General relativistic corrections to $N$-body simulations and the Zel'dovich approximation

TL;DR

This paper demonstrates that standard Newtonian $N$-body simulations inherently include first-order relativistic corrections when initial conditions are set using the Zel'dovich approximation, and introduces a new gauge where these corrections vanish.

Contribution

The authors define the $N$-body gauge, showing how to incorporate relativistic effects into Newtonian $N$-body simulations and clarifying the gauge choice for initial conditions.

Findings

Relativistic corrections affect initial displacements in most gauges.

The $N$-body gauge eliminates first-order relativistic corrections.

Newtonian simulations implicitly include relativistic effects in this gauge.

Abstract

The initial conditions for Newtonian $N$-body simulations are usually generated by applying the Zel'dovich approximation to the initial displacements of the particles using an initial power spectrum of density fluctuations generated by an Einstein-Boltzmann solver. We show that in most gauges the initial displacements generated in this way receive a first-order relativistic correction. We define a new gauge, the $N$-body gauge, in which this relativistic correction vanishes and show that a conventional Newtonian $N$-body simulation includes all first-order relativistic contributions (in the absence of radiation) if we identify the coordinates in Newtonian simulations with those in the relativistic $N$-body gauge.