On stability of exponential cosmological solutions with non-static volume factor in the Einstein-Gauss-Bonnet model

A (n+1)-dimensional gravitational model with Gauss-Bonnet term and cosmological constant term is considered. When ansatz with diagonal cosmological metrics is adopted, the solutions with exponential dependence of scale factors: a_i ~ exp( v^i t), i = 1, ..., n, are analysed for n>3. We study the stability of the solutions with non-static volume factor, i.e. if K(v) = \sum_{k = 1}^{n} v^k \neq 0. We prove that under certain restriction R imposed solutions with K(v)>0 are stable while solutions with K(v)<0 are unstable. Certain examples of stable solutions are presented. We show that the solutions with v^1 = v^2 = v^3 = H>0 and zero variation of the effective gravitational constant are stable if the restriction R is obeyed.


Introduction
This paper is devoted to D-dimensional gravitational model with the so-called Gauss-Bonnet term. It is governed by the action (1.1) where g = g M N dz M ⊗ dz N is the metric defined on the manifold M, dim M = D, |g| = | det(g M N )| and is the quadratic "Gauss-Bonnet term" and Λ is cosmological term. Here α 1 and α 2 are non-zero constants. The appearance of the Gauss-Bonnet term was motivated by string theory [1,2,3]. At present, the so-called Einstein-Gauss-Bonnet (EGB) gravitational model which is governed by the action (1.1) and its modifications are intensively used in cosmology, see [4] - [23] and references therein, e.g. for explanation of accelerating expansion of the Universe following from supernovae (type Ia) observational data [24,25,26].
Here we consider the cosmological solutions with diagonal metrics governed by n scale factors depending upon one variable, where n > 3; D = n + 1. We study the stability of solutions with exponential dependence of scale factors with respect to the synchronous time variable t a i (t) ∼ exp (v i t), (1.3) i = 1, . . . , n. In our analysis we restrict ourselves by a class of perturbations which depend on t and do not disturb the diagonal form of the metric. For possible physical applications solutions describing an exponential isotropic expansion of 3-dimensional flat factor-space, i.e. with v 1 = v 2 = v 3 = H > 0, (1.4) and small enough variation of the effective gravitational constant G are of interest. We remind that G (for 4d metric in Jordan frame, see Remark 4 in Section 4) is proportional to the inverse volume scale factor of the internal space, see [27,28,29] and refs. therein. Due to experimental data, the variation of G is allowed at the level of 10 −13 per year and less. The most stringent limitation on G-dot (coming from the set of ephemerides) was obtained in ref. [30]Ġ /G = (0.16 ± 0.6) · 10 −13 year −1 (1.5) allowed at 95% confidence (2-σ).
Here we reduce the set of cosmological equations to the (mixed) set of algebraic and differential equations f 0 (h) = 0, (1.6) f i (ḣ, h) = 0. (1.7) where h = h(t) = (h i (t) =ȧ i (t)/a i (t)) is the set of so-called "Hubble-like" parameters corresponding to scale factors a i (t); f 0 (h) and f i (ḣ, h) are polynomials of the fourth order in h i ; f i (ḣ, h) are polynomials of the first order inḣ i . The fixed point solutions h i (t) = v i (i = 1, . . . , n) correspond to exponential solutions of (1.3). They obey a set of n + 1 polynomial equations of the fourth order. We analyze the stability of the fixed point solutions by imposing the following restriction which guarantees the local resolution of eqs. (1.7) in the vicinity of the point (0, v) ∈ R 2n : We also impose another restriction on v: which means that the solutions with constant volume scale factor are not considered here. We note that a solution with n k=1 v k = 0 obeying (1.4) gives an enormously big value of the variation of G:Ġ/G = 3H, where H is the Hubble parameter, see Remark 5 in Section 4 below. This value of G-dot contradicts to the observational restrictions, e.g. (1.5). We remind that the present value of H is (6.929 ± 0.157) · 10 −11 year −1 [31] (with 95% confidence level).
The main result of the paper is the following one: fixed point solutions h(t) = v to eqs. (1.6) and (1.7), which obey restrictions (1.8) and (1.9), are stable if and only if n k=1 v k > 0. This result is in agreement with the approach of S. Pavluchenko from ref. [22], see Remark 2 in Section 3 below.
The paper is organized as follows. In Section 2 the equations of motion for Ddimensional EGB model are considered. For diagonal cosmological metrics the equations of motion are equivalent to a set of Lagrange equations corresponding to a certain "effective" Lagrangian. The Lagrange equtions for a certain choice of the lapse function (corresponding to the synchronous time variable) are reduced to the set of eqs. (1.6), (1.7). Section 3 is devoted to analysis of stability of the exponential solutions with constant Hubble-like parameters: here a set of equations for perturbations δh i (t) (obtained in linear approximation) is studied and general solution to these equations is found. The main proposition on stability of exponential solutions (Proposition 2) is proved. In Section 4 some examples of stable cosmological solutions with exponential behavior of scale factors are presented.
The integrand in (1.1), when the metric (2.2) is substituted, reads as follows are respectively the components of two metrics on R n [15,16]. The first one is "minisupermetric" -2-metric of pseudo-Euclidean signature and the second one is the Finslerian 4-metric [15,16]. Here we denoteȦ = dA/dt etc. The function f (t) in (2.3) is irrelevant for our consideration (see [15,16]). In derivation of (2.4) the following identities [15,16] were used: Here and in what follows The definitions (2.5) and (2.6) imply The equations of motion corresponding to the action (1.1) have the following form It may be shown (along a line as it was done in [16] for the case Λ = 0) that the field eqs. (2.11) for the metric (2.2) are equivalent to the Lagrange equations corresponding to the Lagrangian L from (2.4).
Thus, eqs. (2.11) read as follows . . , n; and L is defined in (2.4). Now we put γ = 0. By introducing "Hubble-like" variables h i =β i , eqs. (2.14) and (2.15) may be rewritten as follows where α = α 1 /α 2 , and i = 1, . . . , n. Thus, we are led to the autonomous system of the first-order differential equations on h 1 (t), . . . , h n (t) (see [15,16] for Λ = 0). Due to (2.16) we have In what follows we will use instead of (2.16), (2.17) an equivalent set of equations: (2.16) and We note that the following identity is valid This identity may be proved by using two relations:

