On stable exponential cosmological solutions in the EGB model with a   $\Lambda$-term in dimensions D = 5,6,7,8

D.M. Chirkov; A.V. Toporensky

arXiv:1706.08889·gr-qc·July 16, 2018

On stable exponential cosmological solutions in the EGB model with a $\Lambda$-term in dimensions D = 5,6,7,8

D.M. Chirkov, A.V. Toporensky

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TL;DR

This paper analyzes the stability of exponential cosmological solutions in a higher-dimensional Einstein-Gauss-Bonnet model with a cosmological constant, applying a known criterion to solutions up to 8 dimensions.

Contribution

It applies a stability criterion to all known exponential solutions in D=5 to 8 dimensions within the EGB model with a cosmological term.

Findings

01

Most solutions are stable under the criterion.

02

Some discrete solutions are unstable.

03

The stability criterion is broadly applicable across dimensions.

Abstract

A $D$ -dimensional Einstein-Gauss-Bonnet (EGB) flat cosmological model with a cosmological term $Λ$ is considered. We focus on solutions with exponential dependence of scale factor on time. Using previously developed general analysis of stability of such solutions done by V.D.Ivashchuk (2016) we apply the criterion from that paper to all known exponential solutions up to dimensionality 7+1. We show that this criterion which guarantees stability of solution under consideration is fulfilled for all combination of coupling constant of the theory except for some discrete set.

Tables5

Table 1. Table 1: Stability of solutions (d=4)

$(v^{1}, v^{2}, v^{3}, v^{4})$	Solution	$det (L)$	Stability condition
$(H, H, h, h)$	$\begin{matrix} H^{2} = \frac{1}{4} (2 Λ + \frac{1}{2 α} \pm \sqrt{{(2 Λ + \frac{1}{2 α})}^{2} - \frac{1}{α^{2}}}), h = - \frac{1}{4 α H} \\ {\begin{matrix} Λ > \frac{1}{4 α} \\ α > 0 \end{matrix} or {\begin{matrix} Λ > - \frac{3}{4 α} \\ α < 0 \end{matrix} \end{matrix}$	$\frac{256 {(H^{2} α + 1 / 4)}^{4}}{H^{4} α^{2}}$	$\begin{matrix} {\begin{matrix} H \in (- \frac{1}{\sqrt{4 α}}; 0) \cup \\ \cup (\frac{1}{\sqrt{4 α}}; + \infty) \\ α > 0 \end{matrix} \\ or \\ H > 0 if α < 0 \end{matrix}$
$(H, H, h, h)$	$H^{2} = h^{2} = - \frac{1}{4 α}, Λ = - \frac{3}{4 α}, α < 0$	0	—
$(H, H, H, h)$	$H^{2} = - \frac{1}{4 α}, h \in ℝ, Λ = - \frac{3}{4 α}, α < 0$	0	—
$(H, H, H, H)$	$\begin{matrix} H^{2} = \frac{1}{4 α} (- 1 + \sqrt{1 + \frac{4 α Λ}{3}}), α > 0, Λ > 0 \\ H^{2} = \frac{1}{4 α} (- 1 - \sqrt{1 + \frac{4 α Λ}{3}}), α < 0, Λ < - \frac{3}{4 α} \\ H^{2} = \frac{1}{4 α} (- 1 + \sqrt{1 + \frac{4 α Λ}{3}}), α < 0, 0 < Λ < - \frac{3}{4 α} \\ Vacuum solution: H^{2} = - \frac{1}{2 α}, α < 0 \end{matrix}$	$- 12288 {(H^{2} α + 1 / 4)}^{4}$	$H > 0$
$(H, H, H, H)$	$H^{2} = - \frac{1}{4 α}, Λ = - \frac{3}{4 α}, α < 0$	0	—

Table 2. Table 2: Stability of solutions (d=5)

$(v^{1}, v^{2}, v^{3}, v^{4}, v^{5})$	Solution	$det (L)$	Stability condition
$(H, H, - H, - H, h)$	$H^{2} = \frac{1}{4 α}, h \in ℝ, Λ = \frac{1}{4 α}, α > 0$	0	—
$(H, H, H, H, h)$	$H^{2} = - \frac{1}{12 α}, h \in ℝ, Λ = - \frac{5}{12 α}, α < 0$	0	—
$(H, H, H, H, H)$	$\begin{matrix} H^{2} = \frac{1}{12 α} (- 1 + \sqrt{1 + \frac{12 α Λ}{5}}), α > 0, Λ > 0 \\ H^{2} = \frac{1}{12 α} (- 1 - \sqrt{1 + \frac{12 α Λ}{5}}), α < 0, Λ < - \frac{5}{12 α} \\ H^{2} = \frac{1}{12 α} (- 1 + \sqrt{1 + \frac{12 α Λ}{5}}), α < 0, 0 < Λ < - \frac{5}{12 α} \\ Vacuum solution: H^{2} = - \frac{1}{6 α}, α < 0 \end{matrix}$	$- 128 {(12 H^{2} α + 1)}^{5}$	$H > 0$
	$H^{2} = - \frac{1}{12 α}, α < 0, Λ = - \frac{5}{12 α}$	0	—

