On Chaplygin models in $f(G)$ gravity

Fidele Twagirayezu1, Abraham Ayirwanda1, Albert Munyeshyaka2, Solange
Mukeshimana3, Joseph Ntahompagaze1, and Leon Fidele Ruganzu Uwimbabazi3

1Department of Physics, College of Science and Technology, University of
Rwanda, Rwanda
2Department of Physics, Mbarara University of Science and Technology, Mbarara,
Uganda
3Department of Mathematics, College of Science and Technology, University of
Rwanda, Rwanda.

Correspondence<EMAIL_ADDRESS>
Abstract

The current work treats cosmological perturbation in a mixture of standard
matter, Chaplygin gas as well as Gauss-bonnet fluids using a 1+3 covariant
approach in the context of modified $f(G)$ gravity. We define the gradient
variables to obtain linear perturbation equations. After scalar and redshift
transformations, we consider both an original Chaplygin and generalized
Chaplygin gas models under Gauss-bonnet gravity. For pedagogical purposes, the
consideration of polynomial $f(G)$ gravity model was used to solve the
perturbation equations for short- and long- wavelength modes and investigate
the late time evolution. The numerical solutions were obtained. The results
show that the energy overdensity perturbations decay with an increase in
redshift. The treatment recovers GR results under limiting cases.

keywords: Cosmology; Chaplygin gas; Covariant perturbations; Dark energy
PACS numbers: 04.50.Kd, 98.80.-k, 95.36.+x, 98.80.Cq; MSC 2020: 83F05, 83D05

##  1 Introduction

Recent studies in modern cosmology have revealed information that the present
universe is undergoing a cosmic acceleration [1, 2, 3]. Such an acceleration
according to standard model of cosmology is caused by a fluid with a negative
pressure [4, 5, 6]. This model is built on the Einstein’s theory of relativity
known as General Relativity (GR), which was so far a fundamental and a
successful theory of gravity for several years [7]. However, GR suffers to
explain the recent observed cosmic acceleration at least without invoking the
cosmological constant [8, 9, 10]. Recently, it was shown that one can consider
Chaplygin gas model, a model with a negative pressure by construction, to
employ it under cosmological set up. This can help to study and possibly
explain the late cosmic acceleration as well as other cosmological features
raised through modern observational cosmology [11]. Several works have treated
the Chaplygin gas model in GR [12, 13, 14], in $f(R)$ gravity [15, 17], $R$
being Ricci scalar, in $f(T)$ gravity [18], $T$ being the torsion scalar, to
name a few. It was shown that the Chaplygin gas models can contribute to the
understanding of the evolution of the universe through perturbation analysis
[11]. There are different types of Chaplygin gas models. For instance, the
original and generalized Chaplygin gas model was treated in [19] where the
scalar field contribution were discussed. Other exploration of the same model
was previously done in [16, 15, 17, 20].

In the present work, a mixture of a Chaplygin gas fluid, standard matter fluid
as well as Gauss-bonnet fluid, as non-interacting fluids, is considered using
a $1+3$ covariant approach. The motivation behind the choice of $1+3$
covariant approach is that the perturbation equations that describe the
dynamical evolution of the universe and predicting the large scale structures
formation in late time are easily found using that approach rather than metric
perturbation. In addition to that, the advantage to consider the $1+3$
covariant formalism over the metric formalism relies in the fact that it does
not leave physical modes in the evolution of perturbations [21, 22]. The 1+ 3
covariant approach being the way of dividing the space-time into foliated
hyper-surfaces and a perpendicular 4-vector-field, it was introduced by Ellis
and Bruni in 1989 [23] focusing on GR. Progressively after the recent
discovery of cosmic acceleration, it gained much attention to extent that it
was employed to different modified gravity theories [24, 29].

The next part of this paper is organized as follows: in Section 2, the
presentation of the mathematical framework where one describes the $1+3$
approach, defines the vector gradients and linear evolution of the variables,
is done. The section 3 covers the linear evolutions equations, and presents
the scalar perturbation equations in redshift space. In Section 4, we presents
the asymptotic description of the system with the consideration of both long-
and short- wavelength modes in both dust and radiation epochs respectively and
present the numerical results of the perturbation equations by considering
both $GR$ limit and the system of matter chaplygin gas and Gauss-Bonnet
fluids. The section 5 is devoted to discussions and conclusions. The adopted
spacetime signature is $(-+++)$ and unless stated otherwise, we have used the
convention $8\pi G=c=1$, where $G$ is the gravitational constant and $c$ is
the speed of light.

##  2 Mathematical Framework

In this section, we present the mathematical aspects to describe the cosmic
evolution. In this regard, we first of all define the vector gradient
variables of individual fluid and find how it evolves. The action for modified
Gauss-Bonnet gravity is given as [25, 26, 30, 27, 31, 28]

$\displaystyle S=\frac{1}{2}\int
d^{4}x\sqrt{-g}\Big{[}R+f(G)+2\mathcal{L}_{m}\Big{]},$ (1)

with Kappa ($\kappa$) is assumed to equals to $1$ and $R$ is the Ricci scalar
and $G$ is the Gasuu-Bonnet parameter. The modified Einstein equation becomes

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=T^{m}_{\mu\nu}+\frac{1}{2}g^{\mu\nu}f-2f^{\prime}RR^{\mu\nu}+4f^{\prime}R^{\mu}_{\lambda}R^{\nu\lambda}-2f^{\prime}R^{\mu\nu\sigma\tau}R^{\lambda\sigma\tau}_{b}$
$\displaystyle-4f^{\prime}R^{\mu\lambda\sigma\nu}R_{\lambda\sigma}+2R\bigtriangledown^{\mu}\bigtriangledown_{\nu}f^{\prime}-2Rg^{\mu\nu}\bigtriangledown^{2}f^{\prime}-4R^{\nu\lambda}\bigtriangledown_{\lambda}\bigtriangledown^{\mu}f^{\prime}$
$\displaystyle-4R^{\mu\lambda}\bigtriangledown_{\lambda}\bigtriangledown^{\nu}f^{\prime}+4R^{\mu\nu}\bigtriangledown^{2}f^{\prime}+4g^{\mu\nu}R^{\lambda\sigma}\bigtriangledown_{\lambda}\bigtriangledown_{\sigma}f^{\prime}-4R^{\mu\lambda\nu\sigma}\bigtriangledown_{\lambda}\bigtriangledown_{\sigma}f^{\prime},$
(2)

where $f\equiv f(G)$ and $f^{\prime}=\frac{\partial f}{\partial G}$ and
$T^{m}_{\mu\nu}$ is the energy momentum tensor of the matter fluid (photons,
baryons, cold dark matter, and light neutrinos) with $G$ is given as
$R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}$,
$R^{\mu\nu}$ is the Ricci tensor and $R^{\mu\nu\alpha\beta}$ is the Riemann
tensor. The energy-momentum tensor of matter fluid forms is given by
$T_{\mu\nu}=\rho
u_{\mu}u_{\nu}+ph_{\mu\nu}+q_{\mu}u_{\nu}+q_{\nu}u_{\mu}+\pi_{\mu\nu}$ where
$\rho$, $p$, $q_{\mu}$ and $\pi_{\mu\nu}$ are the energy density, isotropic
pressure, heat flux and anisotropic pressure of the fluid respectively. The
quantities $\pi_{\mu\nu}$, $q^{\mu}$, $\rho$ and $p$ are reffered to as
dynamical quantities. The quantities $\sigma_{\mu\nu}$, $\varpi_{\mu\nu}$,
$\theta$ and $A_{\mu}$ are reffered as kinematical quantities. The
consideration of standard matter fluids (dust, radiation, etc), Chaplygin gas
fluid and Gauss-Bonnet contributions leads us to define the total energy
density, isotropic pressures, as

$\rho_{t}=\rho_{m}+\rho_{ch}+\rho_{G},~{}p_{t}=p_{m}+p_{ch}+p_{G},$ (3)

where [11, 30, 31, 32, 33, 34]

