###### Abstract

We explore some cosmological features of the newly suggested 4D Gauss-Bonnet
gravity through two different models assuming a varying cosmological constant
$\Lambda(t)$. Observational constraints, such as the cosmic transit and the
flat curvature, have been considered in constructing the models. The cosmology
in the current work has been probed using a given scale factor derived from
the desired cosmic behavior which is the inverse of the usual viewpoint.
Several models for $\Lambda(t)$ have been proposed in the literature, we use
two different ansatze of varying $\Lambda$ and compare the obtained result. We
have found that the second ansatz for the varying $\Lambda$, expressed in
terms of the Hubble parameter $H$, gives better results than the first one.
The sound speed causality condition along with all nonlinear energy conditions
are satisfied for the case of $\Lambda(H)$. The cosmographic parameters have
been investigated.

Some cosmological features of 4D Gauss-Bonnet gravity with varying
cosmological constant
Nasr Ahmed1,2, Ajab A. Alfreedi1 & Alaa A. Alzulaibani1

1 Mathematics and Statistics Department, Faculty of Science, Taibah
University, KSA.
2 Astronomy Department, National Research Institute of Astronomy and
Geophysics, Helwan, Cairo<EMAIL_ADDRESS>

PACS: 04.50.-h, 98.80.-k, 65.40.gd
Keywords: Modified gravity, cosmology, dark energy.

##  1 Introduction and motivation

The late-time cosmic acceleration has been a major challenging problem in
cosmology and theoretical physics since its discovery in 1998 [1]. One popular
idea to explain this acceleration within the framework of general relativity
is the unknown dark energy which can be simply represented by the cosmological
constant. However, in order for the cosmological constant to play this role,
its value must be incredibly small. The value of the cosmological constant
expected from particle physics, through vacuum energy, is approximately 50
orders of magnitude larger than the observed value extracted from astrophysics
assuming general relativity. A possible explanation is that an alternative
modified gravity theory is required on cosmological scale with no need to
assume dark energy [3, 4].

Several modified gravity theories have been developed as an alternative to
dark energy [8]-[14], a good review has been given in [5]. Examples include
$f(R)$ gravity [15] where $R$ is the Ricci scalar, Gauss-Bonnet gravity [16]
and f(T) gravity [17] where $T$ is the torsion scalar. In Gauss-Bonnet
gravity, the Ricci scalar $R$ in the action has been replaced by the Gauss-
Bonnet term
$G=R^{2}-4R^{\mu\nu}R_{\mu\nu}+R^{\mu\nu\rho\delta}R_{\mu\nu\rho\delta}$.
$f(R,T)$ gravity [6, 7], is a generalization of $f(R)$ gravity where $T$ is
the trace of the energy momentum tensor. Dynamical scalar fields is another
explanation approach [18]-[21] where scalar fields play the role of dark
energy, the quintessence model is the most studied one.

Modified gravity theories can be constrained by observations. The
gravitational wave observation of GW170817 and the corresponding gamma ray
burst (GRB 170817 A) put strong constraints on the viability of dark energy
models in modified gravity [22, 23]. The viability of the scalar-tensor
theories of gravity and Milgrom’s modified Newtonian daynamics (MOND) with
this new observations has been discussed in [24, 25].

Gauss-Bonnet gravity is a modified gravity theory lives in high dimensions
with its action given by [26, 27]

$S=\int d^{D}x\sqrt{-g}\left(\frac{R-2\Lambda}{16\pi G}+\alpha~{}{\cal
G}\right)+S_{m}$ (1)

Where ${\cal G}$ is the Gauss-Bonnet invariant defined as

${\cal
G}=R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}-4R^{\mu\nu}R_{\mu\nu}+R^{2}$ (2)

$\alpha$ is a constant, $S_{m}$ is the matter action, $R$ is the Ricci scalar
and $G$ is Newton’s gravitational constant. A new Gauss-Bonnet 4D theory has
been introduced in [28] by rescaling $\alpha\rightarrow\frac{\alpha}{D-4}$.
Observational constraints have been placed on the free parameter $\alpha$ by
some groups and here we use $\alpha=(1.2\pm 5.2)\times 10^{-17}$ [27, 29].
The modified Einstein equations in D dimensions are given by

