On the accuracy of N-body simulations at very large scales

TL;DR

This paper analyzes the accuracy of Newtonian N-body simulations at very large cosmological scales, showing they remain accurate even beyond the particle horizon with minimal relativistic corrections.

Contribution

It demonstrates that Newtonian simulations accurately reproduce particle trajectories at super-horizon scales, with relativistic corrections being negligible for practical purposes.

Findings

Relativistic corrections are below 10^{-5} at high redshift and below 10^{-3} at present.

Newtonian simulations match relativistic predictions well on large scales.

Differences in real space feature positions are below 10^{-6}.

Abstract

We examine the deviation of Cold Dark Matter particle trajectories from the Newtonian result as the size of the region under study becomes comparable to or exceeds the particle horizon. To first order in the gravitational potential, the general relativistic result coincides with the Zel'dovich approximation and hence the Newtonian prediction on all scales. At second order, General Relativity predicts corrections which overtake the corresponding second order Newtonian terms above a certain scale of the order of the Hubble radius. However, since second order corrections are very much suppressed on such scales, we conclude that simulations which exceed the particle horizon but use Newtonian equations to evolve the particles, reproduce the correct trajectories very well. The dominant relativistic corrections to the power spectrum on scales close to the horizon are at most of the order of $\sim 10^{-5}$ at $z=49$ and $\sim 10^{-3}$ at $z=0$. The differences in the positions of real space features are affected at a level below $10^{-6}$ at both redshifts. Our analysis also clarifies the relation of N-body results to relativistic considerations.