Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions: High-dimensional $\Lambda$-term case

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 {\it 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 $D\geqslant 4$ cases have slightly different dynamics due to the different structure of the equations of motion. We analytically study 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 constraint on the value of Gauss-Bonnet coupling $\alpha$ and the $\Lambda$-term -- this regime appears in two regions on $(\alpha, \Lambda)$ plane: $\alpha < 0$, $\Lambda > 0$, $\alpha\Lambda \leqslant 1/2$ and $\alpha > 0$, $\alpha\Lambda \leqslant (3D^2 - 7D + 6)/(4D(D-1))$, including 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 constraints ($\alpha$, $\Lambda$) even further: $\alpha > 0$, $D \geqslant 2$ with $(3D^2 - 7D + 6)/(4D(D-1)) \geqslant \alpha \Lambda \geqslant - (D+2)(D+3)(D^2 + 5D + 12)/(8(D^2 + 3D + 6)^2)$.