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda
g_{\mu\nu}+\frac{\alpha}{D-4}(4R_{\mu\alpha\beta\sigma}R_{\nu}^{~{}\alpha\beta\sigma}-8R_{\mu\alpha\nu\beta}R^{\alpha\beta}-8R_{\mu\alpha}R_{\nu}^{~{}\alpha}+4RR_{\mu\nu}-g_{\mu\nu}{\cal
G})=8\pi GT_{\mu\nu}.$ (3)

The modified Friedmann equations for a spatially flat 4D FRW universe can be
written as

$\frac{\dot{a}^{2}}{a^{2}}+2\alpha\frac{\dot{a}^{4}}{a^{4}}=\frac{8\pi
G}{3}(\rho_{b}+\rho_{cdm}+\rho_{r})+\frac{\Lambda}{3},$ (4)

$\rho_{b},\rho_{cdm}$ and $\rho_{r}$ are the energy densities of baryons, CDM
and radiation respectively. The conservation equation is

$\dot{\rho_{i}}+3H(p_{i}+\rho_{i})=0.$ (5)

In the current work we assume a time-dependent cosmological constant
$\Lambda(t)$. A new cosmological constant model has been introduced in [37] in
which the ansatz

$\Lambda=\frac{\Lambda_{Pl}}{\left(t/t_{Pl}\right)^{2}}\propto\frac{1}{t^{2}}$
(6)

has been proposed. The cosmological constant $\Lambda\propto\frac{1}{t^{2}}$
starts at the Plank time as $\Lambda_{Pl}=\sim M_{Pl}^{2}$ and allows the
value $\Lambda_{0}\sim 10^{-120}M_{Pl}^{2}$ for the present epoch. Some other
general models for the varying cosmological constant have been introduced (see
[30, 31, 32] and references therein. The following ansatz for $\Lambda(H)$,
with $H$ the Hubble constant, has been first introduced in [33]

$\Lambda(H)=\lambda+\alpha H+3\beta H^{2}$ (7)

where $\lambda$, $\alpha$ and $\beta$ are real constants. It was found in [30,
34, 35, 36] that the case for $\lambda=0$ doesn’t fit with observational data
while the case of $\lambda\neq 0$ behaves like $\Lambda$CDM model at late-
time. Some other models for varying $\Lambda$ in terms of $H$ are [30]

$\displaystyle\Lambda(H)=\beta H+3H^{2}+\delta H^{n},\,\,\,\,\,\,n\in
R-\left\\{0,1\right\\}$ (8)
$\displaystyle\Lambda(H,\dot{H},\ddot{H})=\alpha+\beta H+\delta
H^{2}+\mu\dot{H}+\nu\ddot{H}.$ (9)

In the current work, we first follow [37] and use $\Lambda(t)=\frac{C}{t^{2}}$
as an ansatz for the varying $\Lambda$ where $C$ is a constant. This allows
the cosmological constant to have a very tiny positive value at the current
epoch and late-times as suggested by observations [1, 2] (Figure 2h). We then
re-investigate the same model using the ansatz (7): $\Lambda(H)=\lambda+\alpha
H+3\beta H^{2}$ and compare the results of the two models.

##  2 Cosmological solutions and the ad hoc approach

In the current work, we probe the cosmology of 4D Gauss-Bonnet gravity through
an ad hoc approach by using a given scale factor derived from the desired
cosmic behavior [38] which is inverse of the usual viewpoint. The choice of
the scale factor is restricted by satisfying some observational requirements
mainly for the deceleration and jerk parameters. The evolution of these two
parameters should lead to a cosmic deceleration-acceleration transit along
with a flat $\Lambda CDM$ model at late-time. This ad hoc approach for the
scale factor and scalar fields has been used extensively in the literature to
probe the cosmology of several gravity theories [38, 39, 40, 41, 42, 43, 44,
45, 46].

