Multi-fractal analysis and lacunarity spectrum of the dark matter haloes in the SDSS-DR7

• We analysed the dark matter in Seventh Data Release of the Sloan Digital Sky Survey.
• From the initial sample with 412,468 galaxies, 339,505 dark matter haloes were used.
• We found the multifractal and the lacunarity spectrum as radial distance function.
• The dark matter set did not achieve at the physical dimension of the space.

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 412,468 galaxies and 339,505 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 universe.