Compactification of Anisotropies in Einstein-Scalar-Gauss-Bonnet Cosmology

TL;DR

This work analyzes how anisotropies evolve in Einstein-Scalar-Gauss-Bonnet cosmologies with a scalar field coupled to the Gauss-Bonnet term, focusing on a 4D Bianchi I background and extending to 5D. By performing a minisuperspace reduction and a dynamical-systems analysis, the authors show that late-time solutions generically tend toward locally symmetric states where at least two scale factors share the same evolution, or toward fully isotropic configurations; the presence of a cosmological constant can prevent isotropization in 4D. In 5D, the coupling yields a richer set of asymptotic splittings, with 2+2 splitting stable in pure GB, while the scalar coupling enables 3+1 and 2+1+1 patterns, especially under $\Lambda>0$ or $\Lambda=0$ respectively. Overall, the results illustrate that GB-scalar couplings can drive compactification-like behavior and isotropization toward subspaces, providing insights for higher-dimensional cosmologies and their anisotropy dynamics.

Abstract

We investigate the evolution of anisotropies in Einstein-Gauss-Bonnet theory with a scalar field coupled to the Gauss-Bonnet term. Specifically, we examine the simplest scenario in which the scalar field lacks a kinetic term, and its kinetic contribution arises from an integration by parts of the Gauss-Bonnet scalar. We consider four- and five-dimensional anisotropic spacetimes, focusing on Bianchi I and extended Bianchi I geometries. Our study reveals that the asymptotic solutions correspond to locally symmetric spacetimes where at least two scale factors exhibit analogous behavior or, alternatively, to isotropic configurations where all scale factors evolve identically. Additionally, we discuss the effects of a cosmological constant, finding that the presence of the cosmological constant does not lead to an isotropic universe.

Figures (7)

• Figure 1: Phase-space portrait for the dynamical system (\ref{['ds.01']}), (\ref{['ds.02']}) in the compactified variables $Y_{+}$ and $Y_{-}$. With red are marked the points $P_{1}$, $P_{2}$, $P_{3}$ and $P_{4}$, with blue the points $Q_{4}$ and $Q_{5}^{\pm}$. With green are marked the rest of the stationary points at the infinity. The black circle denotes the infinity line, while the two orange circles describe define the geometric space of points $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$ and $Q_{4},~Q_{5}^{\pm}$. We observe that for initial conditions inside the area defined by the geometric space (magenta) which connect the stationary points $P_{2}$, $P_{3}$, $P_{4}$ and $Q_{4},~Q_{5}^{\pm}$ the future attractor is always the isotropic FLRW universe. This geometric space is defined by three ellipses.
• Figure 2: Phase-space portrait for the three-dimensional dynamical system (\ref{['ai.15']}), (\ref{['ai.16']}) on the space $\left( x,\Sigma_{\pm}\right)$. Left Fig. is the two-dimensional surface with $x=\frac{1}{6}$, while right Fig. is the three-dimensional phase-space portrait. Blue points are the stationary points at the finite regime. Gree and purple vectors represent the trajectoris with $\Sigma_{+}=0,~\Sigma_{-}=0$.
• Figure 3: Qualitative evolution of the dynamical variables $\left( x,\Sigma_{\pm}\right)$ and for the normalized functions$~\left( \frac{H_{1}}{H},\frac{H_{2}}{H},\frac{H_{3}}{H}\right)$ for various sets of initial conditions such that the dynamical system reaches infinity.
• Figure 4: 5D Spacetime: Qualitative evolution for the four Hubble functions$~\left( \frac{H_{1}}{H},\frac{H_{2}}{H},\frac{H_{3}}{H},\frac{H_{4}}{H}\right)$, for the scalar field parameter $\left( x,y\right)$ and the deceleration parameter $q$, as they are given by the solution of the dynamical system (\ref{['bb.00']}) and (\ref{['bb.01']}) for different set of initial conditions and $\Omega_{\Lambda}=0$. We observe that the dynamical behaviour of the physical space is different from the four-dimensional spacetime
• Figure 5: 5D Spacetime: Qualitative evolution for the four Hubble functions$~\left( \frac{H_{1}}{H},\frac{H_{2}}{H},\frac{H_{3}}{H},\frac{H_{4}}{H}\right)$, for the scalar field parameter $\left( x,y\right)$ and the deceleration parameter $q$, as they are given by the solution of the dynamical system (\ref{['bb.00']}) and (\ref{['bb.01']}) for different set of initial conditions and $\Omega_{\Lambda }\neq0$. We observe that the dynamical behaviour of the physical space is different from the four-dimensional spacetime