Useful relations
In what follows we use the definitions Here we deal with the ansatz which contain two Hubble parameters where H appears m-times and h appears l-times, n = m + l. In what follows we adopt the following agreement for indices: µ, ν, . . . = 1, . . . , m; α, β, . and We also denote We note that S ij = S ji and S ii = 0. For isotropic case (2.27) we obtain For the the ansatz (2.29) we obtain (2.37) Here we denote: S µν = S HH for µ = ν; S µα = S αµ = S Hh ; S αβ = S hh for α = β.

Polynomial equations for solutions with constant h i
Let us consider the following solutions to eqs. (2.16) and (2.21) with constant v i , which correspond to the solutions where β i 0 are constants, i = 1, . . . , n. In this case we obtain the metric (2.2) with the exponential dependence of scale factors where L i is defined in (2.19), i = 1, . . . , n. For n > 3 this is the set of forth-order polynomial equations.
Here and in what follows we use relations (2.7), (2.8) and the following formulas . . , n, with some real numbers C 0 , C 1 , C 2 and C 3 = 0. Let us consider the cubic equation Any number v i obeys this equation and hence at most three numbers among v i may be different. Thus, we are led to a contradiction. The proposition is proved. The case Λ = 0 was considered earlier in [15,16].
Remark 1. In pure Einsten case (α = 0) with Λ > 0 we get two exponential solutions with v 1 = . . . = v n = H and n(n−1)H 2 = 2Λ > 0; solution with H > 0 is an attractor for cosmological solutions with diagonal metrics, as t → +∞, see [32] and [33] (for ϕ = 0). Thus in this case (α = 0) we have an isotropisation for t → +∞, while for t → +0 we have Kasner-like behaviour of scale factors near the singularity: In the case of EGB model with Λterm we have for certain Λ and α isotropic exponential solutions with v 1 = . . . = v n = H (see Section 4 below), but we also may have partially anisotropic (PA) solutions, which , and also solutions with n i=1 v i = 0 may take place. For n i=1 v i = 0 (and certain Λ and α) one may obtain examples of totally anisotropic exponential solutions with non-coinciding parameters among v 1 , . . . , v n . Some of the exponential PA solutions are stable (see below) and they are attractors of certain subclasses of general solutions. The appearance of three (or less) independent scale factors in the model under consideration is a feature of exponential (e.g. attractor) solutions, when restriction n i=1 v i = 0 is imposed. We also note that the metric (2.40) may be analyzed on symmetries (apparent or hidden) by using the results of ref. [34], i.e. Killing vectors, isometry group, coset structure G/H etc, may be presented. The Proposition 2 may be also generalized to the Lovelock case [35] . . , m, α = m + 1, . . . , n). Due to (2.44) we have For m > 1 and l > 1 the quadratic form has the signature (−, +). Due to mH + lh = 0 the set of eqs. (2.46) is equivalent to another set of equations where

5)
We remind that v i = G ij v j , L i (v) = 2v i − 4 3 αG ijks v j v k v s and i, j, k, s = 1, . . . , n. We put the following restriction on the matrix L = (L ij (v)) i.e. the matrix L should be invertible.
Here we restrict ourselves by exponential solutions (2.40) with non-static volume factor, which is proportional to exp( n i=1 v i t), i.e. we put Then we get from eq. (2.42) Due to definition (2.19) we have and hence We rewrite relation (3.6) as Due to L i (v) = L 1 and (3.2) we get 14) or, equivalently, We remind that m × m matrix (G µν ) and l × l matrix (G αβ ) are invertible only for m > 1 and l > 1, respectively. Remark 2. Recently, in ref. [22] a criterion for stability of fixed point solutions in the model under consideration (and its extension to the Lovelock case) was used. In our notations (see Introduction) it reads:  (2.22) and (3.13)), it gives a correct result since in this case h) with λ = 0, e.g. for the choice λ = −1 used in [22]. We also note that in our notations 2Λ = Λ P , where Λ P is the Λ-term from ref. [22].

