On exponential cosmological type solutions in the model with Gauss-Bonnet term and variation of gravitational constant

A D-dimensional gravitational model with Gauss-Bonnet term is considered. When ansatz with diagonal cosmological type metrics is adopted, we find solutions with exponential dependence of scale factors (with respect to"synchronous-like"variable) which describe an exponential expansion of"our"3-dimensional factor-space and obey the observational constraints on the temporal variation of effective gravitational constant G. Among them there are two exact solutions in dimensions D = 22, 28 with constant G and also an infinite series of solutions in dimensions D \ge 2690 with the variation of G obeying the observational data.


Introduction
Here we deal with D-dimensional gravitational model with the Gauss-Bonnet term. The action reads 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 standard Gauss-Bonnet term. Here α 1 and α 2 are non-zero constants. Earlier the appearance of the Gauss-Bonnet term was motivated by string theory [1,2,3,4,5]. At present, the (so-called) Einstein-Gauss-Bonnet (EGB) gravitational model and its modifications are intensively used in cosmology, see [6,7] (for D = 4), [8,9,10,11,12,13,14,15,17] and references therein, e.g. for explanation of accelerating expansion of the Universe following from supernovae (type Ia) observational data [18,19,20]. Certain exact solutions in multidimesional EGB cosmology were obtained in [8]- [17] and some other papers.
Here we are dealing with the cosmological type solutions with diagonal metrics (of Bianchi-I-like type) governed by n scale factors depending upon one variable, where n > 3. Moreover, we restrict ourselves by the solutions with exponential dependence of scale factors (with respect to "synchronous-like" variable τ ) a i (τ ) ∼ exp (h i τ ), (1.3) i = 1, . . . , n; D = n + 1, with the aim to find solutions describing an exponential isotropic expansion of 3-dimensional flat factor-space, i.e. with h 1 = h 2 = h 3 = H > 0, (1.4) and small enough variation of the effective gravitational constant G, which is proportional to the inverse volume scale factor of the internal space, i.e. see [21,22,23,24,25] and references therein. We call h i =ȧ i /a i as "Hubble-like" parameter corresponding to i-th subspace.
In cosmological model under consideration with anisotropic "internal space", we get for dimensionless parameter of temporal variation of G the following relation from (1.5) and (1.6 where H is the Hubble parameter.
As for the experimental data, the variation of the gravitational constant is allowed at the level of 10 −13 per year and less. We use the following constraint on the magnitude of the dimensionless variation of the gravitational constant which comes from the most stringent limitation on G-dot obtained by the set of ephemerides [26]Ġ allowed at 95% confidence (2-σ) level and the present value of the Hubble parameter [27] (which characterizes the rate of expansion of the observable Universe) H 0 = (67, 80 ± 1, 54) km/s M pc −1 = (6.929 ± 0, 157) · 10 −11 year −1 , (1.10) with 95% confidence level. It shold be noted that the original result for H 0 in [27] (for the Planck best-fit cosmology including external data set) was presented at 68% confidence (1-σ) level. In restriction (1.8) we use the lower allowed value for H 0 in (1.10) in order to obtain the confidence level more than 95% . Thus, we are seeking here the cosmological solutions which obey the relations (1.3)-(1.8), listed above.
The paper is organized as follows. In Section 2 the equations of motion for Ddimensional EGB model are considered. For diagonal cosmological type metrics the equations of motion are equivalent to a set of Lagrange equations corresponding to certain "effective" Lagrangian [29,30] (see also [9,28]). In Section 3 some cosmological solutions with exponential behavior of scale factors satisfying the restriction (1.8) are obtained for two isotropic factor spaces and positive value of α = α 2 /α 1 .
2 The cosmological type model and its effective Lagrangian
For physical applications we put M 1 = M 2 = M 3 = R, while M 4 , ..., M n will be considered to be compact ones (i.e. coinciding with S 1 ).
The integrand in (1.1), when the metric (2.2) is substituted, reads as follows and are respectively the components of two metrics on R n [29,30]. The first one is the wellknown "minisupermetric" -2-metric of pseudo-Euclidean signature: and the second one is the Finslerian 4-metric: where < ., . > and < ., ., ., . > are respectively 2-and 4-linear symmetric forms on R n . Here we denoteȦ = dA/du etc. The function f (u) in (2.3) is irrelevant for our consideration (see [29,30]).

The equations of motion
The equations of motion corresponding to the action (1.1) have the following form (2. 16) It was shown in [30] that the field equations (2.14) for the metric (2.2) are equivalent to the Lagrange equations corresponding to the Lagrangian L from (2.4).

