Gravitational Dynamics of an Infinite Shuffled Lattice: Particle Coarse-grainings, Non-linear Clustering and the Continuum Limit

TL;DR

This paper investigates the gravitational evolution of infinite shuffled lattice particle distributions, comparing coarse-grained systems with original ones, and explores the continuum limit where a Vlasov-like description becomes valid.

Contribution

It introduces a coarse-graining approach to analyze gravitational clustering and clarifies conditions under which continuum descriptions are applicable.

Findings

Coarse-grainings effectively reproduce two-point correlation functions over certain scales.

Differences between original and coarse-grained systems grow with increased coarse-graining scale.

A finite-time coarse-graining approach better captures the self-similar evolution.

Abstract

We study the evolution under their self-gravity of infinite ``shuffled lattice'' particle distributions, focussing specifically on the comparison of this evolution with that of ``daughter'' particle distributions, defined by a simple coarse-graining procedure. We consider both the case that such coarse-grainings are performed (i) on the initial conditions, and (ii) at a finite time with a specific additional prescription. In numerical simulations we observe that, to a first approximation, these coarse-grainings represent well the evolution of the two-point correlation properties over a significant range of scales. We note, in particular, that the form of the two-point correlation function in the original system, when it is evolving in the asymptotic ``self-similar'' regime, may be reproduced well in a daughter coarse-grained system in which the dynamics are still dominated by two-body (nearest neighbor) interactions. Using analytical results on the early time evolution of these systems, however, we show that small observed differences between the evolved system and its coarse-grainings at the initial time will in fact diverge as the ratio of the coarse-graining scale to the original inter-particle distance increases. The second coarse-graining studied, performed at a finite time in a specified manner, circumvents this problem. It also makes more physically transparent why gravitational dynamics from these initial conditions tends toward a ``self-similar'' evolution. We finally discuss the precise definition of a limit in which a continuum (specifically Vlasov-like) description of the observed linear and non-linear evolution should be applicable.