D维EGB模型中的指数宇宙学解：α1, α2与H, h探究

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"这篇论文是关于Einstein-Gauss-Bonnet (EGB) 模型在没有宇宙项（Λ=0）情况下的指数宇宙学解的深入研究。研究集中在D维空间中，其中有两个比例因子的指数演化，分别由两个独立的哈勃参数H和h控制，它们对应于不同维度的因子空间。论文作者为V.D.Ivashchuk、A.A.Kobtsev，发表在Eur.Phys.J.C期刊2019年第79卷，文章编号824，DOI为10.1140/epjc/s10052-019-7329-8。" 在EGB理论中，宇宙学解是宇宙演化的重要数学表示，它描述了宇宙尺度因子随时间的变化规律。在本研究中，模型包含了两个非零常数α1和α2，它们影响着宇宙的动态行为。这里的指数宇宙学解指的是比例因子随时间呈指数增长或衰减的模型，这与实际观测到的宇宙加速膨胀相符合。哈勃参数H和h分别与两个因子空间的扩张率相关，这两个空间的维度分别为m和l，且m和l都大于2，总维度D等于1+m+l。作者指出，H和h不相等（h≠H），并且mH + lh不等于0，这是为了确保模型的复杂性和多样性，使得解不会简化成单一的指数形式。对于α=α2/α1的值，当α>0时，研究发现存在两个真实的宇宙学解，即（H1，h1）和（H2，h2）。这两个解具有不同的指数关系，其中h1/H1小于-m/l，而h2/H2则小于0。这个结果表明，随着α值的不同，宇宙的扩张模式会呈现出两种不同的行为。当α<0时，论文并未详细给出具体的解，但可以推断，这可能导致完全不同的宇宙动力学，可能包括减速膨胀或者不同的稳定性特性。这种对α的正负值的区分，揭示了EGB模型中的相变或宇宙行为的显著变化。这篇研究通过重新审视D维EGB模型中的指数宇宙学解，提供了对多元宇宙结构动态的新洞察，尤其是在没有宇宙常数的情况下。这样的工作有助于我们更好地理解宇宙的早期状态以及可能存在的不同宇宙演化路径，对于广义相对论和量子引力理论的交叉研究具有重要意义。

Eur. Phys. J. C (2019) 79:824

https://doi.org/10.1140/epjc/s10052-019-7329-8

Regular Article - Theoretical Physics

Exponential cosmological solutions with two factor spaces in EGB

model with  = 0 revisited

V. D. Ivashchuk

1,2,a

,A.A.Kobtsev

Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street,

Moscow 117198, Russian Federation

Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Street, Moscow 119361, Russian Federation

Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Troitsk 142190, Russian Federation

Received: 12 April 2019 / Accepted: 20 September 2019 / Published online: 8 October 2019

Abstract We study exact cosmological solutions in D-

dimensional Einstein–Gauss–Bonnet model (with zero cos-

mological term) governed by two non-zero constants: α

and

. We deal with exponential dependence (in time) of two

scale factors governed by Hubble-like parameters H > 0

and h, which correspond to factor spaces of dimensions

m > 2 and l > 2, respectively, and D = 1 + m + l.

We put h = H and mH + lh = 0. We show that for

α = α

/α

> 0 there are two (real) solutions with two sets of

Hubble-like parameters: (H

, h

) and (H

, h

), which obey:

< −m/l < h

< 0, while for α<0 the (real)

solutions are absent. We prove that the cosmological solu-

tion corresponding to (H

, h

) is stable in a class of cosmo-

logical solutions with diagonal metrics, while the solution

corresponding to (H

, h

) is unstable. We present several

examples of analytical solutions, e.g. stable ones with small

enough variation of the effective gravitational constant G,

for (m, l) = (9, l > 2), (12, 11), (11, 16), (15, 6).

1 Introduction

Currently, the Einstein–Gauss–Bonnet (EGB) model and

related theories, see [1–12] and Refs. therein, are under inten-

sive studies in cosmology, aimed at explanation of accel-

erating expansion of the Universe [13,14]. Here we study

the EGB model with zero cosmological term in D dimen-

sions (D = n +1). This model contains Gauss–Bonnet term,

which arises in (super)string theory as a correction to the

(super)string effective action (e.g. heterotic one) [15–17].

The model is governed by two nonzero constants α

and α

which correspond to Einstein and Gauss–Bonnet terms in

the action, respectively. In this paper we continue our stud-

e-mail: ivashchuk@mail.ru

ies of the EGB cosmological model from Ref. [8]. We deal

with diagonal metrics governed by n > 3 scale factors and

consider the following ansatz for scale factors a

(t) (t is syn-

chronous time variable): a

(t) = ··· = a

(t) = exp(Ht)

and a

m+1

(t) =···=a

m+l

(t) = exp(ht ), where n = m +l,

m > 2, l > 2. We put here H > 0 in order to describe expo-

nential accelerated expansion of 3d subspace with Hubble

parameter H [18].

In contrary to our earlier publication [8], where a lot

of numerical solutions with small enough value of varia-

tion of the effective gravitational constant G were found,

here we put our attention mainly to the search of analyti-

cal exponential solutions with two factor spaces of dimen-

sions m and l. Here we show that the anisotropic cosmo-

logical solutions under consideration with two Hubble-like

parameters H > 0 and h obeying restrictions h = H ,

mH + lh = 0 do exist only if α = α

/α

> 0. In this

case we have two solutions with Hubble-like parameters:

> 0, h

< 0) and (H

> 0, h

< 0), respectively, such

that x

= h

< −m/l < x

= h

. By using results

of Refs. [10,11] (see also approach of Ref. [9]) we show that

the solutions with Hubble-like parameters (H

, h

) are stable

(in a class of cosmological solutions with diagonal metrics),

while those corresponding to (H

, h

) are unstable.

Here we also present examples of analytical solutions for:

(i) m = l; (ii) m = 3, l = 4; (iii) m = 9, l > 2; (iv) m = 12,

l = 11; (v) m = 11, l = 16 and (vi) m = 15, l = 6. It

should be noted that analytical solutions in cases (iii) and

(iv) were considered numerically in Ref. [8] in a context

of solutions with a small (enough) variation of G (in Jordan

frame, see Ref. [20]), e.g. obeying the most severe restrictions

on variation of G from Ref. [19]. The stable solutions with

zero variation of G in cases (v) and (vi) were found earlier

in [8], while the stability of these solutions was proved in

Ref. [10].

123

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