# Growth of perturbations in higher dimensional Gauss-Bonnet FRW cosmology

Ahmad Sheykhi<EMAIL_ADDRESS>Department of Physics, College of
Sciences, Shiraz University, Shiraz 71454, Iran  Biruni Observatory, College
of Sciences, Shiraz University, Shiraz 71454, Iran  Bita Farsi
<EMAIL_ADDRESS>Department of Physics, College of Sciences, Shiraz
University, Shiraz 71454, Iran

###### Abstract

We explore the influences of the higher-order Gauss-Bonnet (GB) correction
terms on the growth of perturbations at the early stage of a
$(n+1)$-dimensional Friedmann-Robertson-Walker (FRW) universe. Considering a
cosmological constant in the FRW background, we study the linear perturbations
by adopting the spherically symmetric collapse (SC) formalism. In light of the
modifications that appear in the field equations, we disclose the role of the
GB coupling constant $\alpha$, as well as the extra dimensions $n>3$ on the
growth of perturbations. It, in essence, is done by defining a dimensionless
parameter $\tilde{\beta}=(n-2)(n-3)\alpha H_{0}^{2}$ in which $H_{0}$ is the
Hubble constant. We find that the matter density contrast starts growing at
the early stages of the universe and, as the universe expands, it grows faster
compared to the standard cosmology. Besides, in the framework of GB gravity,
the growth of matter perturbations in higher dimensions is faster than its
standard counterpart $(n=3)$. Further, in the presence of $\alpha$, the growth
of perturbations increases as it increases. This is an expected result, since
the higher order GB correction terms increase the strength of the gravity and
thus support the growth of perturbations. For the existing cosmological model,
we also investigate the behavior of quantities such as density abundance,
deceleration, and the jerk parameter. Finally, we study the imprint of the GB
parameter and the higher dimensions in the evolution of the mass function of
the dark matter halos.

Gauss-Bonnet gravity; Extra-dimensions; Spherically symmetric collapse; Matter
density contrast; Growth function; Mass function

##  I Introduction

It is a general belief that general relativity (GR) is the most prosperous
gravity theory for describing the physical and cosmic phenomena over a wide
range of energies from large scales to small scales [1]. However, it is not a
perfect theory based on observational limitations and theoretical
considerations. Specifically, two dark clouds of modern physics, dark matter
(DM) [2] and dark energy (DE) [3, 4, 5] can not be well explained in the
framework of GR plus $\Lambda$-cold DM ($\Lambda$CDM). This implies that the
underlying gravity theory governing the gravitational dynamics of the universe
may not be GR and could be an alternative gravitational scenario, which can
help understand the dark sector better at least. This motivates physicists to
pay attention to modified theories of gravity. Although the motivations behind
these alternatives to GR may come from mathematics, philosophy, and
observation, almost all lead to generalizations of Einstein’s theory. Among
all these possibilities Einstein-Gauss-Bonnet (EGB) gravity which initially
proposed by Lanczos [6, 7], and subsequently generalized by Lovelock [8, 9]
has a special place. Lovelock’s theorem, in essence, lets us find all possible
field equations that have consistent conservation and symmetry properties,
just like Einstein’s field equations. Features that make EGB a plausible
candidate for generalizing GR include the ghost-free gravitation propagator
[10], natural generalization with Einstein, and cosmological terms [11]. The
presence of the GB term leads to second-order field equations, which are free
of Ostrogradski instabilities since it is a unique term that is quadratic in
the curvature. It is intriguing to note that EGB theory is backed by a
fundamental theory, namely string theory, meaning that at the classical limit,
Einstein’s equations are subject to leading-order corrections, which typically
arise from higher-order curvature terms in the action. Based on Lovelock’s
theorem, the GB term should appear beyond four dimensions since Einstein’s
equations in four dimensions are the most general set of field equations that
satisfy all conditions proposed in Lovelock’s theorem. It means that to find
the contribution of the GB term in cosmology, one must explore it in higher-
dimensional cosmology 111It is important to note one comment here. Recently,
Glavan and Lin, in an attempt to directly introduce the GB term in four-
dimensional gravity, have proposed a covariant-modified gravity well-known as
4DEGB, which bypasses the requirements of Lovelock’s theorem [12]. Although
our focus throughout this manuscript is on the role of extra dimensions GB, it
can be helpful to mention a few of the research done on some aspects of 4DEGB
theory that in recent years have been investigated in the framework of
cosmology [13, 14, 15, 16, 17, 18, 19, 20, 21] and the black hole (compact
objects) physics [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In this
regard, a review paper has also been written that provides a comprehensive
discussion of 4DEGB [34]. [35, 36, 37].

Could there be additional spatial dimensions beyond the three that we know?
Theoretically, there is no concrete a priori reason that forbids us to pursue
this fundamental question (for overviews of diverse theories with extra
dimensions and their physical consequences see e.g.,[38, 39, 40]). One of the
well-known reasons for this is that phenomena that need very different
justifications in three-dimensional space are expected to be nothing more than
manifestations of simpler theories in higher-dimensional manifolds [38]. But
the main issue is how to adapt this idea to the real universe i.e., the three-
dimensionality of space. Both our daily experience and the experiments of
particle and space physics clearly show that we live in a four-dimensional
universe-three for space and one for time. Besides, in light of the joint
detection of gravitational waves and electromagnetic signals [41] and recent
Event Horizon Telescope data [42], the possibility of the extra dimension(s)
in the real universe can not be ruled out. Therefore, to justify the extra
dimensions and search for them, we have to explain where are these extra
dimensions. The commonly answer is that the extra dimensions are compactified
one a very small scale. But this answer, is not the end of the story and gives
rise to another question: how come they became compact? The answer to this
question is not straightforward. A solution known as spontaneous
compactification is the first attempt to handle this fundamental issue [43,
44]. In Refs. [45, 46], one can find similar solutions with approaches closer
to cosmology. In this direction, a more natural way to achieve compactified
extra dimensions is dynamical cementification (see Refs. [47, 48, 49, 50]
involving different approaches and setups).

It is well-known for a long time that extra-dimensional theories can, in the
appropriate limit, behave like a standard four-dimensional spacetime with
additional field content derived from the imprint of the extra dimensions.
Extra dimensions have particular applications and consequences in the context
of cosmology, including questions on the nature of DM and DE. Theories of
matter existing in hidden extra dimensions, or the extra-dimensional effects
on the dynamics of the observed four-dimensional universe could potentially
shed light on these issues. Similarly, early-universe phenomena such as
inflation could be driven by extra-dimensional effects, or at least are
allowed to occur in the presence of extra-dimensions [51, 52, 53, 54, 55, 56,
57, 58, 59]. Considering the extra-dimensional GB in cosmology can be a source
for the generation of primordial cosmic magnetic fields [60]. Recently the GB
cosmology with extra spacetime dimensions has been investigated [61]. Some
aspects of the dynamical compactification scenario, where stabilization of
extra dimensions occurs due to presence the GB term and non-zero spatial
curvature, have been explored in [62].

For all mentioned above, it becomes obvious that investigation the structure
formation at the early stages of the universe in the context of GB cosmology
with extra dimensions is well motivated. In this work we disclose the effects
of the GB correction terms on the gravity side, as well as the extra
dimensions on the growth of perturbations at the early stages of the universe.
Since the GB correction terms contribute to the dynamical field equations in
more than four spacetime dimensions, thus, in order to explore the role of the
GB correction terms on the growth of perturbation, we need to consider higher
spacetime dimensions. We shall use the SC formalism [63] to examine the growth
of perturbations and structure formation. In this approach one considers a
uniform and spherical symmetric perturbation in an expanding background and
describes the growth of perturbations in a spherical region using the same
Friedmann equations for the underlying theory of gravity [64, 65, 66, 67, 68].

The outline of this paper is as follows. In Sec. II, we provide a review on GB
gravity and derive the corresponding Friedmann equations in the context of
(n+1)-dimensional GB cosmology. In Sec. III, using the spherically collapse
approach, we explore the growth of matter perturbation in the background of
the GB cosmology in higher dimensions. In section IV, we investigate the
effects of the GB parameter and extra dimensions on the mass function of the
dark matter halos. We finish with conclusions and discussions in the last
section.

##  II Modified Friedmann Equations higher-dimensional GB Cosmology

The action of the GB gravity in $(n+1)$-dimensional spacetime, and in the
presence of cosmological constant $\Lambda$, can be written as [69]

$\displaystyle
S_{EGB}=\frac{1}{2\kappa_{n+1}^{2}}\int{d^{n+1}x\sqrt{-g}\left(R-2\Lambda+\alpha\mathcal{L}_{GB}\right)}+S_{m},$
(1)

where $\alpha$ is called the GB coupling constant which has the dimension
$[\alpha]=[length]^{2}$,
$\mathcal{L}_{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}$
is the GB Lagrangian, and $S_{m}$ denotes the action of matter. In light of
the string theory evaluations, it is well known that the GB coupling constant
should be positive, i.e., $\alpha>0$ (see [70, 71]). The field equations can
be derived by varying the above action with respect to the metric. One finds

$\displaystyle\kappa_{n+1}^{2}T_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}R+\Lambda
g_{\mu\nu}-\alpha\Big{\\{}\dfrac{1}{2}g_{\mu\nu}\mathcal{L}_{GB}$
$\displaystyle-2RR_{\mu\nu}+4R_{\mu\gamma}R^{\gamma}_{\
\nu}+4R_{\gamma\delta}R^{\gamma\ \delta}_{\ \mu\
\nu}-2R_{\mu\gamma\delta\lambda}R_{\nu}^{\ \gamma\delta\lambda}\Big{\\}}.$ (2)

In what follows we work in the units where $\hslash=c=\kappa_{n+1}=1$. The
line elements of a spatially flat $(n+1)$-dimensional Friedmann-Robertson-
Walker (FRW) metric for the homogeneous and isotropic universe is as follows
222For the sake of simplicity, we employ a higher dimensional FRW metric with
the assumption that the extra spatial dimensions are also homogeneous and
isotropic. In this direction, it can be helpful to remember that exists
another cosmological setting based on Kaluza-Klein theory which considers a
spacetime with manifold $M^{4}\times T^{d=n-3}$. The line element of this
manifold, is given by  $\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}d\Omega^{2}\right)+b(t)^{2}\gamma_{ab}dy^{a}dy^{b},~{}~{}~{}K=0,\pm
1$ In this ansatz metric, there is a generalized term in the form
$b(t)^{2}\gamma_{ab}dy^{a}dy^{b}$, in addition to the spatial manifold labeled
by the coordinates $(r,\Omega)$. It includes the extra spatial coordinates
$y^{a}$, the compact manifold described by the metric $\gamma_{ab}$, and the
scale factor of extra dimensions $b(t)$. Overall, the underlying spacetime is
no longer isotropic since $a(t)\neq b(t)$. The underlying compact manifold is
assumed to be maximally symmetric whose Riemann tensor for the metric
$\gamma_{ab}$ is read as
$R_{abcd}=k(\gamma_{ac}\gamma_{bd}-\gamma_{ad}\gamma_{bc})$. Despite the
absence of observational support for the flatness spatial curvature of the
extra dimensions, interestingly, in Ref. [72] (see also [36]), considering the
contribution of the compact manifold corresponding to the extra dimensions in
the metric, it is shown that it should be flat (i.e., $k=0$) to avoid some
unphysical properties of energy density and perfect fluid pressure. It means
that, in case of extending the assumption of flatness spatial curvature to
higher dimensions, no conflict will arise.

$ds^{2}=-dt^{2}+a^{2}(t)\left(dr^{2}+r^{2}d\Omega^{2}_{n-1}\right),$ (3)

where
$d\Omega^{2}_{n-1}=d\theta_{1}^{2}+Sin^{2}\theta_{1}d\theta_{2}^{2}...+Sin^{2}\theta_{n-2}d\theta_{n-1}^{2}$,
represents the line element of a $(n-1)$-dimensional sphere. Substituting
metric (3) in the gravitational field equations (2), and assuming the matter
content of the universe is in the form of perfect fluid, one get the
corresponding Friedmann equations as [73]

$\displaystyle
H^{2}+\tilde{\alpha}H^{4}=\dfrac{2}{n(n-1)}\left(\rho_{m}+\Lambda\right),$ (4)
$\displaystyle\left(1+2\tilde{\alpha}H^{2}\right)\dot{H}=-\dfrac{1}{n-1}\rho_{m},$
(5)

where $\tilde{\alpha}=(n-2)(n-3)\alpha$, $H\equiv\dot{a}/a$ is the Hubble
parameter, $\rho_{m}$ is the energy density of both baryonic and DM. Notice
that in the limiting case where $n=3,\tilde{\alpha}=0$, Eqs. (4) and (5)
reduce to the Friedmann equations in standard cosmology, meaning that in four
dimensions the GB coupling constant does not contribute to the dynamic
equations. Therefore in four dimensions, $(n=3)$, we should replace
$\alpha\rightarrow\alpha/(n-3)$. Of course, in case of coupling to the scalar
field is not the case and leaves some non-trivial effects in four-dimension
[54, 55]. Concerning the higher dimensions, the cases $n=4$, and $n=5$, are
more popular than others, especially $n=4$, which results in certain
mathematical simplifications (e.g., see [74, 75]). Moreover, the continuity
equation in $(n+1)$-dimensions can be written as

$\dot{\rho}_{m}+nH(\rho_{m}+p_{m})=0.$ (6)

The energy density of the pressureless matter ($p_{m}=0$) can be obtained as
$\rho_{m}=\rho_{m,0}a^{-n}$. Therefore, with the following dimensionless
parameters:

$\displaystyle\tilde{\beta}\equiv\tilde{\alpha}H_{0}^{2}=(n-2)(n-3)\alpha
H_{0}^{2},$ (7) $\displaystyle\Omega_{m}=\frac{2\rho_{m}}{n(n-1)H^{2}},$ (8)
$\displaystyle\Omega_{\Lambda}=\frac{2\Lambda}{n(n-1)H^{2}}.$ (9)

One can rewrite Eq. (4) as

$E^{2}(z)+\tilde{\beta}E^{4}(z)=\Omega_{m,0}(1+z)^{n}+\Omega_{\Lambda,0},$
(10)

where $E(z)=H(z)/H_{0}$. This equation governs the evolution of the
homogeneous universe in the context of an $(n+1)$-dimensional GB gravity.
Moreover, at the present-time($z=0$), Eq. (10), reduces to

$\Omega_{m,0}+\Omega_{\Lambda,0}=1+\tilde{\beta}.$ (11)

Notice that when $\tilde{\beta}\rightarrow 0$, the standard equation is
recovered. By extending the flatness assumption to higher dimensions (see
Footnote 2), thereby, the relation $\Omega_{m}+\Omega_{\Lambda}\simeq 1$ holds
in all dimensions. This implies that the value of the dimensionless coupling
parameter, $\tilde{\beta}$, should be very small. The stringent upper bounds
obtained in light of cosmological settings for the dimensionless parameter
$\alpha H_{0}^{2}$ also confirm that it is merely a perturbative constant [54,
55]. We do so throughout our analysis.

Solving Eq. (10) with respect to $E(z)$ at a given redshift $z$, and
considering the branch where we have a real value of $E(z)$, yields

$E^{2}(z)=\dfrac{H^{2}(z)}{H_{0}^{2}}=\dfrac{1}{2\tilde{\beta}}\left[\sqrt{X(z)}-1\right],$
(12)

where

$X(z)\equiv
1+4\tilde{\beta}\left[\Omega_{m,0}(1+z)^{n}+\Omega_{\Lambda,0}\right].$ (13)

In general for any $z$ we have:

$\Omega_{m}(z)+\Omega_{\Lambda}(z)=1+\tilde{\beta}E^{2}(z).$ (14)

The Hubble expansion rate can be obtained via Eqs. (8), (9) ,(12), (13). We
find

$\displaystyle
H^{2}(z)=\dfrac{H_{0}^{2}}{2\tilde{\beta}}\left[\sqrt{X(z)}-1\right]$
$\displaystyle=\dfrac{2}{n(n-1)}(\rho_{m}+\Lambda)\left(1-\dfrac{2\tilde{\alpha}}{n(n-1)}(\rho_{m}+\Lambda)\right)$
(15)

where we have expanded $X(z)$ and only kept the linear term of
$\tilde{\beta}$, since $\tilde{\beta}\ll 1$ is very small.

The evolution of the normalized Hubble parameter versus $z$ for different
values of $n$ is plotted in Fig. 1. As we can see, in GB cosmology the Hubble
parameter with higher dimensions are larger than lower dimensions model,
implying that in lower dimensions model, our Universe expands slower.

Figure 1: The behavior of the normalized Hubble rate $E(z)$ for different
values of $n$ in GB cosmology, where we have taken $\tilde{\beta}=10^{-16}$.

Also we have plotted this normalized Hubble parameter for different values of
$\tilde{\beta}$ and $n$ in Fig. 2. As we can see, in GB cosmology the Hubble
parameter at high redshifts decreases with increasing the parameter
$\tilde{\beta}$ in higher dimensions, but vice versa at low redshifts.

Figure 2: The behavior of the normalized Hubble rate $E(z)$ for different
values of $(n,\tilde{\beta})$.

In Fig. 3, we have plotted the evolution of the density abundance
$\Omega_{m}$, defined as

$\displaystyle\Omega_{m}$ $\displaystyle\equiv$
$\displaystyle\dfrac{2\rho_{m}}{n(n-1)H^{2}}=\dfrac{2\tilde{\beta}(1+\tilde{\beta}-\Omega_{\Lambda,0})(1+z)^{n}}{\left[\sqrt{X(z)}-1\right]}.$
(16)

As we can see from Fig. 3, the matter density abundance with different
dimensions has the same behavior, i.e., all graphs are reduced by decreasing
$z$. In addition for higher dimensions ($n$ parameter), the density abundance
drops faster.

Figure 3: The evolution of the matter density abundance as a function of
redshift z for different values of $n$, where we have taken
$\tilde{\beta}=10^{-16}$.

Also we can see, the matter density abundance increases with increasing the
parameter $\tilde{\beta}$ in higher dimensions.
In a similar way, the evolution of the density abundance $\Omega_{\Lambda}$ is
given by

$\displaystyle\Omega_{\Lambda}$ $\displaystyle\equiv$
$\displaystyle\dfrac{2\Lambda}{n(n-1)H^{2}}=\dfrac{2\tilde{\beta}\Omega_{\Lambda,0}}{\left[\sqrt{X(z)}-1\right]}.$
(17)

Figure 4: The evolution of the DE density abundance as a function of redshift
z for different values of $n$, where we have taken $\tilde{\beta}=10^{-16}$.

In Figs. 4, and 5 we plot the evolution of the DE density abundance in various
dimensions and for different values of $\Omega_{\Lambda,0}$ parameter. It is
seen that the DE density abundance $\Omega_{\Lambda}$ increases by decreasing
$z$. From Fig. 4, we see that for a fixed value of redshift parameter $z$, the
value of the density abundance decreases with increasing the spacetime
dimensions.

Figure 5: The evolution of the DE density abundance as a function of redshift
z for different values of $\Omega_{\Lambda,0}$, where we have taken
$n=5,\tilde{\beta}=10^{-16}$.

The deceleration parameter in terms of the redshift can be written as

$\displaystyle
q=-1-\dfrac{\dot{H}}{H^{2}}=-1+\dfrac{(1+z)}{H(z)}\dfrac{dH(z)}{dz}$
$\displaystyle=-1+\dfrac{n\
\tilde{\beta}}{\sqrt{X(z)}}\dfrac{(1+z)^{n}(1+\tilde{\beta}-\Omega_{\Lambda,0})}{\left[\sqrt{X(z)}-1\right]}.$
(18)

We have plotted the behavior of the deceleration parameter $q(z)$ for
different dimensions in Fig. 6. We observe that for lower dimensions the
universe experiences a transition from a decelerating phase ($q>0$) to an
accelerating phase ($q<0$), at redshift around $z=z_{tr}$. It is seen that
$z_{tr}$ depends on the spacetime dimension and decreases with increasing $n$.

Figure 6: The behavior of the deceleration parameter $q(z)$ as a function of
redshift for different $n$. Here we have taken $\tilde{\beta}=10^{-16}$.