$\displaystyle\rho_{G}=\frac{1}{2}(f^{\prime}G-f)-24f^{\prime\prime}\dot{G},$
(4) $\displaystyle
p_{G}=\frac{1}{2}(f-f^{\prime}G)+\frac{G\dot{G}}{3H}f^{\prime\prime}+4H^{2}\ddot{G}f^{\prime\prime}+4H^{2}\dot{G}^{2}f^{\prime\prime\prime},$
(5)
$\displaystyle\rho_{ch}=\Big{[}A+\frac{C}{a^{3(1+\alpha)}}\Big{]}^{\frac{1}{1+\alpha}},$
(6) $\displaystyle p_{ch}=-\frac{A}{(\rho_{ch})^{\alpha}}.$ (7)

We assume a spatially flat Friedman-Robert-Walker(FRD) universe,

$ds^{2}=-dt^{2}+a^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$ (8)

so that the equation corresponding to the Friedmann equation is modified as

$\displaystyle
3H^{2}=\frac{1}{2}\left(Gf^{\prime}-f-24\dot{G}H^{3}f^{\prime\prime}\right)+\Big{[}A+\frac{C}{a^{3(1+\alpha)}}\Big{]}^{\frac{1}{1+\alpha}}+\rho_{m}\;,$
(9) $\displaystyle G=24H^{2}(\dot{H}+H^{2})\;,$ (10) $\displaystyle
R=6(\dot{H}+2H^{2})\;,$ (11)

where $A$, $C$ are arbitrary constants, $H=\frac{\dot{a}}{a}$ is the Hubble
parameter and $a$ is the scale factor. The energy density and pressure in the
modified Gauss-Bonnet gravity are presented as

$\displaystyle\rho_{t}=3H^{2}\;,$ (12) $\displaystyle
p_{t}=-(3H^{2}+2\dot{H})\;.$ (13)

###  2.1 The 1+3 Covariant approach in the context of $f(G)$ gravity

The $1+3$ covariant decomposition is a framework used in describing the linear
evolution of the cosmological perturbations [35, 41]. In this approach, a
fundamental observer divides space-time into hyper-surfaces and a
perpendicular 4-velocity field vector where $1+3$ indicates the number of
dimensions involved in each slice [36].That is to mean that manifold geometry
of the GR is discribed in four dimensinal spacs (ie,. time and space). One of
the importance of the $1+3$ covariant approach is to identity a set of
covariant variables which describe the inhomogeneity and anisotropy of the
universe [37]. In this context, we define a four-vector coordinateas function
of cosmological time ($x^{\mu}=x^{\mu}(\tau)$) that labels the comoving
distance along a world-line and the corresponding velocitiy given as :

$u^{\mu}=\frac{dx^{\mu}}{d\tau}$ (14)

The projection tensor, $h_{\alpha\beta}$ into the three dimensional and
orthogonal to $u^{\mu}$, satisfy the following condition:

$\displaystyle h_{\alpha\beta}=g_{\alpha\beta}+u_{\alpha}u_{\beta}\Rightarrow
h^{\alpha}_{\beta}h^{\beta}_{\gamma}=h^{\alpha}_{\gamma},$ (15) $\displaystyle
h^{\alpha}_{\alpha}=3,h_{\alpha\beta}u^{\beta}=0.$ (16)

The covariant derivative of the four-velocitiy in terms of its kinematic
quantities [38] is given by:

$\displaystyle\tilde{\bigtriangledown}_{a}u_{b}=\frac{1}{3}h_{ab}\tilde{\theta}+\tilde{\sigma}_{ab}+\tilde{\omega}_{ab}-u_{a}{\dot{\tilde{u}}}_{b}$
(17)

Where $\theta$, $\tilde{\sigma}_{ab}$, $\tilde{\omega}_{ab}$,
$\dot{\tilde{u}}_{b}$, are: the volume expansion,sheartensor, vorocity tensor
and four-acceleration respectively. The Hubble parameter is related to
$\theta$ as $\theta=3H$. Assume the fluids in our consideration are
irrotational (ie.,$\tilde{\omega}_{ab}=0$) and shear-free (i.e
$\tilde{\sigma}_{ab}=0$), the rate of expansion is given by the Raychaudhuli
and conservation equations as:

$\displaystyle\dot{\theta}=-\frac{\theta^{2}}{3}-\frac{1}{2}(\rho_{t}+p_{t})+\tilde{\bigtriangledown}^{a}\dot{u}_{a},$
(18) $\displaystyle\dot{\rho}=-\theta(\rho+p),$ (19)
$\displaystyle\tilde{\bigtriangledown}_{a}p-(\rho_{t}+p_{t})\dot{u}_{a}=0.$
(20)

###  2.2 General fluids description

In this part, we assume a non-interacting matter fluids with both chaplygin
gas and Gauss-Bonnet fluids in the entire Universe where the growth of the
energy overdensity fluctuations contribute to the large scale structure
formation. We consider an homogenous and expanding (FRW) cosmological
background where it is possible to define the spatial gradient of the Gauge
invariant variables as
$D^{m}_{a}=\frac{a\tilde{\nabla}^{a}\rho_{m}}{\rho_{m}},Z_{a}=a\tilde{\nabla}^{a}\theta,D_{a}^{ch}=\frac{a\tilde{\nabla}_{a}\rho_{ch}}{\rho_{ch}}\;,\\\
{D}^{G}_{a}=\frac{a\tilde{\nabla_{a}}\rho_{G}}{\rho_{G}},\mathcal{G}_{a}=a\tilde{\nabla}_{a}{G},\mathsf{G}_{a}=a\tilde{\nabla}_{a}\dot{G}\;$.
The subscipts $m$, $G$ and $ch$ stand for matter, Gauss-Bonnet fluid,
Chaplygin gas fluid contributions respectively. All the defined gradient
variables will be considered to develop a system of cosmological perturbation
equations in the context of chaplygin-gas assisted $f(G)$ gravity.

##  3 Linear evolution equations

Taking the first derivative with respect to time of the defined gradient
variables, we get:

