Dynamical analysis in regularized $4D$ Einstein-Gauss-Bonnet gravity with non-
minimal coupling

Bilguun Bayarsaikhan${}^{\text{a,b,}}$111e-mail<EMAIL_ADDRESS>Sunly
Khimphun${}^{\text{c,}}$222e-mail<EMAIL_ADDRESS>Phearun
Rithy${}^{\text{c,}}$333e-mail<EMAIL_ADDRESS>and Gansukh
Tumurtushaa${}^{\text{d,}}$444e-mail<EMAIL_ADDRESS>${}^{a}\,$Institute for Theoretical Physics, ELTE Eotvos Lorand University,
Pazmany Peter setany 1/A, H-1117 Budapest, Hungary
${}^{b}\,$Institute of Physics & Technology, Mongolian Academy of Sciences,
Ulaanbaatar 13330, Mongolia
${}^{c}\,$Graduate School of Science, Royal University of Phnom Penh, Phnom
Penh, Cambodia 12150
${}^{d}\,$Department of Science Education, Jeju National University, Jeju,
63243, Korea

We investigate the regularized four-dimensional Einstein-Gauss-Bonnet ($4D$
EGB) gravity with a non-minimal scalar coupling function, which is an
extension of the regularized $4D$ EGB theory. By introducing non-minimal
coupling to the Gauss-Bonnet term, we demonstrate the additional contribution
to the dynamical equations which is otherwise absent in the dimensionally-
regularized theory. Furthermore, we analyze the stability of the system by
using the dynamical system approach based on fixed points. Then, we consider
the time evolution to investigate the history of the universe and constraint
with observational data to obtain the cosmological parameters of the model.

##  1 Introduction

Einstein’s theory of gravity predicts the accelerating expansion of the late-
time universe, which has been confirmed by the cosmological observations from
type Ia Supernovae (SNIa) [1, 2]. The expansion of the universe can be
explained by the simplest model of the cosmological constant that currently
gives the best fit with the current observational data from Cosmic Microwave
Background (CMB) radiation, the measurement of Type Ia Supernovae (SNIa),
Baryon Acoustic Oscillation (BAO), and $H_{0}$ measurement [3, 4]. However,
some issues remain for the cosmological constant or $\Lambda$CDM model as
encountered from the observations [3, 5, 6] and a seriously theoretical
inconsistency [7, 8], and for a review, one may refer to [9, 10]. In these
regards, several modified theories of gravity have been proposed to refine or
provide alternatives. Without exhaustion, we list down some of the theories
which deviate from the standard cosmological model: scalar-tensor theory [11,
12, 13, 14, 15], vector-tensor [16], tensor-tensor theory [17, 18, 19],
massive gravity [20, 21, 22], higher-order derivative theory, in particular,
the f(R)-gravity for cosmology [23, 24, 25, 26], and higher-dimensional and
string-motivated theory [27, 28, 29, 30, 42, 43]. For a nice review, one is
referred to [31]. Also, there are many dark energy models under these
frameworks such as Chevallier Polarski Linder (CPL) model [32], Holographic
Dark Energy model and its constraint [33, 34, 35, 36, 37], Generalized
holographic cosmology via AdS/CFT constraint with low redshift [38], and
Chaplygin gas model [39, 40] to name a few. One can refer to [41] for model
comparison using observational data from Planck 2015.

It is well-known that Lovelock’s theorem is a natural generalization to
general relativity, which satisfies the diffeomorphism and local Lorentz
invariance in $D$-dimensional space-time and, in particular, leads to second-
order field equations [42, 44, 45]. A special case of this is the second order
in Lanczos-Lovelock gravity Lagrangian density so-called Einstein-Gauss-Bonnet
gravity. Consequently, one possible modification of General Relativity is the
$D\geq 5$ Gauss-Bonnet (GB) gravity which satisfies the properties in
Lovelock’s theorem, including the ghost-free [46], natural generalization with
Einstein, and cosmological terms [47]. For its study of higher dimensions in
cosmology, one can refer to [48, 49, 50]. However, the GB term in $4D$ is
topologically invariant, so it does not modify Einstein’s theory since it does
not contribute to gravitational dynamics. Nevertheless, the GB term can
contribute to the $4D$ dynamical equations if one introduces a coupling
function of a scalar field to the GB term [51, 52]. Due to the extra scalar
degree of freedom, this consideration leads to many gravity studies, including
cosmic acceleration in inflationary models of the early universe, and tested
against the observational data [53, 54, 55, 56, 57, 58, 59].

Recently, the $4D$ EGB model proposed in [60] introduces a scaling coupling
constant $\alpha\rightarrow\alpha/(D-4)$, where $D$ is the space-time
dimensions, and considers the limit $D\rightarrow 4$. This model gives rise to
nontrivial contributions to the gravitational dynamics due to the extra
contribution in the equations of motion (EoM) from the GB term as the
consequence of the scaling coupling constant without introducing extra degrees
of freedom. Notice that in this model the space-time was assumed to be
continuous [61]. To be less exhaustive, there is an interesting work in
cosmology adopting this model with the observational constraints which can
resolve the coincidence problem [62] and also indicate that the re-scaling
coupling constant of the model still needs the help of the cosmological
constant to explain the accelerated expansion. For a review of the model, one
may refer to [63].

Due to its richness and peculiar consequences, comments and arguments on this
novel $4D$ EGB theory have been raised [64, 65, 66, 67, 68, 69, 70], and the
diffeomorphism invariant regularization is also considered in [71, 72, 73, 74]
to point out that the theory advertised in [60] is the subclass of Horndeski
theory. However, the arguments from Horndeski theory still rely on a $2+1$
degrees of freedom where higher dimensions are compactified and where in [60]
the space-time is based on the D-dimensional direct product and $D\to 4$
limit, and further investigation into this regularized tensor-scalar theory
leading to a strong-coupling scalar field is studied in [73, 75] and without
the strong coupling field in [76]. Another investigation into the regularized
$4D$ EGB theory is that looking at diffeomorphism invariant property reveals
the inconsistency in the regularized $4D$ EGB theory [67, 69, 70].
Nevertheless, the consistent study of $D\to 4$ EGB theory can be achieved up
to spatially diffeomorphism invariance by using Hamiltonian formalism so-
called minimally modified gravity theory; otherwise, an extra degree of
freedom is required [77, 78]. Therefore, we will proceed by introducing an
additional degree of freedom but still exploiting the regularization scheme
which is expected to render the finite term that contributes to the dynamics.
That is, we will scale $\alpha\rightarrow\alpha/(D-4)$ and take the limit
$D\to 4$ after introducing the non-minimal coupling function to the GB term.
In this regard, we are interested in generalizing this concept by combining
the role of the non-minimal coupling function often used in ordinary $4D$
gravity with the scaling coupling constant advocated in the regularized $4D$
EGB gravity together. This seems to be a redundant consideration at first
because the role of scaling coupling is to render the term of the overall
factor ($D-4$) coming from the GB term so that these higher curvature terms
become relevant, which is also the role of the non-minimal coupling function.
As a result, in the current study, we accept the necessity of the extra scalar
degree of freedom in the framework of regularized $4D$ EGB.

