Connection between Newtonian simulations and general relativity

TL;DR

The paper addresses how to extract general-relativistic observables from Newtonian N-body simulations on horizon-scale by formulating the evolution in the linearized conformal Newtonian gauge and showing that nonrelativistic matter remains well-described by Newtonian dynamics. It demonstrates that the Newtonian potential φ_N matches the conformal Newtonian potential φ_N and that particle coordinates must be shifted by a relativistic initial displacement δx_in, yielding a simple dictionary x_N = x_sim + δx_in, v_N = v_sim, φ_N = φ_sim. It then provides explicit mappings to observable coordinates z_obs, θ_obs, and φ_obs by integrating along the light cone, incorporating lensing and ISW-like effects, thus enabling GR-consistent predictions from standard large-volume simulations. Overall, the work justifies using Newtonian simulations to probe horizon-scale clustering and outlines concrete procedures to obtain GR-corrected and observable quantities from simulation outputs.

Abstract

On large-scales, comparable to the horizon, the observable clustering properties of galaxies are affected by various general relativistic effects. To calculate these effects one needs to consistently solve for the metric, densities and velocities in a specific coordinate system or gauge. The method of choice for simulating large-scale structure is numerical N-body simulations which are performed in the Newtonian limit. Even though one might worry that the use of the Newtonian approximation would make it impossible to use these simulations to compute properties on very large-scales, we show that the simulations are still solving the dynamics correctly even for long modes and we give formulas to obtain the position of particles in the conformal Newtonian gauge given the positions computed in the simulation. We also give formulas to convert from the output coordinates of N-body simulations to the observable coordinates of the particles.