$\displaystyle\dot{D}_{m}^{a}=-\left(1+w_{m}+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\frac{G\dot{G}}{\theta^{2}}\right)Z{a}+\frac{(1+w_{m})}{(1+w_{t})}\frac{\rho_{m}}{\rho_{t}}\omega_{m}\theta
D_{m}^{a}+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}D_{chp}^{a}$
$\displaystyle+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\rho_{chp}a\nabla^{a}w_{chp}+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}(\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+G\dot{G}f^{\prime\prime\prime}+\frac{\dot{G}f^{\prime\prime}}{G^{2}})\mathcal{G}_{a}$
$\displaystyle+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}G_{a}$
(21)
$\displaystyle\dot{\Delta_{ch}}=-[1+\frac{G\dot{G}}{\theta(1+w_{t})\rho_{t}}](1+w_{ch})Z+\frac{\theta(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{m}\rho_{m}\Delta_{m}+(1+\rho_{ch})\frac{\theta(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}\Delta_{ch}$
$\displaystyle-a\theta\tilde{\nabla}_{a}w_{ch}+\frac{\theta(1+w_{ch})}{(1+w_{t})\rho_{t}}[\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+G\dot{G}f^{\prime\prime}+\frac{\dot{G}f^{\prime\prime\prime}}{\theta^{2}}]\mathsf{G}_{a}+\frac{\theta(1+w_{ch})}{(1+w_{t})\rho_{t}}Gf^{\prime\prime}\mathsf{G},$
(22)
$\displaystyle\dot{\mathsf{G}}_{a}=(\frac{\dddot{G}}{\dot{G}}-\frac{\ddot{G}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta})\mathsf{G}_{a}-\frac{\ddot{G}\dot{G}{G}}{(1+w_{t})\rho_{t}\theta^{2}}Z_{a}-\frac{\ddot{G}}{(1+w_{t})\rho_{t}}w_{m}\rho_{m}D_{a}^{m}-\frac{\ddot{G}\rho_{ch}w_{ch}D_{a}^{ch}}{(1+w_{t})\rho_{t}}$
$\displaystyle-\frac{\ddot{G}\rho_{ch}a\nabla_{a}w_{ch}}{(1+w_{t})\rho_{t}}-\frac{\ddot{G}}{2(1+w_{t})\rho_{t}}(1-f^{\prime}-Gf^{\prime\prime})\mathcal{G}_{a}-\frac{\ddot{G}\dot{G}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\mathcal{G}_{a}$
(23)
$\displaystyle\dot{\mathcal{G}}_{a}=(1-\frac{\dot{G}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta})\mathsf{G}_{a}+\frac{\dot{G}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z_{a}-\frac{\dot{G}w_{m}\rho_{m}D_{a}^{m}}{(1+w_{t})\rho_{t}}-\frac{\dot{G}\rho_{ch}w_{ch}D_{a}^{ch}}{(1+w_{t})\rho_{t}}-\frac{\dot{G}\rho_{ch}a\nabla_{a}w_{ch}}{(1+w_{t})\rho_{t}}+$
$\displaystyle\Big{[}\frac{\dot{G}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{\dot{G}^{2}}{2(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}_{a},$
(24)
$\displaystyle\dot{Z}_{a}=-\Big{[}\frac{1}{2}(1+3w_{m})\rho_{m}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{m}\rho_{m}}{(1+w_{t})\rho_{t}}\Big{]}D^{m}_{a}$
$\displaystyle-\Big{[}\frac{1}{2}(1+3w_{chp})\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}D^{chp}_{a}$
$\displaystyle-\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}a\tilde{\bigtriangledown}_{a}w_{chp}$
$\displaystyle+\Big{[}1-3f^{\prime\prime}-12H^{3}\dot{G}f^{iv}+12H\Big{(}H(\ddot{G}f^{\prime\prime\prime}+\frac{\dddot{G}}{G}f^{\prime\prime}+\dot{G}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{[}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{]}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+G\dot{G}f^{\prime\prime\prime}+\frac{\dot{G}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}_{a}$
$\displaystyle+\Big{(}24\dot{G}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}\Big{)}\mathbf{G}_{a}$
$\displaystyle+\Big{[}8H\left(f^{\prime\prime}\ddot{G}+f^{\prime\prime\prime}\dot{G}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}H\dot{G}\right)-\frac{2}{3}\theta-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{G\dot{G}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{]}Z_{a}$
$\displaystyle-\frac{1}{(1+w_{t})\rho_{t}}\bigtriangledown^{2}\Big{[}w_{m}\rho_{m}D^{m}_{a}+w_{chp}\rho_{chp}D^{chp}_{a}+\rho_{chp}a\tilde{\bigtriangledown}^{a}w_{chp}\Big{]}$
$\displaystyle-\frac{1}{(1+w_{t})\rho_{t}}\bigtriangledown^{2}\Big{[}(\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+G\dot{G}f^{\prime\prime\prime}+\frac{\dot{G}f^{\prime\prime}}{G^{2}})\mathcal{G}_{a}+\frac{Gf^{\prime\prime}}{\theta}G_{a}-\frac{G\dot{G}}{\theta^{2}}Z_{a}\Big{]},$
(25)
$\displaystyle\dot{D}^{G}_{a}=-\Big{[}\frac{4H\theta}{\rho_{G}}\Big{(}2H\dot{G}f^{\prime\prime\prime}+(2\dot{H}-2H^{2})f^{\prime\prime}\Big{)}-\theta\Big{(}4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}$
$\displaystyle-H^{2})\dot{f}^{\prime}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}_{a}+4H\Big{[}H(\ddot{G}f^{\prime\prime\prime}+\frac{f^{\prime\prime}\dddot{G}}{\dot{G}}+\dot{G}^{2}f^{\prime\prime\prime})+(2\dot{H}-H^{2})f^{\prime\prime}$
$\displaystyle+\theta\Big{(}4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime}\Big{)}\Big{(}\frac{\frac{1}{2}+G\dot{G}f^{\prime\prime}+\frac{\dot{G}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}_{a}+\frac{1}{3\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}\ddot{G}+f^{\prime\prime\prime}\dot{G}^{2})$
$\displaystyle-8Hf^{\prime\prime}\dot{G}(\frac{G}{12H^{3}}+3H)+(2\dot{H}-H^{2})f^{\prime\prime}\dot{G}-3\Big{(}4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime}\Big{)}$
$\displaystyle+\theta(4H^{2}\ddot{f}^{\prime}-4H(2\dot{H}-H^{2})\dot{f}^{\prime})(\frac{G\dot{G}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})G\dot{G}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z_{a}$
$\displaystyle+\frac{\theta}{{\rho_{G}}}\Big{(}4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime}\Big{)}D^{G}_{a}$
$\displaystyle+\theta\Big{(}-4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime}\Big{)}\frac{w_{m}\rho_{m}(1-f^{\prime}-Gf^{\prime\prime})}{(1+w_{t}\rho_{t}\rho_{G})}D^{m}_{a}$
$\displaystyle+\theta\Big{(}4H^{2}\ddot{f}^{\prime}+4H(2\dot{H}-H^{2})\dot{f}^{\prime}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{a}}\Big{)}a\tilde{\nabla}_{a}w_{ch}$
(26)

We have used

$\dot{u}_{a}=-\frac{\tilde{\nabla}^{a}p_{t}}{\rho_{t}+p_{t}}$ (27)
$a\dot{u}_{a}=-\frac{1}{(1+w_{t})\rho_{t}}[w_{m}\rho_{m}D^{a}_{m}+w_{chp}\rho_{chp}D^{a}_{chp}+a\rho_{chp}\tilde{\nabla}^{a}w_{chp}+\tilde{\nabla}^{a}p_{G}]\;,$
(28)

These linear vector evolution equations (eq. 21-26) can be represented in
redshift space after applying scalar and harmonic decomposition methods. It
can be shown that the scalar perturbation equations for the matter, Chaplygin
gas and Gauss-Bonnet energy densities in redshift space evolve as [42, 43, 34,
41]

$\begin{split}&\Delta^{\prime}_{m}=\frac{1}{(1+z)H}\left(1+w_{m}+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\frac{G\big{(}-(1+z)H)\big{)}G^{\prime}}{\theta^{2}}\right)Z\\\
&-\frac{1}{(1+z)H}\frac{(1+w_{m})}{(1+w_{t})}\frac{\rho_{m}}{\rho_{t}}\omega_{m}\theta\Delta_{m}+\frac{1}{(1+z)H}\alpha\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}\Delta_{chp}\\\
&+\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}\mathsf{G}-\frac{1}{(1+z)H}\frac{(1+w_{m})}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})G(1+z)HG^{\prime}f^{\prime\prime\prime}\\\
&-\frac{(1+z)HG^{\prime}f^{\prime\prime}}{G^{2}}\Big{)}\mathcal{G}\end{split}$
$\displaystyle
Z^{\prime}=-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{m})\rho_{m}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{m}\rho_{m}}{(1+w_{t})\rho_{t}}-\frac{w_{m}\rho_{m}}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{]}\Delta_{m}+$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{chp})\rho_{chp}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}$
$\displaystyle\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}(\alpha+1)w_{chp}-\frac{\big{(}w_{chp}\rho_{chp}-(\alpha+1)w_{chp}\big{)}}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{]}\Delta_{chp}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}1-3f^{\prime\prime}-12H^{3}b_{1}f^{iv}+12H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{b_{3}}{b_{1}}f^{\prime\prime}+b_{1}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}$
$\displaystyle-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{(}24b_{1}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\frac{Gf^{\prime\prime}}{\theta}\Big{)}\mathbf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}8H\left(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}Hb_{1}\right)-\frac{2}{3}\theta-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}$
$\displaystyle+\frac{k^{2}}{a^{2}(1+w_{t})\rho_{t}}\frac{Gb_{1}}{\theta^{2}}\Big{]}Z$
(29)
$\displaystyle\mathcal{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}1-\frac{b_{1}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}-\frac{1}{(1+z)H}\frac{b_{1}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z+\frac{1}{(1+z)H}\frac{b_{1}w_{m}\rho_{m}\Delta_{m}}{(1+w_{t})\rho_{t}}$
$\displaystyle-\frac{1}{(1+z)H}\frac{\alpha
b_{1}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Delta_{ch}+\Big{(}\frac{1}{2(1+z)H}\frac{b_{1}}{(1+w_{t})\rho_{t}}\Big{(}f^{\prime}+Gf^{\prime\prime}-1\Big{)}$
$\displaystyle+\frac{1}{2(1+z)H}\frac{b_{1}^{2}}{(1+w_{t})\rho_{t}}\Big{(}Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}}\Big{)}\Big{)}\mathcal{G}$
(31)
$\displaystyle\mathsf{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}\frac{b_{3}}{b_{1}}-\frac{b_{1}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}b_{1}{G}}{(1+w_{t})\rho_{t}\theta^{2}}\Big{]}Z+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}}{(1+w_{t})\rho_{t}}w_{m}\rho_{m}\Big{]}\Delta_{m}$
$\displaystyle-\Big{[}\frac{1}{(1+z)H}\frac{\alpha
b_{2}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Big{]}\Delta_{ch}-\frac{1}{(1+z)H}\Big{[}\frac{b_{2}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{b_{2}b_{1}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}$
(32)
$\displaystyle\Delta^{\prime}_{G}=-\frac{1}{(1+z)H}\Big{[}-\frac{4H\theta}{\rho_{G}}\Big{(}2Hb_{1}f^{\prime\prime\prime}+(2c_{1}-2H^{2})f^{\prime\prime}\Big{)}$
$\displaystyle+\theta\Big{(}4H^{2}f^{\prime\prime}b_{2}+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}4H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{f^{\prime\prime}b_{2}}{b_{2}}+b_{2}^{2}f^{\prime\prime\prime})+(2c_{1}-H^{2})f^{\prime\prime}\Big{)}$
$\displaystyle+\theta\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1}))+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{(}\frac{\frac{1}{2}+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}+$
$\displaystyle-\frac{1}{3((1+z)H)\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2})-8Hf^{\prime\prime}b_{1}(\frac{G}{12H^{3}}+3H)$
$\displaystyle+(2c_{1}-H^{2})f^{\prime\prime}b_{1}-3\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}$
$\displaystyle+\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})-4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})(\frac{Gb_{1}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z$
$\displaystyle-\frac{3}{(1+z)\rho_{G}}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Delta_{G}$
$\displaystyle-\frac{3}{(1+z)}\Big{(}-4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\frac{w_{m}\rho_{m}(1-f^{\prime}-Gf^{\prime\prime})}{(1+w_{t}\rho_{t}\rho_{G})}\Delta_{m}$
$\displaystyle+\frac{3}{(1+z)}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}w_{ch}(\alpha+1)\Delta_{chp}$
(33)
$\displaystyle\Delta^{{}^{\prime}}_{ch}=\frac{1}{(1+z)H}[1+\frac{Gb_{1}}{\theta(1+w_{t})\rho_{t}}](1+w_{ch})Z-\frac{1}{(1+z)H}\frac{\theta(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{m}\rho_{m}\Delta_{m}$
$\displaystyle-\frac{3}{(1+z)}\Big{[}(1+\rho_{ch})\frac{(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}-(1+\alpha)w_{ch}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{\theta(1+w_{ch})}{(1+z)(1+w_{t})H\rho_{t}}[\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}]\mathcal{G}$
$\displaystyle-\frac{\theta(1+w_{ch})Gf^{\prime\prime}}{(1+z)(1+w_{t})H\rho_{t}}\mathsf{G}\;,$
(34)

