Cosmology with positive and negative exponential potentials

TL;DR

This paper analyzes cosmological models with scalar fields and exponential potentials, identifying conditions for stable solutions and their implications for universe evolution, including scenarios like ekpyrotic and pre big bang.

Contribution

It provides a phase-plane analysis of scalar field cosmologies with positive and negative exponential potentials, revealing stability conditions and late-time behaviors.

Findings

Power-law solutions exist for flat positive or steep negative potentials.

Negative potential solutions tend to recollapse unless stabilized.

Kinetic-dominated solutions are attractors during collapse.

Abstract

We present a phase-plane analysis of cosmologies containing a scalar field $\phi$ with an exponential potential $V \propto \exp(-\lambda \kappa \phi)$ where $\kappa^2 = 8\pi G$ and $V$ may be positive or negative. We show that power-law kinetic-potential scaling solutions only exist for sufficiently flat ($\lambda^2<6$) positive potentials or steep ($\lambda^2>6$) negative potentials. The latter correspond to a class of ever-expanding cosmologies with negative potential. However we show that these expanding solutions with a negative potential are to unstable in the presence of ordinary matter, spatial curvature or anisotropic shear, and generic solutions always recollapse to a singularity. Power-law kinetic-potential scaling solutions are the late-time attractor in a collapsing universe for steep negative potentials (the ekpyrotic scenario) and stable against matter, curvature or shear perturbations. Otherwise kinetic-dominated solutions are the attractor during collapse (the pre big bang scenario) and are only marginally stable with respect to anisotropic shear.