Cosmological Newtonian limits on large spacetime scales

Abstract

We establish the existence of $1$-parameter families of $\epsilon$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $\Lambda >0$ and a linear equation of state $p={\epsilon }^{2}K\rho$, $0<K\le 1/3$, for the parameter values $0<\epsilon <{\epsilon }_0$. These solutions exist globally on the manifold $M=(0,1]\times {\mathbb{R}}^3$, are future complete, and converge as $\epsilon \searrow 0$ to solutions of the cosmological Poisson-Euler equations. They represent inhomogeneous, nonlinear perturbations of a FLRW fluid solution where the inhomogeneities are driven by localized matter fluctuations that evolve to good approximation according to Newtonian gravity.

74 pages. Agrees with published version. arXiv admin note: text overlap with arXiv:1701.03975

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• Recent developments in mathematical aspects of relativistic fluids
• The Stability of Relativistic Fluids in Linearly Expanding Cosmologies
• On the stability of relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1$
• Future instability of FLRW fluid solutions for linear equations of state $p=K\rho$ with $1/3<K<1$
• 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 $0\le K<1/3$
• Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes
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• Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant
• Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability
• Blowups and longtime developments with near-boundary mass accretions of irregularly-shaped Euler--Poisson dominated molecular clouds in astrophysics