Examples of stable exponential cosmological solutions with three factor spaces in EGB model with a $\Lambda$-term

TL;DR

This paper presents three stable exponential cosmological solutions within the Einstein-Gauss-Bonnet model with a cosmological constant, describing accelerated expansion of 3D space and potential variations in gravitational constant.

Contribution

It provides explicit stable solutions with exponential scale factors in a higher-dimensional EGB model, including conditions for small gravitational constant variation.

Findings

Three stable exponential solutions with specific subspace dimensions.

Two solutions allow small variation of the effective gravitational constant.

Solutions correspond to different total spacetime dimensions.

Abstract

We deal with Einstein-Gauss-Bonnet model in dimension $D$ with a $\Lambda$-term. We obtain three stable cosmological solutions with exponential behavior (in time) of three scale factors corresponding to subspaces of dimensions $(l_0,l_1,l_2) = (3, 4, 4), (3, 3, 2), (3, 4, 3)$ and $D = 12, 9, 11$, respectively. Any solution may describe an exponential expansion of $3$-dimensional subspace governed by Hubble parameter $H$. Two of them may also describe a small enough variation of the effective gravitational constant $G$ (in Jordan frame) for certain values of $\Lambda$.

Equations

$$S={\int }_M{d}^{D}z\sqrt{|g|}\{{\alpha }_1(R[g]-2\Lambda )+{\alpha }_2{\mathcal{L}}_2[g]\}.$$

$${\mathcal{L}}_2[g]=R_{MNPQ}{R}^{MNPQ}-4R_{MN}{R}^{MN}+R^2$$

$$M=\mathbb{R}\times M_1\times \dots \times M_n$$

$$g=-dt\otimes dt+\sum _{i=1}^n{B}_i{e}^{2v^{i}t}d{y}^i\otimes d{y}^i,$$

$$E=G_{ij}{v}^i{v}^j+2\Lambda -\alpha G_{ijkl}{v}^i{v}^j{v}^k{v}^l=0,$$

$$Y_i=\left[2G_{ij}{v}^j-\frac{4}{3}\alpha G_{ijkl}{v}^j{v}^k{v}^l\right]\sum _{i=1}^n{v}^i-\frac{2}{3}{G}_{ij}{v}^i{v}^j+\frac{8}{3}\Lambda =0,$$

$$G_{ij}={\delta }_{ij}-1,G_{ijkl}=G_{ij}{G}_{ik}{G}_{il}{G}_{jk}{G}_{jl}{G}_{kl}.$$

$$v=(\stackrel{k_0}{\overbrace{h_0,\dots ,h_0}},\stackrel{k_1}{\overbrace{h_1,\dots ,h_1}},\stackrel{k_2}{\overbrace{h_2,\dots ,h_2}}),$$

$$h_0\ne h_1,h_0\ne h_2,h_1\ne h_2,S_1=k_0{h}_0+k(h_1+h_2)\ne 0.$$

$$\alpha >0.$$

$$\bar{v}=(\underset{l_0=3}{\underbrace{H,H,H}},\underset{l_1}{\underbrace{H_1,\dots ,H_1}},\underset{l_2}{\underbrace{H_2,\dots ,H_2}}),$$

$$g=-dt\otimes dt+\sum _{i=1}^n{\bar{B}}_i{e}^{2{\bar{v}}^{i}t}d{\bar{y}}^i\otimes d{\bar{y}}^i,$$

$$g=-dt\otimes dt+e^{2Ht}\sum _{i=1}^{3}d{x}^i\otimes d{x}^i$$

$$+e^{2H_{1}t}\sum _{i=4}^{3+l_1}d{x}^i\otimes d{x}^i+e^{2H_{2}t}\sum _{i=4+l_1}^{3+l_1+l_2}d{x}^i\otimes d{x}^i,$$

