Exponential cosmological solutions with two factor spaces in EGB model with $\Lambda = 0$ revisited

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 $h \neq H$ and $m H + l h \neq 0$. 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 $(m, l) = (9, l > 2), (12, 11), (11,16), (15, 6)$.