Another quantity which is helpful in understanding the phase transitions of
the universe is called the jerk parameter. This is a dimensionless quantity
obtained by taking the third derivative of the scale factor with respect to
the cosmic time, provides a comparison between different DE models and the
$\Lambda$CDM $(j=1)$ model. The jerk parameter is defined as

$j=\dfrac{1}{aH^{3}}\dfrac{d^{3}a}{dt^{3}}=q(2q+1)+(1+z)\dfrac{dq}{dz}.$ (19)

For the $\Lambda$CDM model, the value of $j$ is always unity. A
non-$\Lambda$CDM model occurs if there is any deviation from the value of
$j=1$. From Fig. 7 we observe that in the context of GB cosmology, the jerk
parameter is larger than $\Lambda$CDM in higher dimensions.

Figure 7: The evolution of jerk parameter with respect to redshift for
different values of $n$ parameter, Here we have taken
$\tilde{\beta}=10^{-16}$.

##  III Growth Of Perturbations in GB Cosmology

We consider a universe filled with pressureless matter, $(p_{m}=0)$. In this
case Eq. (6) reads

$\dot{\rho}_{m}+nH\rho_{m}=0,$ (20)

which has a solution of the form $\rho_{m}=\rho_{m,0}a^{-n}$, where
$\rho_{m,0}$ is the energy density at the present time. In order to study the
growth of perturbations, we consider a spherically symmetric perturbed cloud
of radius $a_{p}$, and with a homogeneous energy density $\rho_{m}^{c}$. The
SC model describes a spherical region with a top-hat profile and uniform
density so that at any time $t$, we can write
$\rho_{m}^{c}(t)=\rho_{m}(t)+\delta\rho_{m}$ [66]. If $\delta\rho_{m}>0$ this
spherical region will eventually collapse under its own gravitational force
and if $\delta\rho_{m}<0$ it will expand faster than the average Hubble
expansion rate, thus generating a void. In other words, $\delta\rho_{m}$ is
positive in overdense region and it is negative in underdense regions. In
fact, when the universe is in the matter dominated era, denser regions expand
slower than the entire universe. Therefore if their density is enough large,
they eventually collapse and create gravitational constraints systems like
clusters [76]. Similar to Eq. (20), the conservation equation for spherical
perturbed region can be written as

$\dot{\rho}_{m}^{c}+nh\rho_{m}^{c}=0,$ (21)

where $h=\dot{a}_{p}/a_{p}$ is the local expansion rate of the spherical
perturbed region of radius $a_{p}$ (subscript $p$ refers to the perturbed). In
order to study the evolution of perturbations, we define a useful and
dimensionless quantity called density contrast as [76]

$\delta_{m}=\dfrac{\rho_{m}^{c}}{\rho_{m}}-1=\dfrac{\delta\rho_{m}}{\rho_{m}},$
(22)

where $\rho_{m}^{c}$ is the energy density of spherical perturbed cloud and
$\rho_{m}$ is the background density. Taking the derivative of Eq.(22) with
respect to the cosmic time and using Eq.(20) and Eq.(21), we obtain

$\displaystyle\dot{\delta}_{m}=n(1+\delta_{m})(H-h),$ (23)
$\displaystyle\ddot{\delta}_{m}=n(\dot{H}-\dot{h})(1+\delta_{m})+\frac{\dot{\delta}_{m}^{2}}{1+\delta_{m}},$
(24)

where the dot denotes the derivative with respect to time. Combining Eqs. (4),
(II), and expanding $X(z)$ (only to the linear term of $\tilde{\beta}$), we
arrive at

$\displaystyle\dfrac{\ddot{a}}{a}$ $\displaystyle=$
$\displaystyle\dfrac{(2-n)}{n(n-1)}\rho_{m}+\dfrac{4\tilde{\alpha}}{n^{2}(n-1)}\rho_{m}^{2}+\dfrac{4\tilde{\alpha}(n-2)}{n^{2}(n-1)^{2}}\rho_{m}\Lambda$
(25)
$\displaystyle+\dfrac{2}{n(n-1)}\Lambda-\dfrac{4\tilde{\alpha}}{n^{2}(n-1)^{2}}\Lambda^{2}.$

According to SC model, a homogeneous sphere of uniform density with radius
$a_{p}$ can itself be modeled using the same equations that govern the
evolution of the universe, with scale factor $a$ [77]. Therefore, we can write
for the spherical perturbed cloud with radius $a_{p}$, an equation similar to
Eq.(25), namely

$\displaystyle\dfrac{\ddot{a}_{p}}{a_{p}}$ $\displaystyle=$
$\displaystyle\dfrac{(2-n)}{n(n-1)}\rho_{m}^{c}+\dfrac{4\tilde{\alpha}}{n^{2}(n-1)}(\rho_{m}^{c})^{2}+\dfrac{4\tilde{\alpha}(n-2)}{n^{2}(n-1)^{2}}\rho_{m}^{c}\Lambda$
(26)
$\displaystyle+\dfrac{2}{n(n-1)}\Lambda-\dfrac{4\tilde{\alpha}}{n^{2}(n-1)^{2}}\Lambda^{2}.$

In general, one may expect $\tilde{\alpha}$ differ inside and outside of the
spherical region. However, for simplicity here we propose they are similar,
namely $\tilde{\alpha}^{c}=\tilde{\alpha}$. This can be easily understood if
we assume the role of the GB term at the early stages of the universe is very
weak, thus we can expect the same values for $\tilde{\alpha}$ inside and
outside of the spherical region.

Combining Eqs. (22), (25) and (26), yields

$\displaystyle\dot{H}-\dot{h}$ $\displaystyle=$
$\displaystyle\dfrac{(2-n)}{n(n-1)}\rho_{m}\delta_{m}-\dfrac{8\tilde{\alpha}}{n^{2}(n-1)}\rho_{m}^{2}\delta_{m}$
(27)
$\displaystyle-\dfrac{4\tilde{\alpha}(n-2)}{n^{2}(n-1)^{2}}\rho_{m}\Lambda\delta_{m}-H^{2}+h^{2},$

where we have expanded the second term and only kept the linear term of
$\delta_{m}$. This is due to the fact that we work in the linear regime where
$\delta_{m}<1$.

Substituting Eq. (27) into Eq. (24) and using Eq. (23), we can find the second
order differential equation for the density contrast $\delta_{m}$ in the
linear regime as

$\displaystyle\ddot{\delta}_{m}+2H\dot{\delta}_{m}-\dfrac{(n-2)}{(n-1)}\rho_{m}\delta_{m}$
$\displaystyle+\dfrac{8\tilde{\alpha}}{n(n-1)}\rho_{m}^{2}\delta_{m}+\dfrac{4\tilde{\alpha}(n-2)}{n(n-1)^{2}}\rho_{m}\Lambda\delta_{m}=0.$
(28)

In order to study the evolution of the density contrast $\delta_{m}$ in terms
of the redshift parameter, $1+z=1/a$, we first replace the time derivatives
with the derivatives with respect to the scale factor $a$. It is a matter of
calculations to show that

$\dot{\delta}_{m}=\delta_{m}^{\prime}aH,\quad\ddot{\delta}_{m}=\delta_{m}^{\prime\prime}a^{2}H^{2}+a\left(H^{2}+\dot{H}\right)\delta^{\prime}_{m},$
(29)

where the prime stands for the derivative respect to the scale factor $a$.
Using Eqs.(II) and (25), we get