$$\mathrm{var}=\frac{\dot{G}}{GH}=-\frac{l_1{H}_1+l_2{H}_2}{H}.$$

$$-0,63\cdot 10^{-3}<\frac{\dot{G}}{GH}<1,13\cdot 10^{-3}.$$

$$-0,42\cdot 10^{-13}y{r}^{-1}<\dot{G}/G<0,75\cdot 10^{-13}y{r}^{-1}$$

$$H_0=(67,3\pm 2,4)km/sMp{c}^{-1}=(6,878\pm 0,245)\cdot 10^{-11}y{r}^{-1},$$

$$H=h_0=\frac{\sqrt{15}}{20\sqrt{\alpha }}\sqrt{1+\sqrt{480\lambda -79}}>0,$$

$$H_{1}=h_{1}=-\frac{\sqrt{15}}{60\sqrt{\alpha}}\Biggl{(}\sqrt{1+\sqrt{480\lambda-79}}-2\sqrt{4-\sqrt{480\lambda-79}}\Biggr{)},$$

$$H_{2}=h_{2}=-\frac{\sqrt{15}}{60\sqrt{\alpha}}\Biggl{(}\sqrt{1+\sqrt{480\lambda-79}}+2\sqrt{4-\sqrt{480\lambda-79}}\Biggr{)}.$$

$$S_1=\frac{\sqrt{15}}{60\sqrt{\alpha }}\sqrt{1+\sqrt{480\lambda -79}}.$$

$$\frac{79}{480}<\lambda <\frac{19}{96}.$$

$$\mathrm{var}=8/3,$$

$$H=h_{2}=-\frac{\sqrt{15}}{60\sqrt{\alpha}}\Biggl{(}\sqrt{1+2\sqrt{60\lambda-11}}-\sqrt{5}\sqrt{5-2\sqrt{60\lambda-11}}\Biggr{)}>0,$$

$$H_{1}=h_{1}=-\frac{\sqrt{15}}{60\sqrt{\alpha}}\Biggl{(}\sqrt{1+2\sqrt{60\lambda-11}}+\sqrt{5}\sqrt{5-2\sqrt{60\lambda-11}}\Biggr{)},$$

$$H_2=h_0=\frac{1}{\sqrt{15\sqrt{\alpha }}}\sqrt{1+2\sqrt{60\lambda -11}},$$

$$S_1=\frac{\sqrt{15}}{30\sqrt{\alpha }}\sqrt{1+2\sqrt{60\lambda -11}}.$$

$$\frac{11}{60}<\lambda <\frac{2209}{8640}.$$

$$|\mathrm{var}|<\delta$$

$$|\lambda -{\lambda }_1|<{ϵ}_1.$$

Full text

Keywords: Gauss-Bonnet, variation of G, accelerated expansion of the Universe

1 Introduction

Here we study a $D$-dimensional Einstein-Gauss-Bonnet (EGB) gravitational model which contains Gauss-Bonnet term and $\Lambda$-term. As it is well-known, the Gauss-Bonnet term appeared in (super)string theory in the next to leading order correction (in slope parameter) to the effective action [1]-[3]. Currently, the EGB model and its extensions, see [4]-[19] and references therein, are under a wide studying in cosmology aimed at possible explanation of accelerating expansion of the Universe (i.e. in a context of the so-called dark energy problem) [20, 21, 22]. Here we restrict ourselves by non-singular cosmological solutions with three scale factors $a_{i}=\exp(H_{i}t)$, $i=0,1,2$, where $t$ is a (synchronous) time variable. We find three exact solutions with three different Hubble-like parameters $H_{0}>0$, $H_{1}$ and $H_{2}$, which correspond to three subspaces of dimensions $3$, $l_{1}$ and $l_{2}$ with $(l_{1},l_{2})=(4,4),(3,2),(4,3)$. Here we put $H_{0}=H$, where $H$ is the Hubble parameter [23]. Due to results of ref. [16] (see also [13, 15]), the solutions are stable in a class of cosmological solutions with diagonal metrics. It is shown that two obtained solutions may describe a small enough variation of the effective gravitational constant $G$ (in Jordan frame) [24, 25, 26, 27] when cosmological constant $\Lambda$ obeys certain restrictions.

