Exponential potentials and cosmological scaling solutions

TL;DR

This work analyzes a spatially flat FRW universe containing a barotropic fluid and a scalar field with an exponential potential $V(\phi)=V_0 e^{-{\lambda}\kappa{\phi}}$ using a phase-plane approach. It shows two main late-time behaviors: a scalar-field-dominated attractor when $\lambda^2<3\gamma$ and a scaling solution with fixed fractional energy density $\Omega_\phi=3\gamma/\lambda^2$ for $\lambda^2>3\gamma$, while the fluid-dominated fixed point is unstable for $\gamma>0$. A crucial nucleosynthesis bound $\lambda^2>20$ ensures the scalar field does not dominate too early, implying a relic-density problem for typical models; inflation, in standard forms, does not generally alleviate this unless extreme or nonstandard histories are invoked. The results reveal that exponential potentials can yield non-negligible cosmic energy fractions and dictate stringent constraints on their cosmological viability. The study emphasizes the need for care in integrating such fields into early- and late-time cosmology, particularly regarding relic abundances and structure formation.

Abstract

We present a phase-plane analysis of cosmologies containing a barotropic fluid with equation of state $p_γ= (γ-1) ρ_γ$, plus a scalar field $φ$ with an exponential potential $V \propto \exp(-λκφ)$ where $κ^2 = 8πG$. In addition to the well-known inflationary solutions for $λ^2 < 2$, there exist scaling solutions when $λ^2 > 3γ$ 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 $V(φ)/ρ_γ\to 0$ at late times, are always unstable (except for the cosmological constant case $γ= 0$). The relative energy density of the fluid and scalar field depends on the steepness of the exponential potential, which is constrained by nucleosynthesis to $λ^2 > 20$. We show that standard inflation models are unable to solve this `relic density' problem.

Figures (2)

• Figure 1: Regions of $(\gamma,\lambda)$ parameter space, as identified in the text. Solutions to the left of the dotted line are inflationary.
• Figure 2: The phase plane for $\gamma = 1$, $\lambda = 1$. The late-time attractor is the scalar field dominated solution with $x=\sqrt{1/6}$, $y=\sqrt{5/6}$.