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Cosmological Post-Newtonian Expansions to Arbitrary Order

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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 \({\epsilon=v_T/c}\) \({(0< \epsilon < \epsilon_0)}\), 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 \({M\cong [0,T)\times \mathbb {T}^3}\), and converge as \({\epsilon \searrow 0}\) to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter \({\epsilon}\) to any specified order with expansion coefficients that satisfy \({\epsilon}\)-independent (nonlocal) symmetric hyperbolic equations.

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  1. School of Mathematical Sciences, Monash University, Melbourne, Vic, 3800, Australia

    Todd A. Oliynyk

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  1. Todd A. Oliynyk
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Correspondence to Todd A. Oliynyk.

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Communicated by P. T. Chruściel

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Oliynyk, T.A. Cosmological Post-Newtonian Expansions to Arbitrary Order. Commun. Math. Phys. 295, 431–463 (2010). https://doi.org/10.1007/s00220-009-0931-0

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