case read

$\Omega^{\prime}_{D}=\frac{g_{1dt}\Omega_{\mathrm{D}}^{2}+g_{2dt}\Omega_{\mathrm{D}}}{\Omega_{\mathrm{D}}-4+2\beta},$
(45)

$\Omega^{\prime}_{m}=\frac{g_{1mt}\Omega_{m}^{2}+g_{2mt}\Omega_{m}+6b^{2}\left(\frac{\Omega_{\mathrm{D}}}{2}-2+\beta\right)\Omega_{\mathrm{D}}}{\Omega_{\mathrm{D}}-4+2\beta},$
(46)

where

$\displaystyle g_{1dt}$ $\displaystyle=$
$\displaystyle-6b^{2}\beta+9b^{2}-8\beta+13,$ $\displaystyle g_{2dt}$
$\displaystyle=$
$\displaystyle\left(-6b^{2}\Omega_{m}-2\Omega_{m}+8\right)\beta+9b^{2}\Omega_{m}+4\Omega_{m}-13,$
$\displaystyle g_{1mt}$ $\displaystyle=$
$\displaystyle-6\left(b^{2}+\frac{1}{3}\right)\left(\beta-2\right).$
$\displaystyle g_{2mt}$ $\displaystyle=$
$\displaystyle\left(\left(-6\beta+15\right)\Omega_{\mathrm{D}}+6\beta-12\right)b^{2}$
$\displaystyle+$
$\displaystyle\left(-8\beta+13\right)\Omega_{\mathrm{D}}+2\beta-4.$

The admitted fixed points in this case are listed in Tab.(2). The interesting
point is that not in BGDE nor in TGDE the Barrow and Tsallis parameters
($\Delta$ and $\beta$) does not affect the number and existence of the fixed
points. However, a closer look reveals the impacts of the models parameters on
the stability and consequently on destination of the universe.

fixed point | $\Omega_{D}$ | $\Omega_{m}$ | $q$ | Nature
---|---|---|---|---
$P_{1}$ | 0 | 0 | $>0$ | saddle for $\beta>\frac{13}{8}$
|  |  |  | unstable for $\beta<\frac{13}{8}$
$P_{2}$ | 0 | 1 | $>0$ | saddle for $\beta<\frac{3}{2}$
|  |  |  | stable for $\frac{3}{2}<\beta<2$
$P_{3}$ | $-b^{2}+1$ | $b^{2}$ | $<0$ | stable for $\beta<\frac{3}{2}+\frac{1+b^{2}}{8}$
Table 2: The admitted fixed points of TGDE in presence of an interaction
term($Q=3b^{2}H(\rho_{D}+\rho_{m})$). One should note that in this table we
assumed $\beta<2$ according to [38] Figure 9: Phase space evolution of the
TGDE model. We set $b^{2}=0.01$ and take $\beta=1.1$. The solid red line
identifies the universe with $\Omega_{D0}=0.69$ and $\Omega_{m0}=0.30996$

Once again the evolution pattern of the system starts from
$P_{1}:\Omega_{\mathrm{D}}=0,\Omega_{m}=0$ which corresponds to a radiation
dominated unstable phase of decelerated expansion. $\frac{13}{8}<\beta<2$
leads to a saddle point while for $\beta<\frac{13}{8}$, $P_{1}$ is an unstable
point. In both cases the system at this point shows a transient character. At
this point the effective EoS parameter and deceleration parameter are
$w_{eff}=-b^{2}$ and $q=\frac{3b^{2}-2\beta+1}{-4+2\beta}$.

The second fixed point($P_{2}:\Omega_{D}=0,\Omega_{m}=1$) provides a matter
dominated epoch. The stability analysis
($\lambda_{1}=-3b^{2}-1,\lambda_{2}=-\frac{3}{2}\frac{2\beta-3}{\beta-2}(b^{2}-1)$)
reveals that for $\frac{3}{2}<\beta<2$ and $b^{2}<1$, this point has an stable
nature and the universe will not transits to a future attractor. However if
$\beta<3/2$, then this point will be a saddle point which has an unstable
character and leaves the chance of transition to future attractor. Thus
necessity of transition from a DM to DE dominated epoch puts a new constraint
on $\beta$. However one should note that this constraint is obtained in the
case $Q=3b^{2}H(\rho_{D}+\rho_{m})$.
$P_{3}:\Omega_{D}=-b^{2}+1,\Omega_{m}=b^{2}$ is another fixed point which
corresponds to a DE dominated epoch. Due to briefness we do not mention the
exact form of eigenvalues in this case. The eigenvalues show that this point
could be stable for $\beta<(\frac{3}{2}+\frac{1+b^{2}}{8})$ which this
condition is already satisfied with the unstability condition of $P_{2}$. Thus
if the model evolves in to a DE dominated epoch it would be stable at this
point. At this stable point one can obtain $w_{eff}=-1=q$ which predicts a
future big rip singularity as destination of the universe. To see the validity
of above description we plotted phase space diagram in the TGDE case in
Fig.(9).

As result we found that for GDE in Barrow and Tsallis cosmology the system
could result in a correct timeline order of evolution with suitable choice of
free parameters $\delta$ and $\beta$. The resulting condition($\beta<3/2$)
from dynamical system analysis belongs to the allowed region($\beta<2$),
mentioned in the literature.

###  4.2 Numerical description of TGDE

Figure 10: Evolution of $\Omega_{D}$ parameter for GDE in Tsallis cosmology.
In upper panel we set $b^{2}=0.02$ while in lower panel we take $\beta=0.9$.

In this section we also discuss a numerical description of the TGDE. To this
end, we plot the figures in allowed region of free parameters obtained from
the dynamical system analysis. According to Fig. 10, TGDE contribution to the
cosmic content at present and future is not very sensitive to $\beta$. However
larger values of $\beta$, result in a more contribution at early epochs of
universe. This result maybe could taken as an early DE component which is
interesting in curing the cosmic tensions. On the other side lower part of
Fig.10 shows that increasing $b^{2}$ results in larger values of $\Omega_{D}$
at far future.

Figure 11: Evolution of $w_{D}$ parameter for GDE in Tsallis cosmology. In
upper panel we set $b^{2}=0.02$ while in lower panel we take $\beta=0.9$.

Fig. 11 indicates that with increasing the non-extensive nature (increasing
$\beta$) of the model decreases the value of $w_{D}$. Besides introducing
interaction between dark components, $w_{D}$ will cross the phantom line,
indicating a future big rip singularity fate of universe, which is similar to
the GDE behavior in standard Friedmann cosmology[67].

Figure 12: Evolution of $q$ parameter for GDE in Tsallis cosmology. In upper
panel we set $b^{2}=0.02$ while in lower panel we take $\beta=0.9$.

The behavior of $q$ is depicted against $z$ in Fig.12. According to the upper
panel of this figure, transition to an accelerated expansion could happen
during $0\leq z\leq 1$. Since the transition to a acceleration phase according
to observation should happen around $z\sim 0.6$ [89, 90] thus the non-
extensive parameter maybe should be close to unity. From lower panel one finds
that increasing $b^{2}$ leads to an earlier phase transition from deceleration
to acceleration. In this figure the phase transition occurs at the
observational favored interval $0.7\leq z\leq 1$ for $\beta=0.9$ and
$b^{2}\in[0,0.03]$. One can also find in both panels of the Fig.12 that the
deceleration parameter approaches $q=-1$ which resembles a phase of de-Sitter
expansion.

We also would like to explore the stability of the GDE in Tsallis cosmology.
The squared sound speed $(v^{2}_{s})$ in this case in presence of an
interaction between dark components read

$\displaystyle v^{2}_{s}=\frac{1}{\left(\Omega_{D}-4+2\beta\right)^{2}}\times$
(47)
$\displaystyle[\left(3-2\beta\right)\Omega^{2}_{D}+\left(-4b^{2}\beta+7b^{2}+2\beta-3\right)\Omega_{D}$
$\displaystyle-4b^{2}\left(-2+\beta\right)^{2}].$

Figure 13: Evolution of $v^{2}_{s}$ parameter for GDE in Tsallis cosmology. We
set $b^{2}=0.02$ in upper panel and take $\beta=0.9$ in the other panel.

In Figs.13 we depicted $v^{2}_{s}$ for different choices of $\beta$ and the
coupling constant $b^{2}$. This figure reveals that increasing $\beta$(or
$b^{2}$) leads more negative values for $v^{2}_{s}$. Further this figure shows
that altering the non-extensive parameter $\beta$ can not change the unstable
nature of the model and the TGDE is still suffering the stability issue.

Figure 14: In this figure we depicted the evolution of $r$ parameter against
redshift. In upper panel we set $b^{2}=0.02$ and in the lower part we take
$\beta=0.9$.

Figure 15: In this figure we depicted the evolution of $s$ parameter against
redshift. In upper panel we set $b^{2}=0.02$ and in the lower part we take
$\beta=0.9$.

Figure 16: In this figure we demonstrated the $\\{r,s\\}$ evolution. In upper
panel we set $b^{2}=0.02$ and in the lower part we take $\beta=0.9$ .

