Scaling of voids in the large scale distribution of matter

TL;DR

This paper introduces a new method for defining and identifying voids in galaxy distributions using stochastic geometry, and applies it to fractal and cosmological simulations to analyze their scaling properties and transitions to homogeneity.

Contribution

It proposes a novel void-finder based on Delaunay and Voronoi tessellations and demonstrates its effectiveness on fractal and cosmological galaxy models.

Findings

Void rank ordering scales with Zipf's law in fractals.

Transition to homogeneity is observable in void distributions.

Bifractal mock galaxy samples show realistic void scaling behavior.

Abstract

Voids are a prominent feature of the galaxy distribution but their quantitative study is hindered by the lack of a precise definition of what constitutes a void. Here we propose a definition of voids in point distributions that uses methods of discrete stochastic geometry, in particular, Delaunay and Voronoi tessellations, and we construct a new void-finder. We then apply the void-finder to scaling point distributions. First, we find the voids of pure fractals with a transition to homogeneity and show that the rank ordering of the voids also scales (Zipf's law) and, in addition, shows the transition to homogeneity. However, a pure fractal is arguably not a good model of the galaxy distribution, so we construct from a cosmological $N$-body simulation a bifractal mock galaxy sample representing two galaxy populations, which we identify as "wall" and "field" galaxies. The wall galaxy distribution fits a pure fractal with a transition to homogeneity and, furthermore, the rank ordering of its voids shows a scaling range with the right slope plus a transition to homogeneity.