$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\dfrac{1}{n-1}\rho_{m}+\dfrac{4\tilde{\alpha}\Lambda}{n^{2}(n-1)^{2}}$
(30)
$\displaystyle+\dfrac{4\tilde{\alpha}}{n(n-1)^{2}}\rho_{m}^{2}+\dfrac{4\tilde{\alpha}}{n^{2}(n-1)}\Lambda\rho_{m}.$

Therefore Eq. (28), after using Eqs. (29) and (30), can be written as

$\displaystyle\delta^{\prime\prime}_{m}+\dfrac{3}{2a}\delta^{\prime}_{m}-\dfrac{(n^{2}-4\tilde{\alpha}\Lambda)}{n^{2}(n-1)}\dfrac{\rho_{m}}{aH^{2}}\delta^{\prime}_{m}+\dfrac{4\tilde{\alpha}\Lambda}{n^{2}(n-1)^{2}}\dfrac{1}{aH^{2}}\delta^{\prime}_{m}$
$\displaystyle+\dfrac{4\tilde{\alpha}}{n(n-1)^{2}}\dfrac{\rho_{m}^{2}}{aH^{2}}\delta^{\prime}_{m}-\dfrac{(n^{2}-n-4\tilde{\alpha}\Lambda)(n-2)}{n(n-1)^{2}}\dfrac{\rho_{m}}{a^{2}H^{2}}\delta_{m}$
$\displaystyle+\dfrac{8\tilde{\alpha}}{n(n-1)}\dfrac{\rho_{m}^{2}}{a^{2}H^{2}}\delta_{m}=0.$
(31)

Since we are working in the linear regime, we neglect $O(\delta_{m}^{2})$ and
$O({\delta^{\prime}_{m}}^{2})$. Combining Eqs.(II) and (III), we arrive at

$\displaystyle\delta^{\prime\prime}_{m}+\dfrac{3}{a}\delta^{\prime}_{m}-\dfrac{1}{a\
\Gamma}\Bigg{\\{}\dfrac{(n^{2}-4\tilde{\alpha}\Lambda)\rho_{m}}{{2n}}-\dfrac{2\tilde{\alpha}\Lambda}{n(n-1)}$
$\displaystyle-\dfrac{2\tilde{\alpha}\rho_{m}^{2}}{(n-1)}\Bigg{\\}}\delta^{\prime}_{m}-\dfrac{1}{a^{2}\
\Gamma}\Bigg{\\{}\dfrac{(n^{2}-n-4\tilde{\alpha}\Lambda)(n-2)\rho_{m}}{{2(n-1)}}$
$\displaystyle-(4\tilde{\alpha}\rho_{m}^{2})\Bigg{\\}}\delta_{m}=0.$ (32)

where

$\Gamma=(\rho_{m}+\Lambda)\left[1-\dfrac{2\tilde{\alpha}}{n(n-1)}(\rho_{m}+\Lambda)\right].$
(33)

It should be noted that in the limiting case where $n=3$ and $\Lambda=0$, Eq.
(32) reduces to

$\displaystyle\delta^{\prime\prime}_{m}+\dfrac{3}{2a}\delta^{\prime}_{m}-\frac{3\delta_{m}}{2a^{2}}=0,$
(34)

which is the result obtained in standard cosmology [63]. In other words, in
the absence of the GB term, the perturbed equation for the density contrast,
$\delta_{m}$, in the linear regime, coincides with the corresponding equation
in standard cosmology.

Since Eq.(32) has no analytic solution, we numerically plot, in Fig. 8, the
matter density contrast for different values of $\tilde{\beta}$ parameter in
various dimensions. We observe that in the framework of GB gravity in five
dimensions, the density contrast of matter starts growing from its initial
value and, as the universe expands, the matter density contrast grows up
faster and deviates from the standard model. Indeed, the growth of matter
perturbations in five spacetime dimensions is faster comparing to the standard
four dimensional cosmology. This means that in a universe with extra
dimensions, the structures forms sooner. Besides, with increasing $\alpha$
($\tilde{\beta}$) parameter, the matter perturbations grows faster. This is an
expected result, since the higher order GB correction terms increase the
strength of the gravity and thus support the growth of perturbations

Figure 8: The evolution of the matter density contrast for different values of
$(\tilde{\beta},n)$, where dotted-line for $(10^{-5},4)$, dashed-line for
$(0,4)$ and solid-line for $(0,3(\Lambda CDM))$. We have chosen
$\delta_{m}(z_{i})=0.0001$ , $z_{i}=400$.

We can investigate the growth rate of matter perturbations which is given by
the growth function as [77]

$f(a)=\dfrac{dlnD}{dlna},\ \ \ \ \
D(a)=\dfrac{\delta_{m}(a)}{\delta_{m}(a=1)}.$ (35)

Let us note that in the absence of the GB term ($\tilde{\alpha}=0$), the
growth function is a constant of unity. In Fig. 9 we have plotted the growth
function in terms of the redshift parameter. We observe that in the framework
of GB gravity, the growth function in higher dimensions grows faster than
four-dimensional GB and the growth function increases with increasing $n$.

Figure 9: The evolution of the growth function with $\tilde{\beta}=10^{-6}$
and for different values of $n$ parameter, where dashed-line for $n=3$,
dotted-line for $n=4$ and solid-line for $n=5$.

Figure 10: The evolution of the growth function for different values of
$\tilde{\beta}$ parameter in 5D, where dotted-line for
$\tilde{\beta}=10^{-5}$, solid-line for $\tilde{\beta}=2\times 10^{-5}$ and
dashed-line for $\tilde{\beta}=3\times 10^{-5}$.

From Fig. 10, we can see that the growth function increases with increasing
$\tilde{\beta}$ parameter in higher-dimensional GB cosmology.

##  IV Halo abundance in GB cosmology

This section is devoted to study the distribution of the number density of
collapsed objects of a given mass range in the framework of GB cosmology. The
collapsed objects which, in essence, are the main source of large-scale
structure formation of the universe are called the DM halos. Besides, the
baryonic matter due to the gravitational attraction follows the DM
distribution. In this way, tracing the distribution of DM haloes becomes
possible by looking at the distribution of galaxy clusters. In order to
investigate the number density of distribution of the collapsed objects or the
galaxy clusters along the redshift, a semi-analytic approach known as the
Press-Schechter formalism is commonly employed [78]. Usually, the matter
density field in the mathematical formulations of the halo mass function
should be enjoyed the Gaussian distribution.