In this works, we study the extension of regularized $4D$ EGB theory by
introducing the non-minimal coupling to the GB term and redefining the
coupling function

$\xi(\phi)\to\xi^{(D-4)}(\phi),$

with flat FRW metric in D-dimensions then take the limit $D\to 4$. The
motivation of the form of coupling function is obvious so that divergence of
the terms associated to the scaling of $\alpha$ remains well-defined in the
EoM. From this consideration, it is obvious that the action could be ambiguous
but not ill-defined since this happens to the original regularized $4D$ EGB
where its action might be investigated further [71, 72, 73, 74]; this issue is
not our concern here. From this, we apply the dynamical system approach (DSA)
[79, 80, 81, 82, 83] with the exponential of the potential and the coupling
function of the scalar field into the dynamical equations of the model. We
find the fixed points and analyze the stability of the autonomous system
equations including the consideration of the evolution of the phase space
universe. Particularly we consider the evolution of the cosmological
parameters as the function of the red-shift and constraint on the potential
parameter with the observational bounds from CMB, BAO, SnIa, and $H_{0}$
measurement.

This article is organized as follows. In section 2, we introduce our setup and
derive the EoM for our model. As a result, there appear two types of
additional contributions in the $4D$ EoM each coming from the non-minimal
coupling of the scalar field to the GB term and the scaling coupling constant
of the GB term. Thus, it is imperative for us to study their respective role
and their dynamic evolution throughout the cosmic history of the universe. To
investigate the dynamical evolution of the universe for our model, we rewrite
the background EoM in the autonomous system form in section 3 and apply the
dynamical system approach, which gives a robust description of the cosmic
history based on the existence of critical points and their stability. In
section 4, we obtain not only the critical fixed points of the system but also
consider the stability of the universe at each point. We solve the dynamical
equations for a broader time scale to better understand how the aforementioned
GB contributions evolve, especially at the late time, and provide our
numerical result with their implications. A summary of our results in this
paper and a conclusion are given in section 5.

##  2 The setup and Equations of Motion

In this work, we consider the action in $D$-dimensional space-time as follow,

$S=\int\mathrm{d}^{D}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}R-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)-\frac{\alpha}{2}\xi(\phi)\mathcal{G}\right]+S_{\text{m,r}},$
(1)

where $R$, $\alpha$, $V(\phi)$, $\xi(\phi)$ are Ricci scalar, a coupling
constant, potential function and the coupling function of scalar field
respectively.

$\mathcal{G}\equiv
R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,,$

is GB term, $S_{\text{m,r}}$ represents the standard matter and radiation
components, $\kappa^{2}=8\pi G$, and $G$ is Newtonian constant. From this
action, after obtaining the EoM, we re-scale $\alpha\to\alpha/(D-4)$ and then
take the limit $D\rightarrow 4$. The problem arises when we introduce
$\xi(\phi)$ couples to GB term since the expected finite terms from
$\xi(\phi)$ which appear in the EoM contain no factor of ($D-4$). As a result,
$1/(D-4)$ from the scaling $\alpha$ will diverge. We propose the redefining
the coupling function

$\xi(\phi)\rightarrow\xi^{(D-4)}(\phi),$

first and then take the limit $D\rightarrow 4$. Therefore, denominator
$1/(D-4)$ will be canceled out with the overall factor ($D-4$) coming from the
redefined coupling function. From the least-action principle, the field
equations after scaling $\alpha$ and redefining $\xi(\phi)$ are

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\kappa^{2}\left(T^{(\phi)}_{\mu\nu}+T^{\text{(m,r)}}_{\mu\nu}+T^{\text{(GB)}}_{\mu\nu}\right),$
(2) $\displaystyle\Box\phi-
V_{\phi}-\frac{\alpha}{2(D-4)}(D-4)\xi^{(D-5)}\xi_{\phi}\mathcal{G}=0,$ (3)

where $\Box\equiv\nabla_{\mu}\nabla^{\mu}$, $V_{\phi}=dV/d\phi$,
$\xi_{\phi}=d\xi/d\phi$, and $T^{(\phi)}_{\mu\nu},\leavevmode\nobreak\
T^{\text{(m,r)}}_{\mu\nu},\leavevmode\nobreak\ T^{\text{(GB)}}_{\mu\nu}$ are
stress-energy-momentum tensor of scalar field, GB term and matter term
respectively, which we find

