Stable exponential cosmological solutions with 3- and l-dimensional factor spaces in the Einstein–Gauss–Bonnet model with a $$\Lambda $$Λ-term

A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological term $$\Lambda $$Λ is studied. We assume the metrics to be diagonal cosmological ones. For certain fine-tuned $$\Lambda $$Λ, we find a class of solutions with exponential time dependence of two scale factors, governed by two Hubble-like parameters $$H >0$$H>0 and h, corresponding to factor spaces of dimensions 3 and $$l > 2$$l>2, respectively and $$D = 1 + 3 + l$$D=1+3+l. The fine-tuned $$\Lambda = \Lambda (x, l, \alpha )$$Λ=Λ(x,l,α) depends upon the ratio $$h/H = x$$h/H=x, l and the ratio $$\alpha = \alpha _2/\alpha _1$$α=α2/α1 of two constants ($$\alpha _2$$α2 and $$\alpha _1$$α1) of the model. For fixed $$\Lambda , \alpha $$Λ,α and $$l > 2$$l>2 the equation $$\Lambda (x,l,\alpha ) = \Lambda $$Λ(x,l,α)=Λ is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals (the example $$l =3$$l=3 is presented). For certain restrictions on x we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. A subclass of solutions with small enough variation of the effective gravitational constant G is considered. It is shown that all solutions from this subclass are stable.


Introduction
In this paper we study a D-dimensional gravitational model with Gauss-Bonnet term and cosmological term , i.e. we deal with the so-called Einstein-Gauss-Bonnet model (in short, EGB-, or more precisely EGB -model). The so-called Gauss-Bonnet term appeared in string theory as a correction to the string effective action [1][2][3][4][5].
At the moment there is a certain interest to Einstein-Gauss-Bonnet (EGB) gravitational model and its modificaa e-mail: ivashchuk@mail.ru tions, see  and Refs. therein. They are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernovae (type Ia) observational data [31][32][33].
Here we consider the cosmological solutions with diagonal metrics. They are governed by n = 3 + l > 5 scale factors which depend upon the synchronous time variable. We deal with solutions which have exponential dependence of scale factors. We present a class of such solutions with two scale factors, which correspond to factor spaces of dimensions 3 and l > 2, and are described by two Hubble-like parameters H > 0 and h, respectively. Here the total dimension is D = 1 + 3 + l . Any of these solutions is presented in parametrized form: the cosmological constant is finetuned, it depends upon the ratio h/H = x, l and a ratio two coupling constants. Any solution describes an exponential expansion of 3d factor space with Hubble parameter H > 0 [34].
Here we study the stability of the solutions in a class of cosmological solutions with diagonal metrics and single out a subclass of stable solutions. Our analysis is based on earlier results of Refs. [25,26] (see also the approach of Ref. [23]).
We also consider a subclass of solutions which correspond to a small enough variation of the effective gravitational constant G in the Jordan frame [35,36] (see also [37][38][39] and Refs. therein). We show that all these solutions are stable.

The setup
The action of the model has the following form where g = g M N dz M ⊗ dz N is a smooth metric defined on a smooth manifold M, dim M = D, |g| = | det(g M N )|, is the cosmological term, R[g] is scalar curvature and is the standard Gauss-Bonnet term. Here α 1 , α 2 are nonzero constants.
Our choice of the manifold is as follows We deal with the metric where B i > 0 are constants, i = 1, . . . , n, and M 1 , . . . , M n are one-dimensional manifolds (e.g. R or S 1 ) and n > 3, D = n + 1.
Equations of motion for the action (2.1) give us the set of polynomial equations [25] Here we use the notations from Refs. [17,18].
which are, respectively, the components of two metrics (2metric and 4-metric) on R n . For n > 3 we get a set of forth-order polynomial equations.
In what follows we deal with anisotropic solutions. The isotropic solutions with v 1 = · · · = v n = H , α < 0 (and n > 3) were considered in Refs. [17,18] and [20] for = 0 and = 0, respectively. As it was shown in Refs. [17,18] there are no more than three different numbers among v 1 , . . . , v n when = 0. This is valid also in the case = 0, when the additional restriction n i=1 v i = 0 is imposed [26].