The comoving number density of the gravitationally collapsed objects
(equivalent to galaxy clusters) at a certain redshift $z$, with mass from $M$
to $M+dM$, is given by the following analytical formula [79]

$\dfrac{dn(M,z)}{dM}=-\dfrac{\rho_{m,0}}{M}\dfrac{d\ln\sigma(M,z)}{dM}f(\sigma(M,z)),$
(36)

where $\rho_{m,0}$, $\sigma(M,z)$ and $f(\sigma)$, respectively, denote the
present matter mean density of the universe, the rms of density fluctuation in
a sphere of radius $r$ surrounding a mass $M$, and the mathematical mass
function proposed by Press and Schechter [78], as follow

$f_{PS}(\sigma)=\sqrt{\dfrac{2}{\pi}}\dfrac{\delta_{c}(z)}{\sigma(M,z)}\exp\left[-\dfrac{\delta_{c}^{2}(z)}{2\sigma^{2}(M,z)}\right].$
(37)

Subscript ”PS”, refers to Press and Schechter. Note that $\delta_{c}(z)$ in
the mass function above is the critical density contrast above which
structures collapse. By serving the linerised growth factor
$D(z)=\delta_{m}(z)/\delta_{m}(z=0)$, as well as the rms of density
fluctuation at a fixed length $r_{8}=8h^{-1}Mpc$, one can express
$\sigma(M,z)$ as

$\sigma(z,M)=\sigma(0,M_{8})\left(\dfrac{M}{M_{8}}\right)^{-\gamma/3}D(z),$
(38)

where the index $\gamma$ reads as [80, 81]

$\gamma=(0.3\Omega_{m,0}h+0.2)\left[2.92+\dfrac{1}{3}\log\left(\dfrac{M}{M_{8}}\right)\right],$
(39)

and $M_{8}=6\times 10^{14}\Omega_{M}^{(0)}h^{-1}M_{\bigodot}$ is the mass
inside a sphere of radius $r_{8}$ ($M_{\bigodot}$ is the solar mass) [82].

Figure 11: The evolution of mass function for objects with mass
$M=10^{13}(h^{-1}M_{\bigodot})$, and different values of $(\tilde{\beta},n)$,
where dotted-line for $(10^{-7},4)$, dashed-line for $(10^{-5},4)$ and solid-
line for $(10^{-5},5)$.

In Fig. 11, we display the redshift evolution of mass function,
$dn/dM(1/Mpc^{3})$, of objects with mass $10^{13}h^{-1}M_{\bigodot}$ for
different values of $\tilde{\beta}$ in various dimensions. This figures
explicitly shows that in each dimension, as the parameter $\tilde{\beta}$
increases, the halo abundance grows rapidly at lower redshifts. Besides, for a
given GB parameter,the mass function with increasing the spacetime dimension.
In other words, the halo abundance is formed later in lower dimensions.

To sum up, in this section we saw that the GB parameter as well as the extra
dimensions influence the evolution of the mass function of the dark matter
halos, therefore comparing to standard $\Lambda$CDM, our model can predict
different results for clustering of galaxy clusters.

##  V Conclusions and discussions

The growth of perturbations at the early stages of the universe and the
formation of galaxies and structures, due to the gravitational collapse, is
still an open question in modern cosmology. It is instructive to explore how
this phenomena occurs in different gravity theories. In the present work, we
have explored the gravitational collapse of matter at the early universe when
the higher order corrections on the gravity side are present in the action. We
have investigated the evolution of the matter perturbations in the context of
GB gravity in a flat universe filled with DM and DE (cosmological constant)
for different values of the model parameters. We have employed the SC
formalism in order to examine the perturbations and worked in the linear
regime for the matter density contrasts $\delta_{m}$ as well as the GB
coupling ${\tilde{\beta}}=(n-2)(n-3)\alpha H_{0}^{2}$. We observe that the
density contrast has similar behavior for different values of $\tilde{\beta}$
parameter; that is, it starts from its initial value and then the growth of
perturbations increases with increasing $\tilde{\beta}$ parameter, which
reveals the influences of $\tilde{\beta}$ in GB cosmology. Interestingly
enough, we found that the growth of perturbations increases with increasing
$\alpha$. This is an expected result, since the higher order GB correction
terms increase the strength of the gravity and thus support the growth of
perturbations. Besides, the matter density contrast $\delta_{m}$ in small
redshifts grows faster in higher dimensions. Physically, this means that the
structures form faster in a universe with extra dimensional spacetime. We have
also studied the evolution of the density parameters. We observed that the
evolution of the matter density abundance ($\Omega_{m}$) and DE density
abundance ($\Omega_{\Lambda}$) for different values of $\tilde{\beta}$
decrease at low redshifts. We found out that the density abundance of matter,
consist of baryonic and DM, drops slower for smaller values of $\tilde{\beta}$
in higher dimensional GB cosmology. From the evolution of the deceleration
parameter, we see that the universe experiences a phase transition from
decelerated phase to an accelerated phase around redshift $z_{tr}$. We saw
that $z_{tr}$ is smaller in higher dimensions, which means that our universe
experiences this phase transition later in higher dimensions. Also we observed
that in the framework of GB gravity, the growth function increases in higher
dimensions, and also increases with increasing $\tilde{\beta}$ parameter in
higher-dimensional GB cosmology. Also we found that, the abundance of halo
dark matter in higher dimensions start to grow in high reshifts, so we can
conclude that large scale structure(LSS) grows faster in this universe.

###### Acknowledgements.

We are grateful to M. Khodadi for useful discussion and valuable comments.