$\displaystyle T^{(\phi)}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(g^{\rho\sigma}\partial_{\rho}\phi\partial_{\sigma}\phi+2V\right),$
(4) $\displaystyle T^{(\text{m,r})}_{\mu\nu}$
$\displaystyle=(\rho+P)U_{\mu}U_{\nu}+Pg_{\mu\nu},$ (5) $\displaystyle
T^{(\text{GB})}_{\mu\nu}=$
$\displaystyle\frac{\alpha}{(D-4)}\Biggl{\\{}\xi^{(D-4)}\left(-4R_{\mu}\leavevmode\nobreak\
^{\alpha}R_{\nu\alpha}+2RR_{\mu\nu}-4R^{\alpha\beta}R_{\mu\alpha\nu\beta}+2R_{\mu}\leavevmode\nobreak\
^{\alpha\beta\gamma}R_{\nu\alpha\beta\gamma}-\frac{1}{2}g_{\mu\nu}\mathcal{G}\right)$
$\displaystyle+(D-4)(D-5)\xi_{\phi}^{2}\xi^{(D-6)}\Biggl{[}g_{\mu\nu}\left(2R\nabla^{\alpha}\phi\nabla_{\alpha}\phi-4R_{\alpha\beta}\nabla^{\alpha}\phi\nabla^{\beta}\phi\right)-4R_{\mu\nu}\nabla^{\alpha}\phi\nabla_{\alpha}\phi$
$\displaystyle+4R_{\mu\alpha\nu\beta}\nabla^{\alpha}\phi\nabla^{\beta}\phi+4R_{\nu\alpha}\nabla^{\alpha}\phi\nabla_{\mu}\phi+4R_{\mu\alpha}\nabla^{\alpha}\phi\nabla_{\nu}\phi-2R\nabla_{\mu}\phi\nabla_{\nu}\phi\Biggr{]}+(D-4)\xi^{(D-5)}$
$\displaystyle\quad\times\Biggl{[}\xi_{\phi}\Bigl{(}-4R_{\mu\nu}\Box\phi+2R_{\mu\alpha\nu\beta}\nabla^{\beta}\nabla^{\alpha}\phi+2R_{\mu\beta\nu\alpha}\nabla^{\beta}\nabla^{\alpha}\phi+g_{\mu\nu}\Bigl{(}2R\Box\phi-4R_{\alpha\beta}\nabla^{\beta}\nabla^{\alpha}\phi\Bigr{)}$
$\displaystyle+4R_{\nu\alpha}\nabla_{\mu}\nabla^{\alpha}\phi+4R_{\mu\alpha}\nabla_{\nu}\nabla^{\alpha}\phi-2R\nabla_{\nu}\nabla_{\mu}\phi\Bigr{)}+\xi_{\phi\phi}\Bigl{(}-4R_{\mu\nu}\nabla^{\alpha}\phi\nabla_{\alpha}\phi+4R_{\mu\alpha\nu\beta}\nabla^{\alpha}\phi\nabla^{\beta}\phi$
$\displaystyle+g_{\mu\nu}\Bigl{(}2R\nabla^{\alpha}\phi\nabla_{\alpha}\phi-4R_{\alpha\beta}\nabla^{\alpha}\nabla^{\beta}\phi\Bigr{)}+4R_{\nu\alpha}\nabla^{\alpha}\phi\nabla_{\mu}\phi+4R_{\mu\alpha}\nabla^{\alpha}\phi\nabla_{\nu}\phi-2R\nabla_{\mu}\phi\nabla_{\nu}\phi\Bigr{)}\Biggr{]}\Biggr{\\}},$
(6)

where $U_{\mu}=(1,0,0,0)$ is four-velocity and $\rho,\leavevmode\nobreak\ P$
are the energy density and pressure in perfect fluid respectively. The
homogeneous and isotropic universe spatially flat $k=0$ of Friedmann-
Robertson-Walker (FRW) metric in $D$-dimensions is given

$\begin{split}\text{d}s^{2}&=-\text{d}t^{2}+a^{2}(t)(\text{d}\chi_{1}^{2}+\text{d}\chi_{2}^{2}+\text{d}\chi_{3}^{2}+...+\text{d}\chi_{D-1}^{2}),\end{split}$
(7)

where $a(t)$ is a scale factor. Notice that above line element is a
$D$-dimensional product space. By solving the field equations with flat FRW
metric in arbitrary dimensions we can write the EoM in general dimensions for
the components $tt$, $ii$, and scalar field equation as the following

$\displaystyle\frac{(D-2)(D-1)}{2\kappa^{2}}H^{2}=\frac{1}{2}\dot{\phi}^{2}+V+\rho_{\text{m}}+\rho_{\text{r}}+2\alpha(D-3)(D-2)(D-1)\dot{\xi}\xi^{(D-5)}H^{3}$
$\displaystyle\qquad+\frac{1}{2}\alpha(D-3)(D-2)(D-1)\xi^{(D-4)}H^{4},$ (8)
$\displaystyle-\left[\frac{(D-3)(D-2)}{2\kappa^{2}}H^{2}+\frac{(D-2)}{\kappa^{2}}\frac{\ddot{a}}{a}\right]=\frac{1}{2}\dot{\phi^{2}}-V+p_{\text{r}}-4\alpha(D-3)(D-2)\dot{\xi}\xi^{(D-5)}H\frac{\ddot{a}}{a}$
$\displaystyle\qquad-2\alpha(D-3)(D-2)\xi^{(D-4)}H^{2}\frac{\ddot{a}}{a}-2\alpha(D-3)(D-2)H^{2}\left[(D-5)\dot{\xi}^{2}\xi^{(D-6)}+\ddot{\xi}\xi^{(D-5)}\right]$
$\displaystyle\qquad\qquad-2\alpha(D-4)(D-3)(D-2)\dot{\xi}\xi^{(D-5)}H^{3}-\frac{1}{2}\alpha(D-3)(D-2)(D-5)\xi^{(D-4)}H^{4},$
(9) $\displaystyle
0=\ddot{\phi}+V_{\phi}+(D-1)\dot{\phi}H+2\alpha(D-3)(D-2)(D-1)\xi_{\phi}\xi^{(D-5)}H^{2}\frac{\ddot{a}}{a}$
$\displaystyle\qquad+\frac{\alpha}{2}(D-4)(D-3)(D-2)(D-1)\xi_{\phi}\xi^{(D-5)}H^{4},$
(10)

where $V_{\phi}=dV/d\phi$, $H\equiv\dot{a}/a$, denote the derivative of
potential with respect to scalar field and Hubble’s parameter respectively. A
dot mean derivative with respect to the cosmic time, $\rho_{\text{m}}$ and
$\rho_{\text{r}}$ are the energy density of matter corresponding to pressure
$p_{\text{m}}=0$ and radiation corresponding to
$p_{\text{r}}=\rho_{\text{r}}/3$ respectively. Then, determining the limit
$D\to 4$ for the Eqs. (8)-(10), we derive the explicit dynamical equations in
$4$-dimensions as

$\displaystyle
3H^{2}=\kappa^{2}\left(\rho_{\text{m}}+\rho_{\text{r}}+\frac{1}{2}\dot{\phi}^{2}+V+12\alpha\frac{\dot{\xi}}{\xi}H^{3}+3\alpha
H^{4}\right),$ (11) $\displaystyle
2\dot{H}+3H^{2}=-\kappa^{2}\left[\frac{\rho_{\text{r}}}{3}+\frac{\dot{\phi}^{2}}{2}-V-8\alpha\frac{\dot{\xi}}{\xi}H(\dot{H}+H^{2})-3\alpha
H^{4}-4\alpha
H^{2}\dot{H}-4\alpha\left(-\frac{\dot{\xi}^{2}}{\xi^{2}}+\frac{\ddot{\xi}}{\xi}\right)H^{2}\right],$
(12)
$\displaystyle\ddot{\phi}+V_{\phi}+3\dot{\phi}H+12\alpha\frac{\xi_{\phi}}{\xi}H^{2}\left(\dot{H}+H^{2}\right)=0,$
(13)

where $\xi_{\phi}=d\xi/d\phi$, $\xi_{\phi\phi}=d^{2}\xi/d\phi^{2}$ and
$\dot{\xi}=\xi_{\phi}\dot{\phi}$ and in (11), the quartic term in Hubble
parameter is the term that appears in [60], but our equations add the cubic
terms in Hubble parameter and terms associated with $\xi(\phi)$ due to the
regularization of the non-minimal coupling function to the GB term. These
additional terms might give explanations of the history of the universe from
the early to the late-time universe. The continuity equation of radiation
density is

