Relativistic-Newtonian correspondence of the zero-pressure but weakly nonlinear cosmology

TL;DR

This paper demonstrates that, in a flat Friedmann universe with zero-pressure and weak nonlinearity, relativistic perturbations match Newtonian results up to second order, validating the use of Newtonian simulations on large cosmological scales.

Contribution

It proves the relativistic-zero pressure fluid perturbations coincide with Newtonian results to second order, including the presence of a cosmological constant, except for gravitational waves.

Findings

Relativistic and Newtonian perturbations agree to second order.

Newtonian hydrodynamics are valid on all cosmological scales.

Gravitational wave contributions are the only relativistic correction.

Abstract

It is well known that couplings occur among the scalar-, vector-, and tensor-type perturbations of Friedmann world model in the second perturbational order. Here, we prove that, except for the gravitational wave contribution, the relativistic zero-pressure irrotational fluid perturbed to second order in a flat Friedmann background coincides exactly with the Newtonian result. Since we include the cosmological constant, our results are relevant to currently favoured cosmology. As we prove that the Newtonian hydrodynamic equations are valid in all cosmological scales to the second order, our result has an important practical implication that one can now use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches and even goes beyond the horizon. That is, our discovery shows that, in the zero-pressure case, except for the gravitational wave contribution, there are no relativistic correction terms even near and beyond the horizon to the second-order perturbation.