Cosmological dynamics of spatially flat Einstein\u2013Gauss\u2013Bonnet models in various dimensions: high-dimensional Lambda -term case

Abstract

In this paper we perform a systematic study of spatially flat [(3+D)+1]-dimensional Einstein–Gauss–Bonnet cosmological models with Lambda -term. We consider models that topologically are the product of two flat isotropic subspaces with different scale factors. One of these subspaces is three-dimensional 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, D=3 and the general Dge 4 cases have slightly different dynamics due to the different structure of the equations of motion. We analytically study the equations of motion in both cases and describe all possible regimes with special interest on the realistic regimes. Our analysis suggests that the only realistic regime is the transition from high-energy (Gauss–Bonnet) Kasner regime, which is the standard cosmological singularity in that case, to the anisotropic exponential regime with expanding three and contracting extra dimensions. Availability of this regime allows us to put a constraint on the value of Gauss–Bonnet coupling alpha and the Lambda -term – this regime appears in two regions on the (alpha , Lambda ) plane: alpha < 0, Lambda > 0, alpha Lambda le -3/2 and alpha > 0, alpha Lambda le (3D^2 - 7D + 6)/(4D(D-1)), including the entire Lambda < 0 region. The obtained bounds are confronted with the restrictions on alpha and Lambda from other considerations, like causality, entropy-to-viscosity ratio in AdS/CFT and others. Joint analysis constrains (alpha , Lambda ) even further: alpha > 0, D ge 2 with (3D^2 - 7D + 6)/(4D(D-1)) ge alpha Lambda ge - (D+2)(D+3)(D^2 + 5D + 12)/(8(D^2 + 3D + 6)^2).