Paradoxes of cosmological physics in the beginning of the 21-st century

In the history of cosmology physical paradoxes played important role for development of contemporary world models. Within the modern standard cosmological model there are both observational and conceptual cosmological paradoxes which stimulate to search their solution. Confrontation of theoretical predictions of the standard cosmological model with the latest astrophysical observational data is considered. A review of conceptual problems of the Friedmann space expending models, which are in the bases of modern cosmological model, is discussed. The main paradoxes, which are discussed in modern literature, are the Newtonian character of the exact Friedmann equation, the violation of the energy conservation within any comoving local volume, violation of the limiting recession velocity of galaxies for the observed high redshift objects. Possible observational tests of the nature of the cosmological redshift are discussed

Fractal Approach to Large-Scale Galaxy Distribution

We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. Angular correlation function was used as a general instrument for the structure analysis. It was realized later that a normalization condition for the reduced correlation function estimator results in distorted values for both R_{hom} and fractal dimension D. Moreover, according to a theorem on projections of fractals, galaxy angular catalogues can not be used for detecting a structure with the fractal dimension D>2. For this 3-d maps are required, and indeed modern extensive redshift-based 3-d maps have revealed the ``hidden fractal dimension of about 2, and have confirmed superclustering at scales even up to 500 Mpc (e.g. the Sloan Great Wall). On scales, where the fractal analysis is possible in completely embedded spheres, a power--law density field has been found. The fractal dimension D =2.2 +- 0.2 was directly obtained from 3-d maps and R_{hom} has expanded from 10 Mpc to scales approaching 100 Mpc. In concordance with the 3-d map results, modern all sky galaxy counts in the interval 10^m - 15^m give a 0.44m-law which corresponds to D=2.2 within a radius of 100h^{-1}_{100} Mpc. We emphasize that the fractal mass--radius law of galaxy clustering has become a key phenomenon in observational cosmology.