𝚲

-term solution

Vacuum solution

(*) 192 ​ H^{6} ​ α^{3} - 112 ​ H^{4} ​ α^{2} + (64 ​ Λ ​ α + 4) ​ H^{2} ​ α - 1 = 0

(* *) h = - \frac{4 ​ H^{2} ​ α + 1}{8 ​ H ​ α}

It is easy to check that when

α > 0

Eqs. (

*

)-(

*, *

) has at least one solution for any

Λ

; when

α < 0

Eqs. (

*

)-(

*, *

) has at least one solution iff

Λ ⩾ - \frac{5}{12 ​ α}

. In this case

det (L) = - \frac{248832 ​ {(H^{2} ​ α - 1 / 4)}^{3} ​ {(H^{2} ​ α + 1 / 12)}^{5}}{H^{6} ​ α^{3}}

One can easily check that

det (L) = 0

iff

H^{2} = \frac{1}{4 ​ α}

or

H^{2} = - \frac{1}{12 ​ α}

;

H^{2} = \frac{1}{4 ​ α}

being the solution of Eqs. (

*

)-(

*, *

) is a particular case of the family

(H, H, - H, - H, h)

;

H^{2} = - \frac{1}{12 ​ α}

being the solution of Eqs. (

*

)-(

*, *

) is a particular case of the family

(H, H, H, H, h)

. A solution of Eqs. (

*

)-(

*, *

) is stable if

3 ​ H + 2 ​ h > 0 \Leftrightarrow \frac{16 ​ H^{2} ​ α - 1}{4 ​ H ​ α} > 0 \Leftrightarrow

\Leftrightarrow {\begin{matrix} H > 0, α < 0 \\ H > \frac{1}{\sqrt{16 ​ α}} ​ or - \frac{1}{\sqrt{16 ​ α}} < H < 0, α > 0 \end{matrix}

H_{1} = H_{2} = H_{3} = H, H_{4} = H_{5} = ξ ​ H

H^{2} = - \frac{ξ^{2} + 6 ​ ξ + 3}{12 ​ α ​ ξ ​ (3 ​ ξ + 2)} |_{ξ = \frac{\sqrt[3]{10}}{3} - \frac{\sqrt[3]{100}}{6} - \frac{2}{3} \approx - 0.722}, α < 0

In this case

\begin{matrix} (⋆) det (L) = - 1769472 \cdot \\ \cdot (H^{4} α^{2} (ξ^{2} + \frac{2}{3} ξ + \frac{1}{3}) - \frac{H^{2} ​ α}{36} (ξ^{2} - 10 ξ - 3) + \frac{1}{72}) \cdot \\ \cdot (ξ ​ (ξ + 2) ​ H^{2} ​ α + \frac{1}{4}) \cdot (H^{2} ​ α + \frac{1}{12}) \end{matrix}

It is easy to check that solution (

⋆

) does not turn determinant

L

to zero, so this solution is stable if

H > 0

.

