Abstract
We study exact cosmological solutions in D-dimensional Einstein–Gauss–Bonnet model (with zero cosmological term) governed by two non-zero constants: alpha _1 and alpha _2 . We deal with exponential dependence (in time) of two scale factors governed by Hubble-like parameters H >0 and h, which correspond to factor spaces of dimensions m >2 and l > 2, respectively, and D = 1 + m + l. We put h ne H and mH + l h ne 0. We show that for alpha = alpha _2/alpha _1 > 0 there are two (real) solutions with two sets of Hubble-like parameters: (H_1, h_1) and (H_2, h_2), which obey: h_1/ H_1< - m/l< h_2/ H_2 < 0, while for alpha < 0 the (real) solutions are absent. We prove that the cosmological solution corresponding to (H_2, h_2) is stable in a class of cosmological solutions with diagonal metrics, while the solution corresponding to (H_1, h_1) is unstable. We present several examples of analytical solutions, e.g. stable ones with small enough variation of the effective gravitational constant G, for (m, l) = (9, l >2), (12, 11), (11,16), (15, 6).
Highlights
In contrary to our earlier publication [8], where a lot of numerical solutions with small enough value of variation of the effective gravitational constant G were found, here we put our attention mainly to the search of analytical exponential solutions with two factor spaces of dimensions m and l
By using results of Refs. [10,11] we show that the solutions with Hubble-like parameters (H2, h2) are stable, while those corresponding to (H1, h1) are unstable
By using the ansatz with diagonal cosmological metrics, we have studied a class of solutions with exponential time dependence of two scale factors, governed by two Hubblelike parameters H > 0 and h, corresponding to submanifolds of dimensions m > 2 and l > 2, respectively, with D = 1 + m + l
Summary
In contrary to our earlier publication [8], where a lot of numerical solutions with small enough value of variation of the effective gravitational constant G were found, here we put our attention mainly to the search of analytical exponential solutions with two factor spaces of dimensions m and l. We show that the anisotropic cosmological solutions under consideration with two Hubble-like parameters H > 0 and h obeying restrictions h = H , m H + lh = 0 do exist only if α = α2/α1 > 0. In this case we have two solutions with Hubble-like parameters: (H1 > 0, h1 < 0) and (H2 > 0, h2 < 0), respectively, such that x1 = h1/H1 < −m/l < x2 = h2/H2. It should be noted that analytical solutions in cases (iii) and (iv) were considered numerically in Ref. The stable solutions with zero variation of G in cases (v) and (vi) were found earlier in [8], while the stability of these solutions was proved in Ref. [10]
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