Stable exponential cosmological type solutions with three factor spaces in EGB model with a $\Lambda$-term

We study a $D$-dimensional Einstein-Gauss-Bonnet model which includes the Gauss-Bonnet term, the cosmological term $\Lambda$ and two non-zero constants: $\alpha_1$ and $\alpha_2$. Under imposing the metric to be diagonal one, we find cosmological type solutions with exponential dependence of three scale factors in a variable $u$, governed by three non-coinciding Hubble-like parameters: $H \neq 0$, $h_1$ and $h_2$, obeying $m H + k_1 h_1 + k_2 h_2 \neq 0$, corresponding to factor spaces of dimensions $m>1$, $k_1>1$ and $k_2>1$, respectively, and depending upon sign parameter $\varepsilon = \pm 1$, where $\varepsilon = 1$ corresponds to cosmological case and $\varepsilon = - 1$ - to static one). We deal with two cases: i) $m<k_1<k_2$ and ii) $1<k_1 = k_2 = k$, $k \neq m$. We show that in both cases the solutions exist if $\varepsilon \alpha = \varepsilon \alpha_2 / \alpha_1>0$ and $\alpha \Lambda>0$ satisfies certain (upper and lower) bounds. The solutions are defined up to solutions of certain polynomial master equation of order four (or less) which may be solved in radicals. In case ii) explicit solutions are presented. In both cases we single out stable and non-stable solutions as $u \to \pm \infty$. The case $H = 0$ is also considered.


Introduction
In this semi-review article, which generalizes our previous work [1], we deal with the so-called Einstein-Gauss-Bonnet (EGB) gravitational model in dimensions D > 7, which contains Gauss-Bonnet term and cosmological term Λ. The model also includes two non-zero constants: α 1 and α 2 , corresponding to Einstein and Gauss-Bonnet terms, respectively. It is well-known that the equations of motion for this model are of the second order (as it appears in General Relativity). The so-called Gauss-Bonnet term has appeared in (super)string theory as a second order correction in curvature to the effective (super)string effective action [2,3].
At present, EGB gravitational model, e.g. with cosmological term, and its modifications [4]- [25] , are under intensive studyies in cosmology and astrophysics, aimed at solution of dark energy problem, i. e. possible explanation of accelerating expansion of the Universe, which follows from supernovae (type Ia) observational data [26,27], and search of possible local manifestation of dark energy (related to black holes, wormholes etc).
Here we consider the cosmological type solutions with exponential dependence of scale factors (upon u-variable) and obtain a class of solutions with three scale factors, governed by three non-coinciding Hubble-like parameters: H, h 1 and h 2 , corresponding to factor spaces of dimensions m > 1, k 1 > 1 and k 2 > 1, respectively (D = 1 + m + k 1 + k 2 ). Here we impose the following restriction S 1 = mH + k 1 h 1 + k 2 h 2 = 0, excluding the solutions with constant volume factor and addressing us to a classification theorem which tells us that for generic anisotropic exponential solutions with Hubble-like parameters h 1 , . . . , h n obeying S 1 = n i=1 h i = 0 the number of different (real) numbers among h 1 , . . . , h n may be 1, or 2, or 3 [21]. The main goal of this paper is to extend the results of Ref. [1] to a class of cosmological type solutions, which include static ones (with ε = −1).
We also study (in Section 5) the stability of the solutions for u → ±∞ in a class of cosmological type solutions with diagonal metrics by using an extension of results of Refs. [21,1] (see also approach of Ref. [18]) and single out the subclasses of stable/non-stable solutions.
We note that the exponential cosmological type solutions with two noncoinciding Hubble-like parameters H = 0 and h obeying S 1 = mH + lh 1 = 0 with m > 2, l > 2 were studied earlier in Ref. [29]. In that case there were two sets of solutions obeying: a) εα > 0, αΛ < λ + (m, l) and b) εα < 0, It should be noted that recently EGB models were used for constricting certain 4-dimensional gravitational models (so-called 4DEGB theories, e.g. belonging to Horndeski class) by using ideas of Glavan-Lin rescaling [30] and/or dimensional reductions. These 4D modified models of gravity are (at the moment) under intensive study and have numerous applications in gravitational physics and cosmology, for a review see Ref. [31].

