  We study the asymptotic stability of de Sitter spacetime with respect to
non-linear perturbations, by considering second order perturbations of a flat
Robertson-Walker universe with dust and a positive cosmological constant. Using
the synchronous comoving gauge we find that, as in the case of linear
perturbations, the non-linear 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.
