Effects of spatial curvature and anisotropy on the asymptotic regimes in Einstein–Gauss–Bonnet gravity

Abstract In this paper we address two important issues which could affect reaching the exponential and Kasner asymptotes in Einstein–Gauss–Bonnet cosmologies—spatial curvature and anisotropy in both three- and extra-dimensional subspaces. In the first part of the paper we consider the cosmological evolution of spaces that are the product of two isotropic and spatially curved subspaces. It is demonstrated that the dynamics in $$D=2$$ D=2 (the number of extra dimensions) and $$D \ge 3$$ D≥3 is different. It was already known that for the $$\Lambda $$ Λ -term case there is a regime with “stabilization” of extra dimensions, where the expansion rate of the three-dimensional subspace as well as the scale factor (the “size”) associated with extra dimensions reaches a constant value. This regime is achieved if the curvature of the extra dimensions is negative. We demonstrate that it takes place only if the number of extra dimensions is $$D \ge 3$$ D≥3 . In the second part of the paper we study the influence of the initial anisotropy. Our study reveals that the transition from Gauss–Bonnet Kasner regime to anisotropic exponential expansion (with three expanding and contracting extra dimensions) is stable with respect to breaking the symmetry within both three- and extra-dimensional subspaces. However, the details of the dynamics in $$D=2$$ D=2 and $$D \ge 3$$ D≥3 are different. Combining the two described effects allows us to construct a scenario in $$D \ge 3$$ D≥3 , where isotropization of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions