Realistic compactification in spatially flat vacuum cosmological models in cubic Lovelock gravity: High-dimensional case

We investigate possible regimes in spatially flat vacuum cosmological models in cubic Lovelock gravity. The spatial section is a product of three- and extra-dimensional isotropic subspaces. This is the second paper of the series and we consider D=5 and general D>=6 cases here. For each D case we found critical values for $\alpha$ (Gauss-Bonnet coupling) and $\beta$ (cubic Lovelock coupling) which separate different dynamical cases and study the dynamics in each region to find all regimes for all initial conditions and for arbitrary values of $\alpha$ and $\beta$. The results suggest that for D>=3 there are regimes with realistic compactification originating from `generalized Taub' solution. The endpoint of the compactification regimes is either anisotropic exponential solution (for $\alpha > 0$, $\mu \equiv \beta/\alpha^2 < \mu_1$ (including entire $\beta < 0$)) or standard Kasner regime (for $\alpha > 0$, $\mu > \mu_1$). For D>=8 there is additional regime which originates from high-energy (cubic Lovelock) Kasner regime and ends as anisotropic exponential solution. It exists in two domains: $\alpha > 0$, $\beta < 0$, $\mu \leqslant \mu_4$ and entire $\alpha > 0$, $\beta > 0$. Let us note that for D>=8 and $\alpha > 0$, $\beta < 0$, $\mu < \mu_4$ there are two realistic compactification regimes which exist at the same time and have two different anisotropic exponential solutions as a future asymptotes. For D>=8 and $\alpha > 0$, $\beta > 0$, $\mu < \mu_2$ there are two realistic compactification regimes but they lead to the same anisotropic exponential solution. This behavior is quite different from the Einstein-Gauss-Bonnet case. There are two more unexpected observations among the results -- all realistic compactification regimes exist only for $\alpha > 0$ and there is no smooth transition from high-energy Kasner regime to low-energy one with realistic compactification.Comment: 34 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1804.0693

Dynamics of the cosmological models with perfect fluid in Einstein-Gauss-Bonnet gravity: low-dimensional case

In this paper we performed investigation of the spatially-flat cosmological models whose spatial section is product of three- ("our Universe") and extra-dimensional parts. The matter source chosen to be the perfect fluid which exists in the entire space. We described all physically sensible cases for the entire range of possible initial conditions and parameters as well as brought the connections with vacuum and $\Lambda$-term regimes described earlier. In the present paper we limit ourselves with $D=1, 2$ (number of extra dimensions). The results suggest that in $D=1$ there are no realistic compactification regimes while in $D=2$ there is if $\alpha > 0$ (the Gauss-Bonnet coupling) and the equation of state $\omega < 1/3$, the measure of the initial conditions leading to this regime is increasing with growth of $\omega$ and reaches its maximum at $\omega \to 1/3 - 0$. We also describe some pecularities of the model, distinct to the vacuum and $\Lambda$-term cases -- existence of the isotropic power-law regime, different role of the constant-volume solution and the presence of the maximal density for $D = 2$, $\alpha < 0$ subcase and associated features.Comment: 35 pages, 9 figure

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

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 cosmological evolution of spaces being the product of two isotropic and spatially curved subspaces. It is demonstrated that the dynamics in $D=2$ (the number of extra dimensions) and $D \geqslant 3$ is different. It was already known that for the $\Lambda$-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 reach constant value. This regime is achieved if the curvature of the extra dimensions is negative. We demonstrate that it take place only if the number of extra dimensions is $D \geqslant 3$. In the second part of the paper we study the influence of initial anisotropy. Our study reveals that the transition from Gauss-Bonnet Kasner regime to anisotropic exponential expansion (with expanding three 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 $D=2$ and $D \geqslant 3$ are different. Combining the two described affects allows us to construct a scenario in $D \geqslant 3$, where isotropisation of outer and inner subspaces is reached dynamically from rather general anisotropic initial conditions.Comment: 22 pages, 3 figure

Realistic compactification in spatially flat vacuum cosmological models in cubic Lovelock gravity: Low-dimensional case

In this paper we begin to perform systematical investigation of all possible regimes in spatially flat vacuum cosmological models in cubic Lovelock gravity. We consider the spatial section to be a product of three- and extra-dimensional isotropic subspaces, with the former considered to be our Universe. As the equations of motion are different for $D=3, 4, 5$ and general $D \geqslant 6$ cases, we considered them all separately. Due to the quite large amount different subcases, in the current paper we consider only $D=3, 4$ cases. For each $D$ case we found values for $\alpha$ (Gauss-Bonnet coupling) and $\beta$ (cubic Lovelock coupling) which separate different dynamical cases, all isotropic and anisotropic exponential solutions, and study the dynamics in each region to find all possible regimes for all possible initial conditions and any values of $\alpha$ and $\beta$. The results suggest that in both $D$ cases the regimes with realistic compactification originate from so-called "generalized Taub" solution. The endpoint of the compactification regimes is either anisotropic exponential (for $\alpha > 0$, $\mu \equiv \beta/\alpha^2 < \mu_1$ (including entire $\beta < 0$)) or standard low-energy Kasner regime (for $\alpha > 0$, $\mu > \mu_1$), as it is compactification regime, both endpoints have expanding three and contracting extra dimensions. There are two unexpected observations among the results -- all realistic compactification regimes exist only for $\alpha > 0$ and there is no smooth transition between high-energy and low-energy Kasner regimes, the latter with realistic compactification.Comment: 36 pages, 10 figure