Next we explore the statefinder analysis for this case. To this end we plotted
r and s against redshift in Figs.14-16. According to these figures we find
that the model can catch the point ${1,0}$ in the future while at present time
the model have a significant distance from the $\Lambda-CDM$. Looking at $r-z$
and $s-z$ diagram, one finds that evolutionary behavior of the model is
evidently sensitive to $\beta$($b^{2}$) parameter specially at past epochs
while at present and future era this sensitivity fades and all models with
different values of $\beta$ ($b^{2}$) approaches a same pattern.

##  5 Growth of Density Perturbations in Barrow and Tsallis cosmology

Evolution of density perturbations is of special interest in any modified
cosmology and the issue deserves a full covariant study. Since covariant
perturbation theory is beyond the scope of this work, we just present a semi-
Newotonian approach here. On this way we follow the Jeans analysis in a two
component(GDE and CDM) universe[91]. Here, we take the governing equation of
perturbation in a flat FRW universe as[91]

$\ddot{\delta}+2H\dot{\delta}+\left(\frac{v_{s}^{2}k^{2}}{a^{2}}-4\pi
G\rho\right)\delta=0,$ (48)

where $\delta=\frac{\delta\rho}{\rho}$ is density contrast,
$v_{s}^{2}=\frac{\partial P}{\partial\rho}$(where P and $\rho$ are cosmic
fluids pressure and energy density respectively) and $k$ denotes the comoving
wave number. Here, we consider evolution of perturbations in large scales and
ignore $k^{2}$ term(which belongs to kinetic pressure) in above equation.
Beside, we need to use the first Friedmann equation in Barrow or Tsallis
cosmology to disclose the effect of GDE and non-extensive nature of the model
with respect to standard cosmology. To make the system of equations
consistent, we also bring the evolution equation of the GDE component. We
recast all of these equations versus the redshift parameter $z$. The resulting
equations in Barrow cosmology read

$\displaystyle H^{2}$
$\displaystyle(1+z)^{2}\delta^{\prime\prime}+((1+z)\frac{H^{\prime}}{H}-1)H^{2}(1+z)\delta^{\prime}$
(49) $\displaystyle-$ $\displaystyle\frac{3}{2}H^{2-\Delta}=0,$ $\displaystyle
H^{\prime}$ $\displaystyle=\frac{3H}{(1+z)(2-\Delta)}(1+\Omega_{D}w_{D}),$
(50) $\displaystyle\Omega^{\prime}$
$\displaystyle=-\frac{(1-\Delta)}{(1+z)(\Omega_{D}-2+\Delta)}3\Omega_{D}(b^{2}+\Omega_{D}-1),$
(51)

where prime denotes derivatives with respect to $z$ and $w_{D}$ could be
replaced from corresponding Eqs.(26,42). Unfortunately an analytic solution
for this system of equations is not available and we ought to try a numerical
task in this case. On this way we take
$\Omega_{D}(0)=0.69,H(0)=68\frac{km}{Mpcs},b^{2}=0.01$ and
$\delta(100)=0.0001$. A same process is followed for the Tsallis case and the
results are depicted against redshift in Fig.(17). The figure reveals that the
density contrast $\delta$ is sensitive to the non-extensive parameter $\Delta$
and $\beta$. The figure reveals that in both TGDE and BGDE the structure
formation has a slower rate with respect to the GDE in standard cosmology.
Thus we can conclude that the non-extensivity effect can slow the structure
formation process.

Figure 17: Evolution of the density contrast($\delta$) is depicted against
redshift for both BGDE and TGDE. In this figure we set $b^{2}=0.01$ for both
models and also assumed $\delta(100)=0.0001$.

##  6 Closing remarks

Based on the correspondence between the laws of thermodynamics and the
gravitational field equations and using the modified Barrow/Tsallis entropy
expression, we have explored the cosmological consequences of the modified
Friedmann equations for a universe filled with pressureless matter and DE.
Here, we have considered the GDE which has been widely investigated in the
literature. At first, we performed a dynamical system analysis to see if the
model can present a correct chronological evolution and found that the TGDE
and BGDE can exhibit such a result for suitable range of free
parameters($0<\Delta<1$ for BGDE and $\beta<3/2$ for TGDE). One important
achievement of this analysis with respect to the GDE in standard cosmology, is
appearance of a radiation dominated phase of expansion for the choice
$Q=3b^{2}H(\rho_{D}+\rho_{m})$ of interaction term. One should note that this
phase of expansion is not available for GDE in standard cosmology[86]. It is
also worthy to mention that higher limit Tsallis parameter($\beta<3/2$), is a
result of seeking an unstable phase of matter dominated in TGDE. Next, we have
explored the impacts of Barrow/Tsallis parameters on the evolutionary
description of the cosmological parameters through a numerical task. A close
look at $w_{D}$, $q$ and $\Omega_{D}$ reveals that for both BGDE and TGDE
models the transition from a deceleration to an accelerated phase can be
achieved during the history of the universe. In Barrow/ Tsallis cosmology,
larger values of $\Delta(\beta)$ result in a delay in the time of
transition($z_{tr}$). We observed that in both BGDE and TGDE, in the non-
interacting regime, the models lie in the quintessence region while adding a
chance of interaction lead the models to cross the phantom line. We also
considered the models in presence of an interaction term between DM and the
GDE. We observed that in both models increasing the $b^{2}$ leads to and
earlier transition to the accelerated expansion. The squared sound speed
analysis reveals that in BGDE(TGDE), the model is unstable against
perturbations and changing the non-extensive parameter $\Delta(\beta)$ can not
survive the model from instability issue. We see a same result for increasing
$b^{2}$ in both models. It is worthy to mention that in the non-interacting
case the models of interest tend to $v^{2}_{s}=0$ which is the border line of
stability at far future.

We have also considered the so-called statefinder parameters $\\{r,s\\}$ which
are based on higher derivatives of the scale factor and is a suitable tool in
discriminating DE models specially with respect to $\Lambda CDM$. In both
underlying models we found out that the models are significantly distinct from
$\Lambda CDM$ at high redshift epochs. Besides, comparing the figures, $r-z$
and $s-z$, of both models reveal that these models show a significantly
different patterns of evolution. Moreover, we saw that both BGDE and TGDE will
catch the point ${r=1,s=0}$.

We have also explored the impacts of the non-extensive parameters on the
density perturbations through a Jeans analysis and found the pattern of
density contrast($\delta$) evolution in TGDE/BGDE which differs from that of
GDE in standard cosmology. We observed that the non-extensivity make a
dilation in the structure formation process.

## References

  * [1] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009–1038.  arXiv:astro-ph/9805201, doi:10.1086/300499.
  * [2] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, C. J. Hogan, S. Jha, R. P. Kirshner, et al., ”observational evidence from supernovae for an accelerating universe and a cosmological constant”, The Astronomical Journal 116 (3) (1998) 1009.
  * [3] S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae, Astrophys. J. 517 (1999) 565–586.  arXiv:astro-ph/9812133, doi:10.1086/307221.
  * [4] D. N. Spergel, et al., First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters, Astrophys. J. Suppl. 148 (2003) 175–194.  arXiv:astro-ph/0302209, doi:10.1086/377226.
  * [5] S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D 11 (2002) 483–492.  arXiv:gr-qc/0201033, doi:10.1142/S0218271802002025.
  * [6] Y. Sobouti, An f(r) gravitation instead of dark matter, Astron. Astrophys. 464 (2007) 921, [Erratum: Astron.Astrophys. 472, 833 (2007)].  arXiv:0704.3345, doi:10.1051/0004-6361:20077452.
  * [7] T. P. Sotiriou, V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82 (2010) 451–497.  arXiv:0805.1726, doi:10.1103/RevModPhys.82.451.
  * [8] C. G. Boehmer, T. Harko, F. S. N. Lobo, Dark matter as a geometric effect in f(R) gravity, Astropart. Phys. 29 (2008) 386–392.  arXiv:0709.0046, doi:10.1016/j.astropartphys.2008.04.003.
  * [9] G. Cognola, E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastiani, S. Zerbini, A Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion, Phys. Rev. D 77 (2008) 046009.  arXiv:0712.4017, doi:10.1103/PhysRevD.77.046009.
  * [10] S. Nojiri, S. D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505 (2011) 59–144.  arXiv:1011.0544, doi:10.1016/j.physrep.2011.04.001.
  * [11] S. Capozziello, M. De Laurentis, Extended Theories of Gravity, Phys. Rept. 509 (2011) 167–321.  arXiv:1108.6266, doi:10.1016/j.physrep.2011.09.003.
  * [12] S. D. Odintsov, V. K. Oikonomou, Unification of Inflation with Dark Energy in $f(R)$ Gravity and Axion Dark Matter, Phys. Rev. D 99 (10) (2019) 104070.  arXiv:1905.03496, doi:10.1103/PhysRevD.99.104070.
  * [13] S. D. Odintsov, V. K. Oikonomou, $f(R)$ Gravity Inflation with String-Corrected Axion Dark Matter, Phys. Rev. D 99 (6) (2019) 064049.  arXiv:1901.05363, doi:10.1103/PhysRevD.99.064049.
  * [14] C. Lanczos, Elektromagnetismus als natürliche eigenschaft der riemannschen geometrie, Zeitschrift für Physik 73 (1932) 147–168.
URL https://api.semanticscholar.org/CorpusID:124628257