with $k=\frac{2\pi a}{\lambda}$, $k$ being the wave number and $\lambda$, the
wavelength of perturbations. The parameters $b_{1}$, $b_{2}$, $b_{3}$ and
$c_{1}$ are presented in Appendix. According to eq. 7, the vector term
$a\tilde{\bigtriangledown}_{a}w_{ch}$ presents its scalar part as
$a\tilde{\bigtriangledown}_{a}w_{ch}=-w_{ch}(1+\alpha)\Delta_{ch}$. In GR
limits, with normal form of matter, one can obtain a closed system of first-
order perturbation equations which is easier to find the analytical solutions.
However, the linear perturbation equations are coupled system of first -order
equations for the density fluctuations of matter, chaplygin and Gauss-Bonnet
fluids which are more complicated to find the analytical solutions. We
therefore considered short wavelength($\frac{k^{2}}{a^{2}H^{2}}\gg 1$) and
long wavelength($\frac{k^{2}}{a^{2}H^{2}}\ll 1$) limits of the perturbation to
analyse the large scale structure implications of the numerical results in the
redshift space.

##  4 Asymptotic Analysis

In this section , we analyse the evolution of the perturbation equations in
both long-wavelength and short-wavelength regimes together with the
consideration that the universe is dominated by dust.

###  4.1 Matter density fluctuations in GR limits

In this part, we analyse the behavior of energy overdensity fluctuations for
dust fluid in GR limits for both long and short wavelength modes for the case
$f(G)=G$ and no contribution from the chaplygin gas fluid. We also define the
normalised energy density contrast for energy density fluid as

$\delta(z)=\frac{\Delta^{k}_{m}(z)}{\Delta(z_{0})},$

where $\Delta(z_{0})$ is the matter energy density at the initial redshift,
hereafter $z_{0}=1100$. If we assume that the Universe is dominated mainly by
dust fluid, the equation of state parameter becomes $w=0$. Consequently eq. 3
through to eq. 34 become

$\displaystyle\Delta^{\prime}_{m}=\frac{1}{(1+z)H}Z,$ (35) $\displaystyle
Z^{\prime}=\frac{\rho_{d}}{2(1+z)H},$ (36)
$\displaystyle\mathcal{G}^{\prime}=0,\mathsf{G}^{\prime}=0,\Delta^{\prime}_{G}=0,\Delta^{\prime}_{ch}=0,$
(37)

which admit the numerical solutions presented in fig. 1 for dust dominated
universe. The matter energy overdensity fluctuations $\delta(z)$ decay with
increase in redshift. Throughout all the plots, we rescaled the $\delta(z)$ to
make it readable.

Figure 1: Plot of matter energy overdensity perturbations versus redshift of
Eq. 35 using eq. 36-37 for GR limit ($f(G)=G$) for a dust($w=0$) dominated
Universe for $\Delta_{in}=10^{-5}$, $Z_{in}=10^{-5}$.

###  4.2  Energy overdensity fluctuations in the context of $f(G)$ gravity
approach

For pedagogical purpose, we consider a polynomial $f(G)$ model presents in
[34] given by

$f(G)=\frac{\alpha_{1}}{G^{2}}+\alpha_{2}G^{\frac{1}{2}},$ (38)

We choose this $f(G)$ model for a quantitative analysis of the derived
perturbation equations eq. 3 through to eq. 34, it is a viable model which is
compatible with cosmological observations and it is a viable model accounts
for the late-time acceleration of the universe without the need for dark
energy. The Hubble parameter in redshift space for a dust dominated universe
($w=0$) is presented as $H(z)=\frac{2m}{3}(1+z)^{\frac{3}{2m}}$, for GR
limits, we set $m=1$. The Gauss-Bonnet term is given by
$G=24H^{2}\left(H^{2}+\dot{H}\right)$ which in redshift space is presented as
$G=\frac{64}{9}m^{3}\Big{(}\frac{2m}{3}-1\Big{)}(1+z)^{\frac{6}{m}}$. We
assume that the dynamics of the universe is driven by the power-law scale
factor of the form $a(t)=t^{\frac{2m}{3(1+w)}}$ and we consider
$H=\frac{\dot{a}}{a}$. We analyse the perturbation equations in both short-
and long- wavelength modes.

####  4.2.1 Short-Wavelength mode

In this subsection, we first consider the short-wavelength regime where
$\frac{k^{2}}{a^{2}}>>1$, with $k=\frac{2\pi a}{\lambda}$ and $\lambda$ is the
wavelength of the mode. Applying this approximation and considering the dust
dominated universe, it means $w_{m}=0$, eq. 3 through to eq. 34 are
represented as