Table 4. Table 3: d=6: special points of solutions

Family of solutions	with $(H, h, Λ)$ such that $det (L) = 0$	overlap to
$(H, H, H, h, h, h)$	$(\pm \frac{1}{\sqrt{- 24 α}}, \mp \frac{5}{\sqrt{- 24 α}}, - \frac{77}{16 α})$	nothing
	$(\pm \frac{1}{\sqrt{3 α}}, \mp \frac{7}{\sqrt{12 α}}, - \frac{143}{4 α})$	nothing
	$(\pm \frac{1}{\sqrt{3 α}}, \mp \frac{1}{\sqrt{12 α}}, \frac{1}{4 α})$	$(H, H, H, h, h, 𝕙)$
	$(\pm \frac{1}{\sqrt{12 α}}, \mp \frac{1}{\sqrt{3 α}}, \frac{1}{4 α})$	$(H, H, H, h, h, 𝕙)$
	$(\pm \frac{1}{\sqrt{- 24 α}}, \pm \frac{1}{\sqrt{- 24 α}}, - \frac{5}{16 α})$	$(H, H, H, H, H, h)$
$(H, H, H, H, h, h)$	$(\pm \frac{1}{\sqrt{12 α}}, \mp \frac{1}{\sqrt{3 α}}, \frac{1}{4 α})$	$(H, H, H, h, h, 𝕙)$
$(H, H, H, H, h, h)$	$(\pm \frac{1}{\sqrt{- 24 α}}, \pm \frac{1}{\sqrt{- 24 α}}, - \frac{5}{16 α})$	$(H, H, H, H, H, h)$
$(H, H, H, H, H, H)$	$(\pm \frac{1}{\sqrt{- 24 α}}, - \frac{5}{16 α})$	$(H, H, H, H, H, h)$

Table 5. Table 4: d=7: special points of solutions

Family of solutions	with $(H, h, 𝕙, Λ)$ such that $det (L) = 0$	overlap to
$(H, H, H, h, h, 𝕙, 𝕙)$	$(\pm \frac{1}{\sqrt{8 α}}, \mp \frac{1}{\sqrt{8 α}}, \mp \frac{1}{\sqrt{8 α}}, \frac{3}{16 α})$	$(H, H, H, - H, - H, - H, h)$
$(H, H, H, h, h, 𝕙, 𝕙)$	$\begin{matrix} (\pm \frac{1}{\sqrt{24 α}}, \mp \frac{\sqrt{3}}{\sqrt{8 α}}, \pm \frac{1}{\sqrt{24 α}}, \frac{13}{48 α}) \\ (\pm \frac{1}{\sqrt{24 α}}, \pm \frac{1}{\sqrt{24 α}}, \mp \frac{\sqrt{3}}{\sqrt{8 α}}, \frac{13}{48 α}) \end{matrix}$	$(H, H, H, H, h, h, 𝕙)$
$(H, H, H, H, h, h, h)$	$(\pm \frac{1}{\sqrt{- 40 α}}, \pm \frac{7}{\sqrt{- 40 α}}, - \frac{2409}{400 α})$	nothing
	$(\pm \frac{1}{\sqrt{8 α}}, \mp \frac{5}{\sqrt{8 α}}, - \frac{285}{16 α})$	nothing
	$(\pm \frac{1}{\sqrt{24 α}}, \mp \frac{\sqrt{3}}{\sqrt{8 α}}, \frac{13}{48 α})$	$(H, H, H, H, h, h, 𝕙)$
	$(\pm \frac{1}{\sqrt{- 40 α}}, \mp \frac{1}{\sqrt{- 40 α}}, - \frac{21}{80 α})$	$(H, H, H, H, H, H, h)$
	$(\pm \frac{1}{\sqrt{8 α}}, \mp \frac{1}{\sqrt{8 α}}, \frac{3}{16 α})$	$(H, H, H, - H, - H, - H, h)$
$(H, H, H, H, H, h, h)$	$(\pm \frac{1}{\sqrt{24 α}}, \mp \frac{\sqrt{3}}{\sqrt{8 α}}, \frac{13}{48 α})$	$(H, H, H, H, h, h, 𝕙)$
$(H, H, H, H, H, h, h)$	$(\pm \frac{1}{\sqrt{- 40 α}}, \pm \frac{1}{\sqrt{- 40 α}}, - \frac{21}{80 α})$	$(H, H, H, H, H, H, h)$
$(H, H, H, H, H, H, H)$	$(\pm \frac{1}{\sqrt{- 40 α}}, - \frac{21}{80 α})$	$(H, H, H, H, H, H, h)$

Equations53

S = \int_{M} d^{D} z ∣ g ∣ {α_{1} (R [g] - 2Λ) + α_{2} L_{2} [g]},

S = \int_{M} d^{D} z ∣ g ∣ {α_{1} (R [g] - 2Λ) + α_{2} L_{2} [g]},

L_{2} = R_{M N P Q} R^{M N P Q} - 4 R_{M N} R^{M N} + R^{2}

L_{2} = R_{M N P Q} R^{M N P Q} - 4 R_{M N} R^{M N} + R^{2}

M = R \times M_{1} \times \dots \times M_{n}

M = R \times M_{1} \times \dots \times M_{n}

g = - d t \otimes d t + i = 1 \sum n B_{i} e^{2 v^{i} t} d y^{i} \otimes d y^{i},

g = - d t \otimes d t + i = 1 \sum n B_{i} e^{2 v^{i} t} d y^{i} \otimes d y^{i},

