Dynamical analysis in regularized 4DEinstein-Gauss-Bonnet gravity with non-minimal coupling Bilguun Bayarsaikhanab1 Sunly Khimphunc2 Phearun Rithyc3 and Gansukh Tumurtushaad4

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Dynamical analysis in regularized 4DEinstein-Gauss-Bonnet gravity

with non-minimal coupling

Bilguun Bayarsaikhana,b,1, Sunly Khimphunc,2, Phearun Rithyc,3, and Gansukh Tumurtushaad,4

aInstitute for Theoretical Physics, ELTE Eotvos Lorand University, Pazmany Peter setany 1/A,

H-1117 Budapest, Hungary

bInstitute of Physics & Technology, Mongolian Academy of Sciences, Ulaanbaatar 13330, Mongolia

cGraduate School of Science, Royal University of Phnom Penh, Phnom Penh, Cambodia 12150

dDepartment of Science Education, Jeju National University, Jeju, 63243, Korea

Abstract

We investigate the regularized four-dimensional Einstein-Gauss-Bonnet (4DEGB) gravity with

a non-minimal scalar coupling function, which is an extension of the regularized 4DEGB 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.

1e-mail : ph.bilguun@gmail.com

2e-mail : khimphun.sunly@rupp.edu.kh

3e-mail : phearunrain0@gmail.com

4e-mail : gansukh@jejunu.ac.kr

arXiv:2210.06717v1 [gr-qc] 13 Oct 2022

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

H0measurement [3, 4]. However, some issues remain for the cosmological constant or ΛCDM 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 partic-

ular, 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≥5 Gauss-Bonnet (GB) gravity which sat-

isﬁes 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 4Dis topologically invariant, so it does not modify Ein-

stein’s theory since it does not contribute to gravitational dynamics. Nevertheless, the GB term can

contribute to the 4Ddynamical 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 4DEGB model proposed in [60] introduces a scaling coupling constant α→α/(D−

4), where Dis the space-time dimensions, and considers the limit D→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 4DEGB

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→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 4DEGB theory is that

looking at diﬀeomorphism invariant property reveals the inconsistency in the regularized 4DEGB

theory [67, 69, 70]. Nevertheless, the consistent study of D→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 α→

α/(D−4) and take the limit D→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 4Dgravity with the scaling coupling constant

advocated in the regularized 4DEGB 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 4DEGB.

In this works, we study the extension of regularized 4DEGB theory by introducing the non-

minimal coupling to the GB term and redeﬁning the coupling function

ξ(φ)→ξ(D−4)(φ),

with ﬂat FRW metric in D-dimensions then take the limit D→4. The motivation of the form of

coupling function is obvious so that divergence of the terms associated to the scaling of αremains 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 4DEGB 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 H0measurement.

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 4DEoM 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=ZdDx√−g1

2κ2R−1

2gµν ∂µφ∂νφ−V(φ)−α

2ξ(φ)G+Sm,r,(1)

where R,α,V(φ), ξ(φ) are Ricci scalar, a coupling constant, potential function and the coupling

function of scalar ﬁeld respectively.

G ≡ R2−4Rµν Rµν +RµνρσRµνρσ ,

is GB term, Sm,r represents the standard matter and radiation components, κ2= 8πG, and Gis

Newtonian constant. From this action, after obtaining the EoM, we re-scale α→α/(D−4) and then

take the limit D→4. The problem arises when we introduce ξ(φ) couples to GB term since the

expected ﬁnite terms from ξ(φ) which appear in the EoM contain no factor of (D−4). As a result,

1/(D−4) from the scaling αwill diverge. We propose the redeﬁning the coupling function

ξ(φ)→ξ(D−4)(φ),

ﬁrst and then take the limit D→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 αand redeﬁning ξ(φ) are

Rµν −1

2gµν R=κ2T(φ)

µν +T(m,r)

µν +T(GB)

µν ,(2)

