Splitting into two isotropic subspaces as a result of cosmological evolution in Einstein-Gauss-Bonnet gravity

TL;DR

This paper studies the evolution of a flat anisotropic universe in Einstein-Gauss-Bonnet gravity, showing that cosmological evolution often results in a split into two isotropic subspaces, with outcomes including singular, oscillatory, or exponential solutions.

Contribution

It demonstrates that in Einstein-Gauss-Bonnet cosmology, the universe can naturally evolve into a split into two isotropic subspaces, a phenomenon previously not well understood.

Findings

Three possible evolution outcomes: singular, oscillatory, exponential.

Evolution often leads to a split into two isotropic subspaces.

Such splitting occurs even when de Sitter solutions are present.

Abstract

We consider numerically dynamics of a flat anisotropic Universe in Einstein-Gauss-Bonnet gravity with positive $\Lambda$ in dimensionalities 5+1 and 6+1. We identify three possible outcomes of the evolution, one singular and two nonsingular. First nonsingular outcome is oscillatory. Second is the known exponential solution. The simplest version of it is the isotropic de Sitter solution. In Gauss-Bonnet cosmology there exist also anisotropic exponential solutions. When an exponential solution being an outcome of cosmological evolution has two different Hubble parameters, the evolution leads from initially totally anisotropic stage to a sum of two isotropic subspaces. We show that such type of evolution is rather typical and possible even in the case when de Sitter solution also exists.

Tables

Table 1. Table 1 : The percentage ratio of solutions of different classes for α = 1 , Λ = 1 ; D = 5 formulae-sequence 𝛼 1 formulae-sequence Λ 1 𝐷 5 \alpha=1,\,\Lambda=1;\;D=5

	{2,-3}	isotropic	non-standard singularity
(2-3)	34.6	0	65.4
(3-2)	8.6	11.5	79.9
(4-1)	5.3	69.1	25.6

Table 2. Table 2 : The percentage ratio of solutions of different classes for α = − 1 , Λ = 0.5 ; D = 5 formulae-sequence 𝛼 1 formulae-sequence Λ 0.5 𝐷 5 \alpha=-1,\,\Lambda=0.5;\;D=5

	isotropic	non-standard singularity
(1-4)	0	100.0
(2-3)	0	100.0
(3-2)	0	100.0
(4-1)	37.0	63.0

Table 3. Table 3 : The percentage ratio of solutions of different classes for α = − 1 , Λ = 1 ; D = 5 formulae-sequence 𝛼 1 formulae-sequence Λ 1 𝐷 5 \alpha=-1,\,\Lambda=1;\;D=5

	{3,2}	non-standard singularity
(1-4)	0	100.0
(2-3)	94.9	5.1
(3-2)	77.4	22.6
(4-1)	44.0	56.0

Table 4. Table 4 : The percentage ratio of solutions of different classes for α = 1 , Λ = 10 ; D = 6 formulae-sequence 𝛼 1 formulae-sequence Λ 10 𝐷 6 \alpha=1,\,\Lambda=10;\;D=6

	{2,-4}	isotropic	oscillations		$\begin{array}{c}\text{non-standard}\\ \text{singularity}\end{array}$
			{1,1,-1,-1,-1,-1}	{1,1,1,1,-1,-1}
(2-4)	0.1	0	5.5	0	94.4
(3-3)	1.3	0	0	0	98.7
(4-2)	0.4	11.0	0	0.2	88.4
(5-1)	0.5	94.5	0	0	5.0
(6-0)	2.7	67.6	0	0	29.7

Table 5. Table 5 : The percentage ratio of solutions of different classes for α = 0.1 , Λ = 2.5 ; D = 6 formulae-sequence 𝛼 0.1 formulae-sequence Λ 2.5 𝐷 6 \alpha=0.1,\,\Lambda=2.5;\;D=6

