On the stability of relativistic perfect fluids with linear equations of state $p=Kρ$ where $1/3<K<1$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929281672675328 |
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| author | Marshall, Elliot Oliynyk, Todd A. |
| author_facet | Marshall, Elliot Oliynyk, Todd A. |
| contents | 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 $p=Kρ$ 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 $\frac{\partial_{x}ρ}ρ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_06982 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the stability of relativistic perfect fluids with linear equations of state $p=Kρ$ where $1/3<K<1$ Marshall, Elliot Oliynyk, Todd A. General Relativity and Quantum Cosmology Analysis of PDEs 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 $p=Kρ$ 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 $\frac{\partial_{x}ρ}ρ$ 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. |
| title | On the stability of relativistic perfect fluids with linear equations of state $p=Kρ$ where $1/3<K<1$ |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs |
| url | https://arxiv.org/abs/2209.06982 |