Reduction to an autonomous system of first order differential equations
Now we put γ = 0 and denote u = τ , where τ is a "synchronous-like" variable. By introducing "Hubble-like" variables h i =β i , the eqs. (2.17) and (2.18) may be rewritten as follows Thus, we are led to the autonomous system of the first-order differential equations on h 1 (τ ), ..., h n (τ ) [30,30].
Here and in what follows we use relations (2.10), (2.11) and the following formulas

Solutions with constant h i
In this paper we deal with the following solutions to equations (2.19) and (2.20) 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 For any non-trivial solution v we have n i=1 v i = 0 (otherwise one gets from (2.28) For n = 1: H is arbitrary and for n = 2, 3: H = 0. When n > 3, the non-zero solution to eq. (2.29) exists only if α < 0 and in this case [29,30] In cosmological case (w = −1) this solution takes place when α 2 /α 1 = α < 0. The isotropic solution for n > 3 gives rise to a very large value ofĠ/G = (n − 3)H, which is forbidden by observational restrictions.
It was shown in [29,30] that there are no more than three different numbers among v 1 , ..., v n .
The first inequality H > 0 in the mandatory condition is necessary for a description of accelerated expansion of 3-dimensional subspace, which may describe our Universe, while the second inequality h < 0 excludes an enormous (of Hubble's order) variationĠ/G for h ≥ 0 and m > 3.
The first desirable condition means that the volume scale factor of the internal space This condition is a sort of weak extension of a possible restriction for m = 3 coming from the unobservability of the "internal space" for all τ > τ 0 . It is also desirable since the negative value of the parameter Int is more probable due to more probable positive value ofĠ/G = −Int, see (1.9).
The second desirable condition may be also rewritten by using the parameter V ar = Here we consider the simplest case, when internal spaces (apart the expansion factors) are flat. The consideration of curved internal spaces will drastically change the equations of motion and may break the existence of solutions with exponential dependence of scale factors. Anyway the inclusion into consideration of curved internal spaces may be worth but need a special treatment, which may be done in a separate work.

The dependence of "Hubble-like" parameters on m and l
The total dimension of the considered space is D = n + 1 = (m + l) + 1, where we have m dimensions expanding with the Hubble parameter H > 0 and l dimensions contracting with the "Hubble-like" parameter h < 0.
According to this, we rewrite the set of polynomial equations (2.27), (2.28), using the formulas (2.22), (2.23), as follows: Here we put for simplicity α = ±1 but keep in mind that general α-dependent solution has the following form Due to these relations the parameterĠ/(GH) does not depend upon |α| and hence our simplification is a reasonable one. For any solution (H, h) with α = ±1 we can find a proper α which will be in agreement with the present value of the Hubble parameter H 0 (see (1.10)) Our numerical analysis (based on Maplesoft Maple) show us that (generically) there are 11 solutions of these equations (for m > 3 and l ≥ 3).
I) The first to mention is, obviously, the zero solution H 1 = h 1 = 0. II) Two other solutions are isotropic ones: We are led to pure imaginary isotropic solutions obeying (2.29). For example, when m = 9 and l = 6 we receive . We are led to isotropic solutions (2.30). When m = 9 and l = 6 the solutions are: III) For α = ±1 the remaining eight solutions are roots of the following two equations which are given by Maple:  It can be seen that none of the solutions for α = −1 satisfies our mandatory conditions written in the beginning of this section.
As for α = 1, the solutions III.1.b and III.1.d are real and H > 0, h < 0. It can be verified that in these cases Int = (m − 3)H + lh < 0. Now we have to calculate the variation of the gravitational constant. For m = 9 and l = 6 we get: The first variation is lower: V ar 5 < V ar 7 , for m = 9 and l = 6. This inequality seems to take place for any m > 3 and l ≥ 3. At the moment a rigorous proof of this fact is absent while certain numerical calculations support it. Anyway, here we will focus on the solution (H 5 , h 5 ), which we consider as a more interesting (for our applications) then (H 7 , h 7 ). Further we will write H and h instead of H 5 and h 5 in common case.
We can plot behaviour of parameters H, h, Int and V ar, for example, keeping fixed m = 8 and m = 10 and raising l from 5 to 100 by 5. See Figure 1 and Figure 2.

The limiting values of H, hl, Int and V ar for fixed m ≤ 9
When m ≤ 9 the internal space parameter Int remains negative, that means that the first desirable condition is satisfied for any l. The variation of G parameter is monotonically decreasing with the increase of l. Moreover, we get finite limits for H and hl as l → +∞. In this subsection we obtain these and other limits (for Int and V ar) for fixed m ≤ 9. Now let us rewrite (3.8) and (3.9) for H and hl keeping only the terms with higher degrees of l: Solving these equations we find the limiting values: (3.14) We fill the Table 1