###  2.1 Model 1

The following hyperbolic ansatz leads to a desired behavior of the
deceleration and jerk parameters for $0<n<1$ [47]

$a(t)=A\sinh^{\frac{1}{n}}(\eta t)$ (10)

The expressions for the deceleration parameter $q$, the jerk parameter $j$,
the cosmic pressure $p$, the energy density $\rho$ are

$q=-\frac{\ddot{a}a}{\dot{a}^{2}}=\frac{-\cosh^{2}(\eta t)+n}{\cosh^{2}(\eta
t)},~{}~{}~{}j=\frac{\dddot{a}}{aH^{3}}=1+\frac{2n^{2}-3n}{\cosh^{2}(\eta
t)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (11)
$\displaystyle\rho=-\frac{-6\alpha
g^{4}\eta^{4}t^{2}-3g^{2}\eta^{2}t^{2}n^{2}f^{2}+Cn^{4}f^{4}}{t^{2}n^{4}f^{4}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(12) $\displaystyle p=-\frac{1}{g\eta
t^{3}n^{4}f^{4}}\left(24f^{2}g^{3}n\alpha
t^{3}n^{5}+6f^{4}gn^{3}\eta^{3}t^{3}-6f^{2}g^{3}n^{3}\eta^{3}t^{3}+2n^{5}Cf^{5}\right.$
(13) $\displaystyle\left.-24n\alpha t^{3}\eta^{5}g^{5}+18\alpha
t^{3}n^{5}g^{5}+9n^{2}t^{3}\eta^{3}f^{2}g^{3}-3Cn^{4}\eta tf^{4}g\right)$

Where $f=\sinh(\eta t)$ and $g=\cosh(\eta t)$. The plots of $q$ and $j$ are
shown in figure 1 for different values of $n$. We will consider $n=2$ where
$q$ changes sign from positive to negative, and $j$ tends to $+1$ at late-time
as expected for a flat $\Lambda$CDM universe.

The figure shows that $q$ varies in the range from $-1$ to $+1$ where cosmic
transit should happen at $\ddot{a}=0$ ( $q=0$ ). Here we get
$t_{q=0}=\frac{1}{2\eta}\ln(3+2\sqrt{2})$ which gives $t\approx 0.88$ for
$\eta=1$. The Hubble parameter plotted as a function of the redshift in
Fig.1(c). Cosmic pressure $p$ changes sign from positive to negative in
corresponding to the sign flipping of the deceleration parameter $q$. Taking
into account the Dark Energy assumption as a component of negative pressure
that gives the effect of a repulsive gravity, cosmic pressure should be
positive in the early-time decelerating era and negative in the late-time
decelerating era [50]. After expressing the equation of state parameter
$\omega$ in terms of the redshif $z$ using $a=\frac{1}{1+z}$, we find that
$\omega(z)=-1$ at the current epoch where $z=0$ as suggested by observations
(Fig.1(f)). The evolution of $\omega$ against cosmic time (Fig.1(g)) shows a
Quintessence dominated universe where $-1\leq\omega\leq 0.4$ with no crossing
to the phantom divide line $\omega=-1$. The no-go theorem introduced in [48]
forbids $\omega$ of a single perfect fluid in FRW universe to cross the
phantom divide line, this no-go theorem doesn’t hold in higher dimensions
[49].

####  2.1.1 Energy and Causality Conditions

The classical linear energy conditions [51, 52] and the new nonlinear energy
conditions (ECs) [53, 54, 55, 56] have been plotted in Fig.1 (h) & (i). The
classical linear ECs (“ the null $\rho+p\geq 0$; weak $\rho\geq 0$,
$\rho+p\geq 0$; strong $\rho+3p\geq 0$ and dominant $\rho\geq\left|p\right|$
energy conditions ” ) should be replaced by other nonlinear ECs when
semiclassical quantum effects are taken into account [53, 56]. Here we
consider (i) The flux EC (FEC): $\rho^{2}\geq p_{i}^{2}$ [54, 55], first
presented in [54]. (ii) The determinant EC (DETEC): $\rho.\Pi p_{i}\geq 0$
[56]. (ii) The trace-of-square EC (TOSEC): $\rho^{2}+\sum p_{i}^{2}\geq 0$
[56]. According to the strong energy condition (SEC), gravity should always be
attractive. But this ‘highly restrictive’ condition fails when describing the
current cosmic accelerated epoch and during inflation [57, 58]. Here we have a
change of sign in cosmic pressure from positive to negative and, consequently,
the SEC is not expected to be satisfied as indicated in Fig. 1(h). The null
energy condition (NEC) and the dominant energy condition are satisfied all the
time. The NEC is the most fundamental of the ECs and on which many key results
are based such as the singularity theorems [59]. Violation of NEC
automatically implies the violation of all other point-wise energy conditions.
We have found that the condition $0\leq\frac{dp}{d\rho}\leq 1$ is not valid
and so the sound speed causality condition is not satisfied.

