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Nonlinear perturbations of Friedmann-Lemaitre cosmologies with dust and apositive cosmological constant have recently attracted considerable attention.In this paper our first goal is to compare the evolution of the first andsecond order perturbations by determining their asymptotic behaviour at latetimes in ever-expanding models. We show that in the presence of spatialcurvature K or a positive cosmological constant, the density perturbationapproaches a finite limit both to first and second order, but the rate ofapproach depends on the model, being power law in the scale factor if thecosmological 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 thedecaying mode does not die away, i.e. it contributes on an equal footing as thegrowing mode to the asymptotic expression for the density perturbation. On theother 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 whichdiverges; moreover, the second order perturbation diverges faster than thefirst order perturbation, which suggests that the Einstein-de Sitter universeis unstable to perturbations, and that the perturbation series do not convergetowards the future. We conclude that the presence of spatial curvature or acosmological constant stabilizes the perturbations. Our second goal is toderive an explicit expression for the second order density perturbation thatcan be used to study the effects of including a cosmological constant andspatial curvature.

Asymptotic analysis of perturbed dust cosmologies to second order