G_{ij} v^{i} v^{j} + 2Λ - α G_{ij k l} v^{i} v^{j} v^{k} v^{l} = 0,

G_{ij} v^{i} v^{j} + 2Λ - α G_{ij k l} v^{i} v^{j} v^{k} v^{l} = 0,

[2 G_{ij} v^{j} - \frac{4}{3} α G_{ij k l} v^{j} v^{k} v^{l}] i = 1 \sum n v^{i} - \frac{2}{3} G_{ij} v^{i} v^{j} + \frac{8}{3} Λ = 0,

G_{ij} = δ_{ij} - 1, G_{ij k l} = G_{ij} G_{ik} G_{i l} G_{j k} G_{j l} G_{k l}

G_{ij} = δ_{ij} - 1, G_{ij k l} = G_{ij} G_{ik} G_{i l} G_{j k} G_{j l} G_{k l}

K = K (v) = i = 1 \sum n v^{i} \neq = 0.

K = K (v) = i = 1 \sum n v^{i} \neq = 0.

(R) det (L_{ij} (v)) \neq = 0.

(R) det (L_{ij} (v)) \neq = 0.

L = (L_{ij} (v)) = (2 G_{ij} - 4 α G_{ij k s} v^{k} v^{s}) .

L = (L_{ij} (v)) = (2 G_{ij} - 4 α G_{ij k s} v^{k} v^{s}) .

g = - d t \otimes d t + i = 1 \sum n e^{2 β^{i} (t)} d y^{i} \otimes d y^{i},

g = - d t \otimes d t + i = 1 \sum n e^{2 β^{i} (t)} d y^{i} \otimes d y^{i},

E = G_{ij} h^{i} h^{j} + 2Λ - α G_{ij k l} h^{i} h^{j} h^{k} h^{l} = 0,

E = G_{ij} h^{i} h^{j} + 2Λ - α G_{ij k l} h^{i} h^{j} h^{k} h^{l} = 0,

Y_{i} = \frac{d L _{i}}{d t} + (j = 1 \sum n h^{j}) L_{i} - \frac{2}{3} (G_{s j} h^{s} h^{j} - 4Λ) = 0,

L_{i} = L_{i} (h) = 2 G_{ij} h^{j} - \frac{4}{3} α G_{ij k l} h^{j} h^{k} h^{l},

L_{i} = L_{i} (h) = 2 G_{ij} h^{j} - \frac{4}{3} α G_{ij k l} h^{j} h^{k} h^{l},

h^{i} (t) = v^{i} + δ h^{i} (t),

h^{i} (t) = v^{i} + δ h^{i} (t),

K (v) = k = 1 \sum n v^{k} > 0

K (v) = k = 1 \sum n v^{k} > 0

C_{i} (v) δ h^{i} = 0,

C_{i} (v) δ h^{i} = 0,

L_{ij} (v) δ \dot{h}^{j} = B_{ij} (v) δ h^{j},

C_{i} (v) = 2 v_{i} - 4 α G_{ij k s} v^{j} v^{k} v^{s},

C_{i} (v) = 2 v_{i} - 4 α G_{ij k s} v^{j} v^{k} v^{s},

B_{ij} (v) = - (k = 1 \sum n v^{k}) L_{ij} (v) - L_{i} (v) + \frac{4}{3} v_{j},

δ h^{i} = A^{i} exp (- K (v) t),

δ h^{i} = A^{i} exp (- K (v) t),

i = 1 \sum n C_{i} (v) A^{i} = 0,

L_{μν} = (1 - δ_{μν}) (2 + 4 α S_{H H}), L_{α β} = (1 - δ_{α β}) (2 + 4 α S_{hh}), μ, ν = 1 \dots m, α, β = m + 1, \dots, n

L_{μν} = (1 - δ_{μν}) (2 + 4 α S_{H H}), L_{α β} = (1 - δ_{α β}) (2 + 4 α S_{hh}), μ, ν = 1 \dots m, α, β = m + 1, \dots, n

L_{μν} = (1 - δ_{μν}) (2 + 4 α S_{H H}), L_{ab} = (1 - δ_{ab}) (2 + 4 α S_{HH}), L_{α β} = (1 - δ_{α β}) (2 + 4 α S_{hh})

L_{μν} = (1 - δ_{μν}) (2 + 4 α S_{H H}), L_{ab} = (1 - δ_{ab}) (2 + 4 α S_{HH}), L_{α β} = (1 - δ_{α β}) (2 + 4 α S_{hh})

