General Relativistic N-body simulations in the weak field limit

TL;DR

This paper introduces a formalism for simulating N-body systems under general relativity in the weak field limit, capturing key nonlinear effects and enabling more accurate cosmological modeling including dark energy and modified gravity.

Contribution

It develops a tractable formalism for relativistic N-body simulations that incorporates nonlinear Einstein gravity effects and demonstrates its feasibility through plane-symmetric test cases.

Findings

Deviations from Newtonian simulations are small in tested scenarios.

The scheme accurately estimates the backreaction correction to the background.

The approach is adaptable for dark energy and modified gravity models.

Abstract

We develop a formalism for General Relativistic N-body simulations in the weak field regime, suitable for cosmological applications. The problem is kept tractable by retaining the metric perturbations to first order, the first derivatives to second order and second derivatives to all orders, thus taking into account the most important nonlinear effects of Einstein gravity. It is also expected that any significant "backreaction" should appear at this order. We show that the simulation scheme is feasible in practice by implementing it for a plane-symmetric situation and running two test cases, one with only cold dark matter, and one which also includes a cosmological constant. For these plane-symmetric situations, the deviations from the usual Newtonian N-body simulations remain small and, apart from a non-trivial correction to the background, can be accurately estimated within the Newtonian framework. The correction to the background scale factor, which is a genuine "backreaction" effect, can be robustly obtained with our algorithm. Our numerical approach is also naturally suited for the inclusion of extra relativistic fields and thus for dark energy or modified gravity simulations.