Stable exponential cosmological solutions with zero variation of G and three different Hubble-like parameters in the Einstein–Gauss–Bonnet model with a $$\Lambda $$Λ-term

We consider a D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term $$\Lambda $$Λ. We restrict the metrics to diagonal cosmological ones and find for certain $$\Lambda $$Λ a class of solutions with exponential time dependence of three scale factors, governed by three non-coinciding Hubble-like parameters $$H >0$$H>0, $$h_1$$h1 and $$h_2$$h2, corresponding to factor spaces of dimensions $$m > 2$$m>2, $$k_1 > 1$$k1>1 and $$k_2 > 1$$k2>1, respectively, with $$k_1 \ne k_2$$k1≠k2 and $$D = 1 + m + k_1 + k_2$$D=1+m+k1+k2. Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameter H and zero variation of the effective gravitational constant G. We prove the stability of these solutions in a class of cosmological solutions with diagonal metrics.


Introduction
In this paper we consider a D-dimensional gravitational model with Gauss-Bonnet term and cosmological term Λ. The so-called Gauss-Bonnet term appeared in string theory as a first order correction (in α ′ ) to the effective action [1]- [4].
We note that at present the Einstein-Gauss-Bonnet (EGB) gravitational model and its modifications, see [5]- [28] and refs. therein, are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernova (type Ia) observational data [29,30,31].
In ref. [28] we were dealing with the cosmological solutions with diagonal metrics governed by n > 3 scale factors depending upon one variable, which is the synchronous time variable. We have restricted ourselves by the solutions with exponential dependence of scale factors and have presented a class of such solutions with two scale factors, governed by two Hubble-like parameters H > 0 and h < 0, which correspond to factor spaces of dimensions m > 3 and l > 1, respectively, with D = 1 + m + l and (m, l) = (6, 6), (7,4), (9,3). Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameters H > 0 [32] and has a constant volume factor of (m − 3 + l)-dimensional internal space, which implies zero variation of the effective gravitational constant G either in a Jordan or an Einstein frame [33,34]; see also [35,36,37] and refs. therein. These solutions satisfy the most severe restrictions on variation of G [38]. We have studied the stability of these solutions in a class of cosmological solutions with diagonal metrics by using results of refs. [24,26] (see also approach of ref. [22]) and have shown that all solutions, presented in ref. [28], are stable. It should be noted that two special solutions for D = 22, 28 and Λ = 0 were found earlier in ref. [21]; in ref. [24] it was proved that these solutions are stable. Another set of six stable exponential solutions, five in dimensions D = 7, 8, 9, 13 and two for D = 14, were considered earlier in [27].
In this paper we extend the results of ref. [28] to the case of solutions with three non-coinciding Hubble-like parameters. The structure of the paper is as follows. In Section 2 we present a setup. A class of exact cosmological solutions with diagonal metrics is found for certain Λ in Section 3. Any of these solutions describes an exponential expansion of 3-dimensional subspace with Hubble parameter H and zero variation of the effective gravitational constant G. In Section 4 we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. Certain examples are presented in Section 5.

The cosmological model
The action of the model reads is the standard Gauss-Bonnet term and α 1 , α 2 are nonzero constants. We consider the manifold with the metric where B i > 0 are arbitrary constants, i = 1, . . . , n, and M 1 , . . . , M n are one-dimensional manifolds (either R or S 1 ) and n > 3. The equations of motion for the action (2.1) give us the set of polynomial equations [24] are, respectively, the components of two metrics on R n [16,17]. 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 Λ = 0 and n > 3 the set of equations (2.4) and (2.5) has an isotropic solution v 1 = · · · = v n = H only if α < 0 [16,17]. This solution was generalized in [19] to the case Λ = 0.
It was shown in [16,17] that there are no more than three different numbers among v 1 , . . . , v n when Λ = 0. This is valid also for Λ = 0 if n i=1 v i = 0 [26].