(H, H, H, h, h, \mathds h) : H^{2} = \frac{1}{12 α}, h = - 2 H, \mathds h \in R, Λ = \frac{1}{4 α}, α > 0

(H, H, H, h, h, \mathds h) : H^{2} = \frac{1}{12 α}, h = - 2 H, \mathds h \in R, Λ = \frac{1}{4 α}, α > 0

(H, H, H, H, H, h) : H^{2} = - \frac{1}{24 α}, h \in R, Λ = - \frac{5}{16 α}, α < 0

(H, H, H, H, H, h) : H^{2} = - \frac{1}{24 α}, h \in R, Λ = - \frac{5}{16 α}, α < 0

(H, H, H, H, h, h, \mathds h) : H^{2} = \frac{1}{24 α}, h = - 3 H, \mathds h \in R, Λ = \frac{13}{48 α}, α > 0

(H, H, H, H, h, h, \mathds h) : H^{2} = \frac{1}{24 α}, h = - 3 H, \mathds h \in R, Λ = \frac{13}{48 α}, α > 0

(H, H, H, - H, - H, - H, h) : H^{2} = \frac{1}{8 α}, h \in R, Λ = \frac{3}{16 α}, α > 0

(H, H, H, - H, - H, - H, h) : H^{2} = \frac{1}{8 α}, h \in R, Λ = \frac{3}{16 α}, α > 0

(H, H, H, H, H, H, h) : H^{2} = - \frac{1}{40 α}, h \in R, Λ = - \frac{21}{80 α}, α < 0

(H, H, H, H, H, H, h) : H^{2} = - \frac{1}{40 α}, h \in R, Λ = - \frac{21}{80 α}, α < 0

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Full text

On stable exponential cosmological solutions in the EGB model with a $\Lambda$ -term

in dimensions $D=5,6,7,8$

Chirkov D.M

Bauman Moscow State Technical University, 2-ya Baumanskaya ul., 5, Moscow, 105005, Russia

Toporensky A.V

Sternberg Astronomical Institute, Lomonosov Moscow State University, Universitetsky Prospect, 13, Moscow 119991, Russia

Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia

Abstract

A $D$ -dimensional Einstein-Gauss-Bonnet (EGB) flat cosmological model with a cosmological term $\Lambda$ is considered. We focus on solutions with exponential dependence of scale factor on time. Using previously developed general analysis of stability of such solutions done by V.D.Ivashchuk (2016) we apply the criterion from that paper to all known exponential solutions upto dimensionality 7+1. We show that this criterion which guarantees stability of solution under consideration is fulfilled for all combination of coupling constant of the theory except for some discrete set.

I Introduction

Lovelock gravity Low can be considered as the most conservative modification of General Relativity (GR) in the sense that equation of motion of this theory are second order differential equations (the same as in GR) in contrast to other metric theories, usually leading to fourth order equations (though some other approaches with the same property exist, for example, Palatini version of $f(R)$ theory or $f(T)$ theory). Usually increasing of the order of equations leads to a variety of new solutions, some of them without a GR limit (for example, famous "false radiation" vacuum isotropic solution in $f(R)$ ), Lovelock gravity being the second order theory is free from this solution. However, due to rather complicated equation of motion Lovelock theory can also contain some solutions without a GR analog. One of such example is exponential solutions in anisotropic cosmology.

In GR there is unique vacuum solution for a flat anisotropic Universe – the Kasner solution (which, strictly speaking is a one-dimensional set of solutions). Scale factors of this solution have a power-law behavior in time. A version of power-law solution is known for general Lovelock gravity. However, when higher Lovelock terms (starting from the Gauss-Bonnet term) are taken into account, new type of solution with exponential time behavior of scale factors (i.e. constant Hubble parameters) appears. Such solutions exist also for non-vacuum Universe, and belong to two different cases. If matter content is different from the cosmological constant, the solution exists only in a very special case of the Universe with a constant volume. For the matter in the form of a cosmological constant there is no restriction on the volume. In this latter case all exponential solutions appear to be a subject of rather strict condition: a space is divided into a restricted number (for Gauss-Bonnet theory, maximum three) of isotropic subspaces. The fact that this division is not introduced "by hand" and appears naturally from equations of motion makes exponential solution interesting for model building in multidimensional cosmology. Any application should be preceded by studies of stability. Stability of exponential solution have been considered recently in several papers, and, in particulary, it was shown that in Einstein - Gauss-Bonnet theory the necessary condition for stability is volume increasing. As for sufficient condition, a special algebraic relation should be satisfied.

