Exponential potentials and cosmological scaling solutions

TL;DR

This paper analyzes cosmological models with an exponential scalar field potential, identifying scaling solutions as late-time attractors and discussing constraints from nucleosynthesis, with implications for inflation models.

Contribution

It provides a phase-plane analysis revealing the existence and stability of scaling solutions in exponential potential cosmologies, and discusses their implications for late-time universe behavior.

Findings

Scaling solutions exist for mbda^2 > 3 mma.

Scaling solutions are the unique late-time attractors.

Nucleosynthesis constrains mbda^2 > 20.

Abstract

We present a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $p_\gamma = (\gamma-1) \rho_\gamma$, plus a scalar field $\phi$ with an exponential potential $V \propto \exp(-\lambda \kappa \phi)$ where $\kappa^2 = 8\pi G$. In addition to the well-known inflationary solutions for $\lambda^2 < 2$, there exist scaling solutions when $\lambda^2 > 3\gamma$ in which the scalar field energy density tracks that of the barotropic fluid (which for example might be radiation or dust). We show that the scaling solutions are the unique late-time attractors whenever they exist. The fluid-dominated solutions, where $V(\phi)/\rho_\gamma \to 0$ at late times, are always unstable (except for the cosmological constant case $\gamma = 0$). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to $\lambda^2 > 20$. We show that standard inflation models are unable to solve this `relic density' problem.