Spherically Symmetric N-body Simulations with General Relativistic Dynamics

TL;DR

This paper presents a spherically symmetric N-body simulation framework within General Relativity, capable of handling relativistic velocities, large gradients, and shell-crossings, providing more accurate results than Newtonian models for cosmological structures.

Contribution

The authors develop a relativistic N-body simulation method that accurately models collisionless particles in spherical symmetry, surpassing Newtonian approaches and enabling studies of bound objects and modified gravity effects.

Findings

The scheme accurately reproduces leading-order post-Newtonian corrections.

It can handle shell-crossings, unlike fluid models.

Configurations with angular momentum suggest bound objects for further study.

Abstract

Within a cosmological context, we study the behaviour of collisionless particles in the weak field approximation to General Relativity, allowing for large gradients of the fields and relativistic velocities for the particles. We consider a spherically symmetric setup such that high resolution simulations are possible with minimal computational resources. We test our formalism by comparing it to two exact solutions: the Schwarzschild solution and the Lema\^itre-Tolman-Bondi model. In order to make the comparison we consider redshifts and lensing angles of photons passing through the simulation. These are both observable quantities and hence are gauge independent. We demonstrate that our scheme is more accurate than a Newtonian scheme, correctly reproducing the leading-order post-Newtonian correction. In addition, our setup is able to handle shell-crossings, which is not possible within a fluid model. Furthermore, by introducing angular momentum, we find configurations corresponding to bound objects which may prove useful for numerical studies of the effects of modified gravity, dynamical dark energy models or even compact bound objects within General Relativity.