  Nonlinear perturbations of Friedmann-Lemaitre cosmologies with dust and a
positive cosmological constant have recently attracted considerable attention.
In this paper our first goal is to compare the evolution of the first and
second order perturbations by determining their asymptotic behaviour at late
times in ever-expanding models. We show that in the presence of spatial
curvature K or a positive cosmological constant, the density perturbation
approaches a finite limit both to first and second order, but the rate of
approach depends on the model, being power law in the scale factor if the
cosmological constant is positive but logarithmic if it is zero and and K<0.
Scalar perturbations in general contain a growing and a decaying mode. We find,
somewhat surprisingly, that if the cosmological constant is positive the
decaying mode does not die away, i.e. it contributes on an equal footing as the
growing mode to the asymptotic expression for the density perturbation. On the
other hand, the future asymptotic regime of the Einstein-de Sitter universe
(for which the cosmological constant and the spatial curvature are both zero)
is completely different, as exemplified by the density perturbation which
diverges; moreover, the second order perturbation diverges faster than the
first order perturbation, which suggests that the Einstein-de Sitter universe
is unstable to perturbations, and that the perturbation series do not converge
towards the future. We conclude that the presence of spatial curvature or a
cosmological constant stabilizes the perturbations. Our second goal is to
derive an explicit expression for the second order density perturbation that
can be used to study the effects of including a cosmological constant and
spatial curvature.
