The Poisson equation at second order in relativistic cosmology

TL;DR

This paper derives a second order relativistic Poisson equation in cosmology, linking curvature and density perturbations, and introduces a framework for initial conditions in structure formation simulations considering primordial non-Gaussianity.

Contribution

It presents the relativistic second order Poisson equation in a specific gauge and applies it to incorporate primordial non-Gaussianity into initial conditions for N-body simulations.

Findings

Derived the second order relativistic Poisson equation.

Provided expressions for initial conditions including non-Gaussianity.

Facilitated the study of primordial non-Gaussianity effects in structure formation.

Abstract

We calculate the relativistic constraint equation which relates the curvature perturbation to the matter density contrast at second order in cosmological perturbation theory. This relativistic "second order Poisson equation" is presented in a gauge where the hydrodynamical inhomogeneities coincide with their Newtonian counterparts exactly for a perfect fluid with constant equation of state. We use this constraint to introduce primordial non-Gaussianity in the density contrast in the framework of General Relativity. We then derive expressions that can be used as initial conditions of N-body codes for structure formation which probe the observable signature of primordial non-Gaussianity in the statistics of the evolved matter density field.