$\displaystyle\dot{\rho_{\text{r}}}+4H\rho_{\text{r}}=0.$ (14)

By using the redefinition of some quantities, we can rewrite Eqs. (11) and
(12) as

$\displaystyle 3H^{2}$
$\displaystyle=\kappa^{2}\left(\rho_{\text{m}}+\rho_{\text{r}}+\rho_{\phi}\right),$
(15) $\displaystyle 2\dot{H}+3H^{2}$
$\displaystyle=-\kappa^{2}\left(\frac{1}{3}\rho_{\text{r}}+p_{\phi}\right),$
(16)

where the energy density and the pressure of the scalar field is defined as

$\displaystyle\rho_{\phi}$
$\displaystyle=\frac{1}{2}\dot{\phi}^{2}+V+12\alpha\frac{\dot{\xi}}{\xi}H^{3}+3\alpha
H^{4},$ (17) $\displaystyle p_{\phi}$
$\displaystyle=\frac{\dot{\phi}^{2}}{2}-V-8\alpha\frac{\dot{\xi}}{\xi}H(\dot{H}+H^{2})-3\alpha
H^{4}-4\alpha
H^{2}\dot{H}-4\alpha\left(-\frac{\dot{\xi}^{2}}{\xi^{2}}+\frac{\ddot{\xi}}{\xi}\right)H^{2}.$
(18)

The components of $\rho_{\phi}$ and $p_{\phi}$ satisfy the conservation
equation and the Eq. (13) can also be rewritten as

$\displaystyle\dot{\rho_{\phi}}+3H(1+w_{\phi})\rho_{\phi}=0,$ (19)

where the equation of state (EoS) for the scalar field can be defined as
$w_{\phi}\equiv p_{\phi}/\rho_{\phi}$, and the effective EoS can also be
defined as

$\displaystyle w_{\text{eff}}\equiv-1-\frac{2\dot{H}}{3H^{2}}.$ (20)

##  3 Dynamical System Approach

To understand the cosmological dynamics of this system, we apply the dynamical
system approach studied in [79, 80, 81, 82, 83]. First, we define the
dimensionless variables in order to conveniently investigate the dimensionless
dynamical equations. Then, we find the fixed points of the autonomous system
equations and analyze the stability. From that, we work further on the time
evolution of cosmological parameters. We can rewrite (11) as

$\displaystyle 1$
$\displaystyle=\frac{\kappa^{2}\rho_{\text{m}}}{3H^{2}}+\frac{\kappa^{2}\rho_{\text{r}}}{3H^{2}}+\frac{\kappa^{2}\dot{\phi}^{2}}{6H^{2}}+\frac{\kappa^{2}V}{3H^{2}}+\frac{4\alpha\kappa^{2}\dot{\xi}H}{\xi}+\alpha\kappa^{2}H^{2},$
(21)

and define the dimensionless variables as

$\displaystyle x$
$\displaystyle=\frac{\kappa\dot{\phi}}{\sqrt{6}H},\leavevmode\nobreak\
\leavevmode\nobreak\ y=\frac{\kappa\sqrt{V}}{\sqrt{3}H},\leavevmode\nobreak\
\leavevmode\nobreak\ \alpha_{\text{GB}}=\sqrt{\alpha}\kappa
H,\leavevmode\nobreak\ \leavevmode\nobreak\
\mu=-\frac{\sqrt{6}\xi_{\phi}}{\kappa\xi},\leavevmode\nobreak\
\leavevmode\nobreak\ \lambda=-\frac{V_{\phi}}{\sqrt{6}\kappa
V},\leavevmode\nobreak\ \leavevmode\nobreak\ \epsilon=-\frac{\dot{H}}{H^{2}}.$
(22)

From (22), the Friedmann equation (11) can be written as

$\displaystyle 1$
$\displaystyle=\Omega_{\text{m}}+\Omega_{\text{r}}+\Omega_{\phi},$ (23)
$\displaystyle\Omega_{\text{m}}$
$\displaystyle=\frac{\kappa^{2}\rho_{\text{m}}}{3H^{2}},$ (24)
$\displaystyle\Omega_{\text{r}}$
$\displaystyle=\frac{\kappa^{2}\rho_{\text{r}}}{3H^{2}},$ (25)
$\displaystyle\Omega_{\phi}$
$\displaystyle=x^{2}+y^{2}+\alpha_{\text{GB}}^{2}-4x\alpha_{\text{GB}}^{2}\mu.$
(26)

We also introduce the variable $N=\text{ln}a$ and $dN=Hdt$ so that by taking
the derivative of these with respect to $N$, we derive the following system of
equations

$\displaystyle\frac{dx}{dN}$
$\displaystyle=\frac{\kappa\ddot{\phi}}{\sqrt{6}H^{2}}-\frac{\kappa\dot{\phi}\dot{H}}{\sqrt{6}H^{3}}=x(\epsilon-\delta),$
(27) $\displaystyle\frac{dy}{dN}$ $\displaystyle=\frac{\kappa
V_{\phi}\dot{\phi}}{2\sqrt{3}\sqrt{V}H^{2}}-\frac{\kappa\sqrt{V}\dot{H}}{\sqrt{3}H^{3}}=y(\epsilon-3x\lambda),$
(28) $\displaystyle\frac{d\alpha_{\text{GB}}}{dN}$
$\displaystyle=\frac{\sqrt{\alpha}\kappa\dot{H}}{H}=-\alpha_{\text{GB}}\epsilon,$
(29) $\displaystyle\frac{d\Omega_{\text{r}}}{dN}$
$\displaystyle=-\frac{2\kappa^{2}\rho_{\text{r}}\dot{H}}{3H^{4}}+\frac{\kappa^{2}\dot{\rho}_{\text{r}}}{3H^{3}}=-4\Omega_{\text{r}}+2\Omega_{\text{r}}\epsilon,$
(30) $\displaystyle\frac{d\mu}{dN}$
$\displaystyle=\frac{\sqrt{6}\xi_{\phi}^{2}\dot{\phi}}{\kappa\xi^{2}H}-\frac{\sqrt{6}\xi_{\phi\phi}\dot{\phi}}{\kappa\xi
H}=x\mu^{2}(1-\Delta),$ (31) $\displaystyle\frac{d\lambda}{dN}$
$\displaystyle=\frac{V_{\phi}^{2}\dot{\phi}}{\sqrt{6}\kappa
V^{2}H}-\frac{V_{\phi\phi}\dot{\phi}}{\sqrt{6}\kappa
VH}=6x\lambda^{2}(1-\Gamma),$ (32)

where

$\displaystyle\Gamma\equiv\frac{V_{\phi\phi}V}{V_{\phi}^{2}},\quad\Delta\equiv\frac{\xi_{\phi\phi}\xi}{\xi_{\phi}^{2}}\,,\quad\delta\equiv-\frac{\ddot{\phi}}{\dot{\phi}H}=3-\frac{3y^{2}\lambda}{x}-\frac{2\alpha_{\text{GB}}^{2}\mu}{x}+\frac{2\alpha_{\text{GB}}^{2}\mu\epsilon}{x}.$
(33)

The autonomous system equations can also be written as

$\displaystyle\frac{dx}{dN}$
$\displaystyle=-3x+3y^{2}\lambda+2\alpha_{\text{GB}}^{2}\mu-2\alpha_{\text{GB}}^{2}\mu\epsilon+x\epsilon,$
(34) $\displaystyle\frac{dy}{dN}$ $\displaystyle=-3xy\lambda+y\epsilon,$ (35)
$\displaystyle\frac{d\alpha_{\text{GB}}}{dN}$
$\displaystyle=-\alpha_{\text{GB}}\epsilon,$ (36)
$\displaystyle\frac{d\Omega_{\text{r}}}{dN}$
$\displaystyle=-4\Omega_{\text{r}}+2\Omega_{\text{r}}\epsilon,$ (37)
$\displaystyle\frac{d\mu}{dN}$ $\displaystyle=x\mu^{2}(1-\Delta),$ (38)
$\displaystyle\frac{d\lambda}{dN}$ $\displaystyle=6x\lambda^{2}(1-\Gamma)\,,$
(39)

where it is worth noting that these equations are invariant under $y\to-y$.
From second Friedmann’s equation (12), we derive

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=$
$\displaystyle{}\frac{1}{2+4\alpha_{\text{GB}}^{2}(-1+2x\mu+2\alpha_{\text{GB}}^{2}\mu^{2})}\Bigl{[}3-3\alpha_{\text{GB}}^{2}-4x\alpha_{\text{GB}}^{2}\mu+8\alpha_{\text{GB}}^{4}\mu^{2}$
$\displaystyle{}+3y^{2}(-1+4\lambda\alpha_{\text{GB}}^{2}\mu)+3x^{2}+4x^{2}\alpha_{\text{GB}}^{2}\mu^{2}(1-\Delta)+\Omega_{\text{r}}\Bigr{]}.$
(40)

Eq. (3) diverges for

$\displaystyle x\equiv
x_{\text{div}}=\frac{-1+2\alpha_{\text{GB}}^{2}-4\alpha_{\text{GB}}^{4}\mu^{2}}{4\alpha_{\text{GB}}^{2}\mu}\,,$
(41)

which is unphysical. To avoid from this unphysical divergence the denominator
must not be zero, _i.e.,_ $x\neq x_{\text{div}}$. From Eqs. (17), (18) and
(20), we derive the EoS parameters for both the scalar field and the effective
as

$\displaystyle w_{\phi}=$
$\displaystyle{}\frac{1}{3(x^{2}+y^{2}+\alpha_{\text{GB}}^{2}-4x\alpha_{\text{GB}}^{2}\mu)(1+4\alpha_{\text{GB}}^{4}\mu^{2}-2\alpha_{\text{GB}}^{2}+4x\alpha_{\text{GB}}^{2}\mu)}\left[\left(-3y^{2}+12y^{2}\alpha_{\text{GB}}^{2}\lambda\mu\right)\right.$
$\displaystyle{}+x^{2}\left.\left(3+4\alpha_{\text{GB}}^{2}\mu^{2}(1-\Delta)\Bigr{)}-4x\alpha_{\text{GB}}^{2}\mu(4+\Omega_{\text{r}})+\alpha_{\text{GB}}^{2}\Bigl{(}3+2\Omega_{\text{r}}-4\alpha_{\text{GB}}^{2}\mu^{2}(1+\Omega_{\text{r}})\right)\right]\,,$
(42) $\displaystyle w_{\text{eff}}$
$\displaystyle\equiv-1-\frac{2\dot{H}}{3H^{2}}=-1+\frac{2\epsilon}{3}.$ (43)

##  4 The Stability of Fixed Points and Numerical Results

In this section, we investigate the stability of the autonomous system
described by Eqs. (34)–(39) for the potential $V(\phi)$ and the coupling
function $\xi(\phi)$ are given as

$\displaystyle
V(\phi)=V_{0}e^{-\sqrt{6}\kappa\lambda\phi},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\xi(\phi)=\xi_{0}e^{-\kappa\mu\phi/\sqrt{6}}.$ (44)

where $\lambda$ and $\mu$ are arbitrary constants. It is straightforward to
show that $\Delta=1$ and $\Gamma=1$ by substituting Eq. (44) into Eq. (33).
Thus, the autonomous system equations Eqs. (34)–(37) are sufficient for
further study. We first obtain fixed points and, then, check the stability of
the system at those fixed points through the linear stability theory, which is
based on analyzing the eigenvalues of the Jacobian matrix for the autonomous
system equations. In particular, a sign of the real part of the eigenvalues of
the Jacobian matrix plays an important role in determining the system’s
stability. If the real parts of all the eigenvalues are negative, it implies
that the fixed point is stable or attractors. In contrast, the real parts of
all the eigenvalues with a positive sign indicating that the fixed point is
unstable or repellers. The fixed point is identified as a saddle point if some
of the eigenvalues are negative while others are positive.