$\displaystyle\Delta^{\prime}_{m}=\frac{1}{(1+z)H}\Big{(}\Big{(}1-\frac{1}{(1+w_{t})\rho_{t}}\Big{)}\frac{(1+z)HGG^{\prime}}{\theta^{2}}\Big{)}Z$
$\displaystyle+\frac{1}{(1+z)H}\alpha\frac{1}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}\Delta_{chp}$
$\displaystyle+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}\mathsf{G}-\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})G(1+z)HG^{\prime}f^{\prime\prime\prime}$
$\displaystyle-\frac{(1+z)HG^{\prime}f^{\prime\prime}}{G^{2}}\Big{)}\mathcal{G}$
(39) $\displaystyle
Z^{\prime}=\frac{1}{(1+z)H}\Big{(}\frac{1}{2}\rho_{d}+\frac{1}{3}\theta^{2}\Big{)}\Delta_{m}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{chp})\rho_{chp}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}$
$\displaystyle\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}(\alpha+1)w_{chp}-\frac{\big{(}w_{chp}\rho_{chp}-(\alpha+1)w_{chp}\big{)}}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{]}\Delta_{chp}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}1-3f^{\prime\prime}-12H^{3}b_{1}f^{iv}+12H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{b_{3}}{b_{1}}f^{\prime\prime}+b_{1}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}$
$\displaystyle-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{(}24b_{1}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\frac{Gf^{\prime\prime}}{\theta}\Big{)}\mathbf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}8H\left(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}Hb_{1}\right)-\frac{2}{3}\theta-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}$
$\displaystyle+\frac{k^{2}}{a^{2}(1+w_{t})\rho_{t}}\frac{Gb_{1}}{\theta^{2}}\Big{]}Z$
(40)
$\displaystyle\mathcal{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}1-\frac{b_{1}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}-\frac{1}{(1+z)H}\frac{b_{1}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z$
$\displaystyle-\frac{1}{(1+z)H}\frac{\alpha
b_{1}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Delta_{ch}+\Big{(}\frac{1}{2(1+z)H}\frac{b_{1}}{(1+w_{t})\rho_{t}}\Big{(}f^{\prime}+Gf^{\prime\prime}-1\Big{)}$
$\displaystyle+\frac{1}{2(1+z)H}\frac{b_{1}^{2}}{(1+w_{t})\rho_{t}}\Big{(}Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}}\Big{)}\Big{)}\mathcal{G}$
(42)
$\displaystyle\mathsf{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}\frac{b_{3}}{b_{1}}-\frac{b_{1}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}b_{1}{G}}{(1+w_{t})\rho_{t}\theta^{2}}\Big{]}Z-\Big{[}\frac{1}{(1+z)H}\frac{\alpha
b_{2}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}\frac{b_{2}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{b_{2}b_{1}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}$
(43)
$\displaystyle\Delta^{\prime}_{G}=-\frac{1}{(1+z)H}\Big{[}-\frac{4H\theta}{\rho_{G}}\Big{[}2Hb_{1}f^{\prime\prime\prime}+(2c_{1}-2H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}f^{\prime\prime}b_{2}+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}4H\Big{[}H(b_{2}f^{\prime\prime\prime}+\frac{f^{\prime\prime}b_{2}}{b_{2}}+b_{2}^{2}f^{\prime\prime\prime})+(2c_{1}-H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1}))+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{(}\frac{\frac{1}{2}+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}+$
$\displaystyle-\frac{1}{3((1+z)H)\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2})-8Hf^{\prime\prime}b_{1}(\frac{G}{12H^{3}}+3H)$
$\displaystyle+(2c_{1}-H^{2})f^{\prime\prime}b_{1}-3\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}$
$\displaystyle+\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})-4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})(\frac{Gb_{1}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z$
$\displaystyle-\frac{3}{(1+z)\rho_{G}}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Delta_{G}$
$\displaystyle+\frac{3}{(1+z)}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}w_{ch}(\alpha+1)\Delta_{chp}$
(44)
$\displaystyle\Delta^{{}^{\prime}}_{ch}=\frac{1}{(1+z)H}[1+\frac{Gb_{1}}{\theta(1+w_{t})\rho_{t}}](1+w_{ch})Z$
$\displaystyle-\frac{3}{(1+z)}\Big{[}(1+\rho_{ch})\frac{(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}-(1+\alpha)w_{ch}\Big{]}\Delta_{ch}-$
$\displaystyle\frac{\theta(1+w_{ch})}{(1+z)(1+w_{t})H\rho_{t}}[\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}]\mathcal{G}-\frac{\theta(1+w_{ch})Gf^{\prime\prime}}{(1+z)(1+w_{t})H\rho_{t}}\mathsf{G}\;$
(45)

The energy density for the chaplygin gas fluid $\Delta_{ch}$ and the energy
density for Gauss-Bonnet fluid $\Delta_{G}$ do not couple with the matter
energy density $\Delta_{m}$. Numerical solutions of eq. 39 through to eq. 45
are presented in fig. 2 for different values of parameter $m$.

Figure 2: Plot of energy density perturbations versus redshift of Eq. 39-45
for different values of $m$ in the dust dominated Universe for a short
wavelength mode $k=100$ and $\alpha=0.2$.

For simplicity we set $f(G)$ to $f$ and considered the initial conditions
$\mathcal{G}_{in}=10^{-5}$, $\mathbf{G}_{in}=10^{-5}$ and
$\Delta_{in}=10^{-5}$ to find numerical solutions. The equations above are
presented for the generalized Chaplygin gas model. During numerical results
computation, where there is $w_{t}$ and $\rho_{t}$ terms, we replaced these
terms by relations of eq. 3-7 and eq. 12-13 throughout all equations. The
original Chaplygin gas model can be obtained by setting $\alpha=1$ in eq. 39
through to eq. 45 and its evolution equations are presented as

$\displaystyle\Delta^{\prime}_{m}=\frac{1}{(1+z)H}\Big{(}\Big{(}1-\frac{1}{(1+w_{t})\rho_{t}}\Big{)}\frac{(1+z)HGG^{\prime}}{\theta^{2}}\Big{)}Z+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}\Delta_{chp}$
$\displaystyle+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}\mathsf{G}-\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})G(1+z)HG^{\prime}f^{\prime\prime\prime}$
$\displaystyle-\frac{(1+z)HG^{\prime}f^{\prime\prime}}{G^{2}}\Big{)}\mathcal{G}$
(46) $\displaystyle
Z^{\prime}=\frac{1}{(1+z)H}\Big{(}\frac{1}{2}\rho_{d}+\frac{1}{3}\theta^{2}\Big{)}\Delta_{m}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{chp})\rho_{chp}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}$
$\displaystyle
2\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}w_{chp}-\frac{\big{(}w_{chp}\rho_{chp}-2w_{chp}\big{)}}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}1-3f^{\prime\prime}-12H^{3}b_{1}f^{iv}+12H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{b_{3}}{b_{1}}f^{\prime\prime}+b_{1}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}$
$\displaystyle-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{(}24b_{1}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}-\frac{1}{(1+w_{t})\rho_{t}}\frac{k^{2}}{a^{2}}\frac{Gf^{\prime\prime}}{\theta}\Big{)}\mathbf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}8H\left(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}Hb_{1}\right)-\frac{2}{3}\theta-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}$
$\displaystyle+\frac{k^{2}}{a^{2}(1+w_{t})\rho_{t}}\frac{Gb_{1}}{\theta^{2}}\Big{]}Z$
(47)
$\displaystyle\mathcal{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}1-\frac{b_{1}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}-\frac{1}{(1+z)H}\frac{b_{1}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z$
$\displaystyle-\frac{1}{(1+z)H}\frac{b_{1}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Delta_{ch}+\Big{(}\frac{1}{2(1+z)H}\frac{b_{1}}{(1+w_{t})\rho_{t}}\Big{(}f^{\prime}+Gf^{\prime\prime}-1\Big{)}$
$\displaystyle+\frac{1}{2(1+z)H}\frac{b_{1}^{2}}{(1+w_{t})\rho_{t}}\Big{(}Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}}\Big{)}\Big{)}\mathcal{G}$
(48)
$\displaystyle\mathsf{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}\frac{b_{3}}{b_{1}}-\frac{b_{1}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}b_{1}{G}}{(1+w_{t})\rho_{t}\theta^{2}}\Big{]}Z-\Big{[}\frac{1}{(1+z)H}\frac{b_{2}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}\frac{b_{2}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{b_{2}b_{1}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}$
(49)
$\displaystyle\Delta^{\prime}_{G}=-\frac{1}{(1+z)H}\Big{[}-\frac{4H\theta}{\rho_{G}}\Big{[}2Hb_{1}f^{\prime\prime\prime}+(2c_{1}-2H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}f^{\prime\prime}b_{2}+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}4H\Big{[}H(b_{2}f^{\prime\prime\prime}+\frac{f^{\prime\prime}b_{2}}{b_{2}}+b_{2}^{2}f^{\prime\prime\prime})+(2c_{1}-H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1}))+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{(}\frac{\frac{1}{2}+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}+$
$\displaystyle-\frac{1}{3((1+z)H)\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2})-8Hf^{\prime\prime}b_{1}(\frac{G}{12H^{3}}+3H)$
$\displaystyle+(2c_{1}-H^{2})f^{\prime\prime}b_{1}-3\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}$
$\displaystyle+\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})-4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})(\frac{Gb_{1}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z$
$\displaystyle-\frac{3}{(1+z)\rho_{G}}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Delta_{G}$
$\displaystyle+\frac{6}{(1+z)}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}w_{ch}\Delta_{chp}$
(50)
$\displaystyle\Delta^{{}^{\prime}}_{ch}=\frac{1}{(1+z)H}[1+\frac{Gb_{1}}{\theta(1+w_{t})\rho_{t}}](1+w_{ch})Z$
$\displaystyle-\frac{3}{(1+z)}\Big{[}(1+\rho_{ch})\frac{(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}-(1+\alpha)w_{ch}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{\theta(1+w_{ch})}{(1+z)(1+w_{t})H\rho_{t}}[\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}]\mathcal{G}$
$\displaystyle-\frac{\theta(1+w_{ch})Gf^{\prime\prime}}{(1+z)(1+w_{t})H\rho_{t}}\mathsf{G}$
(51)

which admit numerical solutions presented in fig. 3.