## References

  * [1] C. M. Will, The Confrontation between General Relativity and Experiment, Living Rev.Rel 09, 03 (2005) [arXiv:gr-qc/0510072].
  * [2] B. L. Young, A survey of dark matter and related topics in cosmology, Front. Phys 12, 121201 (2017).
  * [3] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys 61, 1 (1989) [arXiv:astro-ph/0005265].
  * [4] A. G. Riess et al, Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J 116, 1009 (1998) [arXiv:astro-ph/9805201].
  * [5] S. Perlmutter et al. Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae, Astrophys. J 517, 565 (1999) [arXiv:astro-ph/9812133].
  * [6] C. Lanczos, Elektromagnetismus als naturliche Eigenschaft der Riemannschen Geometrie, Zeitschrift fur Physik 73, 147 (1932).
  * [7] C. Lanczos, A remarkable property of the riemann-christoffel tensor in four dimensions, Annals of Mathematics 39, 842 (1938).
  * [8] D. Lovelock, Divergence-free tensorial concomitants, Aequationes mathematicae 4, 127 (1970).
  * [9] D. Lovelock, The Einstein tensor and its generalizations J. Math. Phys 12, 498 (1971).
  * [10] B. Zumino, Gravity Theories in More Than Four-Dimensions, Phys. Rept 137, 109 (1986).
  * [11] B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156, 315 (1985).
  * [12] D. Glavan and C. Lin, Einstein-Gauss-Bonnet gravity in four-dimensional spacetime, Phys. Rev. Lett. 124, 081301 (2020).
  * [13] S.L. Li, P. Wu and H. Yu, Stability of the Einstein Static Universe in 4D Gauss-Bonnet Gravity, arXiv:2004.02080.
  * [14] K. Aoki, M.A. Gorji and S. Mukohyama, Cosmology and gravitational waves in consistent $D\rightarrow 4$ Einstein- Gauss-Bonnet gravity, J. Cosmol. Astropart. Phys 09, 014 (2020).
  * [15] H. Mohseni Sadjadi, On cosmic acceleration in four di- mensional Einstein-Gauss-Bonnet gravity, Phys. Dark Univ. 30, 100728 (2020).
  * [16] G. Narain and H.Q. Zhang, Cosmic evolution in novel-Gauss Bonnet Gravity, [arXiv:2005.05183].
  * [17] J. Feng, B. Gu, F. Shu, Theorical and observational constraints on regularized 4D Einstein-Gauss-Bonnet gravity, Phys. Rev. D 103, 064002 (2021), [arXiv:2006.16751].
  * [18] M. Garacia and A. Hernandez, Einstein-Gauss-Bonnet gravity: is it compatible with modern cosmology?, [arXiv:2007.06730].
  * [19] K. Aoki, M.A. Gorji, S. Mizuno and S. Mukohyama, Inflationary gravitational waves in consistent $D4$ Einstein-Gauss-Bonnet gravity, J. Cosmol. Astropart. Phys 01, 054 (2021).
  * [20] D. Wang, D. Mota, 4D Gauss-Bonnet gravity: cosmological constraints, $H_{0}$ tension and large scale structure, Physics of the Dark Universe 32,100813 (2021), [arXiv:2103.12358].
  * [21] S. Shahidi and N. Khosravi, Constrained 4D Gauss-Bonnet gravity and its cosmological implications, Eur. Phys. J. C 82, no.3, 269 (2022), [arXiv:2105.02372].
  * [22] R. Kumar and S.G. Ghosh, Rotating black holes in 4D Einstein-Gauss-Bonnet gravity and its shadow, J. Cosmol. Astropart. Phys 07, 053 (2020).
  * [23] Y.P. Zhang, S.W. Wei and Y.X. Liu, Spinning test particle in four-dimensional Einstein-Gauss-Bonnet black holes, Universe. 6, 103 (2020).
  * [24] A. Aragon, R. Becar, P.A. Gonzalez and Y. Vasquez, Perturbative and nonperturbative quasinormal modes of 4D Einstein-Gauss-Bonnet black holes, Eur. Phys. J. C 80, 773 (2020).
  * [25] S.J. Yang, J.J. Wan, J. Chen, J. Yang and Y.Q. Wang, Weak cosmic censorship conjecture for the novel 4D charged Einstein-Gauss-Bonnet black hole with test scalar field and particle, Eur. Phys. J. C 80, 937 (2020).
  * [26] Z.C. Lin, K. Yang, S.W. Wei, Y.Q. Wang and Y.X. Liu, Equivalence of solutions between the four-dimensional novel and regularized EGB theories in a cylindrically symmetric spacetime, Eur. Phys. J. C 80, 1033 (2020).
  * [27] S.W. Wei and Y.X. Liu, Extended thermodynamics and microstructures of four-dimensional charged Gauss- Bonnet black hole in AdS space, Phys. Rev. D 101, 104018 (2020).
  * [28] R.A. Konoplya and A. Zhidenko, Black holes in the four-dimensional Einstein-Lovelock gravity, Phys. Rev. D 101, 084038 (2020).
  * [29] R.A. Konoplya and A. Zhidenko, (In)stability of black holes in the 4D Einstein-Gauss-Bonnet and Einstein- Lovelock gravities, Phys. Dark Univ. 30, 100697 (2020).
  * [30] K. Yang, B.M. Gu, S.W. Wei and Y.X. Liu, Born-Infeld black holes in 4D Einstein-Gauss-Bonnet gravity, Eur. Phys. J. C 80, 662 (2020).
  * [31] M. Heydari-Fard and H.R. Sepangi, Thin accretion disks around rotating black holes in 4D Einstein-Gauss-Bonnet gravity, Eur. Phys. J. C 81, 473 (2021).
  * [32] S.A.Hosseini Mansoori, Thermodynamic geometry of the novel 4-D Gauss-Bonnet AdS black hole, Phys. Dark Univ. 31, 100776 (2021).
  * [33] S.W. Wei and Y.X. Liu, Testing the nature of Gauss- Bonnet gravity by four-dimensional rotating black hole shadow, Eur. Phys. J. Plus 136, 436 (2021).
  * [34] P. G. S. Fernandes, P. Carrilho, T. Clifton and D. J. Mulryne, The 4D Einstein–Gauss–Bonnet theory of gravity: a review, Class. Quant. Grav. 39, no.6, 063001 (2022) [arXiv:2202.13908 [gr-qc]].
  * [35] D. Lorenz-Petzold, Higher Dimensional Gauss-Bonnet Cosmologies, Mod. Phys. Lett. A 3, 827 (1988).
  * [36] K. Andrew, B. Bolen and C. A. Middleton, Solutions of Higher Dimensional Gauss-Bonnet FRW Cosmology, Gen. Rel. Grav 39, 2061 (2007) [arXiv:0708.0373 [gr-qc]].
  * [37] K. Atmjeet, I. Pahwa, T. R. Seshadri and K. Subramanian, Cosmological Magnetogenesis From Extra-dimensional Gauss Bonnet Gravity, Phys. Rev. D 89, 063002 (2014) [arXiv:1312.5815 [astro-ph.CO]].
  * [38] J. M. Overduin and P. S. Wesson, Kaluza-Klein gravity, Phys. Rept 283, 303 (1997), [arXiv:gr-qc/9805018].
  * [39] V. A. Rubakov, Large and infinite extra dimensions, Phys. Usp 44, 871 (2001), [arXiv:hep-ph/0104152].
  * [40] P. Brax and C. van de Bruck, Cosmology and brane worlds: A Review, Class. Quant. Grav 20, 201 (2003), [arXiv:hep-th/0303095].
  * [41] L. Visinelli, N. Bolis and S. Vagnozzi, Brane-world extra dimensions in light of GW170817, Phys. Rev. D 97, no.6, 064039 (2018), [arXiv:1711.06628 [gr-qc]].
  * [42] S. Vagnozzi and L. Visinelli, Hunting for extra dimensions in the shadow of M87*, Phys. Rev. D 100, no.2, 024020 (2019), [arXiv:1905.12421 [gr-qc]].
