Cosmological post-Newtonian approximation compared with perturbation theory

TL;DR

This paper compares the first-order post-Newtonian approximation with relativistic linear perturbation theory in cosmology, analyzing gauge conditions and identifying exact correspondences in certain gauges for density, velocity, and gravitational potential.

Contribution

It provides a detailed comparison between 1PN and perturbation theory, clarifying gauge conditions where their equations and solutions coincide in cosmological models.

Findings

Exact correspondence of velocity and gravitational potential in zero-shear gauge with Newtonian equations.

Density perturbation equations in zero-shear and uniform-expansion gauges match 1PN results.

Identification of gauge conditions where 1PN and perturbation theory equations are equivalent.

Abstract

We compare the cosmological first-order post-Newtonian (1PN) approximation with the relativistic cosmological linear perturbation theory in a zero-pressure medium with the cosmological constant. We compare equations and solutions in several different gauge conditions available in both methods. In the PN method we have perturbation equations for density, velocity and gravitational potential independently of the gauge condition to 1PN order. However, correspondences with these 1PN equations are available only in certain gauge conditions in the perturbation theory. Equations of perturbed velocity and the perturbed gravitational potential in the zero-shear gauge exactly coincide with the Newtonian equations which remain valid even to 1PN order (the same is true for perturbed velocity in the comoving gauge), and equations of perturbed density in the zero-shear gauge and the uniform-expansion gauge coincide to 1PN order. We identify other correspondences available in different gauge conditions of the perturbation theory.