Figure 3: Plot of energy density perturbations versus redshift of Eq. 46-51
for different values of $m$ in the dust dominated Universe for short
wavelength mode $k=1000$ in a generalized Chaplygin model $\alpha=1$.

We considered the initial conditions $\mathcal{G}_{in}=10^{-5}$,
$\mathbf{G}_{in}=10^{-5}$ and $\Delta_{in}=10^{-5}$ to find numerical
solutions of Eq. 46 through to Eq. 51. From the plot, the energy overdensity
decay with increase in redshift.

####  4.2.2 Long-Wavelength mode

In this subsection, we consider the long-wavelength regime where
$\frac{k^{2}}{a^{2}}<<1$, this means that all $k$’s terms become negligible.
Applying this approximation, and consider the dust dominated Universe, eq.3
through to eq. 34 are represented as

$\displaystyle\Delta^{\prime}_{m}=\frac{1}{(1+z)H}\Big{(}\Big{(}1-\frac{1}{(1+w_{t})\rho_{t}}\Big{)}\frac{(1+z)HGG^{\prime}}{\theta^{2}}\Big{)}Z$
$\displaystyle+\frac{1}{(1+z)H}\alpha\frac{1}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}\Delta_{chp}$
$\displaystyle+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}\mathsf{G}-\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})G(1+z)HG^{\prime}f^{\prime\prime\prime}$
$\displaystyle-\frac{(1+z)HG^{\prime}f^{\prime\prime}}{G^{2}}\Big{)}\mathcal{G}$
(52) $\displaystyle
Z^{\prime}=\frac{1}{(1+z)H}\Big{(}\frac{1}{2}\rho_{d}+\frac{1}{3}\theta^{2}\Big{)}\Delta_{m}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{chp})\rho_{chp}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}$
$\displaystyle\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}(\alpha+1)w_{chp}\Big{]}\Delta_{chp}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}1-3f^{\prime\prime}-12H^{3}b_{1}f^{iv}+12H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{b_{3}}{b_{1}}f^{\prime\prime}+b_{1}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{(}24b_{1}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}\Big{)}\mathbf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}8H\left(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}Hb_{1}\right)-\frac{2}{3}\theta$
$\displaystyle-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{]}Z$
(53)
$\displaystyle\mathcal{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}1-\frac{b_{1}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}-\frac{1}{(1+z)H}\frac{b_{1}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z$
$\displaystyle-\frac{1}{(1+z)H}\frac{\alpha
b_{1}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Delta_{ch}+\Big{(}\frac{1}{2(1+z)H}\frac{b_{1}}{(1+w_{t})\rho_{t}}\Big{(}f^{\prime}+Gf^{\prime\prime}-1\Big{)}$
$\displaystyle+\frac{1}{2(1+z)H}\frac{b_{1}^{2}}{(1+w_{t})\rho_{t}}\Big{(}Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}}\Big{)}\Big{)}\mathcal{G}$
(54)
$\displaystyle\mathsf{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}\frac{b_{3}}{b_{1}}-\frac{b_{1}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}b_{1}{G}}{(1+w_{t})\rho_{t}\theta^{2}}\Big{]}Z-\Big{[}\frac{1}{(1+z)H}\frac{\alpha
b_{2}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}\frac{b_{2}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{b_{2}b_{1}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}$
(55)
$\displaystyle\Delta^{\prime}_{G}=-\frac{1}{(1+z)H}\Big{[}-\frac{4H\theta}{\rho_{G}}\Big{[}2Hb_{1}f^{\prime\prime\prime}+(2c_{1}-2H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}f^{\prime\prime}b_{2}+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}4H\Big{[}H(b_{2}f^{\prime\prime\prime}+\frac{f^{\prime\prime}b_{2}}{b_{2}}+b_{2}^{2}f^{\prime\prime\prime})+(2c_{1}-H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1}))+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{(}\frac{\frac{1}{2}+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}+$
$\displaystyle-\frac{1}{3((1+z)H)\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2})-8Hf^{\prime\prime}b_{1}(\frac{G}{12H^{3}}+3H)$
$\displaystyle+(2c_{1}-H^{2})f^{\prime\prime}b_{1}-3\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}$
$\displaystyle+\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})-4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})(\frac{Gb_{1}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z$
$\displaystyle-\frac{3}{(1+z)\rho_{G}}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Delta_{G}$
$\displaystyle+\frac{3}{(1+z)}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}w_{ch}(\alpha+1)\Delta_{chp}$
(56)
$\displaystyle\Delta^{{}^{\prime}}_{ch}=\frac{1}{(1+z)H}[1+\frac{Gb_{1}}{\theta(1+w_{t})\rho_{t}}](1+w_{ch})Z$
$\displaystyle-\frac{3}{(1+z)}\Big{[}(1+\rho_{ch})\frac{(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}-(1+\alpha)w_{ch}\Big{]}\Delta_{ch}-$
$\displaystyle\frac{\theta(1+w_{ch})}{(1+z)(1+w_{t})H\rho_{t}}[\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}]\mathcal{G}-\frac{\theta(1+w_{ch})Gf^{\prime\prime}}{(1+z)(1+w_{t})H\rho_{t}}\mathsf{G}$
(57)

Figure 4: Plot of energy density perturbations versus redshift of Eq. 52-57
for different values of $m$ in the dust dominated Universe for long wavelength
mode $k=0$ and $\alpha=0.2$. Figure 5: Plot of energy density perturbations
versus redshift of Eq. 52-57 for different values of $m$ in the dust dominated
Universe for long wavelength mode $k=0.001$ and $\alpha=0.2$.

For simplicity we set $f(G)$ to $f$ and considered the initial conditions
$\mathcal{G}_{in}=10^{-5}$, $\mathbf{G}_{in}=10^{-5}$ and
$\Delta_{in}=10^{-5}$ to find numerical solutions of Eq. 52 through to Eq. 57
which are presented in Fig. 4 through to Fig. 5. From the plots, the energy
overdensity decay with increase in redshift. The Eq. 52 through to Eq. 57 are
presented for the generalized Chaplygin gas model. The original Chaplygin gas
model can be obtained by setting $\alpha=1$ in eq. 52 through to eq. 57 and
its evolution equations are presented as

