Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: Vacuum case

In this paper we perform a systematic study of vacuum spatially flat anisotropic [(3 + D) + 1]-dimensional Einstein-Gauss-Bonnet cosmological models. We consider models that topologically are the product of two flat isotropic submanifolds with different scale factors. One of these submanifolds is three dimensional and represents our 3D space and the other is D dimensional and represents extra dimensions. We consider no Ansatz on the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play and with the symmetry involved, the cases with D = 1, D = 2, D = 3, and D >= 4 have different dynamics due to the different structures of the equations of motion. We analytically analyze equations of motion in all cases and describe all possible regimes. It appears that the only regimes with nonsingular future asymptotes are the Kasner regime in general relativity and exponential regimes. As of the past asymptotes, for a smooth transition only the Kasner regime in Gauss-Bonnet is an option. With this at hand, we are down to only two viable regimes: the "pure" Kasner regime [transition from a high-energy (Gauss-Bonnet) to a low-energy (general relativity) Kasner regime] and a transition from a high-energy Kasner regime to an anisotropic exponential solution. It appears that these regimes take place for different signs of the Gauss-Bonnet coupling a: the "pure" Kasner regime occurs for alpha > 0 at low D and alpha < 0 for high D; the anisotropic exponential regime is reached only for alpha > 0. So if we restrain ourselves with alpha > 0 solutions (which would be the case, say, if we identify alpha with inverse string tension in heterotic string theory), the only late-time regimes are Kasner for D = 1, 2 and anisotropic exponential for D >= 2. Also, low-energy Kasner regimes [a(t) proportional to t(p)] have expansion rates for (3 + 1)-dimensional subspace ("our Universe") ranging from p = 0.5 (D = 1) to p = 1/root 3 approximate to 0.577 (D -> infinity), which contradicts the dust-dominated Friedmann prediction (p = 2/3).