Second order solutions of cosmological perturbation in the matter dominated era

TL;DR

This paper derives second order solutions for cosmological perturbations during the matter dominated era, examining gauge-invariant variables and the Newtonian correspondence of relativistic perturbations.

Contribution

It provides new second order solutions and gauge-invariant combinations, extending the understanding of relativistic and Newtonian perturbation correspondence.

Findings

Second order solutions for density and velocity perturbations are derived.

Gauge-invariant combinations are identified for general fluid contexts.

Relativistic gravitational potential matches Newtonian potential to second order.

Abstract

We present the growing mode solutions of cosmological perturbations to the second order in the matter dominated era. We also present several gauge-invariant combinations of perturbation variables to the second order in most general fluid context. Based on the solutions we study the Newtonian correspondence of relativistic perturbations to the second order. In addition to the previously known exact relativistic/Newtonian correspondence of density and velocity perturbations to the second order in the comoving gauge, here we show that in the sub-horizon limit we have the correspondences for density, velocity and potential perturbations in the zero-shear gauge and in the uniform-expansion gauge to the second order. Density perturbation in the uniform curvature gauge also shows the correspondence to the second order in the sub-horizon scale. We also identify the relativistic gravitational potential which shows exact correspondence to the Newtonian one to the second order.