$\displaystyle\Delta^{\prime}_{m}=\frac{1}{(1+z)H}\Big{(}\Big{(}1-\frac{1}{(1+w_{t})\rho_{t}}\Big{)}\frac{(1+z)HGG^{\prime}}{\theta^{2}}\Big{)}Z$
$\displaystyle+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}w_{chp}\rho_{chp}\Delta_{chp}+\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\frac{Gf^{\prime\prime}}{\theta}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})G(1+z)HG^{\prime}f^{\prime\prime\prime}-\frac{(1+z)HG^{\prime}f^{\prime\prime}}{G^{2}}\Big{)}\mathcal{G},$
(58) $\displaystyle
Z^{\prime}=\frac{1}{(1+z)H}\Big{(}\frac{1}{2}\rho_{d}+\frac{1}{3}\theta^{2}\Big{)}\Delta_{m}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}-\frac{1}{2}(1+3w_{chp})\rho_{chp}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{w_{chp}\rho_{chp}}{(1+w_{t})\rho_{t}}$
$\displaystyle
2\Big{[}\frac{3}{2}\rho_{chp}-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{\rho_{chp}}{(1+w_{t})\rho_{t}}\Big{]}w_{chp}\Big{]}\Delta_{chp}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}1-3f^{\prime\prime}-12H^{3}b_{1}f^{iv}+12H\Big{(}H(b_{2}f^{\prime\prime\prime}+\frac{b_{3}}{b_{1}}f^{\prime\prime}+b_{1}^{2}f^{iv})+2H^{3}f^{\prime\prime}\Big{)}$
$\displaystyle+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{1}{(1+w_{t})\rho_{t}}\Big{(}\frac{1}{2}(1-f^{\prime}-Gf^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime\prime}+\frac{b_{1}f^{\prime\prime}}{G^{2}}\Big{)}\Big{]}\mathcal{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{(}24b_{1}f^{\prime\prime\prime}-12H^{3}f^{\prime\prime\prime}+\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}}\Big{)}\mathbf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}8H\left(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2}+3Hf^{\prime}-\frac{3}{2}f^{\prime\prime\prime}Hb_{1}\right)-\frac{2}{3}\theta$
$\displaystyle-\big{(}\frac{1}{3}\theta^{2}+\frac{1}{2}(1+3w_{t})\rho_{t}\big{)}\frac{Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{]}Z$
(59)
$\displaystyle\mathcal{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}1-\frac{b_{1}Gf^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}-\frac{1}{(1+z)H}\frac{b_{1}^{2}G}{(1+w_{t})\rho_{t}\theta^{2}}Z$
$\displaystyle-\frac{1}{(1+z)H}\frac{b_{1}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Delta_{ch}+\Big{(}\frac{1}{2(1+z)H}\frac{b_{1}}{(1+w_{t})\rho_{t}}\Big{(}f^{\prime}+Gf^{\prime\prime}-1\Big{)}$
$\displaystyle+\frac{1}{2(1+z)H}\frac{b_{1}^{2}}{(1+w_{t})\rho_{t}}\Big{(}Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}}\Big{)}\Big{)}\mathcal{G}$
(60)
$\displaystyle\mathsf{G}^{\prime}=-\frac{1}{(1+z)H}\Big{(}\frac{b_{3}}{b_{1}}-\frac{b_{1}{G}f^{\prime\prime}}{(1+w_{t})\rho_{t}\theta}\Big{)}\mathsf{G}+\Big{[}\frac{1}{(1+z)H}\frac{b_{2}b_{1}{G}}{(1+w_{t})\rho_{t}\theta^{2}}\Big{]}Z-\Big{[}\frac{1}{(1+z)H}\frac{b_{2}\rho_{ch}w_{ch}}{(1+w_{t})\rho_{t}}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}\frac{b_{2}}{2(1+w_{t})\rho_{t}}(f^{\prime}+Gf^{\prime\prime}-1)-\frac{b_{2}b_{1}}{(1+w_{t})\rho_{t}}(Gf^{\prime\prime}+\frac{f^{\prime\prime}}{\theta^{2}})\Big{]}\mathcal{G}$
(61)
$\displaystyle\Delta^{\prime}_{G}=-\frac{1}{(1+z)H}\Big{[}-\frac{4H\theta}{\rho_{G}}\Big{[}2Hb_{1}f^{\prime\prime\prime}+(2c_{1}-2H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}f^{\prime\prime}b_{2}+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{Gf^{\prime\prime}}{\theta(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathsf{G}$
$\displaystyle-\frac{1}{(1+z)H}\Big{[}4H\Big{[}H(b_{2}f^{\prime\prime\prime}+\frac{f^{\prime\prime}b_{2}}{b_{2}}+b_{2}^{2}f^{\prime\prime\prime})+(2c_{1}-H^{2})f^{\prime\prime}\Big{]}$
$\displaystyle+\theta\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1}))+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{(}\frac{\frac{1}{2}+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}\Big{]}\mathcal{G}+$
$\displaystyle-\frac{1}{3((1+z)H)\rho_{G}}\Big{[}\Big{(}2H(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b_{1}^{2})-8Hf^{\prime\prime}b_{1}(\frac{G}{12H^{3}}+3H)$
$\displaystyle+(2c_{1}-H^{2})f^{\prime\prime}b_{1}-3\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}$
$\displaystyle+\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})-4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})(\frac{Gb_{1}\theta^{2}}{(1+w_{t})\rho_{t}\rho_{G}})\Big{)}$
$\displaystyle+3\theta(4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1})\Big{(}\frac{(1-f^{\prime}-Gf^{\prime\prime})Gb_{1}}{\theta^{2}(1+w_{t})\rho_{t}}\Big{)}\Big{]}Z$
$\displaystyle-\frac{3}{(1+z)\rho_{G}}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Delta_{G}$
$\displaystyle+\frac{6}{(1+z)}\Big{(}4H^{2}(f^{\prime\prime}b_{2}+f^{\prime\prime\prime}b^{2}_{1})+4H(2c_{1}-H^{2})f^{\prime\prime}b_{1}\Big{)}\Big{(}\frac{w_{ch}\rho_{ch}}{(1+w_{t})\rho_{t}\rho_{G}}\Big{)}w_{ch}\Delta_{ch}$
(62)
$\displaystyle\Delta^{{}^{\prime}}_{ch}=\frac{1}{(1+z)H}\Big{[}1+\frac{Gb_{1}}{\theta(1+w_{t})\rho_{t}}\Big{]}(1+w_{ch})Z-\frac{3}{(1+z)}\Big{[}(1+\rho_{ch})\frac{(1+w_{ch})}{(1+w_{t})\rho_{t}}w_{ch}\rho_{ch}-2w_{ch}\Big{]}\Delta_{ch}$
$\displaystyle-\frac{\theta(1+w_{ch})}{(1+z)(1+w_{t})H\rho_{t}}\Big{[}\frac{1}{2}(1-f^{\prime}-Gf^{\prime\prime})+Gb_{1}f^{\prime\prime}+\frac{b_{1}f^{\prime\prime\prime}}{\theta^{2}}\Big{]}\mathcal{G}-\frac{\theta(1+w_{ch})Gf^{\prime\prime}}{(1+z)(1+w_{t})H\rho_{t}}\mathsf{G}$
(63)

which admit the numerical solutions presented in fig. 6.

Figure 6: Plot of energy density perturbations versus redshift of Eq. 58-63
for different values of $m$ in the dust dominated Universe for long wavelength
mode $k=0.00001$ in a generalized Chaplygin model $\alpha=1$.

We considered the initial conditions $\mathcal{G}_{in}=10^{-5}$,
$\mathbf{G}_{in}=10^{-5}$ and $\Delta_{in}=10^{-5}$ to find numerical
solutions of Eq. 58 through to Eq. 63. The energy overdensity fluctuations
decay with increase in redshift.

##  5 Discussion and Conclusions

The current treatment is focused on a mixture of matter fluid, chaplygin gas
and Gauss-Bonnet fluid as non-interacting entities to study cosmological
perturbations in $f(G)$ gravity using $1+3$ covariant Gauge invariant
formalism. After presenting the covariant form of field equations and the
kinematic quantities in the presence of Gauss-Bonnet invariant, we define
matter fluids, chaplygin gas fluids and Gauss-Bonnet fluids to derive first
order evolution equations for the defined gradient variables. Since our main
interest lies in the structure formation of the Universe, we use scalar
decomposition method to decompose the evolotion equations into their scalar
parts believed to be responsible for the spherical clustering of large scale
structure. These scalar perturbation equations are considered as input to
study the energy overdensity fluctuations in a dust-Chaplygin gas-Gauss-Bonnet
system by applying harmonic decomposition method. After we transform the
linear perturbation equations into redshift space for further analysis. We
apply both the generalized and original chaplygin gas together with the $f(G)$
model to find numerical results and analyse how the energy density
fluctuations evolve with redshift. We considered the short wavelength limits
of the perturbation equations in the Universe dominated by a mixture of dust-
chaplygin gas-Gauss-Bonnet fluids system and we depict that the energy
overdensity perturbations decay with increase in redshift. From all plots, we
observe that the decay of matter density contrast is fast decaying with
redshift compared to the matter-Chaplygin-Gauss-Bonnet fluids system, since
the universe is believed to become less structured in the accelerated phase of
the universe evolution. We have considered different initial conditions such
as $\mathcal{G}_{in}=10^{-5}$, $\mathbf{G}_{in}=10^{-5}$ and
$\Delta_{in}=10^{-5}$ to find numerical solutions of different perturbation
equations. The use of initial conditions constrained the amplitude of matter
energy densities but does not alter the behaviour of the curves, but the
values of the parameter $m$, $k$ and other considered parameters changes the
behavior of the curves. From all the plots, there is no oscilatory behaviours
observed.

