Cosmological phase-space analysis of $f(G)$-theories of gravity

TL;DR

The paper examines nonlinear Gauss–Bonnet ($f(G)$) gravity in a FLRW background by recasting it as an Einstein–Gauss–Bonnet–scalar field theory using a Lagrange multiplier. It constructs a novel autonomous dynamical system with variables that permit the Hubble rate to change sign, analyzes fixed points and their stability for both flat and curved universes, and identifies de Sitter and scaling solutions as attractors or saddles depending on the potential. For an exponential potential, two attractors emerge in the flat case, but the de Sitter solution generally appears as a saddle, with most trajectories leading to collapse; in the power-law case, a de Sitter attractor can arise on a stable surface for certain parameter ranges. Overall, while topological contributions can yield accelerating epochs, they tend to be disfavored as descriptors of the entire cosmic history, motivating extensions that include additional fields or perturbations to stabilize broader regions of phase space.

Abstract

The impact of topological terms that modify the Hilbert-Einstein action is here explored by virtue of a further $f(G)$ contribution. In particular, we investigate the phase-space stability and critical points of an equivalent scalar field representation that makes use of a massive field, whose potential is function of the topological correction. To do so, we introduce to the gravitational Action Integral a Lagrange multiplier and model the modified Friedmann equations by virtue of new non-dimensional variables. We single out dimensionless variables that permit \emph{a priori} the Hubble rate change of sign, enabling regions in which the Hubble parameter either vanishes or becomes negative. In this respect, we thus analyze the various possibilities associated with a universe characterized by such topological contributions and find the attractors, saddle points and regions of stability, in general. The overall analysis is carried out considering the exponential potential first and then shifting to more complicated cases, where the underlying variables do not simplify. We compute the eigenvalues of our coupled differential equations and, accordingly, the stability of the system, in both a spatially-flat and non-flat universe. Quite remarkably, regardless of the spatial curvature, we show that a stable de Sitter-like phase that can model current time appears only a small fraction of the entire phase-space, suggesting that the model under exam is unlikely in describing the whole universe dynamics, i.e., the topological terms appear disfavored in framing the entire evolution of the universe.

Figures (2)

• Figure 1: Phase-space portrait for the dynamical system (\ref{['cc.19']}), (\ref{['cc.20']}) in the compactified variables $X$ and $\eta$. Equilibrium points at the infinity are marked with red, the source points $A_{1}$ and $A_{4}$ are marked with orange, the saddle points $A_{5}$ and $A_{6}$ with green, and the attractors $A_{2}$ and $A_{5}$ are marked with blue. We observe that the main of the initial conditions lead to a collapsed universe, while there exist only a small area, less than then one quarter, of the phase-space, where the trajectories lead to a future attractor which describes cosmic acceleration.
• Figure 2: Phase-space portrait for the dynamical system (\ref{['cc.19']}), (\ref{['cc.20']}) and (\ref{['cc.15b']}) in the variables $x,\eta~$and $\lambda$ where the family of points $C_{3}$ is marked. Left Fig. is for $n=\frac{1}{5}$ and right Fig. for $n=-\frac{1}{5}$ . It is easy to see that for $n=\frac{1}{5}$, the trajectories lie on the stable surface $C_{3}$, and the de Sitter solution is an attractor.