	{3,-3}	{4,-2}	isotropic	non-standard singularity
(2-4)	0	0	0	100.0
(3-3)	19.1	23.6	43.3	14.0
(4-2)	0	35.3	45.8	18.9
(5-1)	0	0	92.5	7.5
(6-0)	0	0	97.6	2.4

Table 6. Table 6 : The percentage ratio of solutions of different classes for α = − 1 , Λ = 10 ; D = 6 formulae-sequence 𝛼 1 formulae-sequence Λ 10 𝐷 6 \alpha=-1,\,\Lambda=10;\;D=6

	{4,-2}	{4,2}	{3,-3}	non-standard singularity
(1-5)	0	0	0	100.0
(2-4)	0	0.1	0	99.9
(3-3)	24.9	3.0	19.7	52.4
(4-2)	67.4	0.4	0.2	32.0
(5-1)	1.2	0	0	98.8
(6-0)	0	96.9	0.8	2.3

Table 7. Table 7 : The percentage ratio of solutions of different classes for α = − 0.3 , Λ = 1 ; D = 6 formulae-sequence 𝛼 0.3 formulae-sequence Λ 1 𝐷 6 \alpha=-0.3,\,\Lambda=1;\;D=6

	isotropic	non-standard singularity
(1-5)	0	100.0
(2-4)	0	100.0
(3-3)	3.0	97.0
(4-2)	1.5	98.5
(5-1)	39.0	61.0
(6-0)	74.0	26.0

Equations

$$S=\int_{\mathcal{M}}d^{D+1}x\sqrt{|g|}\left\{R-\Lambda+\alpha\bigl{(}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}\bigr{)}\right\},$$

$$d{s}^2=-d{t}^2+\sum _{k=1}^D{a}_k^2(t)d{x}_k^2$$

$$\begin{array}{rl}2\sum _{i\ne j}({\dot{H}}_i+H_i^2)& +2\sum _{\{i>k\}\ne j}{H}_i{H}_k+\\ & +8\alpha \sum _{i\ne j}({\dot{H}}_i+H_i^2)\sum _{\{k>l\}\ne \{i,j\}}{H}_k{H}_l+24\alpha \sum _{\{i>k>l>m\}\ne j}{H}_i{H}_k{H}_l{H}_m=\Lambda ,j=1,\dots ,D\end{array}$$

$$2\sum _{i>j}{H}_i{H}_j+24\alpha \sum _{i>j>k>l}{H}_i{H}_j{H}_k{H}_l=\Lambda$$

$$3{\varphi }^3-7{\varphi }^2+(1+4\alpha \Lambda )\varphi -1=0,\varphi =4\alpha H^2;h=-\frac{H}{2}-\frac{1}{8\alpha H}$$

$$H^2=-\frac{1}{12\alpha }\left[1\pm \sqrt{1+\frac{6\alpha \Lambda }{5}}\right]$$

$$\Lambda =\frac{6(H^4+4H^{3}h+5H^2{h}^2+4H{h}^3+h^4)}{H^2+4Hh+h^2},h=-2H\pm \sqrt{3H^2-\frac{1}{4\alpha }}$$

$$\Lambda =\frac{2(5H^3+6H^{2}h+3H{h}^2+h^3)}{H+h},h=-H-\frac{1}{12\alpha H}$$

$$H^2=-\frac{1}{24\alpha }\left[1\pm \sqrt{1+\frac{8\alpha \Lambda }{5}}\right]$$

$$20736z^4-17280z^3+\left(2736+4032\xi \right)z^2-\left(216+528\xi \right)z+{\left(2\xi +3\right)}^2=0$$

$$z=\frac{5}{24}\pm \frac{\sqrt{1-2\xi }}{6}\pm \frac{1}{24}\sqrt{-\frac{24\xi \sqrt{1-2\xi }-21\sqrt{1-2\xi }+48\xi -24}{\sqrt{1-2\xi }}}$$

$$f_1(H)=0,\text{mbox}where{f}_1(H)=1728H^6{\alpha }^6-432H^4{\alpha }^4+72{\alpha }^4{H}^2\Lambda -1$$

