Anisotropic Spacetimes in $f(G)$-gravity: Bianchi I, Bianchi III and Kantowski-Sachs Cosmologies

TL;DR

This work investigates the evolution of cosmological anisotropies in four-dimensional $f(G)$-gravity for LRS geometries that cover Bianchi I, Bianchi III, and Kantowski-Sachs spacetimes. By introducing a Lagrange multiplier, the authors recast the theory as Einstein-Gauss-Bonnet gravity coupled to a scalar field without a kinetic term, and they formulate a minisuperspace description with dimensionless dynamical variables to study the phase space. Through numerical analysis for exponential potentials, they identify two finite attractors—a Minkowski spacetime and an isotropic, spatially flat, accelerating FLRW solution—while de Sitter behavior appears unstable and Big Rip/Big Crunch singularities occur in the infinite regime. Kasner-like anisotropic solutions are found to be unstable, and Kantowski-Sachs can permit naked-singularity paths, highlighting both the isotropization/flatness implications and the singularity structure of $f(G)$-gravity in these anisotropic cosmologies.

Abstract

We investigate the evolution of cosmological anisotropies within the framework of $f\left(G\right)$-gravity. Specifically, we consider a locally rotationally symmetric geometry in four-dimensional spacetime that describes the Bianchi I, Bianchi III, and the Kantowski-Sachs spacetimes. Within this context, we introduce a Lagrange multiplier which allows us to reformulate the geometric degrees of freedom in terms of a scalar field. The resulting theory is dynamically equivalent to an Einstein-Gauss-Bonnet scalar field model. We normalize the field equations by introducing dimensionless variables. The dynamics of our system is then explored by solving the resulting nonlinear differential equations numerically for various sets of initial conditions. Our analysis reveals the existence of two finite attractors: the Minkowski universe and an isotropic, spatially flat solution capable of describing accelerated expansion. Although de Sitter expansion may be recovered, it appears only as an unstable solution. In addition, the theory suffers from the existence of Big Rip singularities.

Figures (8)

• Figure 1: Bianchi I: Numerical solution of the field equations within the Bianchi I background ($\kappa=0$) for different set of initial conditions. For the numerical simulations we selected $\lambda=-0.1$. For the initial conditions we consider $\Sigma_{0}=0.05$, $\eta_{0}=0.6$ and $x_{0}=\left\{ -0.5,0.01,0.1,0.2,0.3\right\}$.
• Figure 2: Bianchi I: Numerical solution of the field equations within the Bianchi I background ($\kappa=0$) for different set of initial conditions. For the numerical simulations we selected $\lambda=-0.1$. For the initial conditions we consider $\Sigma_{0}=0.05$, $\eta_{0}=0.6$ and $x_{0}=\left\{ 0.51,0.52,0.53,0.54,0.55\right\}$.
• Figure 3: Bianchi I: Numerical solution of the field equations within the Bianchi I background ($\kappa=0$) for different set of initial conditions. For the numerical simulations we selected $\lambda=-0.1$. For the initial conditions we consider $\Sigma_{0}=-0.5$, $\eta_{0}=0.6$ and $x_{0}=\left\{ 0.1,0.3,0.5,0.8,1\right\}$.
• Figure 4: Bianchi I: Numerical solution of the field equations within the Bianchi I background ($\kappa=0$) for different set of initial conditions. For the numerical simulations we selected $\lambda=-0.1$. For the initial conditions we consider $x_{0}=0.2$, $\Sigma_{0}=0.5$ and $\eta_{0}=\left\{ 0.3,0.35,0.4,0.5\right\}$
• Figure 5: Bianchi III: Numerical solution of the field equations within the Bianchi III background for different set of initial conditions. For the numerical simulations we selected $\lambda=-0.1$. For the initial conditions we consider $\Sigma _{0}=0.05$, $\eta_{0}=0.6,\Omega_{\kappa0}=0.1$ and $x_{0}=\left\{ -0.5,-0.1,-0.01,0.1,0.2\right\}$.