Millennium simulation dark matter haloes: multifractal and lacunarity analysis and the transition to homogeneity

Abstract We investigate from a fractal viewpoint the way in which dark matter is grouped at z = 0 in the Millennium dark matter cosmological simulation. Determination of the crossing point to homogeneity in the Millennium Simulation data is described with regard to the behaviour of the fractal dimension and 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. Additionally, we determine the multifractal dimension and the lacunarity spectrum, including their dependence on radial distance. The calculations show a radial distance dependence of all fractal quantities, with heterogeneous clustering of dark matter haloes up to depths of 100 Mpc h −1. Dark matter halo clustering in the Millennium Simulation shows a radial distance dependence, 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 halo distribution. With use of the multifractal dimension and the lacunarity spectrum, the transition to homogeneity at depths between 100 Mpc h −1 and 120 Mpc h −1 for Millennium Simulation dark matter haloes is detected.