The Cosmic No-Hair Theorem and the Nonlinear Stability of Homogeneous Newtonian Cosmological Models

TL;DR

This paper proves the nonlinear stability of homogeneous Newtonian cosmological models with a positive cosmological constant, showing small perturbations decay over time without requiring linearization or symmetry assumptions.

Contribution

It demonstrates the nonlinear stability of homogeneous Newtonian cosmological models with a positive cosmological constant using a novel perturbation approach.

Findings

Small perturbations in expanding models decay over time.

Homogeneous models are nonlinearly stable.

A method to relate inhomogeneous models to homogeneous ones.

Abstract

The validity of the cosmic no-hair theorem is investigated in the context of Newtonian cosmology with a perfect fluid matter model and a positive cosmological constant. It is shown that if the initial data for an expanding cosmological model of this type is subjected to a small perturbation then the corresponding solution exists globally in the future and the perturbation decays in a way which can be described precisely. It is emphasized that no linearization of the equations or special symmetry assumptions are needed. The result can also be interpreted as a proof of the nonlinear stability of the homogeneous models. In order to prove the theorem we write the general solution as the sum of a homogeneous background and a perturbation. As a by-product of the analysis it is found that there is an invariant sense in which an inhomogeneous model can be regarded as a perturbation of a unique homogeneous model. A method is given for associating uniquely to each Newtonian cosmological model with compact spatial sections a spatially homogeneous model which incorporates its large-scale dynamics. This procedure appears very natural in the Newton-Cartan theory which we take as the starting point for Newtonian cosmology.