Second order perturbations of a zero-pressure cosmological medium: Proofs of the relativistic-Newtonian correspondence

TL;DR

This paper proves that, except for gravitational waves, second-order relativistic perturbation equations for a zero-pressure cosmological medium match Newtonian equations, extending Newtonian applicability to all cosmological scales including super-horizon.

Contribution

It demonstrates the exact relativistic-Newtonian correspondence at second order for zero-pressure fluids in a flat universe, including the cosmological constant.

Findings

Relativistic and Newtonian equations coincide to second order for zero-pressure fluids.

The correspondence extends to super-horizon scales, including the cosmological constant.

Gravitational wave equations are also derived to second order.

Abstract

The dynamic world model and its linear perturbations were first studied in Einstein's gravity. In the system without pressure the relativistic equations coincide exactly with the later known ones in Newton's gravity. Here we prove that, except for the gravitational wave contribution, even to the second-order perturbations, equations for the relativistic irrotational zero-pressure fluid in a flat Friedmann background coincide exactly with the previously known Newtonian equations. Thus, to the second order, we correctly identify the relativistic density and velocity perturbation variables, and we expand the range of applicability of the Newtonian medium without pressure to all cosmological scales including the super-horizon scale. In the relativistic analyses, however, we do not have a relativistic variable which corresponds to the Newtonian potential to the second order. Mixed usage of different gauge conditions is useful to make such proofs and to examine the result with perspective. We also present the gravitational wave equation to the second order. Since our correspondence includes the cosmological constant, our results are relevant to currently favoured cosmology. Our result has an important practical implication that one can use the large-scale Newtonian numerical simulation more reliably even as the simulation scale approaches near horizon.