Some of the specific highlights of the present work include:

  * •

We have derived the linear perturbation equations eq. 3-34 in the context of
$f(G)$ gravity, believed to contribute to the large scale structure formation.

  * •

The matter energy density fluctuations couples with the energy density
fluctuations of the chaplygin gas and that of Gauss-Bonnet fluids which then
decouple for the GR limits.

  * •

In the analysis stage, we considered short wavelength and long wavelength
modes for the energy overdensity fluctuations in a Universe dominated by a
mixture of a dust-chaplygin gas-Gauss-Bonnet fluids by combining chaplygin gas
and the $f(G)$ models which can mimics the $\Lambda$CDM for the $f(G)=G$ case.

  * •

We solved the perturbation equations using numerical method and the numerical
results of the energy overdensity fluctuations are presented in Fig. 1 GR
limits and Fig. 2-6 for the combination of chaplygin gas and $f(G)$ model in
the short and long wavelength modes. From these figures the energy overdensity
decay with redshift for the considered range of parameter $m$.

  * •

From the plots, the numerical results of the perturbation equations are
sensible to the parameter $m$. As $m$ changes, we notice an change in the
amplitudes of the energy overdensity fluctuations whih reduces to the GR
limits for $m=1$ but the results do not depend much on the parameter $\alpha$.

  * •

Under the consideration of both original and generalized Chaplygin gas models,
it was observed that there is no significant difference in the behavior of the
results.

In general, the combination of chaplygin gas models and $f(G)$ model offers an
alternative for large scale structure formation scenarios even though there is
a variation in the growth of amplitudes of the energy overdensity fluctuations
which can be beneficial for constraining the parameters of the models by
referring to the observational data which provide different cosmological
scenarios that are consistent with the $\Lambda$CDM limits. We look forward to
undertake this aspect of the task to different chaplygin gas models as well as
$f(G)$ models in the $1+3$ covariant cosmological perturbations framework for
the future work.

## Appendix

## Useful Linearised Differential Identities

For all scalars $f$, vectors $V_{a}$ and tensors that vanish in the
background, $S_{ab}=S_{\langle ab\rangle}$, the following linearised
identities hold:

$\displaystyle\left(\tilde{\nabla}_{\langle
a}\tilde{\nabla}_{b\rangle}f\right)^{.}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}_{\langle
a}\tilde{\nabla}_{b\rangle}\dot{f}-{\textstyle{2\over
3}}\Theta\tilde{\nabla}_{\langle
a}\tilde{\nabla}_{b\rangle}f+\dot{f}\tilde{\nabla}_{\langle a}A_{b\rangle}\;,$
(64) $\displaystyle\varepsilon^{abc}\tilde{\nabla}_{b}\tilde{\nabla}_{c}f$
$\displaystyle=$ $\displaystyle 0\;,$ (65)
$\displaystyle\varepsilon_{cda}\tilde{\nabla}^{c}\tilde{\nabla}_{\langle
b}\tilde{\nabla}^{d\rangle}f$ $\displaystyle=$
$\displaystyle\varepsilon_{cda}\tilde{\nabla}^{c}\tilde{\nabla}_{(b}\tilde{\nabla}^{d)}f=\varepsilon_{cda}\tilde{\nabla}^{c}\tilde{\nabla}_{b}\tilde{\nabla}^{d}f=0\;,$
(66) $\displaystyle\tilde{\nabla}^{2}\left(\tilde{\nabla}_{a}f\right)$
$\displaystyle=$
$\displaystyle\tilde{\nabla}_{a}\left(\tilde{\nabla}^{2}f\right)+{\textstyle{1\over
3}}\tilde{R}\tilde{\nabla}_{a}f\;,$ (67)
$\displaystyle\left(\tilde{\nabla}_{a}f\right)^{\displaystyle{\cdot}}$
$\displaystyle=$ $\displaystyle\tilde{\nabla}_{a}\dot{f}-{\textstyle{1\over
3}}\Theta\tilde{\nabla}_{a}f+\dot{f}A_{a}\;,$ (68)
$\displaystyle\left(\tilde{\nabla}_{a}S_{b\cdots}\right)^{\displaystyle{\cdot}}$
$\displaystyle=$
$\displaystyle\tilde{\nabla}_{a}\dot{S}_{b\cdots}-{\textstyle{1\over
3}}\Theta\tilde{\nabla}_{a}S_{b\cdots}\;,$ (69)
$\displaystyle\left(\tilde{\nabla}^{2}f\right)^{\displaystyle{\cdot}}$
$\displaystyle=$ $\displaystyle\tilde{\nabla}^{2}\dot{f}-{\textstyle{2\over
3}}\Theta\tilde{\nabla}^{2}f+\dot{f}\tilde{\nabla}^{a}A_{a}\;,$ (70)
$\displaystyle\tilde{\nabla}_{[a}\tilde{\nabla}_{b]}V_{c}$ $\displaystyle=$
$\displaystyle-{\textstyle{1\over 6}}\tilde{R}V_{[a}h_{b]c}\;,$ (71)
$\displaystyle\tilde{\nabla}_{[a}\tilde{\nabla}_{b]}S^{cd}$ $\displaystyle=$
$\displaystyle-{\textstyle{1\over 3}}\tilde{R}S_{[a}{}^{(c}h_{b]}{}^{d)}\;,$
(72)
$\displaystyle\tilde{\nabla}^{a}\left(\varepsilon_{abc}\tilde{\nabla}^{b}V^{c}\right)$
$\displaystyle=$ $\displaystyle 0\;,$ (73)
$\displaystyle\tilde{\nabla}_{b}\left(\varepsilon^{cd\langle
a}\tilde{\nabla}_{c}S^{b\rangle}_{d}\right)$ $\displaystyle=$
$\displaystyle{\textstyle{1\over
2}}\varepsilon^{abc}\tilde{\nabla}_{b}\left(\tilde{\nabla}_{d}S^{d}_{c}\right)\;,$
(74) $\displaystyle\text{curlcurl}V_{a}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}_{a}\left(\tilde{\nabla}^{b}V_{b}\right)-\tilde{\nabla}^{2}V_{a}+{\textstyle{1\over
3}}\tilde{R}V_{a}\;,$ (75)

## Used parameters in redshif space

$\displaystyle H=\frac{2m}{3}(1+z)^{\frac{3}{2m}}$ (76)
$\displaystyle\dot{H}=c_{1}=-\frac{2m}{3}(1+z)^{\frac{3}{m}}$ (77)
$\displaystyle\dot{G}=b_{1}=\frac{256}{9}m^{3}\Big{(}1-\frac{2m}{3}\Big{)}(1+z)^{\frac{15}{2m}}$
(78)
$\displaystyle\ddot{G}=b_{2}=1280m^{3}\Big{(}\frac{2m}{3}-1\Big{)}(1+z)^{\frac{9}{m}}$
(79)
$\displaystyle\dddot{G}=b_{3}=\frac{2560}{3}m^{3}\Big{(}1-\frac{2m}{3}\Big{)}(1+z)^{\frac{21}{2m}}$
(80) $\displaystyle\dot{f}=-(1+z)Hf^{\prime}.$ (81)

## Acknowledgments

We thank the anonymous reviewer(s) for the constructive comments towards the
significant improvement of this manuscript. The authors acknowledge the
financial support from UR-SIDA Grant No. DVC-AAR506/2022 . AM acknowledges the
support from International Science Program (ISP) to Rwanda Astrophysics, Space
and Climate Science Research Group (RASCSRG), University of Rwanda grant No:
RWA01 and to East African Astrophysics Research Network(EAARN) grant No:
Afro:05. AM also acknowledges the hospitality of the department of Physics of
the University of Rwanda, where part of this work was conceptualise and
completed.

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