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