$$H=\sqrt{-\frac{1}{24\alpha }\left[1-\sqrt{1+\frac{8\alpha \Lambda }{5}}\right]}$$

$$20736z^4+17280z^3+\left(2736-4032\xi \right)z^2+\left(216-528\xi \right)z+{\left(2\xi -3\right)}^2=0$$

$$z=-\frac{5}{24}\pm \frac{\sqrt{1+2\xi }}{6}\pm \frac{1}{24}\sqrt{\frac{24\xi \sqrt{1+2\xi }+21\sqrt{1+2\xi }-48\xi -24}{\sqrt{1+2\xi }}}$$

$$f_2(z)=0\text{mbox}where{f}_2(z)=1728z^3|\alpha {|}^3-72\Lambda {\alpha }^{2}z+432{\alpha }^2{z}^2+1,z\equiv H^2$$

$$z_{+}=\frac{-1+\sqrt{2\Lambda |\alpha |+1}}{12|\alpha |}>0\text{mbox}foranyvaluesof\alpha ,\Lambda$$

$$f_2^{\min }=-4\xi \sqrt{2\xi +1}+6\xi -2\sqrt{2\xi +1}+3,\xi =\Lambda |\alpha |$$

$$H=\sqrt{\frac{1}{24|\alpha |}\left[1\pm \sqrt{1-\frac{8|\alpha |\Lambda }{5}}\right]}$$

Full text

I Introduction

Cosmological dynamics in Einstein-Gauss-Bonnet (EGB) gravity have been studied intensively during last three decades. This modification of gravity is the simplest example (apart from General Relativity (GR) itself) of Lovelock gravity – the theory which keeps the order of equations of motion the same as in GR Low . In the standard metric formalism the Lovelock gravity is the only generalisation of GR with this property – other modifications like $f(R)$ theory Sotiriou usually lead to 4-th order equations of motion instead of the 2-nd order in GR. Bigger order of equation leads to bigger dimensionality of the corresponding phase spaces of cosmological solutions, and as a result, many new classes of solutions impossible in GR appear in different versions of $f(R)$ as well as other 4-order theories. As for Lovelock gravity, it is reasonable to expect less drastic changes with respect to GR. Indeed, some of rather unusual for GR regimes like isotropic vacuum solution and its modifications iso1 ; iso2 ; iso3 do not present in Lovelock gravity.

However, rather recently a class of solutions with no direct analog have been found in Einstein-Gauss-Bonnet cosmology Iv-10 and later have been generalized to Lovelock gravity ChPavTop1 . In this class scale factors change exponentially with time, so Hubble parameters remain constant. In GR there are no vacuum solutions of this type PT , and the only solution with non-zero cosmological constant $\Lambda>0$ is the de Sitter solution, when all Hubble parameters are equal to the same value. On the contrary, there are anisotropic vacuum solutions in EGB cosmology, as well as anisotropic solutions with non-zero cosmological constant IvErn ; IvKob ; ChPavTop . In this sense, a positive cosmological constant in the whole multidimensional space-time does not lead ultimately to isotropization, as it is in GR.

In the vacuum case a nonsingular cosmological evolution of a flat Universe in EGB gravity is usually directed from Gauss-Bonnet Kasner regime to the classical GR Kasner regime PTop (except for a pathological 4+1 dimension case Ronid which will not be considered here). Since Hubble parameters decrease by absolute values, in the late time of such evolution the influence of Gauss-Bonnet term is negligible. On the contrary, if the cosmological dynamics tends to an exponential solution, the influence of GB term does not disappear at the late time, since Hubble parameters tend to constants. In the present paper we focus on the case with positive $\Lambda$ in the action, so a late-time Kasner GR solution does not exist for our set-up, and we expect that there are exponential solutions which dominate in the future asymptotics of the cosmological evolution.

