Scaling solutions in Robertson-Walker spacetimes

We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p = (gamma - 1)rho and a non-interacting scalar fiel d phi with an exponential potential V (phi) = V(0)e(-kappa phi). We study h omogeneous and isotropic spacetimes with non-zero spatial curvature and fin d three possible asymptotic future attractors in an ever-expanding universe . One is the zero-curvature power-law inflation solution where Omega(phi) = 1 (gamma < 2/3,kappa(2) < 3 gamma and gamma > 2/3,kappa(2) < 2). Another i s the zero-curvature scaling solution, first identified by Wetterich, where the energy density of the scalar field is proportional to that of matter w ith Omega(phi) = 3 gamma/kappa(2) (gamma < 2/3, kappa(2) > 3 gamma). We fin d that this matter scaling solution is unstable to curvature perturbations for gamma > 2/3. The third possible future asymptotic attractor is a soluti on with negative spatial curvature where the scalar field energy density re mains proportional to the curvature with Omega(phi) = 2/kappa(2) (gamma > 2 /3, kappa(2) > 2). We find that solutions with Omega(phi) = 0 are never lat e-time attractors.