Infinite series of solutions for m = 9
Now we consider the case m = 9. We get the following relations. We do not present here exact analytical forms of these solutions in radicals which are bulky ones. For example, the relation for the parameter H, when l = 2680, contains (17 times) the radical √ 32839778319264444823234828568184005. The numerical calculations for fixed m = 9 gives us an evidence of monotonically decreasing behaviour of the function V ar(l) for l ≥ 2680 5 as well as the asymptotical relation: V ar(l) ∼ A/l, as l → +∞, where A > 0. See Figure 3.
Thus, for m = 9 there is an infinite series of admissible cosmological solutions with l = 2680, 2681, ..., which satisfy all the conditions imposed. Any of such solution describes accelerated expansion of the three-dimensional factor space with sufficiently small value of the variation of the effective gravitational constant G. This variation may be arbitrary small for a big enough value of l.  1. for m = 10: the variation of G parameter is minimal for l = 31, see Figure 4, This case is not of particular interest. The radical forms of the solutions are too bulky, so we approximated them. Nevertheless the variation of G parameter is out of the allowed domain and the "internal space" parameter is positive. Figure 4: The variation of G parameter for m = 10.
2. for m = 11: the variation of G and the "internal space" parameters are zero for l = 16, see Figure  5, This case is the first one with zero variation of G. Also, the exact values of "Hubble-like" parameters (H = −2h) in contrast to the previous case have rather simple and compact form. 3. for m = 12: the variation of G parameter is minimal for l = 11, see Figure 6, Here all our four conditions are satisfied. The variation of G parameter is non-zero and the volume of the internal space is decreasing. Figure 6: The variation of G parameter for m = 12.
4. for m = 13: the variation of G parameter is minimal for l = 9 and the "internal space" parameter is positive, see Figure 7, Int ≈ 0.0048967, Var ≈ 0.0240058, for l = 8 the variation of G parameter is slightly higher, but the "internal space" parameter is negative, see Figure 8,  the variation of G parameter is minimal for l = 7, see Figure 8, The variation of G parameter exceeds our limits, and the condition of the volume contraction of the "inner space" is not met. Figure 8: The variation of G parameter for m = 14.
6. for m = 15: the variation of G and the "internal space" parameters are zero for l = 6, see Figure 9, This is the second case with zero variation of G. The exact values of "Hubble-like" parameters (H = − 1 2 h) have simple and compact form. Figure 9: The variation of G parameter for m = 15. Now we will reverse our method and look for the solutions with minimal variation of G for fixed l instead of m. The calculations lead to the following rule: the lesser is l the greater is an appropriate m which gives the minimum of V ar . As we consider l ≥ 3 and for l = 6 the solution with minimal variation of G parameter is already found, we should examine only three cases: 1. for l = 5: the variation of G parameter is minimal for m = 17, see Figure 10, None of the desirable conditions are satisfied. -the variation of G parameter is minimal for m = 20, see Figure 11, The amount of variation is too high and the "internal space" parameter is positive. 3. for l = 3: the variation of G parameter is minimal for m = 28, see Figure 12, The "internal space" parameter is also positive and the variation of G parameter exceeds the limits imposed. Figure 12: The variation of G parameter for l = 3.
It should be noted that for m = 3 and l = 2 the solution with H ≈ 0, 750173 and h ≈ −0, 541715 was found earlier in [16]. For this solution we have a contracting "internal space" but the variation of G is a huge one (Ġ/G is of Hubble parameter order). Recently, an exact analytic form of this solution was obtained in [17].

Conclusions
We have considered the (n + 1)-dimensional Einstein-Gauss-Bonnet (EGB) model. By using the ansatz with diagonal cosmological type metrics, we have found solutions with exponential dependence of scale factors with respect to "synchronous-like" variable τ .
In cosmological case (w = −1) these solutions describe an exponential expansion of "our" 3-dimensional factor-space with the Hubble parameter H > 0 and obey the observational constraints on the temporal variation of the effective gravitational constant G. Any solution describes (m − 3 + l)-dimensional "internal space", which is anisotropic: it is expanding in (m − 3) dimensions with the Hubble rate H > 0 and contracting in l dimensions.
Thus, we have shown that it is possible in the framework of EGB model to describe the accelerated expansion of the three-dimensional factor space with sufficiently small (or even zero) value of the variation of the effective gravitational constant G. For the case w = 1 we have obtained by product a family of static configurations which may be of interset within some other possible applications.
Here we have considered a gravitational model in more than 4-dimensions. In such a case the Gauss-Bonnet term gives non-trivial contributions to the generalized Einstein field equations. In particular we have shown that there are cosmological solutions in agreement with observations when "projected on the (3 + 1)-dimensional physical space-time. For the sake of simplicity, we restrict ourselves to vacuum solutions in multidimensional gravity with the Gauss-Bonnet term . Such ansatz may be considered as a part of a general "geometrical program aimed at the explanation of dark energy in 4-dimensional space, e.g. by using extra dimensions and modified equation of motions just without matter sources. This is a first step. The inclusion of matter sources (e.g. anisotropic fluid) will be the next step, e.g. as a subject of a next publication.