We list all fixed points of our system in Table 1 and present their
eigenvalues, as well as stability analyses of each fixed point, in Table 2. In
Table 1, all the fixed points are arranged into two groups: (i.) the GR fixed
points, including from $A^{\pm}_{1}$ to $A^{\pm}_{6}$, where
$\alpha_{\text{GB}}=0$ and (ii.) the GB fixed points, from $A^{\pm}_{7}$ to
$A^{\pm}_{10}$, where $\alpha_{\text{GB}}\neq 0$. Since the stability of the
GR fixed points is well investigated in the literature [80, 83], we focus on
understanding the stability of the GB fixed points in this work. In Figure 1,
we plot the phase portraits of the autonomous system in the “$x$–$y$” plane.
We only present the positive “$y$”-axis because the background equations are
symmetric under the change of $y\rightarrow-y$. These symmetric properties of
the EoM imply that the $A_{7}^{\pm}$ and $A_{8}^{\pm}$ fixed points in Table 1
are basically the same points.

Points $x$ $y$ $\alpha_{\text{GB}}$ $\Omega_{r}$ $\Omega_{m}$ $\Omega_{\phi}$
Existence $\omega_{\text{eff}}$=$\omega_{\phi}$ $A_{1}^{\pm}$
$\frac{1}{2\lambda}$ $\pm\frac{1}{2\lambda}$ 0 0 $1-\frac{1}{2\lambda^{2}}$
$\frac{1}{2\lambda^{2}}$ $\lambda^{2}>1/2$ 0 $A_{2}^{\pm}$
$\frac{2}{3\lambda}$ $\pm\frac{\sqrt{2}}{3\lambda}$ 0
$1-\frac{2}{3\lambda^{2}}$ 0 $\frac{2}{3\lambda^{2}}$ $\lambda^{2}>2/3$ $1/3$
$A_{3}^{\pm}$ $\pm 1$ 0 0 0 0 1 $\forall\lambda$ 1 $A_{4}$ 0 0 0 0 1 0
$\forall\lambda$ 0 $A_{5}$ 0 0 0 1 0 0 $\forall\lambda$ $1/3$ $A_{6}^{\pm}$
$\lambda$ $\pm\sqrt{1-\lambda^{2}}$ 0 0 0 1 $\lambda^{2}<1$ $-1+2\lambda^{2}$
$A_{7}^{\pm}$ 0 $-\sqrt{\frac{2\mu}{2\mu-3\lambda}}$
$\pm\sqrt{\frac{3\lambda}{3\lambda-2\mu}}$ 0 0 1 $(\mu\geq 0,\lambda<0)$ or
$(\mu\leq 0,\lambda>0)$ $-1$ $A_{8}^{\pm}$ 0
$\sqrt{\frac{2\mu}{2\mu-3\lambda}}$ $\pm\sqrt{\frac{3\lambda}{3\lambda-2\mu}}$
0 0 1 $(\mu\geq 0,\lambda<0)$ or $(\mu\leq 0,\lambda>0)$ $-1$ $A_{9}^{\pm}$
$\frac{3-\sqrt{9-80\mu^{2}}}{20\mu}$ 0
$\pm\sqrt{\frac{6}{3+\sqrt{9-80\mu^{2}}}}$ 0 0 1
$-\frac{3}{4\sqrt{5}}\leq\mu\leq\frac{3}{4\sqrt{5}},\mu\neq 0$ $-1$
$A_{10}^{\pm}$ $\frac{3+\sqrt{9-80\mu^{2}}}{20\mu}$ 0
$\pm\sqrt{\frac{9+3\sqrt{9-80\mu^{2}}}{40\mu^{2}}}$ 0 0 1
$-\frac{3}{4\sqrt{5}}\leq\mu\leq\frac{3}{4\sqrt{5}},\mu\neq 0$ $-1$

Table 1: The fixed points and cosmological parameters of the system Eqs.
(34)–(37) with $\Delta=1$ and $\Gamma=1$.

Points Eigenvalues Stability $A_{1}^{\pm}$
$\left\\{-\frac{3}{2},-1,-\frac{3}{4}\left(1+\sqrt{\frac{4}{\lambda^{2}}-7}\right),-\frac{3}{4}\left(1-\sqrt{\frac{4}{\lambda^{2}}-7}\right)\right\\}$
stable for: $1/2<\lambda^{2}\leq 4/7$, stable-focus for: $\lambda^{2}>4/7$,
saddle for: $\lambda^{2}<1/2$ $A_{2}^{\pm}$
$\left\\{-2,+1,-\frac{1}{2}\left(1+\sqrt{\frac{32}{3\lambda^{2}}-15}\,\right),-\frac{1}{2}\left(1-\sqrt{\frac{32}{3\lambda^{2}}-15}\,\right)\right\\}$
saddle. $A_{3}^{\pm}$ $\left\\{-3,+3,+2,3\pm 3\lambda\right\\}$ saddle.
$A_{4}$ $\left\\{-\frac{3}{2},-\frac{3}{2},+\frac{3}{2},-1\right\\}$ saddle.
$A_{5}$ $\left\\{-2,+2,-1,+1\right\\}$ saddle. $A_{6}^{\pm}$
$\left\\{-3\lambda^{2},3\left(\lambda^{2}-1\right),6\lambda^{2}-3,6\lambda^{2}-4\right\\}$
stable for: $0<\lambda^{2}<1/2$; otherwise saddle. $A_{7}^{\pm}$
$\left\\{-4,-3,-\frac{3}{2}\left(1-\sqrt{1+\frac{32\lambda^{2}\mu^{2}-48\lambda^{3}\mu}{4\mu^{2}-\lambda^{2}(9-36\mu^{2})}}\right),-\frac{3}{2}\left(1+\sqrt{1+\frac{32\lambda^{2}\mu^{2}-48\lambda^{3}\mu}{4\mu^{2}-\lambda^{2}(9-36\mu^{2})}}\right)\right\\}$
stable for: $\text{con}_{1}$, $A_{8}^{\pm}$
$\left\\{-4,-3,-\frac{3}{2}\left(1-\sqrt{1+\frac{32\lambda^{2}\mu^{2}-48\lambda^{3}\mu}{4\mu^{2}-\lambda^{2}(9-36\mu^{2})}}\right),-\frac{3}{2}\left(1+\sqrt{1+\frac{32\lambda^{2}\mu^{2}-48\lambda^{3}\mu}{4\mu^{2}-\lambda^{2}(9-36\mu^{2})}}\right)\right\\}$
stable focus for:
$\frac{3\lambda^{2}}{-8\lambda^{3}+2\sqrt{16\lambda^{6}+17\lambda^{4}+\lambda^{2}}}<\mu<\frac{3\lambda}{2\sqrt{1+9\lambda^{2}}}$
, saddle for: $\mu>\frac{3}{2}\sqrt{\frac{\lambda^{2}}{1+9\lambda^{2}}}$ if
$\lambda<0$ or $\mu<-\frac{3}{2}\sqrt{\frac{\lambda^{2}}{1+9\lambda^{2}}}$ if
$\lambda>0$. $A_{9}^{\pm}$
$\left\\{-\frac{3\lambda}{20\mu}\left(3-\sqrt{9-80\mu^{2}}\right),-4,-3,-3\right\\}$
stable for: $\text{con}_{2}$; otherwise saddle. $A_{10}^{\pm}$
$\left\\{-4,-3,-3,-\frac{3\lambda}{20\mu}\left(3+\sqrt{9-80\mu^{2}}\right)\right\\}$
stable for: $\text{con}_{2}$; otherwise saddle.