###  2.2 Re-investigating Model 1 with $\Lambda(H)=\lambda+\alpha H+3\beta
H^{2}$

Using (7) along with (10) into (4) and (5) we obtain the following expressions
for cosmic pressure and energy density

$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{\eta\alpha}{3}\tanh(\eta
t)+\frac{2\eta^{2}\beta}{n}-\frac{2\eta^{2}}{n}-\coth^{2}(\eta
t)(\frac{8\eta^{4}\alpha}{n^{3}}+\lambda+\frac{3\eta^{2}\beta}{n^{2}}-\frac{2\eta^{2}\beta}{n}-\frac{\eta^{2}}{n^{2}}+\frac{2\eta^{2}}{n})$
$\displaystyle+$ $\displaystyle\coth^{4}(\eta
t)(\frac{8\eta^{4}\alpha}{n^{3}}-\frac{6\eta^{4}\alpha}{n^{4}})-\coth(\eta
t)(\frac{\eta\alpha}{3}-\frac{\eta\alpha}{n})$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle-\lambda-\frac{\eta\alpha}{n}\coth(\eta
t)-\frac{3\eta^{2}}{n^{2}}\coth^{2}(\eta
t)(1-\beta)+\frac{6\eta^{4}\alpha}{n^{4}}\coth^{4}(\eta t)$

The behavior of $p$, $\rho$, $\omega$, energy conditions and the sound speed
has been plotted in fig(2)

(a) $q$

(b) $j$

(c) $H(z)$

(d) $p$

(e) $\rho$

(f) $\omega(z)$

(g) $\omega(t)$

(h) $ECs)$

(i) $nlECs$

Figure 1: The hyperbolic solution using $\Lambda(t)=\frac{C}{t^{2}}$: (a) The
deceleration parameter $q$ changes sign from positive to negative. (b) The
jerk parameter $j=1$ at late-times where the model tends to a flat
$\Lambda$CDM model. (c) The variation of Hubble parameter against redshift.
(d) Cosmic pressure goes from positive to negative. (e) The evolution of
energy density. (f) EoS parameter $\omega=-1$ at $z=0$. (g) The EoS parameter
$\omega(t)$ tends to $-1$ at late-time with no crossing to the phantom divide
line . (h) and (g) shows linear and nonlinear energy conditions. The model
fails to pass the sound speed causality condition. Here $\alpha=7.2\times
10^{-17}$, $A=0.1$, $C=0.01$ and $\eta=1$.

(a) $\rho$

(b) $p$

(c) $\omega(t)$

(d) $ECs$

(e) $nlECs$

(f) $v_{s}=\frac{dp}{d\rho}$

Figure 2: The hyperbolic solution using $\Lambda(H)=\lambda+\alpha H+3\beta
H^{2}$: (a) Cosmic pressure goes from positive to negative. (b) The evolution
of energy density. (c) $\omega(t)$ tends to $-1$ at late-time with no crossing
to the phantom divide line . (d) and (e) shows linear and nonlinear energy
conditions, all the nonlinear ECs are satisfied for this case. (f)The sound
speed causality condition $0\leq C^{2}_{s}=\frac{dp}{d\rho}\leq 1$ is
satisfied for this choice of $\Lambda(H)$ in contrast to the first choice of
$\Lambda(t)=\frac{C}{t^{2}}$ . Here $\alpha=7.2\times 10^{-17}$, $A=0.1$,
$\lambda=\alpha=\beta=0.1$, $n=2$ and $\eta=1$.

##  3 Model 2 (Hybrid solution with $\Lambda(H)=\lambda+\alpha H+3\beta
H^{2}$)

Another ansatz that leads to a desired behavior of the deceleration and jerk
parameters (Fig.2 a,b) is the hybrid one given as [60]

$a(t)=a_{1}t^{\alpha_{1}}e^{\beta_{1}t},$ (15)

where $a_{1}>0$, $\alpha_{1}\geq 0$ and $\beta_{1}\geq 0$ are constants. The
scale factor (15) is a mixture of the power-law and exponential-law
cosmologies and a generalization to both of them. The power-law cosmology is
obtained for $\beta_{1}=0$, and the exponential-law cosmology is obtained for
$\beta_{1}=0$. New cosmologies can be explored for $\alpha_{1}>0$ and
$\beta_{1}>0$. The deceleration and jerk parameter respectively are

