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It has been proposed recently to consider in the framework of cosmology anextension of the semiclassical Einstein's equations in which the Einsteintensor is considered as a random function. This paradigm yields a hierarchy ofequations between the $n$-point functions of the quantum, normal ordered,stress energy-tensor and those associated to the stochastic Einstein tensor.Assuming that the matter content is a conformally coupled massive scalar fieldon de Sitter spacetime, this framework has been applied to compute the powerspectrum of the quantum fluctuations and to show that it is almostscale-invariant. We test the robustness and the range of applicability of thisproposal by applying it to a less idealized, but physically motivated,scenario, namely we consider Friedmann-Robertson-Walker spacetimes which behaveonly asymptotically in the past as a de Sitter spacetime. We show in particularthat, under this new assumption and independently from any renormalizationfreedom, the power spectrum associated to scalar perturbations of the metricbehaves consistently with an almost scale-invariant power spectrum.

Curvature fluctuations on asymptotically de Sitter spacetimes via the semiclassical Einstein's equations