  * [15] C. Lanczos, A remarkable property of the riemann-christoffel tensor in four dimensions, Annals of Mathematics 39 (1938) 842–850.
URL https://api.semanticscholar.org/CorpusID:124477207

  * [16] M. H. Dehghani, Accelerated expansion of the Universe in Gauss-Bonnet gravity, Phys. Rev. D 70 (2004) 064009.  arXiv:hep-th/0404118, doi:10.1103/PhysRevD.70.064009.
  * [17] D. Lovelock, Divergence-free tensorial concomitants, aequationes mathematicae 4 (1969) 127–138.
URL https://api.semanticscholar.org/CorpusID:189832116

  * [18] T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260–1263.  arXiv:gr-qc/9504004, doi:10.1103/PhysRevLett.75.1260.
  * [19] E. P. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 04 (2011) 029.  arXiv:1001.0785, doi:10.1007/JHEP04(2011)029.
  * [20] R.-G. Cai, L.-M. Cao, N. Ohta, Friedmann Equations from Entropic Force, Phys. Rev. D 81 (2010) 061501.  arXiv:1001.3470, doi:10.1103/PhysRevD.81.061501.
  * [21] R.-G. Cai, L.-M. Cao, N. Ohta, Notes on Entropy Force in General Spherically Symmetric Spacetimes, Phys. Rev. D 81 (2010) 084012.  arXiv:1002.1136, doi:10.1103/PhysRevD.81.084012.
  * [22] R. Banerjee, B. R. Majhi, Statistical Origin of Gravity, Phys. Rev. D 81 (2010) 124006.  arXiv:1003.2312, doi:10.1103/PhysRevD.81.124006.
  * [23] Y.-X. Liu, Y.-Q. Wang, S.-W. Wei, A Note on Temperature and Energy of 4-dimensional Black Holes from Entropic Force, Class. Quant. Grav. 27 (2010) 185002\.  arXiv:1002.1062, doi:10.1088/0264-9381/27/18/185002.
  * [24] A. Sheykhi, Entropic Corrections to Friedmann Equations, Phys. Rev. D 81 (2010) 104011.  arXiv:1004.0627, doi:10.1103/PhysRevD.81.104011.
  * [25] L. Modesto, A. Randono, Entropic Corrections to Newton’s Law (3 2010).  arXiv:1003.1998.
  * [26] Y.-F. Cai, J. Liu, H. Li, Entropic cosmology: a unified model of inflation and late-time acceleration, Phys. Lett. B 690 (2010) 213–219.  arXiv:1003.4526, doi:10.1016/j.physletb.2010.05.033.
  * [27] S. H. Hendi, A. Sheykhi, Entropic Corrections to Einstein Equations, Phys. Rev. D 83 (2011) 084012.  arXiv:1012.0381, doi:10.1103/PhysRevD.83.084012.
  * [28] A. Sheykhi, S. H. Hendi, Power-Law Entropic Corrections to Newton’s Law and Friedmann Equations, Phys. Rev. D 84 (2011) 044023.  arXiv:1011.0676, doi:10.1103/PhysRevD.84.044023.
  * [29] C. Tsallis, Possible Generalization of Boltzmann-Gibbs Statistics, J. Statist. Phys. 52 (1988) 479–487.  doi:10.1007/BF01016429.
  * [30] M. L. Lyra, C. Tsallis, Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems, Phys. Rev. Lett. 80 (1998) 53–56.  arXiv:cond-mat/9709226, doi:10.1103/PhysRevLett.80.53.
  * [31] G. Wilk, Z. Wlodarczyk, On the interpretation of nonextensive parameter q in Tsallis statistics and Levy distributions, Phys. Rev. Lett. 84 (2000) 2770.  arXiv:hep-ph/9908459, doi:10.1103/PhysRevLett.84.2770.
  * [32] C. Tsallis, L. J. L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 (2013) 2487.  arXiv:1202.2154, doi:10.1140/epjc/s10052-013-2487-6.
  * [33] M. Tavayef, A. Sheykhi, K. Bamba, H. Moradpour, Tsallis Holographic Dark Energy, Phys. Lett. B 781 (2018) 195–200.  arXiv:1804.02983, doi:10.1016/j.physletb.2018.04.001.
  * [34] M. A. Zadeh, A. Sheykhi, H. Moradpour, K. Bamba, Note on Tsallis holographic dark energy, Eur. Phys. J. C 78 (11) (2018) 940.  arXiv:1806.07285, doi:10.1140/epjc/s10052-018-6427-3.