Table 2: The eigenvalues and the stability analyses of fixed points the system
Eqs. (34)–(37)

.

Table 2 shows that the system is stable at the $A^{\pm}_{7,8}$ fixed points if
the following conditions are satisfied:

$\displaystyle\text{con}_{1}$
$\displaystyle=\left\\{\begin{array}[]{@{}lr@{}}0<\mu\leq\frac{3\lambda^{2}}{-8\lambda^{3}+2\sqrt{16\lambda^{6}+17\lambda^{4}+\lambda^{2}}},&\text{for
}\lambda<0\\\
-\frac{3\lambda^{2}}{8\lambda^{3}+2\sqrt{16\lambda^{6}+17\lambda^{4}+\lambda^{2}}}\leq\mu<0,&\text{for
}\lambda>0\end{array}\right.\,,$ (47)

and also at the $A_{9}^{\pm}$ and $A_{10}^{\pm}$ fixed points if the following
conditions are satisfied:

$\displaystyle\text{con}_{2}$
$\displaystyle=\left\\{\begin{array}[]{@{}lr@{}}-\frac{3}{4\sqrt{5}}\leq\mu<0\,,&\text{for
}\lambda<0\\\ 0<\mu\leq\frac{3}{4\sqrt{5}}\,,&\text{for
}\lambda>0\end{array}\right.\,.$ (50)

For obtaining Eq. (47), we take the existence conditions of the
$A^{\pm}_{7,8}$ points into account, which implies the $\mu$ and $\lambda$
parameters should have the opposite signs, see the existence column of
$A^{\pm}_{7,8}$ in Table 1. The stability conditions in Eqs. (47) and (50) put
constraints on the $\\{\lambda,\mu\\}$ values. _For example_ , from Eq. (47),
one can calculate that the largest (smallest) value for $\mu$ is
$\mu_{\text{max}}\simeq 0.2271$ ($\mu_{\text{min}}\simeq-0.2271$) when
$\lambda\simeq-0.3357$ ($\lambda\simeq 0.3357$). In both the small and the
large $|\lambda|$ limit, the range of $\mu$ shrinks and eventually approaches
to zero from both the negative and the positive sides. On the other hand, from
Eq. (50), we see that the range of $\mu$ is determined depending on the sign
of $\lambda$, but not on the amplitude. By taking these constraints on the
numerical values of $\\{\lambda,\mu\\}$ into account, we plot the phase
portraits of our dynamical system in Figure 1.

Figure 1: Phase space portraits of the autonomous system on a $x$ vs. $y$
plane. The acceleration of the universe is possible within the gray-shaded
region. The green and red lines are particular attractor trajectories toward
the fixed points $A_{7,8}^{\pm}$, $A_{9}^{\pm}$, and $A_{10}^{\pm}$ when their
stability conditions are satisfied. The dashed lines at $x=x_{\text{div}}$
indicate the divergence of dynamical equations, see Eq. (41).

The blue points in the figure indicate the GB fixed points; namely the
$A_{7,8}^{\pm}$, $A_{9}^{\pm}$, and $A_{10}^{\pm}$ fixed points, and the
numerical values of $\\{\lambda,\mu\\}$ are given such a way that most
trajectories, including the red and green ones, are attracted to these stable
points. The gray shaded regions represent the region where the EoS parameter
$\omega_{\text{eff}}=\omega_{\phi}<-1/3$ such that the accelerated expansion
of the universe is possible. The dashed lines show the divergence lines given
in Eq. (41).

The right column of Figure 2 shows the time evolution of
$\Omega_{i}(z)(i=r,m,\phi)$ and the EoS parameter of the scalar field
$\omega_{\phi}(z)$, as well as the effective EoS of the system
$\omega_{\text{eff}}(z)$, while the left one shows the time evolution of the
autonomous variables, $\\{x(z),y(z),\alpha_{\text{GB}}(z)\\}$. In the figure,
from top to down, the numerical value of $\lambda$ changes as
$\lambda=0.1,0.3$, and $0.6$ for the $\mu=0.3$. If we change the $\mu$ value
within the range given in Eqs. (47) and (50) for each given $\lambda$, there
is no noticeable change occurs because the $\mu$ value is too small to make
significant contributions. In other words, we find that the cosmic phase
transitions, as well as the time evolution of the autonomous variables, are
more sensitive to the $\lambda$ value than the $\mu$ value as long as the
stability conditions of the GB fixed points are concerned.

The initial conditions of the autonomous variables are given in such a way
that the universe experiences the right
$\Omega_{r}\rightarrow\Omega_{m}\rightarrow\Omega_{\phi}$ phase transitions.
Thus, the effective EoS parameter starts evolving from
$\omega_{\text{eff}}=1/3$ to a some value $\omega_{\text{eff}}<-1/3$, for
allowing the late-time universe to experience the accelerated expansion. To
replicate the viable cosmic history, the time evolution of
$\\{x,y,\alpha_{\text{GB}}\\}$ must depend on fine-tuned initial conditions:
$y>\alpha_{\text{GB}}>x$ in our case.

Figure 2: Left: The red-shift evolution of the $\Omega_{i}(z)$, where
$i=r,m,\phi$, the EoS parameter $\omega_{\phi}$, and the effective EoS of the
system $\omega_{\text{eff}}$. Right: The red-shift evolution of the $x,y$,
$\alpha_{\text{GB}}$. In both columns, the $\lambda$ value increases from top
to bottom; $\lambda=0.1,0.3,$ and $0.6$, respectively.