Despite the fact that there exist anisotropic exponential solutions in the presence of positive $\Lambda$, it is known that all Hubble parameters can not be different. It have been shown that in EGB gravity it can be no more than 3 different Hubble parameters independently of the number of dimensions Iv-16 . This means that a solution in question represent a division of the space into a warped product of two or three isotropic subspaces. Note, that such a decomposition, in particular case with two isotropic subspaces, one expanding and one contracting, is often appears as a starting point in papers devoted to compactification studies. Assuming only two different scale factors allows to get many results in analytic form despite of big number of dimension. On the contrary, if exponential solutions can be dynamical attractors for initially totally anisotropic geometries, we get this decomposition in a natural way dynamically. To study this possibility, first we need get results on stability of these solutions. The analysis of Pavl-15 ; ErIvKob-16 ; ChT indicate that solutions with no one-dimensional subspaces are locally stable when sum of Hubble-like parameters is positive, except for some special discrete sets of coupling constant. However, local stability is not enough to claim that these solutions are natural attractors for general anisotropic initial conditions. To claim that we need a global analysis. This is the goal of the present paper.

II The set-up

We consider the Einstein-Gauss-Bonnet action in $(D+1)$-dimensional spacetime $\mathcal{M}$:

[TABLE]

where $|g|$ is the determinant of metric tensor, $R,R_{\alpha\beta},R_{\alpha\beta\gamma\delta}$ are the $(D+1)$-dimensional scalar curvature, Ricci tensor and Riemann tensor respectively; $\alpha$ is the coupling constant; $\Lambda$ is the cosmological term.

We introduce the metric ansatz as

[TABLE]

In spatially flat model scale factors $a_{k}(t)$ are defined only up to a constant multiple, so it makes sense to use the Hubble parameters $H_{k}=\frac{\dot{a}_{k}}{a_{k}},\;k=1,\ldots,D$ instead of scale factors.

Equations of motion read

[TABLE]

In addition to the equations of motion we have the constraint equation:

[TABLE]

In what following we consider only stable exponential solutions and assume that $\sum\limits_{i=1}^{D}H_{i}>0$.

III Numerical analysis of solutions for 5+1 and 6+1 dimensional models

For numerical calculations we choose sets of initial values of Hubble parameters with different numbers of positive and negative elements. and denote a set with $p$ positive and $n$ negative elements by $(p-n)$; it implies that initially $p$ dimensions are expanding, $n$ dimensions are contracting. We use this notation to simplify comparing the number of expanding dimensions at the beginning and at the end of integration, in case when the outcome of the dynamics studied is an exponential solution. We will see later that the change in number of expanding dimensions is possible indeed.

In all our numerical experiments initial values of Hubble parameters is chosen at random from (-10;10) with uniform probability; totally we have checked up $\sim 10^{4}$ sets of initial values for each dimensionality. In addition we choose only such initial values of Hubble parameters that sum of them is positive. Since one of the Hubble parameters is defined from the constraint, in some cases we could not find any sets of initial values with a certain number of positive and negative Hubble parameters which obey this condition; for example, for $D=5,\,\alpha>0$ we found no sets of initial values with 1 positive, 4 negative Hubble parameters and initial values with all 5 positive Hubble parameters such that the sum of them is positive. In such cases, the tables demonstrating distribution of results of numerical calculations do not contain the corresponding rows (it is tables 1,4,5).

We consider 5+1 and 6+1 dimensional models; each of these models has several classes of stable exponential solutions with isotropic subspaces. A solution containing $k$-dimensional and $m$-dimensional isotropic subspaces is referred to as $[k,m]$-splitting in what follows. We use braces when we need to specify whether isotropic subspaces are expanded or contracted; if an isotropic subspace is expanded we use positive number; if an isotropic subspace is contracted we use negative number. For example, notation $\{2,-3\}$ means that the space representing by a corresponding solution is a warped product of 3D contracting isotropic subspace and 2D expanding isotropic subspace. Such way, a case with $[k,m]$-splitting includes 4 subcases: $\{k,m\},\{k,-m\},\{-k,m\},\{-k,-m\}$.