2 The cosmological model

The action of the model reads

[TABLE]

Here $g=g_{MN}dz^{M}\otimes dz^{N}$ is the metric defined on the manifold $M$, ${\dim M}=D$, $|g|=|\det(g_{MN})|$, $\Lambda$ is the cosmological term, $R[g]$ is scalar curvature,

[TABLE]

is the Gauss-Bonnet term and $\alpha_{1}$, $\alpha_{2}$ are nonzero constants.

We consider the manifold

[TABLE]

with the metric

[TABLE]

where $B_{i}>0$ are constants, $i=1,\dots,n$, and $M_{1},\dots,M_{n}$ are one-dimensional manifolds (either compact or non-compact ones) and $n>3$.

The equations of motion for the action (2.1) read [15]

[TABLE]

$i=1,\ldots,n$, where $\alpha=\alpha_{2}/\alpha_{1}$. Here [8, 9]

[TABLE]

3 Solutions with three factor spaces

In this section we present a class of solutions to the set of equations (2.4), (2.5) of the following form:

[TABLE]

where $h_{0}$ is the Hubble-like parameter corresponding to an $k_{0}$-dimensional factor space with $k_{0}>1$, $h_{1}$ is the Hubble-like parameter corresponding to a $k$-dimensional factor space and $h_{2}$ ($h_{2}\neq h_{1}$) is the Hubble-like parameter corresponding to another $k$-dimensional factor space, $k>1$.

We consider the ansatz (3.1) with three Hubble parameters $H$, $h_{1}$ and $h_{2}$ which obey the following restrictions:

[TABLE]

We put

[TABLE]

Here we present three examples of solutions to equations (2.4), (2.5) with the restrictions (3.2) imposed. These solutions are obtained by using Mathematica. Any solution contains at least one number $3$ among $k_{0},k_{1},k_{2}$. We renumerate the subspaces in such way that renumerated set $\bar{v}$ has the form

[TABLE]

$H>0$ and the metric (2.3) reads

[TABLE]

where $\bar{v}^{i}=v^{\sigma(i)}$, $\bar{B}_{i}=B_{\sigma(i)}>0$ and $\bar{y}^{i}=y^{\sigma(i)}\equiv x^{i}$, $i=1,\dots,n$, for certain permutation $\sigma\in S_{n}$.

The metric (3.5) may be rewritten as

[TABLE]

where $H>0$ is Hubble parameter and the set of parameters $(H,H_{1},H_{2})$ is obtained from the set $(h_{0},h_{1},h_{2})$ by certain permutation.

The dimensionless parameter of variation of (effective) gravitational constant (in Jordan frame) [24, 25, 26, 27] reads

[TABLE]

Due to the experimental data, the variation of the gravitational constant is allowed at the level of $10^{-13}$ per year and less. Here one may use the following constraint on the magnitude of the dimensionless variation of the effective gravitational constant:

[TABLE]

It comes from the most stringent limitation on $G$-dot obtained by the set of ephemerides [28]

[TABLE]

allowed at 95% confidence (2$\sigma$) level and the present value of the Hubble parameter [23]

[TABLE]

with 95% confidence level ($2\sigma$).

In what follows we denote $\lambda=\Lambda\alpha$, $\alpha>0$.

3.1 The case $k_{0}=3$, $k=4$

Let $k_{0}=3$, $k=4$. Here we put $k_{0}=l_{0}=3$, $l_{1}=k_{1}=4$, $l_{2}=k_{2}=4$ and $(l_{0},l_{1},l_{2})=(3,4,4)$. The solution reads:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here

[TABLE]

We obtain a huge value for the dimensionless variation of $G$

[TABLE]

which does not obey restriction (3.8).

