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Cosmological post-Newtonian expansions to arbitrary order. (English) Zbl 1195.35286

Summary: 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 \({\varepsilon=v_T/c}\) \({(0< \varepsilon < \varepsilon_0)}\), 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 \({\varepsilon \searrow 0}\) to a solution of the cosmological Poisson-Euler equations of Newtonian gravity. Moreover, we establish that these solutions can be expanded in the parameter \({\varepsilon}\) to any specified order with expansion coefficients that satisfy \({\varepsilon}\)-independent (nonlocal) symmetric hyperbolic equations.

MSC:

35Q76 Einstein equations
35C20 Asymptotic expansions of solutions to PDEs
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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