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 that develop into solutions which 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. 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 ${\mathcal{J}}^{+}$, only a linear equation must be solved to construct such data. In the case where the energy density $\hat{\rho }$ is everywhere positive such data can be constructed without solving any differential equation at all.

44 pages

References in corpus (1)

• Investigations of solutions of Einstein's field equations close to lambda-Taub-NUT

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