Stability of fluids in spacetimes with decelerated expansion

TL;DR

This paper proves the nonlinear stability of homogeneous perfect fluid solutions in decelerating cosmological spacetimes under specific conditions relating sound speed and expansion rate, with numerical evidence supporting the sharpness of these conditions.

Contribution

It establishes the stability criteria for fluids in decelerated expansion and simplifies the proof using a universal energy functional applicable to linear expansion.

Findings

Stability depends on a specific inequality between sound speed and expansion rate.

Numerical evidence indicates the stability condition is sharp.

Shock formation occurs for small perturbations in dust and radiation fluids.

Abstract

We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.