We show that at the onset of the radiation-dominated phase, the contribution
of the GB term must be the second largest after the potential energy
contribution of the scalar field, and its time evolution
$\alpha_{\text{GB}}(z)$ experiences a continuous decrease over time. As a
result, the present-day value of $\alpha_{\text{GB}}(z=0)$ is negligible, and
so are the effects of the GB term. On contrary, the initial value of $x$, or
the kinetic energy contribution of the scalar field, starts off as the
smallest in the radiation-dominated phase, but its time evolution experiences
a drastic increment during the matter-dominated phase after a short decrease
towards the end of radiation dominated era, and it eventually catches up with
the potential energy contributions, the $y(z)$, in the present universe. Thus,
from Figure 2, it is explicit that the potential energy contribution of the
scalar field plays the most dominant role throughout the cosmic history of the
universe, and its present-day value is also the largest among the others. For
the given $\mu$, Figure 2 also shows that the present-day value of the EoS
parameter increases as the $\lambda$ value increases, and it eventually
surpasses “$-1/3$”, ending the scalar field-dominated phase. In our case, the
universe experiences the late-time acceleration, driven by the scalar field
coupled to the GB term, if $\lambda<0.746$.

Figure 3: The red-shift evolution of the $\omega_{\phi}$ for different values
of $\lambda$. The background dark- and light-gray shaded regions represent the
observational bounds from the CMB, BAO, SnIa, and $H_{0}$ data using the CPL
parameterization [84]. The colored regions show the different parameter spaces
of $\lambda$.

To put further constraint on the potential parameter $\lambda$ in light of the
observational bounds from CMB, BAO, SnIa, and $H_{0}$ data [84], we plot in
the red-shift evolution of the EoS $\omega_{\phi}(z)$ in Figure 3. In the
figure, the parameter space for $\lambda$ is represented by the different
colors and we adopted the Chevallier-Polarski-Linders (CPL) parameterization
of the EoS that reads [32],

$\omega(z)=\omega_{0}+\frac{z}{1+z}\omega_{a}.$ (51)

The result confirms our observation of Figure 2 that the smaller values of
$\lambda$ are more preferred. For instance, the preferred parameter space for
$\lambda$ is $\lambda\lesssim 0.3166$ to be consistent with the observational
data at the $1\sigma$ level at present. The range is tighter being
$\lambda\lesssim 0.1$ to be consistent with the data for red-shift up to
$z=1$. Thus, now we can provide tighter constraints on the parameters of our
model given in Eq. (44). In order for the universe to be stable at the GB
fixed points and experience the late-time accelerated expansion, the
parameters $\\{\lambda,\mu\\}$ of our model must be within the following
range:

$\displaystyle\text{For
}A^{\pm}_{7,8}:\quad-0.2268\lesssim\mu<0\quad\text{when}\quad 0<\lambda\leq
0.3166\,,$ (52) $\displaystyle\text{For }A^{\pm}_{9,10}:\quad
0<\mu\lesssim\frac{3}{4\sqrt{5}}\quad\text{when}\quad 0<\lambda\leq 0.3166\,.$
(53)

##  5 Conclusions

In this work, we have extended regularized $4D$ EGB gravity by introducing the
non-minimal coupling function and the regularization scheme introduced in
[60]. Along this consideration, a redefinition $\xi(\phi)\to\xi^{(D-4)}(\phi)$
is required so that EoM is well-defined. In the spatially flat space-time
dimensions of FRW metric, we can write the EoM in arbitrary $D$-dimensions and
then take the $D\to 4$ limit. As a solution, we derived the extended version
of EoM obtained in [60] for non-trivial $\xi(\phi)$, which is often associated
with the cubic term in Hubble parameter and could be interesting in a further
study of the cosmic evolution and the observational data constraint.

To explore the cosmological dynamics from this extension, we applied the
dynamical system approach and analyzed the stability of critical fixed points
for given exponential forms of the potential and the coupling function of the
scalar field in Eq. (44). We obtained ten fixed points, of which $A^{\pm}_{1}$
to $A^{\pm}_{6}$ are called the GR fixed points and $A^{\pm}_{7}$ to
$A^{\pm}_{10}$ are called the GB fixed points because their existence is due
to the presence of non-minimal coupling function
$\xi(\phi)\rightarrow\xi^{(D-4)}(\phi)$ and rescaling of the coupling constant
$\alpha\to\alpha/(D-4)$, and they exist and stable under specific conditions
listed in Table 1 and 2. The GB fixed points correspond to the de-Sitter
solution with the EoS parameter $\omega_{\text{eff}}=-1=\omega_{\phi}$ that
admits the late-time accelerating universe under the conditions in Eqs. (47)
and (50). The phase space plots in Figure 1 confirm the stable and
accelerating solutions.

We also present the cosmic history of the universe,
$\Omega_{r}\rightarrow\Omega_{m}\rightarrow\Omega_{\phi}$, in Figure 2 along
with the time evolution of dynamical variables; $x$, $y$, and
$\alpha_{\text{GB}}$, associated with the kinetic and potential energy
contribution of the scalar field and the GB contribution, respectively. Both
kinetic and potential energies of the scalar field play the most important
role in the present day; particularly the potential energy, while the effect
of the GB term is appreciated at the beginning of the radiation-dominated
epoch. However, the GB contribution decreases in time and, eventually, it does
not play many roles in the present universe.

The time evolution of the EoS parameter for different values of $\lambda$, the
potential parameter, is presented in Figure 3 together with the upper bounds
from the observational data. The effective EoS starts evolving from
$\omega_{\text{eff}}=1/3$ to some value $\omega_{\text{eff}}<-1/3$, which
indicates the late-time universe experiences the accelerated expansion. If
$\lambda<0.746$, our model explains the late-time accelerated expansion of the
universe driven by the scalar field coupled to the GB term and the EoS
satisfies the observational constraint by CMB, BAO, SnIa, and $H_{0}$ data if
$\lambda\lesssim 0.1$ at $1\sigma$ C.L.. In conclusion, our model suggests
that the universe can be stable at the GB fixed points and experiences the
late-time accelerated expansion when the values of the model parameters
$\\{\lambda,\mu\\}$ are in the range given in Eqs. (52) and (53).

## Acknowledgments

BB is partially supported by the project ”Extension of the Standard Model and
Cosmology, heavy meson study” (SHUTBIHHZG-2022/168) by the Mongolian Ministry
of Education and Science and Mongolian Science and Technology Foundation. SK
is supported by the Higher Education Improvement Project (HEIP) funded by the
Cambodian Government (IDA Credit No. 6221-KH) and PR is supported by the
Swedish International Development Cooperation Agency (SIDA) through Sweden and
the Royal University of Phnom Penh (RUPP)’s Pilot Research Cooperation
Programme (Sida Contribution No. 11599). GT was supported by Basic Science
Research Program through the National Research Foundation of Korea(NRF) funded
by the Ministry of Education (grant number) (NRF-2022R1I1A1A01053784) and
(NRF-2021R1A2C1005748).

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