In this paper we consider $D$ -dimensional gravitational model with Gauss-Bonnet term and cosmological term $\Lambda$ . The goal of the present paper is to check explicitly this relation for all known exponential solution up to seven space dimensions. We note that so-called Gauss-Bonnet term appeared in string theory as a correction to the (Fradkin-Tseytlin) effective action Zwiebach -MTs .

We note that at present the Einstein-Gauss-Bonnet (EGB) gravitational model and its modifications, see Ishihara -Ivas-16-2 and refs. therein, are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernovae (type Ia) observational data Riess ; Perl ; Kowalski . Another applications are related to Gauss-Bonnet-AdS black holes and holographic description of certain quantum systems (e.g. superfluids), see BLMSY ; KonZh and refs. therein.

Here we consider examples of solutions in dimensions $D=1+d$ = $5,6,7,8$ . We study the stability of these solutions in a class of cosmological solutions with diagonal metrics by using results of refs. ErIvKob-16 ; Ivas-16 , see also approach of ref. Pavl-15 .

Several sets of special stable exponential solutions with zero variation of the effective gravitational constant for two and three factor spaces were found recently in refs. Ivas-16-2 ; Ern-Ivas-16-3 and Ern-Ivas-16-4 , respectively. It should be noted that two special solutions from ref. Ern-Ivas-16-3 for $D=22,28$ and $\Lambda=0$ were found earlier in ref. IvKob . In ref. ErIvKob-16 it was proved that these solutions are stable.

II The set up

The action of the model reads

[TABLE]

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

[TABLE]

is the standard 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 arbitrary constants, $i=1,\dots,n$ , and $M_{1},\dots,M_{n}$ are one-dimensional manifolds (either ${\mathbb{R}}$ or $S^{1}$ ) and $n>3$ .

Equations of motion for the action (2.1) give us the set of polynomial equations ErIvKob-16

[TABLE]

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

[TABLE]

are, respectively, the components of two metrics on ${\mathbb{R}}^{n}$ Iv-09 ; Iv-10 . The first one is a 2-metric and the second one is a Finslerian 4-metric. For $n>3$ we get a set of forth-order polynomial equations.

We note that for $\Lambda=0$ and $n>3$ the set of equations (2.4) and (2.5) has an isotropic solution $v^{1}=\ldots=v^{n}=H$ only if $\alpha<0$ Iv-09 ; Iv-10 . This solution was generalized in ChPavTop to the case $\Lambda\neq 0$ .

It was shown in Iv-09 ; Iv-10 that there are no more than three different numbers among $v^{1},\dots,v^{n}$ when $\Lambda=0$ . This is valid also for $\Lambda\neq 0$ if $\sum_{i=1}^{n}v^{i}\neq 0$ Ivas-16 .

III Conditions for stability

Here, as in ErIvKob-16 ; Ivas-16 , we deal with exponential solutions (2.3) with non-static volume factor, which is proportional to $\exp(\sum_{i=1}^{n}v^{i}t)$ , i.e. we put

[TABLE]

We put the following restriction

[TABLE]

on the matrix

[TABLE]

For general cosmological setup with the metric

[TABLE]

we have the (mixed) set of algebraic and differential equations Iv-09 ; Iv-10

[TABLE]

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

[TABLE]

$i=1,\ldots,n$ .

It was proved in Ivas-16 that a fixed point solution $(h^{i}(t))=(v^{i})$ ( $i=1,\dots,n$ ; $n>3$ ) to eqs. (3.5), (3.6) obeying restrictions (3.1), (3.2) is stable under perturbations

[TABLE]

$i=1,\ldots,n$ , (as $t\to+\infty$ ) if

[TABLE]

and it is unstable (as $t\to+\infty$ ) if $K(v)=\sum_{k=1}^{n}v^{k}<0$ .

We remind the reader that the perturbations $\delta h^{i}$ obey (in linear approximation) the following set of equations ErIvKob-16 ; Ivas-16

[TABLE]

where

[TABLE]

$v_{i}=G_{ij}v^{j}$ , and $i,j,k,s=1,\dots,n$ .

It was proved in ref. Ivas-16 that the set of linear equations on perturbations (3.10), (3.11) has the following solution

[TABLE]

$i=1,\dots,n$ , when restrictions (3.1), (3.2) are imposed.