$q(t)=\frac{\alpha_{1}}{(\beta_{1}t+\alpha_{1})^{2}}-1~{}~{}~{},~{}~{}~{}j={\frac{{\alpha_{1}}^{3}+\left(3\,\beta\,t-3\right){\alpha_{1}}^{2}+\left(3\,{\beta}^{2}{t}^{2}-3\,\beta\,t+2\right)\alpha_{1}+{\beta}^{3}{t}^{3}}{\left(\beta\,t+\alpha_{1}\right)^{3}}}$
(16)

Using (15) in (4) and (5) along with (7) we get the cosmic pressure and energy
density as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{t^{4}}\left[(6\alpha\beta_{1}^{4}+3\beta_{1}^{2}\beta+3\beta_{1}^{2}-\alpha\beta_{1}-\lambda)t^{4}+(24\alpha\alpha_{1}\beta_{1}^{3}-6\alpha_{1}\beta_{1}\beta+6\alpha_{1}\beta_{1}-\alpha_{1}\alpha)t^{3}\right.$
$\displaystyle+$
$\displaystyle\left.(36\alpha\alpha_{1}^{2}\beta_{1}^{2}-3\alpha_{1}^{2}\beta+3\alpha_{1}^{2})t^{2}+24\alpha\beta_{1}\alpha_{1}^{3}t+6\alpha\alpha_{1}^{4}\right]$
$\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{1}{3t^{4}(t\beta_{1}+\alpha_{1})}\left[t^{5}(18\alpha\beta_{1}^{5}-9\beta_{1}^{3}\beta+9\beta_{1}^{3}-3\beta_{1}^{2}\alpha-3\beta_{1}\lambda)\right.$
$\displaystyle+$
$\displaystyle\left.t^{4}(90\alpha\alpha_{1}\beta_{1}^{4}-27\alpha_{1}\beta_{1}^{2}\beta+27\alpha_{1}\beta_{1}^{2}-6\alpha_{1}\beta_{1}\alpha-3\alpha_{1}\lambda)\right.$
$\displaystyle+$
$\displaystyle\left.t^{3}(180\alpha\alpha_{1}^{2}\beta_{1}^{3}-24\alpha\alpha_{1}\beta_{1}^{3}-27\alpha_{1}^{2}\beta_{1}\beta+27\alpha_{1}^{2}\beta_{1}-3\alpha_{1}^{2}\alpha+6\alpha_{1}\beta_{1}\beta-6\alpha_{1}\beta_{1}+\alpha_{1}\alpha)\right.$
$\displaystyle+$
$\displaystyle\left.t^{2}(180\alpha\alpha_{1}^{3}\beta_{1}^{2}-27\alpha\alpha_{1}^{2}\beta_{1}^{2}-9\alpha_{1}^{3}\beta+9\alpha_{1}^{3}+6\alpha_{1}^{2}\beta-6\alpha_{1}^{2})+t(90\alpha\alpha_{1}^{4}\beta_{1}-72\alpha\alpha_{1}^{3}\beta_{1})\right.$
$\displaystyle+$
$\displaystyle\left.18\alpha\alpha_{1}^{5}-24\alpha\alpha_{1}^{4}\right]$

The behavior of $\rho$ and $p$ is plotted in Fig.2. The pressure is always
positive in contrast to the hyperbolic solution where there is a sign flipping
in from positive in the early-time decelerating epoch, to negative in the
late-time accelerating epoch. The validity of the linear and non-linear energy
conditions at early and late-times are shown in (Fig.2 e and f). We have also
found that the sound speed causality condition $0\leq
C^{2}_{s}=\frac{dp}{d\rho}\leq 1$ is not satisfied for this model

(a) $q$ (b) $j$

(c) $p$

(d) $\rho$ (e) $ECs)$

(f) $nlECs$

Figure 3: The hybrid solution using $\Lambda(H)=\lambda+\alpha H+3\beta
H^{2}$: (a) shows the sign flipping of $q$ from positive to negative. (b) The
jerk parametr tends to $1$. (c) $p$ is always positive (d) The evolution of
energy density. (e) and (f) show linear and nonlinear energy conditions. The
sound speed causality condition $0\leq C^{2}_{s}=\frac{dp}{d\rho}\leq 1$ is
not satisfied for this model. Here $\alpha=7.2\times 10^{-17}$,
$\lambda=\alpha=\beta=0.1$, $\alpha_{1}=\beta_{1}=0.$3

##  4 Cosmographic Analysis

The cosmography of the universe has recently become a popular research area
[61, 62] in which cosmological parameters can be expressed in terms of
kinematics only. As a result, cosmographic analysis is model-independent with
no need to an equation of state to investigate cosmic dynamics [63]. The
Taylor expansion of the scale factor $a(t)$ around the present time $t_{0}$ is
written as

$a(t)=a_{0}\left[1+\sum_{n=1}^{\infty}\frac{1}{n!}\frac{d^{n}a}{dt^{n}}(t-t_{0})^{n}\right]$
(18)

The following coefficients of the expansion (18) are described respectively as
the Hubble parameter $H$, deceleration parameter $q$, the jerk $j$, snap $s$,
lerk $l$ and max-out $m$ parameters

$\displaystyle
H=\frac{1}{a}\frac{da}{dt}~{}~{},~{}~{}q=-\frac{1}{aH^{2}}\frac{d^{2}a}{dt^{2}}~{}~{},~{}~{}j=\frac{1}{aH^{3}}\frac{d^{3}a}{dt^{3}}$
(19) $\displaystyle
s=\frac{1}{aH^{4}}\frac{d^{4}a}{dt^{4}}~{}~{},~{}~{}l=\frac{1}{aH^{5}}\frac{d^{5}a}{dt^{5}}~{}~{},~{}~{}m=\frac{1}{aH^{6}}\frac{d^{6}a}{dt^{6}}.$

For the first model we get

$\displaystyle j$ $\displaystyle=$ $\displaystyle
1+\frac{2n^{2}-3n}{\cosh^{2}(\eta t)},$ (20) $\displaystyle s$
$\displaystyle=$ $\displaystyle\frac{1}{\cosh^{4}(\eta t)}\left[\cosh^{4}(\eta
t)-\cosh^{2}(\eta t)(4n^{3}-8n^{2}+6n)-2n^{3}+3n^{2}\right]$ (21)
$\displaystyle l$ $\displaystyle=$ $\displaystyle\frac{1}{\cosh^{4}(\eta
t)}\left[\cosh^{4}(\eta t)-\cosh^{2}(\eta
t)(8n^{4}-20n^{3}+20n^{2}-10n)+16n^{4}-30n^{3}+15n^{2}\right]$ (22)
$\displaystyle m$ $\displaystyle=$ $\displaystyle\frac{1}{\cosh^{6}(\eta
t)}\left[\cosh^{6}(\eta t)+\cosh^{4}(\eta
t)(-16n^{5}+48n^{4}-60n^{3}-15n)\right.$ $\displaystyle+$
$\displaystyle\left.\cosh^{2}(\eta
t)(-88n^{5}+196n^{4}-150n^{3}+45n^{2})-16n^{5}+30n^{4}-15n^{3}\right]$

