Millennium Simulation Dark Matter Haloes: Multi-fractal and Lacunarity Analysis with Homogeneity Transition

TL;DR

This study uses fractal analysis to examine the distribution of dark matter haloes in the Millennium Simulation, identifying a transition to homogeneity around 100-120 Mpc/h through multi-fractal and lacunarity measures.

Contribution

It introduces a detailed fractal and lacunarity analysis of dark matter haloes, revealing the scale of homogeneity transition in the Millennium Simulation data.

Findings

Dark matter haloes exhibit radial distance-dependent clustering.

Lacunarity spectrum indicates formation of clusters and voids.

Homogeneity transition occurs between 100 and 120 Mpc/h.

Abstract

We investigate from the fractal viewpoint the way in which the dark matter is grouped at z = 0 in the Millennium dark matter cosmological simulation. The determination of the cross to homogeneity in the Millennium Simulation data is described from the behaviour of the fractal dimension and the lacunarity. We use the sliding window technique to calculate the fractal mass-radius dimension, the pre-factor F and the lacunarity of this fractal relation. Besides, we determinate the multi-fractal dimension and the lacunarity spectrum, including their dependence with radial distance. This calculations show a radial distance dependency of all the fractal quantities, with heterogeneity clustering of dark matter haloes up to depths of 100 Mpc/h. The dark matter haloes clustering in the Millennium Simulation shows a radial distance dependency, with two regions clearly defined. The lacunarity spectrum for values of the structure parameter q >= 1 shows regions with relative maxima, revealing the formation of clusters and voids in the dark matter haloes distribution. With the use of the multi-fractal dimension and the lacunarity spectrum, the transition to homogeneity at depths between 100 Mpc/h and 120 Mpc/h for the Millennium Simulation dark matter haloes is detected.