  * [35] M. Abdollahi Zadeh, A. Sheykhi, H. Moradpour, Thermal stability of Tsallis holographic dark energy in nonflat universe, Gen. Rel. Grav. 51 (1) (2019) 12\.  doi:10.1007/s10714-018-2497-7.
  * [36] B. D. Pandey, S. K. P, Pankaj, U. K. Sharma, New Tsallis holographic dark energy, Eur. Phys. J. C 82 (3) (2022) 233.  arXiv:2110.13628, doi:10.1140/epjc/s10052-022-10171-w.
  * [37] Q. Huang, H. Huang, J. Chen, L. Zhang, F. Tu, Stability analysis of a Tsallis holographic dark energy model, Class. Quant. Grav. 36 (17) (2019) 175001.  arXiv:2201.12504, doi:10.1088/1361-6382/ab3504.
  * [38] A. Sheykhi, Modified Friedmann Equations from Tsallis Entropy, Phys. Lett. B 785 (2018) 118–126.  arXiv:1806.03996, doi:10.1016/j.physletb.2018.08.036.
  * [39] M. Asghari, A. Sheykhi, Observational constraints on Tsallis modified gravity, Mon. Not. Roy. Astron. Soc. 508 (2) (2021) 2855–2861.  arXiv:2106.15551, doi:10.1093/mnras/stab2671.
  * [40] A. Sheykhi, B. Farsi, Growth of perturbations in Tsallis and Barrow cosmology, Eur. Phys. J. C 82 (12) (2022) 1111.  arXiv:2205.04138, doi:10.1140/epjc/s10052-022-11044-y.
  * [41] S. Basilakos, A. Lymperis, M. Petronikolou, E. N. Saridakis, Alleviating both $H_{0}$ and $\sigma_{8}$ tensions in Tsallis cosmology (8 2023).  arXiv:2308.01200.
  * [42] G. G. Luciano, J. Gine, Baryogenesis in non-extensive Tsallis Cosmology, Phys. Lett. B 833 (2022) 137352.  arXiv:2204.02723, doi:10.1016/j.physletb.2022.137352.
  * [43] J. D. Barrow, The Area of a Rough Black Hole, Phys. Lett. B 808 (2020) 135643\.  arXiv:2004.09444, doi:10.1016/j.physletb.2020.135643.
  * [44] A. Sheykhi, Barrow Entropy Corrections to Friedmann Equations, Phys. Rev. D 103 (12) (2021) 123503.  arXiv:2102.06550, doi:10.1103/PhysRevD.103.123503.
  * [45] E. N. Saridakis, Modified cosmology through spacetime thermodynamics and Barrow horizon entropy, JCAP 07 (2020) 031.  arXiv:2006.01105, doi:10.1088/1475-7516/2020/07/031.
  * [46] E. N. Saridakis, S. Basilakos, The generalized second law of thermodynamics with Barrow entropy, Eur. Phys. J. C 81 (7) (2021) 644.  arXiv:2005.08258, doi:10.1140/epjc/s10052-021-09431-y.
  * [47] E. N. Saridakis, Barrow holographic dark energy, Phys. Rev. D 102 (12) (2020) 123525\.  arXiv:2005.04115, doi:10.1103/PhysRevD.102.123525.
  * [48] S. Srivastava, U. K. Sharma, Barrow holographic dark energy with Hubble horizon as IR cutoff, Int. J. Geom. Meth. Mod. Phys. 18 (01) (2021) 2150014.  arXiv:2010.09439, doi:10.1142/S0219887821500146.
  * [49] P. Adhikary, S. Das, S. Basilakos, E. N. Saridakis, Barrow holographic dark energy in a nonflat universe, Phys. Rev. D 104 (12) (2021) 123519.  arXiv:2104.13118, doi:10.1103/PhysRevD.104.123519.
  * [50] A. Oliveros, M. A. Sabogal, M. A. Acero, Barrow holographic dark energy with Granda–Oliveros cutoff, Eur. Phys. J. Plus 137 (7) (2022) 783.  arXiv:2203.14464, doi:10.1140/epjp/s13360-022-02994-z.
  * [51] F. K. Anagnostopoulos, S. Basilakos, E. N. Saridakis, Observational constraints on Barrow holographic dark energy, Eur. Phys. J. C 80 (9) (2020) 826\.  arXiv:2005.10302, doi:10.1140/epjc/s10052-020-8360-5.
  * [52] M. P. Dabrowski, V. Salzano, Geometrical observational bounds on a fractal horizon holographic dark energy, Phys. Rev. D 102 (6) (2020) 064047.  arXiv:2009.08306, doi:10.1103/PhysRevD.102.064047.
  * [53] G. Veneziano, U (1) without instantons, Nuclear Physics B 159 (1-2) (1979) 213–224.
