Scaling Solutions in Robertson-Walker Spacetimes

Abstract

We investigate the stability of cosmological scaling solutions describing a barotropic fluid with $p=(γ-1)ρ$ and a non-interacting scalar field $ϕ$ with an exponential potential $V(ϕ)=V_0\e^{-κϕ}$. We study homogeneous and isotropic spacetimes with non-zero spatial curvature and find three possible asymptotic future attractors in an ever-expanding universe. One is the zero-curvature power-law inflation solution where $Ω_ϕ=1$ ($γ<2/3,κ^2<3γ$ and $γ>2/3,κ^2<2$). Another is the zero-curvature scaling solution, first identified by Wetterich, where the energy density of the scalar field is proportional to that of matter with $Ω_ϕ=3γ/κ^2$ ($γ<2/3,κ^2>3γ$). We find that this matter scaling solution is unstable to curvature perturbations for $γ>2/3$. The third possible future asymptotic attractor is a solution with negative spatial curvature where the scalar field energy density remains proportional to the curvature with $Ω_ϕ=2/κ^2$ ($γ>2/3,κ^2>2$). We find that solutions with $Ω_ϕ=0$ are never late-time attractors.