Exponential potentials and cosmological scaling solutions

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 $V ∝ \exp(-λ κ φ)$ where $κ^2 = 8π G$. 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.