Study of the fractal structure of the Large-scale matter distribution in the Universe (2015)
The current cosmological view asserts that the universe is homogeneous and isotropic; the observed heterogeneities are local in nature and should vanish at sufficiently large scales. This principle is based firstly on philosophical considerations: the observed universe must be statistically equal to any observer regardless of the point and direction of observation; and secondly, on cosmological observations: mainly isotropy measurements of the cosmic microwave background and clustering analysis of galaxies. This standard cosmological principle constitutes the basis of cosmology as a branch of physics. From the structure of space-time of the universe, to large-scale structure formation, this principle is present in the conceptual foundations, in the statistical information processes and the interpretation of the results. Regardless the success of the physical models based on the standard cosmological principle, there still remains unresolved fundamental questions within the formation of large scale structures in the universe. Is it supported by the observations the standard cosmological principle? And, What is scale of distance from which the homogeneity transition occurs? Already some research groups claim that astrophysical objects are grouped into highly structured hierarchical patterns with self-similarity properties, scale invariance and Hausdorff dimension less than the physical dimension of space, specific characteristics of fractal behaviour, where the standard cosmological principle is tested. Based on these concepts a topological analysis of large scale matter clustering in the universe is preformed from the fractal point of view. First is analysed the way in which dark matter is grouped at redshift z = 0 in the Millennium cosmological simulation. The determination of the homogeneity transition in the Millennium Simulation data is demonstrated from the behaviour of the fractal dimension and the lacunarity. The sliding window technique is used to determine the fractal mass-radius relation in order to find the fractal dimension, the pre-factor F and the lacunarity for the dark matter distribution in this simulation. In addition, the multi-fractal dimension and the lacunarity spectrum, including their dependence on a radial distance is obtained. These calculations demonstrate a radial distance dependency of all the fractal quantities, with heterogeneity clustering of dark matter haloes up to depths of 100 Mpc/h. Second, dark matter halo distribution is used in order to understand the fractal behaviour of the observed universe while avoiding the effects of luminosity selection. The data based on four limited-volume galaxy samples was obtained by Mu~noz-Cuartas and Mueller (2012) on 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, 339505 dark matter haloes were used as input the fractal calculations. Using again the sliding-window technique for dark matter distribution; the multi-fractal dimension and the lacunarity spectrum with its dependence on radial distance are determined in every sample. The dark matter halo clustering in the Millennium simulation shows a radial distance dependency of the calculated quantities with two clearly defined regions. The lacunarity spectrum for values of the structure parameter q 1 shows regions with relative maxima, which reveal the formation of clusters and voids in the distribution of dark matter haloes. Using the multi-fractal dimension spectrum and its complement the lacunarity spectrum, the transition to homogeneity is observed at depths from the centres ranging between 100 Mpc/h and 120 Mpc/h in the simulation. In contrast the homogeneity transition is not observed in the dark matter halo distribution obtained from the SDSS-DR7 limited-volume 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 the parameter values of the analysed structure, at least up to radial distances ordered from 165 Mpc/h from the available centres of each sample. Finally, the density contrast for the spherical collapse of a dark matter which evolves in a non-homogeneous universe is theoretically developed, taking as a basis the cosmological model of a spherically symmetric pressure less dust developed by Georges Lemaître, Richard C. Tolman y Hermann Bondi (LTB model), within the context of the General Relativity Theory. With the purpose of determining the density contrast for a LTB universe, a perturbation with E (r) 0 which evolves inside of a background with E (r) 0 in an expansion process is proposed. It was found a radial function for the density contrast (not a constant value independent of the radial coordinate), that in first-order of approximation reproduces the predicted value by the standard cosmology for a homogeneous universe. In order to find expression for the mass function in a fractal distribution of matter, the Excursion Set Theory is used for a moving barrier in order to find a mass function which depends on the cosmic density field variance and the power spectrum for a fractal distribution of matter. The main results of this thesis have been submitted for publication in international journals; the firrst one published in Monthly Notices of the Royal Astronomical Society Millennium entitled: \Millennium simulation dark matter haloes: multifractal and lacunarity" (Chacon-Cardona and Casas- Miranda, 2012), the second one submitted to the same journal, entitled: \Multi-fractal analysis and lacunarity spectrum of the dark matter haloes in the SDSS-DR7" is in the review process by academic peers, and the third one, entitled: \ Lemaître-Tolman-Bondi inhomogeneous dust: Contrast density for collapse and fractal mass function" submitted to the Physical Review D.