The cosmological model
We start with the model governed by the action (2.1) is the Gauss-Bonnet term and α 1 , α 2 are certain nonzero constants.
In Ref. [13,14] the following proposition was proved: there are no more than three different numbers among v 1 , . . . , v n if Λ = 0. This proposition was generalised in ref. [21] for Λ = 0, when the following condition is imposed In this paper we study solutions to equations (2.4), (2.5) by using the following ansatz:  (2.7) Here H is the Hubble-like parameter which corresponds to an m-dimensional factor space with inequality m > 1 imposed, while h 1 is the Hubble-like parameter which is related to an k 1 -dimensional factor space with k 1 > 1 and h 2 is the Hubble-like parameter assigned to an k 2 -dimensional factor space with k 2 > 1.
In what follows we add additional restrictions to our ansatz (2.7): It was shown in Ref. [22] that the set of (n + 1) polynomial equations (2.4), (2.5) under ansatz (2.7) and restrictions (2.8) obeyed are equivalent to a set of polynomial equations which are of fourth, second and first orders, respectively. Here E is defined in (2.4) and For more general prescription of scheme of reduction of polynomial equations of motion see Ref. [17] (the so-called Chirkov-Pavluchenko-Toporensky trick). Relation (2.10) is a special case of more general relations [22] , i = j, (2.14) i, j = 0, 1, 2, with notation h 0 = H used. Relation (2.8) excludes the following case H = h 1 = h 2 = 0. In the main body of the paper we put As in Ref. [1] we denote In terms of dimensionless parameters the restrictions (2.8) may rewritten as following In what follows we do not consider the case which lead us to the empty set of solutions, since we find for m = k 1 = k 2 > 1 from restriction (2.17): Due to (2.10) and (2.12) we obtain The relation (2.20) is valid for αεP < 0. It can be readily proved that [1] for m > 1, k 1 > 1, k 2 > 1. Indeed [1], The equation (2.9) reads [1] and Owing to eq. (2.18) we get Hence, from eq. (2.29) we get a master equation in x 1 variable (2.31) This polynomial equation is of fourth order or less (this depends upon the value of λ). One may solve it in radicals for all m > 1, k 1 > 1 and k 2 > 1.
Relations (2.23) and (2.30) imply the identity which will be used below.
3 The case k 1 = k 2 In this section we put k 1 = k 2 . We rewrite relation (2.28) as following (3.1) Due to (2.30) we present restrictions (2.17) in the following form [1]
In the limit x 1 → ±∞ we obtain . (3.17) Here we obtain [1] for all k 1 > 1 and k 2 > 1. The definitions of X i imply [1] Here m > 1, k 1 > 1 and k 2 > 1. From this point up to Section 4 we impose the following inequality It was shown in Ref. [1] that and and For (m, k 1 , k 2 ) = (4, 6, 7) the the function λ = f (x 1 ) is presented grafically at Figure 1.
It was proved in Ref. [1] that It was proved in Ref. [1] that for the function f (x 1 ) mentioned above X 3 is the point of absolute maximum and X 1 is the point of absolute minimum, i.e.
for all x 1 ∈ R. We remind that according to (3.2) the points X 1 , X 2 , X 3 , X 4 are forbidden for our analysis. We obtain for all x 1 = X 1 , X 2 , X 3 , X 4 . Let us denote the set of definition of the fuction f for our consideration (−∞, ∞) * ≡ {x|x ∈ R, x = X 1 , X 2 , X 3 , X 4 }.
Since the function f (x 1 ) is continuous one the image of the function f (due to

The analysis of stability
Here we analyse the stability of our solutions along a line as it was done in refs. [20,21,22]. We impose the following restriction det(L ij (v)) = 0, Here one should deal with general cosmological type setup with the metric where ε = ±1, ε i = ±1, i = 1, . . . , n. For the equations of motion we obtain [28] and it is unstable, as u → +∞, if and only if In the limit u → −∞ the stability condition is given by (5.9) while the unstability condition reads as (5.8). These conditions just follow from solutions for perturbations δh i (t) = C i exp(−S 1 (v)u) (C i = const = 0) which are valid in the leading order.
Here a key point is the verification of the relation (5.1). It was fulfilled in Ref. [22] by using first three relations in (2.8) and (2.14) and k 1 > 1, k 2 > 1 and m > 1.
First we consider the case 1 < m < k 1 < k 2 . By using (2.18) we find that for H > 0 the condition (5.8) may be written as or, equivalently, For H < 0 the stability condition (5.8) is following one The non-stability condition (5.9) for u → +∞ reads as (5.12) for H > 0 and as (5.11) for H < 0. These conditions are reversed in case u → −∞.