Examples
Here we consider several examples of exponential solutions and analyse their stability. Let Λ = 0. The trivial solution H = 0 is valid for any α. This is the unique solution for α > 0. For α < 0 we have two non-trivial solutions [15,16] with
This solution was generalized in [19] to the case Λ = 0. Let us consider the case of generic Λ in detail. First, we put α > 0. Then, a solution to eq. (4.1) does exist if and only if Λ ≥ 0. For Λ = 0 we get H = 0, while for Λ > 0 we have two non-zero solutions for H with H 2 > 0: , Now we put α < 0. A solution to eq. (4.1) exists only if Λ ≤ Λ cr , where is the maximum value of the function F (H 2 ) from (4.1). For 0 < Λ < Λ cr (and α < 0) we have two solutions for H 2 (or four solutions for H) which are given by relation .
For Λ = Λ cr and α < 0 we get one solution for H 2 (or two solutions for H): . (4.8) The case Λ = 0 (and α < 0) was mentioned above (two solutions for H 2 , or three -for H). For Λ < 0 (and α < 0) we obtain one solution for H 2 (or two solutions for H): .

Anisotropic solutions with two Hubble parameters
In  It the approximate form this solution was found earlier in [17], in analytic form (different from (4.12), (4.13)) it was obtained in [19].
was found in [21]. This solution describes the zero variation of the effective cosmological constant G.

A subclass of solutions with zero variation of G
The 4d effective gravitational constant is proportional to inverse volume scale factor of the internal space (see [27,28,29]), i.e.
where a i (t) = exp(β i (t)). Remark 4. Here G = G J ef f (t) is four-dimensional effective gravitational constant which appear in (multidimensional analogue of ) the so-called Brans-Dicke-Jordan (or simply Jordan) frame [36]. In this case the physical 4-dimensional metric g (4) is defined as 4-dimensional section of the multidimensional metric g, i.e. g (4) = g (4,J) , where g = g (4,J) + n i=4 a 2 i (t)dy i ⊗ dy i . When the Einstein-Pauli (or simply Einstein) frame is used, we put g (4) = g (4,E) = ( n i=4 a i (t))g (4,J) [36,37] and hence we get the effective gravitational constant to be an exact constant: [36]. For the solutions (2.40) we obtain the following relations which implyĠ Now, let us consider a subclass of cosmological solutions (2.40) which obey restriction (3.7) and describe an exponential isotropic expansion of 3-dimensional flat factor-space with v 1 = v 2 = v 3 = H > 0 with zero variation of G. Then we get from (4.33) K int (v) = 0 and hence K(v) = n i=1 v i = 3H + K int (v) = 3H > 0. According to Proposition 2 any solution from this subclass is stable. Three solutions from the previous subsection: (4.19), (4.23) and (4.27) with m = 3 (and l > 1) belong to this subclass.

Conclusions
We have considered the (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model with the Λ-term. By using the ansatz with diagonal cosmological metrics, we have studyed the stability of solutions with exponential dependence of scale factors a i ∼ exp (v i t), i = 1, . . . , n, with respect to synchronous time variable t in dimension D > 4.
The problem was reduced to the analysis of stability of the fixed point solutions h i (t) = v i to eqs. (2.16) and (2.21), where h i (t) are Hubble-like parameters.
In this paper a set of equations for perturbations δh i was considered (in linear approximation) and general solution to these equations was found. We have proved (in Proposition 2) that the solutions with non-static volume factor, i.e. with K(v) = n k=1 v k = 0, which obey restriction (3.7), are stable if K(v) > 0 while they are unstable if K(v) < 0.
We have also proved (in Proposition 1) that for any exponential solution with v = (v 1 , ..., v n ) there are no more than three different numbers among v 1 , ..., v n , if n i=1 v i = 0. Here we have presented several examples of stable cosmological solutions with exponential behavior of scale factors. Among them the isotropic solution v = (H, . . . , H) and several anisotropic solutions with two Hubble parameters v = (H, . . . , H, h, . . . , h) were considered. The isotropic solution is stable if H > 0 and H = H cr for α < 0 (see (4.8)). For the anisotropic case our examples deal with the Hubble-like parameter H > 0 corresponding to m-dimensional flat subspace with m ≥ 3 and the Hubble-like parameter h corresponding to l-dimensional flat subspace with l > 1. This subclass of (anisotropic) solutions contains the following cases: i) m = 3, l = 2, Λ = 0; ii) m = l = 3, Λ = 0; iii) m = 11, l = 16, Λ = 0; iv) m = 15, l = 6, Λ = 0; v) m ≥ 3, l > 1, Λ > 0. We have also shown that general solutions with v 1 = v 2 = v 3 = H > 0 and zero variation of the effective gravitational constant are stable if the restriction (3.7) is obeyed.