Stable exponential cosmological type solutions with three factor spaces in EGB model with a $\Lambda$-term

TL;DR

This paper finds stable and unstable exponential cosmological solutions with three factor spaces in a higher-dimensional Einstein-Gauss-Bonnet model including a cosmological term, analyzing conditions for their existence and explicit solutions in specific cases.

Contribution

It introduces new exponential solutions with three factor spaces in an EGB model with a $ ext{Lambda}$-term, including stability analysis and explicit solutions for certain parameter cases.

Findings

Solutions exist under specific sign and bound conditions on parameters.

Explicit solutions are provided for the case where two factor spaces are equal.

Stability of solutions varies with the parameters and solution type.

Abstract

We study a $D$-dimensional Einstein-Gauss-Bonnet model which includes the Gauss-Bonnet term, the cosmological term $\Lambda$ and two non-zero constants: $\alpha_1$ and $\alpha_2$. Under imposing the metric to be diagonal one, we find cosmological type solutions with exponential dependence of three scale factors in a variable $u$, governed by three non-coinciding Hubble-like parameters: $H \neq 0$, $h_1$ and $h_2$, obeying $m H + k_1 h_1 + k_2 h_2 \neq 0$, corresponding to factor spaces of dimensions $m > 1$, $k_1 > 1$ and $k_2 > 1$, respectively, and depending upon sign parameter $\varepsilon = \pm 1$, where $\varepsilon = 1$ corresponds to cosmological case and $\varepsilon = - 1$ - to static one). We deal with two cases: i) $m < k_1 < k_2$ and ii) $1< k_1 = k_2 = k$, $k \neq m$. We show that in both cases the solutions exist if $\varepsilon \alpha = \varepsilon \alpha_2 / \alpha_1 > 0$ and $\alpha \Lambda > 0$ satisfies certain (upper and lower) bounds. The solutions are defined up to solutions of certain polynomial master equation of order four (or less) which may be solved in radicals. In case ii) explicit solutions are presented. In both cases we single out stable and non-stable solutions as $u \to \pm \infty$. The case $H = 0$ is also considered.