3.2 The case $k_{0}=2$, $k=3$

Let us put $k_{0}=2$, $k=3$. Here $l_{0}=k_{2}=3$, $l_{1}=k_{1}=3$, $l_{2}=k_{0}=2$, and $(l_{0},l_{1},l_{2})=(3,3,2)$. The solution reads:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here we have the following restriction on $\lambda$

[TABLE]

For $\lambda=\lambda_{1}=213/980$ we get $H=\frac{1}{2\sqrt{35\alpha}}$, $H_{1}=-4H$, $H_{2}=6H$, $S_{1}=3H$, ${\rm var}=0$ in agreement with [18].

In case when the following restriction is imposed:

[TABLE]

there exists $\epsilon_{1}>0$ such that the (3.22) is obeyed if

[TABLE]

This follows from the continuity of the function ${\rm var}={\rm var}(\lambda)$.

3.3 The case $k_{0}=4$, $k=3$

Now we put $k_{0}=4$, $k=3$ and $l_{0}=k_{1}=3$, $l_{1}=k_{0}=4$, $l_{2}=k_{2}=3$, and $(l_{0},l_{1},l_{2})=(3,4,3)$. The solution reads:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here we impose

[TABLE]

For $\lambda=\lambda_{2}=1443/8092$ we get $H=\frac{1}{2\sqrt{119\alpha}}$, $H_{1}=-6H$, $H_{2}=8H$, $S_{1}=3H$ and ${\rm var}=0$ in agreement with [18].

When the restriction (3.22) is imposed there exists $\epsilon_{2}>0$ such that the (3.22) is obeyed if

[TABLE]

4 The analysis of stability

The solutions obey two restrictions:

[TABLE]

and

[TABLE]

where

[TABLE]

Relation (4.2) was proved in ref. [18] by using restrictions from (3.2).

We remind that for general cosmological setup with the metric

[TABLE]

we have the set of equations [15]

[TABLE]

where $h^{i}=\dot{\beta}^{i}$,

[TABLE]

$i=1,\ldots,n$.

Due to results of ref. [16] a fixed point solution $(h^{i}(t))=(v^{i})$ ($i=1,\dots,n$; $n>3$) to eqs. (4.5), (4.6) obeying restrictions (4.1) and (4.2) is stable under perturbations

[TABLE]

$i=1,\ldots,n$, as $t\to+\infty$. This follows from the relations [16]

[TABLE]

($A^{i}$ are constants) $i=1,\dots,n$, which are valid when restrictions (4.1), (4.2) are imposed. Thus, all solutions under consideration are stable.

5 Conclusions

Here we were studying the Einstein-Gauss-Bonnet (EGB) model in $D$ dimensions with the cosmological term $\Lambda$. We have obtained three examples of non-singular solutions with exponential time dependence (with respect to synchronous time variable $t$) of three scale factors, governed by three non-coinciding Hubble-like parameters: $H_{0}>0$, $H_{1}$ and $H_{2}$, which correspond to factor spaces of dimensions $3$, $l_{1}$ and $l_{2}$, respectively, and $D=4+l_{1}+l_{2}$. Here $(l_{1},l_{2})=(4,4),(3,2),(4,3)$. Due to results of ref. [16] all the obtained solutions are stable (as $t\to+\infty$). We put $H_{0}=H$, where $H$ is Hubble parameter. We have shown that two of these solutions with $(l_{1},l_{2})=(3,2),(4,3)$ may describe a small enough (e.g. zero) variation of the effective gravitational constant $G$ (in Jordan frame) for certain chosen values of $\Lambda$.

Acknowledgments

The publication was prepared with the support of the “RUDN University Program 5-100”. It was also partially supported by the Russian Foundation for Basic Research, grant Nr. 16-02-00602.