Scaling solutions inRobertson-Walker spacetimes

Abstract

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() = V0e-. 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),2 < 3 and > (2/3),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 3/2 ( < (2/3),2 > 3). 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.