Sharp asymptotics for Einstein-λ-dust flows

We consider the Einstein-dust equations with positive cosmological constant λ on manifolds with time slices diffeomorphic to an orientable, compact 3-manifold S. It is shown that the set of standard Cauchy data for the Einstein-λ-dust equations on S contains an open (in terms of suitable Sobolev norms) subset of data that develop into solutions which admit at future time-like infinity a space-like conformal boundary J that is C∞ if the data are of class C∞ and of correspondingly lower smoothness otherwise. As a particular case follows a strong stability result for FLRW solutions. The solutions can conveniently be characterized in terms of their asymptotic end data induced on J , only a linear equation must be solved to construct such data. In the case where the energy density ρ̂ is everywhere positive such data can be constructed without solving any differential equation at all.