Future stability of self-gravitating dust balls in an expanding universe

TL;DR

This paper demonstrates the existence and stability of self-gravitating dust configurations in an expanding universe, showing conditions under which solutions are future geodesically complete and resemble de Sitter spacetime.

Contribution

It provides a rigorous analysis of the stability and future behavior of dust solutions in an expanding universe using conformal Einstein equations.

Findings

Solutions can be prescribed at infinity and evolved backward without shocks.

Small density relative to the cosmological constant allows for stable cosmological solutions.

Under certain conditions, solutions are future geodesically complete and smoothly extendable at infinity.

Abstract

We consider a system representing self-gravitating balls of dust in an expanding Universe. It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or singularities. The resulting solution to the Einstein-lambda-dust equations exists for an infinite amount of time in the asymptotic region of the spacetime. Furthermore, we find that if the density is small compared to the Cosmological constant, then it is possible to construct Cosmological solutions to the Einstein constraint equations on a standard Cauchy hypersurface representing self-gravitating balls of dust. If, in addition, the density is assumed to be sufficiently small, then this initial data gives rise to a future geodesically complete solution to the Einstein-lambda-dust equations admitting a smooth conformal extension at infinity which can be regarded as a perturbation of de Sitter spacetime. The main technical tool in this analysis are Friedrichs conformal Einstein field equations for the Einstein-lambda-dust written in terms of a gauge in which the flow lines of the dust are recast as conformal geodesics.