  * [43] F. Muller-Hoissen, Spontaneous compactification with quadratic and cubic curvature terms,Phys. Lett. B 163, 106 (1985).
  * [44] N. Deruelle and L. Farina-Busto, The Lovelock Gravitational Field Equations in Cosmology, Phys. Rev. D 41, 3696 (1990).
  * [45] F. Muller-Hoissen, Dimensionally continued Euler forms: Kaluza-Klein cosmology and dimensional reduction, Class. Quant. Grav 3, 665 (1986).
  * [46] S.A. Pavluchenko, General features of Bianchi-I cosmological models in Lovelock gravity, Phys. Rev. D 80, 107501 (2009).
  * [47] G. A. Mena Marugan, Perturbative formalism of Lovelock gravity, Phys. Rev. D 46, 4340 (1992)
  * [48] E. Elizalde, A.N. Makarenko, V.V. Obukhov, K.E. Osetrin, and A.E. Filippov, Stationary vs. singular points in an accelerating FRW cosmology derived from six-dimensional Einstein-Gauss-Bonnet gravity, Phys. Lett. B 644, 1 (2007) [arXiv:hep-th/0611213].
  * [49] K.I. Maeda and N. Ohta, Inflation from superstring/M-theory compactification with higher order corrections, Phys. Rev. D 71, 063520 (2005) [arXiv:hepth/0411093].
  * [50] K.I. Maeda and N. Ohta, Cosmic acceleration with a negative cosmological constant in higher dimensions, JHEP 2014, 095 (2014) [arXiv:1404.0561].
  * [51] P. Kanti, R. Gannouji, and N. Dadhich, Gauss-Bonnet Inflation, Phys. Rev. D 92, 041302 (2015), [arXiv:1503.01579].
  * [52] P. Wongjun, Casimir Dark Energy, Stabilization of the Extra Dimensions and Gauss-Bonnet Term, Eur. Phys. J. C 75, 6 (2015), [arXiv:1308.4802].
  * [53] S. Nojiri, S. D. Odintsov, and M. Sasaki, Gauss-Bonnet dark energy, Phys. Rev. D 71, 123509 (2005), [arXiv:hep-th/0504052].
  * [54] L. Amendola, C. Charmousis, and S. C. Davis, Constraints on Gauss-Bonnet gravity in dark energy cosmologies, JCAP 0612, 020 (2006), [arXiv:hep-th/0506137].
  * [55] L. Amendola, C. Charmousis, and S. C. Davis, Solar System Constraints on Gauss-Bonnet Mediated Dark Energy, JCAP 0710, 004 (2007), [arXiv:0704.0175].
  * [56] T. Koivisto and D. F. Mota, Gauss-Bonnet Quintessence: Background Evolution, Large Scale Structure and Cosmological Constraints, Phys. Rev. D 75, 023518 (2007) [arXiv:hep-th/0609155].
  * [57] I. P. Neupane, On compatibility of string effective action with an accelerating universe, Class. Quant. Grav. 23, 7493 (2006) [arXiv:hep-th/0602097].
  * [58] B. M. Leith and I. P. Neupane, Gauss-Bonnet cosmologies: Crossing the phantom divide and the transition from matter dominance to dark energy, JCAP 0705, 019 (2007) [arXiv:hep-th/0702002].
  * [59] S. D. Odintsov and V. K. Oikonomou, Viable Inflation in Scalar-Gauss-Bonnet Gravity and Reconstruction from Observational Indices, Phys. Rev. D 98, 044039 (2018) [arXiv:1808.05045].
  * [60] K. Atmjeet, I. Pahwa, T. R. Seshadri and K. Subramanian, Cosmological Magnetogenesis From Extra-dimensional Gauss Bonnet Gravity, Phys. Rev. D 89, no.6, 063002 (2014), [arXiv:1312.5815 [astro-ph.CO]].
  * [61] Dmitry Chirkov, Alex Giacomini, Sergey A. Pavluchenko, Alexey Toporensky, Cosmological solutions in Einstein-Gauss-Bonnet gravity with static curved extra dimensions, [arXiv:2012.03517].
  * [62] Dmitry Chirkov, Sergey A. Pavluchenko, Some aspects of the cosmological dynamics in Einstein-Gauss-Bonnet gravity, Mod. Phys. Lett. A 36, 2150092 (2021) [arxiv:2101.12066].
  * [63] R. Abramo, R. Batista, L. Liberato and R.Rosenfeld, Struture formation in the presene of dark energy perturbations, JCAP 11, 012 (2007) [arXiv:0707.2882].
  * [64] S. Planelles, D. Schleicher, A. Bykov, Large- scale structure formation: from the first non-linear objects to massive galaxy clusters, Space Sci. Rev 51, 93 (2016) [arXiv:1404.3956].
  * [65] A. Ziaie, H. Shabani, S. Ghaffari, Effects of Rastall parameter on perturbation of dark sectors of the Universe, Mod. Phys. Lett. A 36, 2150082 (2021) [arXiv:1909.12085].
  * [66] A. H. Ziaie, H. Moradpour, H. Shabani, Structure Formation in Generalized Rastall Gravity, Eur. Phys. J. Plus 135, 916 (2020), [arXiv:2002.12757].
  * [67] B. Farsi and A. Sheykhi, Structure formation in mimetic gravity, Phys. Rev. D 106, 024053 (2022), [arXiv:2205.04138].
  * [68] A. Sheykhi and B. Farsi, Growth of perturbations in Tsallis and Barrow cosmology, Eur. Phys. J. C 82, 1111 (2022), [arXiv:2205.04138 [gr-qc]].
  * [69] R.G. Cai, Gauss-Bonnet Black Holes in AdS Spaces, Phys. Rev. D 65, 084014, (2002) [arXiv:hep-th/0109133].
  * [70] D. G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett. 55, 2656 (1985)
  * [71] B. C. Paul and S. Mukherjee, Higher dimensional cosmology with Gauss-Bonnet terms and the cosmological constant problem, Phys. Rev. D 42, 2595-2600 (1990)
  * [72] N. Mohammedi, Dynamical compactification, standard cosmology and the accelerating universe, Phys. Rev. D 65, 104018 (2002), [arXiv:hep-th/0202119 [hep-th]].
  * [73] R.G. Cai, S.P. Kim, First Low of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe, JHEP 0502, 050 (2005), [arXiv:hep-th/0501055].
  * [74] S. Hansraj and N. Mkhize, Exact Einstein-Gauss-Bonnet spacetime in six dimensions, Phys. Rev. D 102, no.8, 084028 (2020)
  * [75] R. S. Bogadi, M. Govender and S. Moyo, Five-dimensional Einstein–Gauss–Bonnet gravity compactified and applied to a colour-flavour-locked equation of state, Eur. Phys. J. Plus 138, no.5, 426 (2023).
  * [76] B. Ryden, Introduction to Cosmology (Addison-Wesley Press), San Francisco (2003).
  * [77] P. Peebles, Principles of Physical Cosmology (Princeton University Press), Princeton, NJ (1993).
  * [78] H. William , P. Schechter, Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation, Astrophysical Journal 187,  425 (1974).
  * [79] L. Liberato and R. Rosenfeld, Dark energy parameterizations and their effect on dark halos,Mon. J. Cosmol. Astropart. Phys. 0607,  009 (2006),[astro-ph/0604071].
  * [80] A. Mukherjee, Spherical collapse in a dark energy model with reconstructed effective equation of state, [arXiv:2008.03792].
  * [81] A. Mukherjee, Spherical collapse in DGP braneworld cosmology, [arXiv:2008.08979].
  * [82] P. T. P. Viana and A. R. Liddle, The Cluster Abundance in Flat and Open Cosmologies, Mon. Not. Roy. Astron. So 281,  323 (1996),[astro-ph/9511007].