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

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.

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Cited by in corpus (5)

• Recent developments in mathematical aspects of relativistic fluids
• The Stability of Relativistic Fluids in Linearly Expanding Cosmologies
• Localized big bang stability for the Einstein-scalar field equations
• Past instability of FLRW solutions of the Einstein-Euler-scalar field equations for linear equations of state $p=K\rho$ with $0\le K<1/3$
• Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability