The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

TL;DR

This paper proves the future stability and global regularity of relativistic fluids in linearly expanding cosmological spacetimes with linear equations of state, extending previous results to include rotational perturbations.

Contribution

It introduces a novel transformation of fluid variables enabling the analysis of stability without irrotational restrictions, and applies this to demonstrate stability of Milne spacetime.

Findings

Stability of nonlinear perturbations in relativistic fluids

Global regularity of Milne spacetime under Einstein--Euler equations

Extension to rotational perturbations in cosmological models

Abstract

In this paper we study cosmological solutions to the Einstein--Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly expanding cosmological spacetimes with a linear equation of state $p=K \rho$ for the parameter values $K \in (0,1/3)$. This removes the restriction to irrotational perturbations in earlier work, and relies on a novel transformation of the fluid variables that is well-adapted to Fuchsian methods. We then apply this new transformation to show the global regularity and stability of the Milne spacetime under the coupled Einstein--Euler equations, again with a linear equation of state $p=K \rho$, $K \in (0,1/3)$. Our proof requires a correction mechanism to account for the spatially curved geometry. In total, this is indicative that structure formation in cosmological fluid-filled spacetimes requires an epoch of decelerated expansion.