  * [54] E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269–283.  doi:10.1016/0550-3213(79)90031-2.
  * [55] C. Rosenzweig, J. Schechter, C. G. Trahern, Is the Effective Lagrangian for QCD a Sigma Model?, Phys. Rev. D 21 (1980) 3388.  doi:10.1103/PhysRevD.21.3388.
  * [56] P. Nath, R. L. Arnowitt, The U(1) Problem: Current Algebra and the Theta Vacuum, Phys. Rev. D 23 (1981) 473.  doi:10.1103/PhysRevD.23.473.
  * [57] K. Kawarabayashi, N. Ohta, The Problem of $\eta$ in the Large $N$ Limit: Effective Lagrangian Approach, Nucl. Phys. B 175 (1980) 477–492.  doi:10.1016/0550-3213(80)90024-3.
  * [58] N. Ohta, Dark Energy and QCD Ghost, Phys. Lett. B 695 (2011) 41–44.  arXiv:1010.1339, doi:10.1016/j.physletb.2010.11.044.
  * [59] F. R. Urban, A. R. Zhitnitsky, The cosmological constant from the QCD Veneziano ghost, Phys. Lett. B 688 (2010) 9–12.  arXiv:0906.2162, doi:10.1016/j.physletb.2010.03.080.
  * [60] R.-G. Cai, Z.-L. Tuo, H.-B. Zhang, Q. Su, Notes on Ghost Dark Energy, Phys. Rev. D 84 (2011) 123501.  arXiv:1011.3212, doi:10.1103/PhysRevD.84.123501.
  * [61] A. Sheykhi, M. Sadegh Movahed, Interacting Ghost Dark Energy in Non-Flat Universe, Gen. Rel. Grav. 44 (2012) 449–465.  arXiv:1104.4713, doi:10.1007/s10714-011-1286-3.
  * [62] E. Ebrahimi, A. Sheykhi, Instability of QCD ghost dark energy model, Int. J. Mod. Phys. D 20 (2011) 2369–2381.  arXiv:1106.3504, doi:10.1142/S021827181102041X.
  * [63] M. Malekjani, A. Khodam-Mohammadi, Statefinder diagnosis and the interacting ghost model of dark energy, Astrophys. Space Sci. 343 (2013) 451–461.  arXiv:1202.4154, doi:10.1007/s10509-012-1230-3.
  * [64] Y. Liu, Interacting ghost dark energy in complex quintessence theory, Eur. Phys. J. C 80 (12) (2020) 1204.  arXiv:2201.00658, doi:10.1140/epjc/s10052-020-08786-y.
  * [65] A. Sheykhi, M. S. Hamedan, Holographic Dark Energy in Modified Barrow Cosmology, Entropy 25 (4) (2023) 569.  arXiv:2211.00088, doi:10.3390/e25040569.
  * [66] A. Sheykhi, S. Ghaffari, Note on agegraphic dark energy inspired by modified Barrow entropy, Phys. Dark Univ. 41 (2023) 101241.  arXiv:2304.03261, doi:10.1016/j.dark.2023.101241.
  * [67] R.-G. Cai, N. Ohta, Horizon Thermodynamics and Gravitational Field Equations in Horava-Lifshitz Gravity, Phys. Rev. D 81 (2010) 084061.  arXiv:0910.2307, doi:10.1103/PhysRevD.81.084061.
  * [68] S. A. Hayward, Unified first law of black hole dynamics and relativistic thermodynamics, Class. Quant. Grav. 15 (1998) 3147–3162.  arXiv:gr-qc/9710089, doi:10.1088/0264-9381/15/10/017.
  * [69] O. Bertolami, F. Gil Pedro, M. Le Delliou, Dark Energy-Dark Matter Interaction and the Violation of the Equivalence Principle from the Abell Cluster A586, Phys. Lett. B 654 (2007) 165–169.  arXiv:astro-ph/0703462, doi:10.1016/j.physletb.2007.08.046.
  * [70] C. Wetterich, Cosmology and the Fate of Dilatation Symmetry, Nucl. Phys. B 302 (1988) 668–696.  arXiv:1711.03844, doi:10.1016/0550-3213(88)90193-9.
  * [71] L. Amendola, Scaling solutions in general nonminimal coupling theories, Phys. Rev. D 60 (1999) 043501.  arXiv:astro-ph/9904120, doi:10.1103/PhysRevD.60.043501.
  * [72] L. Amendola, D. Tocchini-Valentini, Stationary dark energy: The Present universe as a global attractor, Phys. Rev. D 64 (2001) 043509.  arXiv:astro-ph/0011243, doi:10.1103/PhysRevD.64.043509.
  * [73] W. Zimdahl, D. Pavon, Interacting quintessence, Phys. Lett. B 521 (2001) 133–138.  arXiv:astro-ph/0105479, doi:10.1016/S0370-2693(01)01174-1.
