N-body methods for relativistic cosmology

TL;DR

This paper introduces a relativistic N-body simulation framework for weak gravitational fields in cosmology, expanding Einstein's equations around a Friedmann-Lemaître background and allowing for various matter and stress-energy sources.

Contribution

It develops a fully general relativistic N-body simulation approach that includes linear metric perturbations and their derivatives, improving upon traditional Newtonian methods.

Findings

Relativistic corrections are small in LambdaCDM cosmology.

The framework can incorporate relativistic particles and modified gravity.

Comparison shows relativistic methods are more robust and flexible.

Abstract

We present a framework for general relativistic N-body simulations in the regime of weak gravitational fields. In this approach, Einstein's equations are expanded in terms of metric perturbations about a Friedmann-Lema\^itre background, which are assumed to remain small. The metric perturbations themselves are only kept to linear order, but we keep their first spatial derivatives to second order and treat their second spatial derivatives as well as sources of stress-energy fully non-perturbatively. The evolution of matter is modelled by an N-body ensemble which can consist of free-streaming nonrelativistic (e.g. cold dark matter) or relativistic particle species (e.g. cosmic neutrinos), but the framework is fully general and also allows for other sources of stress-energy, in particular additional relativistic sources like modified-gravity models or topological defects. We compare our method with the traditional Newtonian approach and argue that relativistic methods are conceptually more robust and flexible, at the cost of a moderate increase of numerical difficulty. However, for a LambdaCDM cosmology, where nonrelativistic matter is the only source of perturbations, the relativistic corrections are expected to be small. We quantify this statement by extracting post-Newtonian estimates from Newtonian N-body simulations.