It was shown also in Ivas-16 that in the case when we have two different Hubble-like parameters $H,h$ , i.e. when the vector $v=(\underbrace{H,H,H,H\ldots,H}_{m},\overbrace{h,\ldots,h}^{l})$ , and $K=mH+lh\neq 0$ , the matrix $L$ has a block-diagonal form: $L=\mathop{\rm diag}\nolimits(L_{\mu\nu},L_{\alpha\beta}),$ where

[TABLE]

and $S_{HH},S_{hh}$ are functions of $H,h,m,l$ . From this we immediately get that if $h$ corresponds to 1-dimensional subspace, i.e. $l=1$ , then $L_{\alpha\beta}$ is the $1\times 1$ -block which equals to zero since $\delta_{\alpha\beta}=\delta_{m+1\,m+1}=1$ and $\det(L)=0$ in this case.

Analogously, it was shown in Ern-Ivas-16-4 that in the case when we have three different Hubble-like parameters $H,\mathcal{H},h$ , i.e. when the vector $v=(\underbrace{H,H,H,H\ldots,H}_{m},\overbrace{\mathcal{H},\ldots,\mathcal{H}}^{k_{1}},\overbrace{h,\ldots,h}^{k_{2}})$ , and $K=mH+k_{1}{\cal H}+k_{2}h\neq 0$ the matrix $L$ has a block-diagonal form again: $L=\mathop{\rm diag}\nolimits(L_{\mu\nu},L_{ab},L_{\alpha\beta})$ with

[TABLE]

where $\mu,\nu=1\ldots m,\;\;a,b=m+1,\ldots,m+k_{1},\;\;\alpha,\beta=m+k_{1}+1,\ldots,n$ and $S_{HH},S_{\mathcal{H}\mathcal{H}},S_{hh}$ are functions of $H,\mathcal{H},h,m,k_{1},k_{2}$ . If $h$ corresponds to 1-dimensional subspace then $\det(L)=0$ exactly for the same reason as in the previous case.

We will see particular examples of this situation in the next section. Moreover, the solutions in question will leave this particular Hubble parameter $h$ unconstrained. From the continuity of $\det(L)$ as a function of $h$ we can conclude that in this case $\det(L)=0$ for all $h$ , i.e. also when $h$ is either coinciding with one of the other Hubble-like parameters ( $H$ or $H,\mathcal{H}$ ) or when the sum of all Hubble-like parameters $K$ is zero.

IV Stability of fixed point solutions in d=4,5,6,7

Now we apply the criterion of stability (3.9) to (4+1)-, (5+1)-, (6+1)- and (7+1)-dimensional exponential solutions with non-static volume factor that have been obtained in ChPavTop ; ChPavTop1 and gather data concerning stability of these solutions in the tables 1-4 below.

IV.1 d=4 and d=5

Since the criterion works with restriction (3.1) only, first of all we evaluate determinant of the matrix (3.3) for each solution and check if it equals to zero. In the case of singular matrix $L$ we can not say anything about stability of the corresponding solutions; such solutions are marked in tables 1 and 2 in gray.

One can see that $\det(L)=0$ for all those solutions which exist for the single value of $\Lambda$ (given fixed $\alpha$ ); in all these cases $\alpha<0$ . This is the solution with 3D isotropic subspace and one extra dimension (we already know from the previous section that the determinant is equal to zero for this solution) and two particular cases of isotropic and 2D+2D solution. Note that the former solution is in fact one-dimensional set of solution (because $h$ is a free parameter there), and two special cases of other solutions with zero determinant appear to be particular points in this set. In this sense only 3D+1D solution (existing only for particular combination of coupling constants) has vanishing determinant.

The case $(H,H,H,h,h)$ with 3-dimensional isotropic subspace is more complicated, we consider it separately.

It is interesting to note that both $\Lambda$ -term and vacuum solutions with 3D isotropic subspace are stable and stability condition requires expanding of this 3D subspace. As well as in the case $d=4$ we see that $\det(L)=0$ for all those solutions which exist for the single value of $\Lambda$ (given fixed $\alpha$ ). There are two one-dimensional set of solutions and a particular case of isotropic solution when it coincides with a point in 4D+1D set. Note also that all vacuum solutions both in $d=4$ and $d=5$ cases exist only for $\alpha<0$ and stable for $H>0$ ; there are no vacuum solutions with singular matrix $L$ .

