Scaling solutions in Robertson-Walker spacetimes

We investigate the stability of cosmological scaling solutions describing a barotropic fluid with p= (-1) and a non-interacting scalar field with an exponential potential V () =V₀^-. 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 _=1 (2/3, ²3). We find that this matter scaling solution is unstable to curvature perturbations for >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 _=2/² (>2/3, ²>2). We find that solutions with _=0 are never late-time attractors.