Cosmic no-hair: nonlinear asymptotic stability of de Sitter universe

We study the asymptotic stability of de Sitter spacetime with respect to nonlinear perturbations, by considering second-order perturbations of a flat Friedmann-Lemaitre-Robertson-Walker universe with dust and a positive cosmological constant. Using the synchronous co-moving gauge we find that, as in the case of linear perturbations, the nonlinear perturbations also tend to constants, asymptotically in time. Analysing curvature and other spacetime invariants, we show, however, that these quantities asymptotically tend to their de Sitter values, thus demonstrating that the geometry is indeed locally asymptotically de Sitter, despite the fact that matter inhomogeneities tend to constants in time. Our results support the inflationary picture of frozen amplitude matter perturbations that are stretched outside the horizon, and demonstrate the validity of the cosmic no-hair conjecture in the nonlinear inhomogeneous settings considered here.