On Stability of Exponential Cosmological Type Solutions with Two Factor Spaces in the Einstein–Gauss–Bonnet Model with a $\boldsymbol{\Lambda}$-Term

Abstract

We study a $D$-dimensional Einstein–Gauss–Bonnet gravitational model including the Gauss-Bonnet term and the cosmological constant $\Lambda$. We find a class of cosmological type solutions with exponential dependence of two scale factors on the variable $u$ (either cosmological time or a spatial coordinate), governed by two Hubble-like parameters $H\neq 0$ and $h$, corresponding to factor spaces of dimensions $m>2$ and $l>2$, respectively, and depending on the sign parameter $\varepsilon=\pm 1$ ($\varepsilon=1$ corresponds to cosmological solutions and $\varepsilon=-1$ to static ones). These solutions are governed by a certain master equation $\Lambda\alpha=\lambda(x)$ and the restriction $\alpha\varepsilon(x-x_{+})(x-x_{-})<0$ ($x_{-}<x_{+}<0$) for the ratio $h/H=x$, where $\alpha=\alpha_{2}/\alpha_{1}$ is the ratio of two constants of the model . The master equation is equivalent to a polynomial equation of either fourth or third order and may be solved in radicals. Imposing certain restrictions on $x$, and we prove the stability of the solutions for $u\to\pm\infty$ in a certain class of cosmological solutions with diagonal metrics.