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

A D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D$$\end{document}-dimensional gravitational model with Gauss–Bonnet term is considered. When an ansatz with diagonal cosmological type metrics is adopted, we find solutions with an exponential dependence of the scale factors (with respect to a “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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}. Among them there are two exact solutions in dimensions D=22,28\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D = 22, 28$$\end{document} with constant G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} and also an infinite series of solutions in dimensions D≥2690\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D \ge 2690$$\end{document} with the variation of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document} obeying the observational data.

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 to the solutions with an exponential dependence of scale factors (with respect to the "synchronous-like" variable τ ) see [21][22][23][24][25] and references therein. We call h i =ȧ i /a i the "Hubble-like" parameter corresponding to the ith subspace. In the cosmological model under consideration with anisotropic "internal space", we get for the dimensionless parameter of the 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:  [27] (for the Planck best-fit cosmology including an external data set) was presented at 68 % confidence (1-σ ) level. In the restriction (1.8) we use the lower allowed value for H 0 in (1.10) in order to obtain the confidence level of more than 95 %. Thus, we are seeking here the cosmological solutions which obey (1.3)-(1.8), listed above.
The paper is organized as follows. In Sect. 2 the equations of motion for the D-dimensional EGB model are considered. For diagonal cosmological type metrics the equations of motion are equivalent to a set of Lagrange equations corresponding to a certain "effective" Lagrangian [29,30] (see also [9,28]). In Sect. 3 some cosmological solutions with an exponential behavior of the scale factors satisfying the restriction (1.8) are obtained for two isotropic factor spaces and a positive value of α = α 2 /α 1 .
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 the 2-metrics on R n [29,30]. The first one is the well-known "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 = d A/du etc. The function f (u) in (2.3) is irrelevant for our considerations (see [29,30]).

The equations of motion
The equations of motion corresponding to the action (1.1) have the following form: M N = 0, (2.14) where 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).
Here and in what follows we use Eqs. (2.10), (2.11), and the following formulas:

Solutions with constant h i
In this paper we deal with the following solutions to (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 the scale factors where w = ±1, ε i = ±1, and B i > 0 are arbitrary constants. For the fixed point v = (v i ) we have the set of polynomial equations The set of equations (2.27) and (2.28) has an isotropic solution v 1 = · · · = v n = H , where (2.29) For n = 1: H is arbitrary and for n = 2, 3: H = 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 . 2. desirable: 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 the order of the Hubble parameter) variationĠ/G for h ≥ 0 and m > 3.
The first desirable condition means that the volume scale factor of the internal space where B > 0 is constant, decreases over time. 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 the more probable positive value ofĠ/G = −I nt; see (1.9).
The second desirable condition may also be rewritten by using the parameter Var = |Ġ G H | = |(m − 3) + lh H |: Here we consider the simplest case, when the internal spaces (apart from 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 an exponential dependence of the scale factors. Anyway, the inclusion into our consideration of curved internal spaces may be worthwhile, but it needs a special treatment, which may be given in a separate work. (3.5) 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/(G H) 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) shows 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 obtain (a) We are led to isotropic solutions (2.   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 Now we have to calculate the variation of the gravitational constant. For m = 9 and l = 6 we get The first variation is lower: Var 5 < Var 7 , for m = 9 and l = 6. This inequality seems to occur 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 more interesting (for our applications) than (H 7 , h 7 ). Further we will write H and h instead of H 5 and h 5 in common cases.
We can plot the behavior of the parameters H , h, Int, and Var, for example, keeping fixed m = 8 and m = 10 and raising l from 5 to 100 by 5. See Figs. 1 and 2 When m ≤ 9 the internal space parameter Int remains negative, which means that the first desirable condition is satisfied for any l. The variation of the 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 Var) 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)  The numerical calculations for fixed m = 9 gives evidence of the monotonically decreasing behavior of the function Var(l) for l ≥ 2680 3 as well as the asymptotical relation: Var(l) ∼ A/l, as l → +∞, where A > 0. See Fig. 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 such solution describes an accelerated expansion of the 3-dimensional factor space with sufficiently small value of the variation of the effective gravitational constant G. This variation may be arbitrarily small for a big enough value of l.
The infinite series of solutions for m = 9 and l = 2680, 2681, . . . starts from the (special) total dimension D = 2690. For D < 2690 and m = 9 the solutions do not obey restriction (1.8) on the variation of G and hence are not of interest for our consideration.

Some solutions for m > 9 with minimal Var-parameter
When m > 9 the internal space parameter Int becomes positive. As l can only be a natural number we should look for a value of l, which gives the minimal magnitude of the variation of G parameter Var. Below we present the calculated This case is not of particular interest. The radical forms of the solutions are too bulky, so we approximated them. Nevertheless the variation of the G parameter is out of the allowed domain and the "internal space" parameter is positive.
This case is the first one with zero variation of G. Also, the exact values of the "Hubble-like" parameters (H = −2h) in contrast to the previous case have rather simple and compact forms. For l = 8 the variation of the G parameter is slightly higher, but the "internal space" parameter is negative, see Fig. 8.   The variation of the G parameter exceeds our limits, and the condition of the volume contraction of the "inner space" is not met.
6. For m = 15: the variation of the G and the "internal space" parameters are zero for l = 6, see Fig. 9, This is the second case with a zero variation of G.
The exact values of the "Hubble-like" parameters (H = − 1 2 h) have simple and compact forms. Now we will reverse our method and look for the solutions with minimal variation of G for fixed l instead of m. The Fig. 9 The variation of the G parameter for m = 15 Fig. 10 The variation of the G parameter for l = 5 calculations lead to the following rule: the lesser is l the greater is an appropriate m which gives the minimum of Var. As we consider l ≥ 3 and for l = 6 the solution with minimal variation of the G parameter is already found, and we should examine only three cases.
1. For l = 5: the variation of the G parameter is minimal for m = 17, see Fig. 10, None of the desirable conditions are satisfied.
2. For l = 4: the variation of the G parameter is minimal for m = 20, see Fig. 11, Fig. 11 Variation of G for l = 4 Fig. 12 The variation of the G parameter for l = 3 Int ≈ 0.004405301, The amount of variation is too high and the "internal space" parameter is positive.
3. For l = 3: the variation of the G parameter is minimal for m = 28, see Fig. 12, The "internal space" parameter is also positive and the variation of the G parameter exceeds the limits imposed.
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 the order of the Hubble parameter). Recently, an exact analytic form of this solution was obtained in [17].

Conclusions
We have considered the (n +1)-dimensional EGB model. By using the ansatz with diagonal cosmological type metrics, we have found solutions with an exponential dependence of the scale factors with respect to the "synchronous-like" variable τ .
In the 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 an (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.
These solutions were found (in numerical or analytical forms) for the following cases: Thus, we have shown that it is possible in the framework of the EGB model to describe the accelerated expansion of the 3-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 as a byproduct a family of static configurations which may be of interest 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 multi-dimensional gravity with the Gauss-Bonnet term. Such an 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 equations of motion just without matter sources. This is a first step. The inclusion of matter sources (e.g. an anisotropic fluid) will be the next step, as a subject of a subsequent publication.