And for the second model

$\displaystyle j$ $\displaystyle=$
$\displaystyle{\frac{{\alpha_{1}}^{3}+\left(3\,\beta\,t-3\right){\alpha_{1}}^{2}+\left(3\,{\beta}^{2}{t}^{2}-3\,\beta\,t+2\right)\alpha_{1}+{\beta}^{3}{t}^{3}}{\left(\beta\,t+\alpha_{1}\right)^{3}}},$
(25) $\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\beta\,t+\alpha_{1}\right)^{4}}\left(\beta^{4}t^{4}+4\alpha_{1}\beta^{3}t^{3}+6\beta^{2}\alpha_{1}(\alpha_{1}-1)t^{2}+(4\alpha_{1}^{2}-12\alpha_{1}+8)\alpha_{1}\beta
t\right.$ $\displaystyle+$
$\displaystyle\left.\alpha_{1}^{4}-6\alpha_{1}^{3}+11\alpha_{1}^{2}-6\alpha_{1}\right)$
$\displaystyle l$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\beta\,t+\alpha_{1}\right)^{5}}\left(\beta^{5}t^{5}+5\alpha_{1}\beta^{4}t^{4}+10\beta^{3}\alpha_{1}(\alpha_{1}-1)t^{3}+10\alpha_{1}\beta_{1}(\beta\alpha_{1}^{2}-3\beta\alpha_{1}+2\beta)t^{2}\right.$
$\displaystyle+$
$\displaystyle\left.5\beta\alpha_{1}(\beta^{3}-6\alpha_{1}^{2}+11\alpha_{1}-6)t+\alpha_{1}^{5}-10\alpha_{1}^{4}+35\alpha_{1}^{3}-50\alpha_{1}^{2}+24\alpha_{1}\right)$
$\displaystyle m$ $\displaystyle=$
$\displaystyle\frac{1}{\left(\beta\,t+\alpha_{1}\right)^{6}}\left(\beta^{6}t^{6}+6\alpha_{1}\beta^{5}t^{5}+15\beta^{4}\alpha_{1}(\alpha_{1}-1)t^{4}+20\alpha_{1}^{2}\beta^{3}(2\alpha_{1}^{2}+\alpha_{1}-3)t^{3}\right.$
$\displaystyle+$
$\displaystyle\left.15\beta^{2}\alpha_{1}(\alpha_{1}^{3}-6\alpha_{1}^{2}+11\alpha_{1}-6)t^{2}+6\beta\alpha_{1}(\alpha_{1}^{4}-10\alpha_{1}^{3}+35\alpha_{1}^{2}-50\alpha_{1}+24)t\right.$
$\displaystyle+$
$\displaystyle\left.\alpha_{1}^{6}-15\alpha_{1}^{5}+85\alpha_{1}^{4}-225\alpha_{1}^{3}+274\alpha_{1}^{2}-120\alpha_{1}\right)$

The value of $s$ is required for the study of the dark energy evolution. In
addition to providing a suitable way to describe models close to $\Lambda
CDM$, The presence of a transition time when cosmic expansion is modified is
shown by the positive sign of $j$. The sign of $q$ also shows if the expansion
is accelerating or decelerating. Despite its advantages, a valuable discussion
of the cosmographic approach’s limitations and downsides has been provided in
[61].

##  5 positive $\Lambda$ and the eternal acceleration problem

The behavior of the deceleration parameter $q$ in both models shows that it
tends to a constant value $-1$ as $t\rightarrow\infty$. The same behavior
happens for all cosmographic parameters $s$, $l$ and $m$ where they tend to a
constant value at late-time. While $\Lambda$ represents the vacuum energy
density and is supposed to has a tiny positive magnitude $\approx 10^{-123}$
[1, 2], It has been suggested that a negative $\Lambda$ can also fit a large
data set and provide a solution to such ’eternal acceleration’ problem which
is a consequence of assuming a positive $\Lambda$ to explain the cosmic
acceleration[64]. The cosmological constant in the current model has a very
small positive value at late-times and there’s nothing to stop the accelerated
expansion.

(a) $s$

(b) $l$

(c) $m$

(d) $s$

(e) $l$

(f) $m$

Figure 4: Evolution of the cosmographic parameters $s$, $l$ and $m$ with
cosmic time for model 1: (a), (b), (c) and for model 2: (d), (e), (f)

##  6 Conclusion

Some cosmological features of the recently suggested 4D Gauss-Bonnet gravity
have been investigated through two different models assuming a varying
cosmological constant. Unlike the usual viewpoint, In each model we have used
a given scale factor derived from the desired cosmic behavior. The first
hyperbolic model has been investigated through two different models of varying
$\Lambda$: The first is $\Lambda(t)=\frac{C}{t^{2}}$ and the second is
$\Lambda(H)=\lambda+\alpha H+3\beta H^{2}$. The second one gives better
results where, for example, the sound speed causality condition along with the
three nonlinear energy conditions are satisfied. The deceleration-acceleration
cosmic transit occurs in with a sign flipping in cosmic pressure from positive
to negative. The deceleration parameter $q$ in both models tends to a constant
value $-1$ as $t\rightarrow\infty$. Also, the jerk parameter also tends to
$+1$ as $t\rightarrow\infty$ in both models. The second hybrid model has been
investigated through $\Lambda(H)=\lambda+\alpha H+3\beta H^{2}$ where we have
got a positive pressure with no sign flipping. Also, the sound speed causality
condition is not satisfied.

## Acknowledgment

We are so grateful to the reviewer for his valuable suggestions and comments
that significantly improved the paper.

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