Solutions with two Hubble-like parameters
In where H is the Hubble-like parameter corresponding to the 3dimensional factor space and h is the Hubble-like parameter corresponding to the l-dimensional factor space, l > 2.
We set for a description of an accelerated expansion of the 3dimensional subspace (which may describe our Universe). The evolution of the l-dimensional internal factor space is described by the Hubble-like parameter h. It is widely known that the 4d effective gravitational constant G = G e f f in the Brans-Dicke-Jordan (or simply Jordan) frame [35] (see also [36]) is proportional to the inverse volume scale factor of the internal space, see [37,39] and references therein.
It follows from Ref. [26] (for a more general scheme see [21]) that if we consider the ansatz (3.1) with two Hubblelike parameters H and h obeying two restrictions imposed we may reduce relations (2.4) and (2.5) to the following set of equations (3.5) Using Eq. (3.5) we get for l > 2 where and Due to restrictions (3.3) we have for x from (3.8) (3.14) From (3.11) we get , where x ± (l) are roots of the quadratic equation P(x, l) = 0. They obey the identities which imply the following inequalities It follows from (3.9) and (3.12) that and hence which does not depend upon l. In this case the Hubble-like parameters read and our ansatz (2.2), (2.3) gives us the product of (a part of) 4-dimensional de-Sitter space and l-dimensional Euclidean space. Let us consider the behaviour of the function λ(x, l) in the vicinity of the points x − (l) and x + (l). Here the following proposition is valid.
In the proof of the Proposition 1 the following lemma is used.
The Lemma is proved in the Appendix A.
The proof of Proposition 1 By using the relation P( for l > 2. Relation (3.28) just follows from (3.27) and (3.30). The Proposition 1 is proved.
Now we study the behaviour of the function λ(x, l) for fixed l and x = x ± (l). First, we find the extremum points which obey ∂ ∂ x λ(x, l) = 0. The calculations give us

32)
x = x ± (l). By using these relations we find the following extremum points x a ≡ 1, (3.33) We also obtain Using relations , and We also obtain for l > 2. Using these relations we find for l > 2 and (3.57) Now we analyze the behaviour of the function λ(x, l) with respect to x for fixed l > 2. We calculate n( , α), which is the number of solutions (in variable x) of the relation α = λ(x, l). In what follows we use relations First, we consider the case α > 0 and x − < x < x + . We keep in mind that the solution x = x d is excluded.
(A) 2 < l < 6. We get x b < x d < x c and λ d < λ c , λ d < λ b . Points x b , x c are points of local maximum and x d is a point of local minimum.
We split this case on two subcases: (A 0 ) l = 3 and (A − ) 3 < l < 6. (3.59) An example of the function λ(x) = (x)α for α > 0 and l = 4 is depicted at Fig. 2. (B) l > 6. We have x d < x b < x c and λ b < λ d < λ c . Points x c and x d are points of local maximum (x c is a point of maximum on interval (x − , x + )) and x b is a point of local (3.60) An example of the function λ(x) = (x)α for α > 0 and l = 12 is depicted at Fig. 3. (B 0 ) l = 6. We have x d = x b < x c and λ b = λ d < λ c . The point x c is a point of maximum on interval (x − , x + ) and x b = x d is a point of inflection. We obtain (3.61) An example of the function λ(x) = (x)α for α > 0 and l = 6 is depicted at Fig. 4. Thus, we see that for α > 0 and small enough value of there exist at least two solutions Now we consider the case α < 0. Here we remember that the solution x = x a is excluded. We get (x)|α| = −λ(x), where x < x − or x > x + . According to the identities (3.23), Eur. Phys. J. C (2018) 78 :100 for l > 2. The graphical representation of functions (x) = (x, α) = λ(x)/α for α = +1, −1, respectively, and l = 3 is given at Fig. 5.

Stability analysis
Here we outline main relations from [25,26], devoted to stability analysis of exponential solutions (2.3) with non-static volume factor [proportional to exp( n i=1 v i t)], which obey For a general cosmological diagonal metric where h i =β i , We set the following restriction on the matrix It was proved in [26] that a fixed point solution (h i (t)) = (v i ) (i = 1, . . . , n; n > 3) to Eqs.
and it is unstable (as t → +∞) if The set of equations for perturbations is presented in Appendix B. For our ansatz with K (v) = 3H + lh and H > 0 the restriction (4.9) is equivalent to the inequality while the restriction (4.10) is equivalent to another inequality It follows from Ref. [26] that for the vector v from (3.1), obeying relations (3.3), the matrix L has a block-diagonal form where L μν = G μν (2 + 4αS H H ), (4.14) and The matrix (4.13) is invertible (for l > 1) if and only if [26] S H H = − 1 2α , (4.18) Now, let us prove that inequalities (4.18), (4.19) are satisfied if and for l > 2. Let us suppose that (4.18) is not satisfied, i.e. S H H = − 1 2α . Then using (3.5) we get which implies due to H − h = 0: which is in contradiction with our restriction (4.20). This contradiction proves the inequality (4.18). Now we suppose that (4.19) is not valid, i.e. S hh = − 1 2α . Then using (3.5) we get which implies due to H − h = 0: which contradicts the restriction (4.21). The contradiction proves the inequality (4.19). Thus, we proved that relations (4.18) and (4.19) are valid. Hence the restriction (4.6) is satisfied for our solutions.
Thus we are led to the following proposition.