Apart from exponential solution the cosmological dynamics in question can end up in a non-standard singularity, when time derivative of Hubble parameters diverge while they themselves are finite. In 6+1 dimensions we have found also one more regime which will be described later in the corresponding section.

III.1 5+1 dimensional model

There are 2 classes of stable exponential solutions in EGB model for $D=5$ ChPavTop :

1. 1.

Solution with [3,2]-splitting ($H_{1}=H_{2}=H_{3}=H,\;H_{5}=H_{6}=h$) is defined by the equations

[TABLE]

Explicit solution for the first of these equations is cumbersome enough, so we do not write it down here; in what following we discuss briefly existence conditions for solutions of this equation. 2. 2.

Isotropic solution with $H_{1}=H_{2}=H_{3}=H_{4}=H_{5}=H_{6}=H$ and

[TABLE]

Solving the latter equation we choose $H>0$ (for stability).

It is easy to check that when $\alpha>0$ there exist both solution with [3,2]-splitting and isotropic solution for any $\Lambda>0$. This is confirmed numerically: table 1 shows what we get starting from initial values with different splittings. We list only solutions existing for the set of coupling studied, zeros in the table do not necessary mean that the corresponding solution can not be reached from the initial conditions indicated, but it can mean that we have insufficient statistics to detect it.

In the case $\alpha<0$ solutions with [3,2]-splitting exist iff $|\alpha|\Lambda>\frac{5}{6}$; isotropic solutions exist under condition $|\alpha|\Lambda<\frac{5}{6}$. Numerical calculations verify these conclusions (see tables 2 and 3).

III.2 6+1 dimensional model

Below we list 3 classes of stable exponential solutions in EGB model for $D=6$ ChPavTop1 :

1. 1.

Solution with [3,3]-splitting IvKob-2 , where $H_{1}=H_{2}=H_{3}=H,\;H_{4}=H_{5}=H_{6}=h$ and

[TABLE] 2. 2.

Solution with [4,2]-splitting IvKob-3 , where $H_{1}=H_{2}=H_{3}=H_{4}=H,\;H_{5}=H_{6}=h$ and

[TABLE] 3. 3.

Isotropic solution with $H_{1}=H_{2}=H_{3}=H_{4}=H_{5}=H_{6}=H$ and

[TABLE]

Solving the latter equation we choose $H>0$ (for stability).

Below we discuss solutions with $\alpha>0$ and $\alpha<0$ separately.

III.2.1 The case $\alpha>0$

First, we find out for what values of $\alpha>0$ and $\Lambda>0$ there exist solutions with [3,3]- and [4,2]-splittings as well as isotropic solution, and are there such $\alpha$ and $\Lambda$ that all these solutions co-exist, i.e. we can reach each of these solutions for different initial values of Hubble parameters but for the same values of $\alpha$ and $\Lambda$.

I. [3,3]-splitting: substituting the expression for $h$ into the equation for $\Lambda$ and doing simple transformations we obtain

[TABLE]

where $z=H^{2}\alpha,\;\xi=\Lambda\alpha$. Note that $z>0$ and $\xi>0$. Solutions of this equation are

[TABLE]

There exists at least one positive $z$ for all $\xi<\frac{1}{2}$.

II. [4,2]-splitting: substituting the expression for $h$ into the equation for $\Lambda$ we get

[TABLE]

This equation has at least one positive root for any values $\alpha$ and $\Lambda$ since $f_{1}(0)=-1$ and $f_{1}(H)\stackrel{{\scriptstyle H\rightarrow+\infty}}{{\longrightarrow}}+\infty$.

III. isotropic solution exists for any $\alpha>0$ and $\Lambda>0$:

[TABLE]

Summing things up we can say that solutions of all three types co-exist for $0<\alpha\Lambda<\frac{1}{2}$; for $\alpha\Lambda\geqslant\frac{1}{2}$ only isotropic solution and solution with [4,2]-splitting exist. This conclusions are confirmed by the results of numerical calculations for $\alpha=1,\,\Lambda=10$ (see table 4) and $\alpha=0.1,\,\Lambda=2.5$ (see table 5).