Solutions with constant G
In this section we present a class of solutions to the set of equations (2.4), (2.5) of the following form: where H is the Hubble-like parameter corresponding to an m-dimensional factor space with m > 2, h 1 is the Hubble-like parameter corresponding to an k 1 -dimensional factor space with k 1 > 1 and h 2 (h 2 = h 1 ) is the Hubblelike parameter corresponding to an k 2 -dimensional factor space with k 2 > 1.
We split the m-dimensional factor space into the product of two subspaces of dimensions 3 and m − 3, respectively. The first one is identified with "our" 3d space while the second one is considered as a subspace of (m−3+k for a description of an accelerated expansion of a 3-dimensional subspace (which may describe our Universe) and also put for a description of a zero variation of the effective gravitational constant G. We remind (the reader) that the effective gravitational constant G = G ef f in the Brans-Dicke-Jordan (or simply Jordan) frame [33] (see also [34]) is proportional to the inverse volume scale factor of the internal space; see [35,36,37] and references therein.
Due to (3.1) "our" 3d space expands isotropically with Hubble parameter H, while the (m − 3)-dimensional part of the internal space expands isotropically with the same Hubble parameter H too. Here, like in ref. [28], we consider for cosmological applications (in our epoch) the internal space to be compact one, i.e. we put in (2.2) M 4 = · · · = M n = S 1 . We put the internal scale factors corresponding to present time t 0 : a j (t 0 ) = B 1/2 j exp(v j t 0 ), j = 4, . . . , n, (see (2.3)) to be small enough in comparison with the scale factor of "our" space for t = t 0 : According to the ansatz (3.1), the m-dimensional factor space is expanding with the Hubble parameter H > 0, while the k i -dimensional factor space is contracting with the Hubble-like parameter h i < 0, where i is either 1 or 2.
Now we consider the ansatz (3.1) with three Hubble parameters H, h 1 and h 2 which obey the following restrictions: (3.4) The first inequality in (3.4) is valid since S 1 = 3H > 0 due to (3.2) and (3.3).
Due to S 1 = mH + k 1 h 1 + k 2 h 2 = 0 the set of eqs. (3.5) is equivalent to the following set of equations The last relation in (3.13) may be omitted since [26]. Using this fact and relations (3.4) and (3.10) we reduce the system (3.13) to the following one . (3.14) Using the identity we reduce the set of equations (3.14) to the equivalent set Here we put Q = Q h 1 h 2 though other choices, Q = Q Hh 1 or Q = Q Hh 2 give us equivalent sets of equations. Thus the set of (n + 1) polynomial equations (2.4), (2.5) under ansatz (3.1) and restrictions (3.4) imposed is reduced to a set (3.16) of three polynomial equations (of fourth, second and first orders). This reduction is a special case of the more general prescription from ref. [20]. Using the condition (3.3) of zero variation of G and the linear equation from (3.16) we obtain for k 1 = k 2 For k 1 = k 2 we get H = 0, which is not appropriate for our consideration. The substitution of (3.17) into relation Q h 1 h 2 = − 1 2α gives us the following relation which implies It may be readily verified that for all m > 2, k 1 > 1, k 2 > 1, k 1 = k 2 and hence our solutions take place for α > 0. The substitution of (3.17) into (3.5) gives us and  (3.25) where P = P (m, k 1 , k 2 ) is defined in (3.19).
The function Λ(m, k 1 , k 2 ) in (3.25) is symmetric with respect to k 1 and k 2 , i.e. Λ(m, k 1 , k 2 ) = Λ(m, k 2 , k 1 ). (3.26) For k 2 = 0 we get a function Λ(m, k 1 , 0) = Λ(m, k 1 ), where Λ(m, k 1 ) was obtained in ref. [28] for the case of two different Hubble-like parameters. It may be readily verified that for k 1 (k) = n 1 k + q 1 and k 2 (k) = n 2 k + q 2 , where k, n 1 > 0, q 1 , n 2 > 0, q 2 are integer numbers, we get as k → +∞ for any fixed m ≥ 3. We note that the limit (3.27) is positive and does not depend upon m. For fixed integer m > 2 and k 2 ≥ 1 we are led to the following limit as k 1 → +∞ and analogous relation (due to (3.26)) for fixed m > 2, k 1 ≥ 1 and k 2 → +∞. It can be easily verified that for these values of m, k 1 we get: Λ(m, ∞, k 2 ) > 0. Relation (3.27) and (3.28) may be used in a context of (1/D)-expansion for large D in the model under consideration, see [25] and refs. therein.

Examples
Here we present several examples of stable solutions under consideration.

The case m =3
Let us consider the case m = 3. From (3.25) we get For (m, k 1 , k 2 ) = (3, 3, 2) we have P = −70, Now we put (m, k 1 , k 2 ) = (3, 4, 2). We obtain P = −120 and According to our analysis from the previous section both solutions are stable. In this case we obtain

Conclusions
We have considered the D-dimensional Einstein-Gauss-Bonnet (EGB) model with the Λ-term and two constants α 1 and α 2 . By using the ansatz with diagonal cosmological metrics, we have found, for certain Λ = Λ(m, k 1 .k 2 ) and α = α 2 /α 1 < 0, a class of solutions with exponential time dependence of three scale factors, governed by three different Hubble-like parameters H > 0, h 1 and h 2 , corresponding to submanifolds of dimensions m > 2, k 1 > 1, k 2 > 1, respectively, with k 1 = k 2 and D = 1 + m + k 1 + k 2 . Here m > 2 is the dimension of the expanding subspace. Any of these solutions describes an exponential expansion of "our" 3dimensional subspace with the Hubble parameter H > 0 and anisotropic behaviour of (m − 3 + k 1 + k 2 )-dimensional internal space: expanding in (m − 3) dimensions (with Hubble-like parameter H) and either contracting in k 1 dimensions (with Hubble-like parameter h 1 ) and expanding in k 2 dimensions (with Hubble-like parameter h 2 ) for k 1 > k 2 or expanding in k 1 dimensions and contracting in k 2 dimensions for k 1 < k 2 . Each solution has a constant volume factor of internal space and hence it describes zero variation of the effective gravitational constant G. By using results of ref. [26] we have proved that all these solutions are stable as t → +∞. We have presented several examples of stable solutions for m = 3, 4, 5.