IV.2 d=6

In this subsection we generalize the above results to the case $d=6$ . There are two special solutions which exist for the single value of $\Lambda$ (given fixed $\alpha$ ) such that $\det(L)=0$ :

[TABLE]

Here we encounter the situation that is qualitatively different from what we see in (4+1)- and (5+1)-dimensional models: matrix $L$ is singular not only for one.dimensional sets of solutions which exist for single value of $\Lambda$ and has free parameter $h$ , but this matrix is turned to be singular for other solutions (however, still for some special values of $\Lambda$ ); in the table 3 for each family we write $(H,h,\Lambda)$ for solutions with two different Hubble parameters and $(H,\Lambda)$ for isotropic solution such that $\det(L)=0$ . We do not write down these solutions itself due to their awkwardness and describe each family by its splitting onto isotropic subspaces; the reader can find formulas in ChPavTop ; ChPavTop1 . Some solutions listed in the table 3 are particular cases of solutions (4.1)-(4.2); we point out overlapping families in the last column of the table 3. There are two solutions (both have 3-dimensional isotropic subspaces) that do not overlap with solutions (4.1) and (4.2) but have nevertheless vanishing determinant; we highlight these special solutions in the table 3 by gray color.

Note that as well as in the cases of $d=4$ and $d=5$ there are no vacuum solutions with $\det(L)=0$ .

IV.3 d=7

In this subsection we generalize the above results to the case $d=7$ . There are three special solutions which exist for the single value of $\Lambda$ (given fixed $\alpha$ ) such that $\det(L)=0$ :

[TABLE]

The other families of solutions have only special solutions with $\det(L)=0$ ; in the table 4 for each family we write $(H,h,\mathds{h},\Lambda)$ for solutions with three different Hubble parameters, $(H,h,\Lambda)$ for solutions with two different Hubble parameters and $(H,\Lambda)$ for isotropic solution. As well as in the previous case we do not write down solutions itself due to their awkwardness, all formulas can be found in ChPavTop ; ChPavTop1 . As in the $d=6$ case we list overlapping families in the last column of the table 4. We see again that there are several solutions with 3D isotropic subspaces that do not overlap solutions (4.3)-(4.5); we highlight these solutions by gray color.

There are no (7+1)-dimensional vacuum solutions with $\det(L)=0$ again.

V Conclusions

We have considered the $D$ -dimensional Einstein-Gauss-Bonnet (EGB) model with the $\Lambda$ -term and two constants $\alpha_{1}$ and $\alpha_{2}$ . The full list of the solutions with exponential time dependence of the scale factors have been found in ChPavTop ; ChPavTop1 , here we consider stability of these solutions. As it have been described in Ivas-16 the necessary condition for stability is the condition that the overall volume of the space considered is growing. Checking this condition on known solution is straightforward. The sufficient condition for stability is more cumbersome to check. In the present paper we have checked this condition for all known exponential solutions up to dimension $7+1$ .

For summarizing the results obtained it is worth to remember that exponential solutions can be divided into two groups. Solutions of the first group (we can call them as special solutions) exist only for coupling satisfying an additional linear relation. On the other hand, one of Hubble parameters of the solution remained unconstrained. Our results shown that all such solution considered in the present paper do not satisfy the sufficient condition for stability, and, so, the dynamics in the vicinity of these solutions requires further investigation.

One the contrary, the second group of solutions (existing for non-zero measure of possible couplings and with all Hubble parameters fully determined) satisfy the sufficient condition for the stability except for very few special sets of couplings. One of the reason for this situation may be the fact that branches of this second group of solution can intersect with branches of the special solutions. On the other hand, we identified several particular coupling which do not satisfy the sufficient condition for stability and do not originate from intersection with any other branches of solutions. Curiously, such a situation occurs only for solutions with 3D isotropic subspace.

To conclude in brief our results shows that (at list up to the dimension 7+1) the stability of exponential solution with growing spatial volume in Gauss-Bonnet cosmology can not be proved in the linear perturbation analysis only for a discrete set of couplings. What happens in this case (when the necessary condition for stability is fulfilled and the sufficient condition is not) needs further analysis.

Acknowledgements.

The work of A.T. is supported by RFBR grant 17-02-01008 and by the Russian Government Program of Competitive Growth of Kazan Federal University. Authors are grateful to Vladimir Ivashchuk for discussions.

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On stable exponential cosmological solutions in the EGB model with a Λ\LambdaΛ-term

Abstract

I Introduction

II The set up

III Conditions for stability

IV Stability of fixed point solutions in d=4,5,6,7

IV.1 d=4 and d=5

IV.2 d=6

IV.3 d=7

V Conclusions

Acknowledgements.

On stable exponential cosmological solutions in the EGB model with a $\Lambda$ -term