Sharp asymptotics for Einstein-$\lambda$-dust flows

TL;DR

This paper analyzes the asymptotic behavior of solutions to Einstein-dust equations with positive cosmological constant, showing stability and smoothness of future conformal boundaries for a broad class of initial data.

Contribution

It establishes the existence of an open set of initial data leading to solutions with smooth conformal boundaries and provides a simplified method for constructing asymptotic data.

Findings

Solutions admit a smooth space-like conformal boundary at future infinity.

A strong stability result is proved for FLRW solutions.

Asymptotic end data can be constructed via a linear equation, simplifying analysis.

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 ${\cal J}^+$ 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 ${\cal J}^+$, 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.