Proposition 2 The cosmological solutions under consider-
We note that for the anisotropic solutions under consideration the points x = x a and x = x d are excluded. Nevertheless, the solutions are defined for special extremal cases, The stability analysis of these special solutions can not be deduced just from the equations for perturbations (see relations (B.3), (B.4) from the Appendix) in the linear approximation. They need a special consideration [30].
We note that for the example l = 3 from Sect. 3 we get stable cosmological solutions for x > −1 and x = −1/2, x = 1.
Let us denote by n + ( , α) the number of non-special stable solutions which are given by Proposition 2 [see item (i)]. By using the results (e.g. figures) from the previous section we find for α > 0: (4.26) (4.27) (4.28) Thus, for α > 0 and small enough value of there exists at least one stable solution with x ∈ (x − , x + ).
For α < 0 we obtain (4.29) Here we use x = x a = 1. Thus, for α < 0 and big enough value of there exists at least one stable solution with x obeying x > x + . The solution with x < x − is unstable.

Solutions with small enough variation of G
The solutions under consideration may be analysed on a variation of the effective gravitational constant G, which is proportional (in the Jordan frame) to the inverse volume scale factor of the anisotropic internal space, i.e.
see [22,[37][38][39] and references therein. From (5.1) we get the following relation for a dimensionless parameter of temporal variation of G: We remind that H > 0 is the Hubble parameter. Due to experimental (or observational) data, the variation of the gravitational constant is allowed at the level of 10 −13 per year and less. In Ref. [22] the following constraint on the value of the dimensionless variation of the effective gravitational constant was used: It comes from the most stringent limitation on G-dot obtained in Ref. [40] (by the set of ephemerides) allowed at 95% confidence (2-σ ) level and the present value of the Hubble parameter [34] H 0 = (67.80 ± 1.54) km/s Mpc −1 = (6.929 ± 0, 157) · 10 −11 year −1 , (5.5) with 95% confidence level. For a given value of δ we get from (5.2) Our solutions are defined if x 0 (δ, l) = x ± (l), (5.7) or, equivalently, if The calculation of quadratic polynomial (3.7) for x = x 0 (δ, l) gives us (5.9) The inequality (5.8) is satisfied due to the bounds (5.3) for any l > 2. Hence, relation (5.7) is valid. Now we analyse the stability of the solutions with small enough variation of G. The main condition for the stability x 0 (δ, l) > x d is satisfied since due to our bounds (5.3).
Other three conditions (see Proposition 2) x 0 (δ, l) = x a , x 0 (δ, l) = x b and x 0 (δ, l) = x c give us which are satisfied due to bounds (5.3) and inequalities: δ a ≤ −3, δ b > 2 and δ c > 1 for l > 2. Thus, we have shown that all (well-defined) solutions under consideration obeying the bounds (5.3) (coming from the physical bounds on variation of G) are stable. We note that the proof of this fact is also valid for less restrictive bounds for δ than (5.3).
For fixed , α and l > 2 the equation (x, l, α) = is equivalent to a polynomial equation of either fourth or third order and hence may be solved in radicals.
Any of solutions describes an exponential expansion of 3d subspace (our space) with the Hubble parameter H > 0 and either contraction or expansion (with Hubble-like parameter h), or stabilization (h = 0) of l-dimensional internal subspace.
By using results of Ref. [26] we have proved that the cosmological solution (under consideration) is stable as t → +∞, if it obey the following restrictions: x > x d = − 3/l, x = x b = − 2 l−2 and x = x c = − 1 l−1 . Here the points x a , x b , x c , x d are points of extremum of the function λ(x, l) = α (x, l, α) for any l > 2.
We have also shown that all (well-defined) solutions with small enough variation of the effective gravitational constant G (in the Jordan frame), obeying physical bounds, are stable.