Exponential potentials and cosmological scaling solutions

Edmund J Copeland; Andrew R Liddle; David Wands

doi:10.1103/PhysRevD.57.4686

paper

Exponential potentials and cosmological scaling solutions

arXiv:gr-qc/9711068 · doi:10.1103/PhysRevD.57.4686

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 $V \propto exp (- λκ ϕ)$ where $κ^{2} = 8 π G$ . In addition to the well-known inflationary solutions for $λ^{2} < 2$ , there exist scaling solutions when $λ^{2} > 3 γ$ 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.

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Exponential potentials and cosmological scaling solutions

Abstract

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