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

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-λ-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-λ-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 Friedrich's conformal Einstein field equations for the Einstein-λ-dust written in terms of a gauge in which the flow lines of the dust are recast as conformal geodesics.

Asymptotic structure and stability of spatially homogeneous space-times with a positive cosmological constant

“…Following the arguments of Section 4.1, we see that for w ≥ 1 3 , i.e. γ ≥ 4 3 we achieve k ≤ −1. So from Proposition 2 the rescaled Weyl curvatures E αβ and H αβ have a finite limit at τ ∞ .…”

“…We close this section by providing an overview in Table 1 of the various matter models considered and the results we are able to deduce for our class of Bianchi space-times. Viscous fluids with (64), (66) and γ ≥ 4 3 .…”

“…Suppose we have a matter model that satisfies the DEC and SEC and has a trace-free energy-momentum tensor 3 . The latter implies that (ρ + p) = 4 3 ρ, i.e. γ = 4 3 in ( 54) and (55).…”

“…The work in [30] also shows the stability for Einstein-dust space-times wide class of non-spatially homogeneous background space-times. A specific application to space-times with self-gravitating dust balls is given in [4].…”

“…• Elastic matter with (62) and γ ≥ 4 3 , • Viscous fluids with (64), ( 66) and γ ≥ 4 3 . Then, each of these space-times satisfies the identities (102)-(105) at conformal infinity.…”