The dark matter halo distribution of the nearby universe is used to study the fractal behaviour in the proximate universe. The data, which is based on four volume-limited galaxy samples was obtained by Muñoz-Cuartas and Mueller (2012) from the Seventh Data Release of the Sloan Digital Sky Survey (SDSS-DR7). In order to know the fractal behaviour of the observed universe, from the initial sample which contains 412468 galaxies and 339505 dark matter haloes were used as input for the fractal calculations. Using this data we use the sliding-window technique for the dark matter distribution and compute the multi-fractal dimension and the lacunarity spectrum and use it to study its dependence on radial distance in every sample. The transition to homogeneity is not observed in the dark matter halo distribution obtained from the SDSS-DR7 volume-limited galaxy samples; in its place the dark matter halo distribution exhibits a persistent multi-fractal behaviour where the measured dimension does not arrive at the value of the physical dimension of the space, for all structure parameter values of the analysed set, at least up to radial distances of the ordered from 165 Mpc/h from the available centres of each sample. Our results and their implications are discussed in the context of the formation of large-scale structures in the

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.

In this work, we develop a statistical analysis of the large-scale clustering of matter in the Universe from the fractal point of view using galaxies from the Ninth Sloan Digital Sky Survey (SDSS) Data Release (DR9). From the total set of galaxies, a magnitude-limited sample of galaxies with redshifts in the range 0 < z < 0.15 was created. The sample covers the largest completely connected area of the celestial sphere within the catalogue, with limits in right ascension of 120 • < α < 240 • and declination 0 • < δ < 60 • , which is a region that includes the largest galactic samples that have been studied from the fractal viewpoint to date. The sample contains 164,168 galaxies.Using the sliding-window technique, the multifractal dimension spectrum and its dependence on radial distance are determined. This generalisation of the concept of fractal dimension is used to analyse large-scale clustering of matter in complex systems. Likewise, the lacunarity spectrum, which is a quantity that complements the characterisation of a fractal set by quantifying how the set fills the space in which it is embedded, is determined.Using these statistical tools, we find that the clustering of galaxies exhibits fractal behaviour that depends on the radial distance for all calculated quantities. A transition to homogeneity is not observed in the calculation of the fractal dimension of galaxies; instead, the galaxies exhibit a multifractal behaviour whose dimensional spectrum does not exceed the physical spatial dimension for radial distances up to 180 Mpc/h from each centre within the sample. Our results and their implications are discussed in the context of the formation of large-scale structures in the Universe.