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

TL;DR

This paper systematically analyzes vacuum flat Einstein-Gauss-Bonnet cosmological models across various dimensions, identifying key regimes and their dependence on the Gauss-Bonnet coupling sign, with implications for universe expansion rates.

Contribution

It provides a comprehensive analytical study of the dynamics of higher-dimensional Einstein-Gauss-Bonnet models without scale factor ansatz, revealing distinct regimes and their dependence on model parameters.

Findings

Only Kasner and exponential regimes have nonsingular future asymptotes.

Transition from high- to low-energy Kasner regimes is possible.

Expansion rates in our universe differ from dust-dominated Friedmann predictions.

Abstract

In this paper we perform a systematic study of vacuum spatially flat ((3+D)+1)-dimensional Einstein-Gauss-Bonnet cosmological models. We consider models which topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is 3D and represents our space and the other is D-dimensional and represents extra dimensions. We consider no ansatz of the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play, the cases with D=1, D=2, D=3 and $D\geqslant 4$ have different dynamics due to different structure of the equations of motion. We analytically study 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 GR as well as exponential regimes. As of the past asymptotes, for a smooth transition only Kasner regime in Gauss-Bonnet is an option. With that at hand, we are down only to two viable regimes -- "pure" Kasner regime (transition from high- to low-energy Kasner regime) and a transition from high-energy Kasner to anisotropic exponential solution. It appears that these regimes take place for different signs of the Gauss-Bonnet coupling $\alpha$: "pure" Kasner regime occur for $\alpha > 0$ at low D and $\alpha < 0$ for high D; anisotropic exponential regime is reached only for $\alpha > 0$. So if we restrain ourselves with $\alpha > 0$ solutions, the only late-time regimes are Kasner for D=1, 2 and anisotropic exponential for $D\geqslant 2$. Also, low-energy Kasner regimes ($a(t)\propto t^p$) have expansion rates for (3+1)-dimensional subspace ("our Universe") ranging from p=0.5 (D=1) to $p=1/\sqrt{3} \approx 0.577$ ($D\to\infty$), which contradicts with dust-dominated Friedmann prediction (p=2/3).