φ−Vφ−α

2(D−4)(D−4)ξ(D−5)ξφG= 0,(3)

where ≡ ∇µ∇µ,Vφ=dV/dφ,ξφ=dξ/dφ, and T(φ)

µν , T (m,r)

µν , T (GB)

µν are stress-energy-momentum

tensor of scalar ﬁeld, GB term and matter term respectively, which we ﬁnd

T(φ)

µν =∂µφ∂νφ−1

2gµν (gρσ∂ρφ∂σφ+ 2V),(4)

T(m,r)

µν = (ρ+P)UµUν+P gµν ,(5)

T(GB)

µν =α

(D−4) (ξ(D−4) −4RµαRνα + 2RRµν −4Rαβ Rµανβ + 2Rµαβγ Rναβγ −1

2gµν G

+ (D−4)(D−5)ξ2

φξ(D−6)"gµν 2R∇αφ∇αφ−4Rαβ ∇αφ∇βφ−4Rµν ∇αφ∇αφ

+ 4Rµανβ ∇αφ∇βφ+ 4Rνα∇αφ∇µφ+ 4Rµα ∇αφ∇νφ−2R∇µφ∇νφ#+ (D−4)ξ(D−5)

×"ξφ−4Rµν φ+ 2Rµανβ ∇β∇αφ+ 2Rµβνα∇β∇αφ+gµν 2Rφ−4Rαβ ∇β∇αφ

+ 4Rνα∇µ∇αφ+ 4Rµα∇ν∇αφ−2R∇ν∇µφ+ξφφ−4Rµν ∇αφ∇αφ+ 4Rµανβ ∇αφ∇βφ

+gµν 2R∇αφ∇αφ−4Rαβ ∇α∇βφ+ 4Rνα∇αφ∇µφ+ 4Rµα∇αφ∇νφ−2R∇µφ∇νφ#),(6)

where Uµ= (1,0,0,0) is four-velocity and ρ, 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

ds2=−dt2+a2(t)(dχ2

1+ dχ2

2+ dχ2

3+... + dχ2

D−1),(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

(D−2)(D−1)

2κ2H2=1

2˙

φ2+V+ρm+ρr+ 2α(D−3)(D−2)(D−1) ˙

ξξ(D−5)H3

2α(D−3)(D−2)(D−1)ξ(D−4)H4,(8)

−(D−3)(D−2)

2κ2H2+(D−2)

κ2

¨a

a=1

φ2−V+pr−4α(D−3)(D−2) ˙

ξξ(D−5)H¨a

−2α(D−3)(D−2)ξ(D−4)H2¨a

a−2α(D−3)(D−2)H2h(D−5) ˙

ξ2ξ(D−6) +¨

ξξ(D−5)i

−2α(D−4)(D−3)(D−2) ˙

ξξ(D−5)H3−1

2α(D−3)(D−2)(D−5)ξ(D−4)H4,(9)

0 = ¨

φ+Vφ+ (D−1) ˙

φH + 2α(D−3)(D−2)(D−1)ξφξ(D−5)H2¨a

+α

2(D−4)(D−3)(D−2)(D−1)ξφξ(D−5)H4,(10)

where Vφ=dV/dφ,H≡˙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, ρmand ρr

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Dynamicalanalysisinregularized4DEinstein-Gauss-Bonnetgravitywithnon-minimalcouplingBilguunBayarsaikhana,b,1,SunlyKhimphunc,2,PhearunRithyc,3,andGansukhTumurtushaad,4aInstituteforTheoreticalPhysics,ELTEEotvosLorandUniversity,PazmanyPetersetany1/A,H-1117Budapest,HungarybInstituteofPhysics&Technology,M...

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Dynamical analysis in regularized 4DEinstein-Gauss-Bonnet gravity with non-minimal coupling Bilguun Bayarsaikhanab1 Sunly Khimphunc2 Phearun Rithyc3 and Gansukh Tumurtushaad4

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