Numerical calculations show the existence of some specific solutions which we call "oscillations". We found oscillations in 6+1 dimensional model with $\alpha>0$ and for large enough values of $\alpha\Lambda$ (see Fig. 1). Numerical calculations show also that despite the overall measure of initial conditions set leading to oscillations is not large, the oscillation regime is stable with respect to small perturbations of initial conditions. The period of oscillations is fixed for fixed coupling constants, while amplitudes can vary depending on initial conditions. Note that there are no oscillations found for $D=5$ case.

III.2.2 The case $\alpha<0$

Analogously to the previous subsection we find out what solutions can exist for negative values of $\alpha$ (as before, we assume that $\Lambda>0$).

I. [3,3]-splitting: substituting the expression for $h$ into the equation for $\Lambda$ we obtain

[TABLE]

where $z=H^{2}\alpha,\;\xi=\Lambda|\alpha|$. Note that $z>0$ and $\xi>0$ again. Solutions of this equation are

[TABLE]

It is easy to check that we have at least one positive $z$ for all $\xi\geqslant\frac{5}{8}$.

II. [4,2]-splitting: substituting the expression for $h$ into the equation for $\Lambda$ we get

[TABLE]

One can see that $f_{2}(0)=1,\;f_{2}(z)\stackrel{{\scriptstyle z\rightarrow+\infty}}{{\longrightarrow}}+\infty$; therefore, in order to the equation $f_{2}(z)=0$ has at least one positive root it is necessary that the function $f_{2}$ has non-positive minimum at a point $z_{+}>0$. It is easy to check that minimum point of the function $f_{2}$ is

[TABLE]

Substituting that $z_{+}$ into $f_{2}$ we obtain

[TABLE]

It is easy to prove that $f_{2}^{\rm min}\leqslant 0$ for all $\xi\geqslant\frac{5}{8}$.

III. isotropic solution exists for such values of $\alpha$ and $\Lambda$ that $|\alpha|\Lambda\equiv\xi\leqslant\frac{5}{8}$

[TABLE]

So, we see that for $|\alpha|\Lambda>\frac{5}{8}$ only [3,3]-splitting and [4,2]-splitting exist (see table 6); for $|\alpha|\Lambda<\frac{5}{8}$ only isotropic solution exists (see table 7).

IV Conclusions

We have considered a cosmological evolution of a flat 5+1 and 6+1 dimensional anisotropic Universe in Gauss-Bonnet gravity. We started from an arbitrary anisotropic initial conditions and study the outcome of corresponding cosmological evolution. Three possible outcomes have been identified. About a half of trajectories traced (we have $\sim 10^{4}$ trajectories for each dimensionality) ends in a non-standard singularity. A few percents of trajectories represent a periodic in Hubble parameters solution which, to our knowledge, have not been described before. And the rest of them (roughly a half of overall number) lead to exponential solutions.

For the studied dimensionality only exponential solutions with one or two different Hubble parameters are stable. In the former case we have a de Sitter solution. In the latter case initially totally anisotropic Universe becomes a warped product of two isotropic subspaces. What follow from our results is that this can happen even if de Sitter solution exists for the particular set of coupling constants – nevertheless, depending on initial conditions, a Universe can "miss" de Sitter stage and split into two isotropic subspaces. We can see also that despite most trajectories have conserved number of expanding and contracting dimensions, nevertheless signs of Hubble parameters can change during cosmological evolution.

Our results show that a situation when a space is splitted into a warped product of isotropic subspaces can be a natural result in cosmological evolution of a flat Universe in Gauss-Bonnet gravity.

Acknowledgments

The work of A.T. is supported by RFBR grant 17-02-01008. Authors are grateful to Sergey Pavluchenko for discussions.