The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

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.

35 pages

References in corpus (4)

• On the stability of relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1$
• Stable cosmologies with collisionless charged matter
• Slowly expanding stable dust spacetimes
• The nonlinear stability of n+1 dimensional FLRW spacetimes

Cited by in corpus (2)

• Localized big bang stability for the Einstein-scalar field equations
• Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability