Avoiding Big Rip Singularities in Phantom Scalar Field theory with Gauss-Bonnet term

TL;DR

The paper studies a phantom scalar field $\phi$ nonminimally coupled to the Gauss-Bonnet term in a spatially flat $FLRW$ universe, a setup that can drive future singularities such as a Big Rip. A dual dynamical-systems approach is employed: a Hubble-normalized formulation and a matter-scalar normalization, with $V(\phi)=V_0 e^{\lambda \phi}$, $f(\phi)=f_0 \phi$, and $g(\phi)=e^{2\beta \phi}$; stationary points are analyzed to reveal the late-time behavior. The Gauss-Bonnet coupling yields de Sitter attractors and forbids any stationary point corresponding to a Big Rip or Big Crunch, and this result persists even when the interaction $g(\phi)$ is absent ($\beta=0$). The conclusions hold for a broad class of $f(\phi)$, $g(\phi)$, and $V(\phi)$ beyond the specific choices studied, underscoring the GB term as a robust stabilizing mechanism for phantom cosmologies.

Abstract

We consider a phantom scalar field coupled to the Gauss-Bonnet scalar within a spatially flat FLRW geometry. Moreover, we assume a nonzero interaction between the scalar field and the matter term. We perform a detailed phase space analysis using two sets of dimensionless variables. Specifically, we introduce dimensionless variables based on the Hubble normalization approach and a new set based on the matter-scalar field normalization. These two sets of variables allow for a comprehensive phase space analysis. This model supports inflationary solutions without the Big Rip or Big Crunch singularities appearing as asymptotic solutions. This outcome is attributed to the presence of the Gauss-Bonnet scalar. The result remains valid even in the absence of the interaction term.

Figures (2)

• Figure 1: "x", "y" Phase-space on the surface $\eta=-1$, assuming $f_{0} \rightarrow 10, \beta \rightarrow \frac{1}{5}, \lambda \rightarrow 1$
• Figure 2: "x", "$\eta$" Phase-space on the surface $y=0$, assuming $f_{0} \rightarrow -10, \beta \rightarrow \frac{1}{5}, \lambda \rightarrow 1$