Conclusions
We have studied the D-dimensional Einstein-Gauss-Bonnet (EGB) model with the Λ-term and two non-zero constants α 1 and α 2 . By dealing with diagonal cosmological type metrics, we have considered a class of solutions with exponential dependence of three scale factors (upon u-variable) for any α = α 2 /α 1 = 0, signature parameter ε = ±1 and generic dimensionless parameter Λα.
More precisely speaking we have described a class of cosmological type solutions with exponential dependence of three scale factors, governed by three non-coinciding Hubble-like parameters H, h 1 and h 2 . These parameters correspond, respectively, to factor spaces of dimensions m > 1, k 1 > 1 and k 2 > 1 (D = 1 + m + k 1 + k 2 ), and obey the following restriction S 1 = mH + k 1 h 1 + k 2 h 2 = 0. We have analyzed two cases: i) m < k 1 < k 2 and ii) 1 < k 1 = k 2 = k = m. This choice does not restrict the generality, since, as it was shown, there are no solutions under consideration for k 1 = k 2 = m.) It was shown that the solutions exist only if, λ = αΛ > 0 and the (dimensionless) parameter λ obeys certain restrictions, e.g. upper and lower bounds for H = 0, which depend upon dimensions m, k 1 and k 2 (Proposition 1). In the case ii) we have presented explicit solutions for all k > 1 and k = m ( Proposition 2).
By using Chirkov-Pavluchenko-Toporensky splitting trick from Ref. [17] we have reduced the problem for H = 0 to master equation on the dimensionless variable x 1 = h 1 /H. This equation is of fourth order (in generic case) or less (depending on λ), and may be solved in radicals for all m > 1, k 1 > 1, k 2 > 1 and λ. The master equation does not depend upon the signature parameter ε = ±1 which is only controlling the sign of α according to inequality αε > 0. Due to bounds obtained λ = αΛ > 0. (This is valid also for H = 0). Hence the solutions under consideration do exist if Λε > 0 , i.e. when Λ > 0 in cosmological case (ε = 1) and Λ < 0 in static case (ε = −1). Here there are no solutions under considerations for Λ = 0 -contrary to the case of two factor spaces [29,32].
Here we have analyzed the stability of solutions as u → ±∞ in a class of cosmological type solutions with diagonal metrics. In both cases ((i) and (ii)) for H = 0 the "islands" of stability and instability were singled out. (The case H = 0 was also analysed.) We have shown that in the case i) the solutions with H > 0 are stable as u → ∞ for x 1 = h 1 /H > X 4 = m−k 2 k 2 −k 1 and unstable as u → ∞ for x 1 < X 4 (see Proposition 3). These conditions should be reversed when we consider the case H > 0, u → −∞ or we deal with H < 0, u → +∞ (see Proposition 3). It was proved that in the case ii) the solutions with H > 0 are stable as u → ∞ for k > m and unstable as u → ∞ for k < m (see Proposition 4). For given choice of asymptotic u → ±∞ the stability condition for H < 0 is equivalent to instability conditions for H > 0 and vice versa.
We have also found that the solution with H = 0 exists only for k 1 = k 2 , αε > 0 and fixed value of εΛ > 0 depending upon k 1 and k 2 . Here we have two opposite in sign solutions for (h 1 , h 2 ) with one solution being stable (u → ±∞) and the second one -unstable depending upon the sign of k 1 − k 2 .
Some cosmological applications of the model (ε = 1), e.g. in context of variation of gravitational constant, where considered in Refs. [33,34,1]. For static case (ε = −1) possible applications of the obtained solutions may be a subject of a further research, aimed at a search of topological black hole solutions (with flat horizon) or wormhole solutions which are coinciding asymptotically (for (u → ±∞)) with our solutions.