# Relativistic initial conditions for N-body simulations

## TL;DR

This paper examines the relativistic consistency of initial conditions in cosmological N-body simulations, demonstrating the validity of the back-scaling method for standard cosmologies and proposing alternatives for non-standard cases.

## Contribution

It provides a relativistic analysis of the back-scaling initial condition method using the Newtonian motion gauge framework and discusses its applicability and limitations.

## Key findings

- Back-scaling initial conditions are consistent with relativistic space-time for LambdaCDM.
- Metric perturbations remain small for standard cosmologies at low redshift.
- Large metric perturbations occur in non-standard cosmologies, indicating the need for alternative approaches.

## Abstract

Initial conditions for (Newtonian) cosmological N-body simulations are usually set by re-scaling the present-day power spectrum obtained from linear (relativistic) Boltzmann codes to the desired initial redshift of the simulation. This back-scaling method can account for the effect of inhomogeneous residual thermal radiation at early times, which is absent in the Newtonian simulations. We analyse this procedure from a fully relativistic perspective, employing the recently-proposed Newtonian motion gauge framework. We find that N-body simulations for LambdaCDM cosmology starting from back-scaled initial conditions can be self-consistently embedded in a relativistic space-time with first-order metric potentials calculated using a linear Boltzmann code. This space-time coincides with a simple "N-body gauge" for z<50 for all observable modes. Care must be taken, however, when simulating non-standard cosmologies. As an example, we analyse the back-scaling method in a cosmology with decaying dark matter, and show that metric perturbations become large at early times in the back-scaling approach, indicating a breakdown of the perturbative description. We suggest a suitable "forwards approach" for such cases.