On the stability of relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1$

TL;DR

This paper investigates the stability of relativistic perfect fluids with linear equations of state on expanding spacetimes, proving stability for non-isotropic solutions across a broad parameter range and exploring complex behaviors of isotropic perturbations through numerical simulations.

Contribution

It extends previous stability results to the full parameter range $1/3<K<1$ for non-isotropic solutions and provides numerical insights into the behavior of isotropic perturbations.

Findings

Stability of non-isotropic solutions for all $1/3<K<1$.

Numerical evidence of steep gradient formation in isotropic perturbations.

Development of unbounded density contrast at finite points in space.

Abstract

For $1/3<K<1$, we consider the stability of two distinct families of spatially homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=K\rho$ on exponentially expanding FLRW spacetimes. The two families are distinguished by one being spatially isotropic while the other is not. We establish the future stability of nonlinear perturbations of the non-isotropic family for the full range of parameter values $1/3<K<1$, which improves a previous stability result established by the second author that required $K$ to lie in the restricted range $(1/3,1/2)$. As a first step towards understanding the behaviour of nonlinear perturbations of the isotropic family, we construct numerical solutions to the relativistic Euler equations under a $\mathbb{T}^2$-symmetry assumption. These solutions are generated from initial data at a fixed time that is chosen to be suitably close to the initial data of an isotropic solution. Our numerical results reveal that, for the full parameter range $1/3<K<1$, the density contrast $\frac{\partial_{x}\rho}{\rho}$ associated to a nonlinear perturbation of an isotropic solution develops steep gradients near a finite number of spatial points where it becomes unbounded at future timelike infinity. This behaviour, anticipated by Rendall in \cite{Rendall:2004}, is of particular interest since it is not consistent with the standard picture for inflation in cosmology.