Nonlinear Stability of the Milne Model with Matter

Abstract

We show that any 3+1-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein-Vlasov system. For the analysis of the Einstein equations we use the constantmean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric L 2 -estimates based on the Sasaki-metric. The resulting estimates on the energy momentum tensor are then upgraded by employing the natural continuity equation for the energy density.

Isotropization of slowly expanding spacetimes

“…The slower expansion rate for the radiative case makes it a priori more difficult to establish sufficiently strong decay estimates. We point out that nonlinear stability results are established for exponential scale factors [29] or polynomial scale factors with significantly higher exponents [2]. Indeed, our analysis requires a more careful treatment of the evolution equation for the shear tensor and metric perturbations.…”

“…The Einstein-deSitter models pose interesting examples of matter-dominated cosmological spacetimes, i.e., spacetimes whose asymptotic behaviour is altered by the presence of matter. While for initial data close to the Milne geometry on hyperbolic spatial manifolds vacuum and non-vacuum future asymptotics are similar [2], on toroidal spatial topologies vacuum asymptotics deviate drastically from the matter dominated regime of the Einstein-deSitter models [26]. It is of essential interest to investigate the stability properties of those model solutions in order to understand whether their behaviour is representative for generic spacetimes with similar initial data.…”

“…This type of energy methods has been used for the Einstein equations in different contexts (cf. [2] and references therein). Here it is used in a context where the behaviour of the matter variables has a strong effect on the geometry (matter dominated regime).…”

Kantowski–Sachs cosmology with Vlasov matter

“…The Einstein-Vlasov system describes spacetimes containing ensembles of selfgravitating, collisionless particles and constitutes an excellent model for the largescale structure of the universe where collisions of particles (modeling galaxies or galaxy clusters) are indeed rare. For the class of cosmological spacetimes certain stability results were resolved in recent years [1,32] (cf [11,18,34] for recent stability results for the Einstein-Vlasov system in the asymptotically flat case).…”

“…The cosmologies for which they succeeded without additional assumptions, and for which the found distributions are non-trivial are: SH but not LRS: Bianchi I. (1) SH and LRS:…”