  * [74] W. Zimdahl, D. Pavón, Scaling cosmology, Gen. Rel. Grav. 35 (2003) 413–422.  arXiv:astro-ph/0210484, doi:10.1023/A:1022369800053.
  * [75] L. P. Chimento, A. S. Jakubi, D. Pavon, W. Zimdahl, Interacting quintessence solution to the coincidence problem, Phys. Rev. D 67 (2003) 083513.  arXiv:astro-ph/0303145, doi:10.1103/PhysRevD.67.083513.
  * [76] B. Wang, Y.-g. Gong, E. Abdalla, Transition of the dark energy equation of state in an interacting holographic dark energy model, Phys. Lett. B 624 (2005) 141–146.  arXiv:hep-th/0506069, doi:10.1016/j.physletb.2005.08.008.
  * [77] B. Wang, C.-Y. Lin, E. Abdalla, Constraints on the interacting holographic dark energy model, Phys. Lett. B 637 (2006) 357–361.  arXiv:hep-th/0509107, doi:10.1016/j.physletb.2006.04.009.
  * [78] G. Mangano, G. Miele, V. Pettorino, Coupled quintessence and the coincidence problem, Mod. Phys. Lett. A 18 (2003) 831–842.  arXiv:astro-ph/0212518, doi:10.1142/S0217732303009940.
  * [79] F. Arevalo, A. P. R. Bacalhau, W. Zimdahl, Cosmological dynamics with non-linear interactions, Class. Quant. Grav. 29 (2012) 235001.  arXiv:1112.5095, doi:10.1088/0264-9381/29/23/235001.
  * [80] M. Baldi, Time dependent couplings in the dark sector: from background evolution to nonlinear structure formation, Mon. Not. Roy. Astron. Soc. 411 (2011) 1077.  arXiv:1005.2188, doi:10.1111/j.1365-2966.2010.17758.x.
  * [81] L. Amendola, Coupled quintessence, Phys. Rev. D 62 (2000) 043511.  arXiv:astro-ph/9908023, doi:10.1103/PhysRevD.62.043511.
  * [82] C. Xu, E. N. Saridakis, G. Leon, Phase-space analysis of teleparallel dark energy, Journal of Cosmology and Astroparticle Physics 2012 (07) (2012) 005.
  * [83] R. C. G. Landim, Coupled dark energy: a dynamical analysis with complex scalar field, Eur. Phys. J. C 76 (1) (2016) 31.  arXiv:1507.00902, doi:10.1140/epjc/s10052-016-3894-2.
  * [84] R. C. G. Landim, Coupled tachyonic dark energy: a dynamical analysis, Int. J. Mod. Phys. D 24 (11) (2015) 1550085.  arXiv:1505.03243, doi:10.1142/S0218271815500856.
  * [85] G. Leon, J. Magaña, A. Hernández-Almada, M. A. García-Aspeitia, T. Verdugo, V. Motta, Barrow Entropy Cosmology: an observational approach with a hint of stability analysis, JCAP 12 (12) (2021) 032.  arXiv:2108.10998, doi:10.1088/1475-7516/2021/12/032.
  * [86] H. Golchin, S. Jamali, E. Ebrahimi, Interacting Dark Energy: Dynamical System Analysis, Int. J. Mod. Phys. D 26 (09) (2017) 1750098.  arXiv:1605.05068, doi:10.1142/S0218271817500985.
  * [87] N. Aghanim, et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [Erratum: Astron.Astrophys. 652, C4 (2021)].  arXiv:1807.06209, doi:10.1051/0004-6361/201833910.
  * [88] V. Sahni, T. D. Saini, A. A. Starobinsky, U. Alam, Statefinder: A New geometrical diagnostic of dark energy, JETP Lett. 77 (2003) 201–206.  arXiv:astro-ph/0201498, doi:10.1134/1.1574831.
  * [89] R. A. Daly, S. G. Djorgovski, K. A. Freeman, M. P. Mory, C. P. O’Dea, P. Kharb, S. Baum, Improved Constraints on the Acceleration History of the Universe and the Properties of the Dark Energy, Astrophys. J. 677 (2008) 1–11.  arXiv:0710.5345, doi:10.1086/528837.
  * [90] E. Komatsu, et al., Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys. J. Suppl. 192 (2011) 18\.  arXiv:1001.4538, doi:10.1088/0067-0049/192/2/18.
  * [91] V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, 2005\.