Sharp Asymptotics for Einstein-{lambda}-Dust Flows

Abstract

We consider the Einstein-dust equations with positive cosmological constant {lambda} 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-{lambda}-dust equations on {S} contains an open (in terms of suitable Sobolev norms) subset of data which develop into solutions that admit at future time-like infinity a space-like conformal boundary {{mathcal J}^+} that is {C^{infty}} if the data are of class {C^{infty}} and of correspondingly lower smoothness otherwise. The class of solutions considered here comprises non-linear perturbations of FLRW solutions as very special cases. It can conveniently be characterized in terms of asymptotic end data induced on {{mathcal J}^+}. These data must only satisfy a linear differential equation. If the energy density is everywhere positive they can be constructed without solving differential equations at all.