Cosmological post-Newtonian expansions to arbitrary order

TL;DR

This paper constructs solutions to Einstein-Euler equations that depend on the parameter  and show convergence to Newtonian gravity solutions, providing a systematic way to expand solutions to any order in .

Contribution

It proves the existence of parameter-dependent solutions to Einstein-Euler equations that converge to Newtonian solutions and can be expanded to arbitrary order in the parameter .

Findings

Solutions depend smoothly on  and converge to Newtonian solutions as  

Solutions can be expanded in  to any order with coefficients satisfying hyperbolic equations

Provides a rigorous framework for post-Newtonian expansions to arbitrary order

Abstract

We prove the existence of a large class of one parameter families of solutions to the Einstein-Euler equations that depend on the singular parameter $\ep=v_T/c$ $(0<\ep < \ep_0)$, where $c$ is the speed of light, and $v_T$ is a typical speed of the gravitating fluid. These solutions are shown to exist on a common spacetime slab $M\cong [0,T)\times \Tbb^3$, and converge as $\ep \searrow 0$ to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter $\ep$ to any specified order with expansion coefficients that satisfy $\ep$-independent (nonlocal) symmetric hyperbolic equations.