Clustering statistics in cosmology

The main tools in cosmology for comparing theoretical models with the observations of the galaxy distribution are statistical. We will review the applications of spatial statistics to the description of the large-scale structure of the universe. Special topics discussed in this talk will be: description of the galaxy samples, selection effects and biases, correlation functions, Fourier analysis, nearest neighbor statistics, Minkowski functionals and structure statistics. Special attention will be devoted to scaling laws and the use of the lacunarity measures in the description of the cosmic texture.Comment: 15 pages, 7 figures, uses spie.cls (included). This paper will be published in SPIE Proceedings Vol. 4847, 2002, "Astronomical Data Analysis II," J.-L. Stark and F. Murtagh, eds., and is made available as an electronic preprint with permission of SPI

Cosmologists in the dark

We review the present status of cosmological discoveries and how these confirm our modern cosmological model, but at the same time we try to focus on its weaknesses and inconsistencies with an historical perspective, and foresee how the on-going big cosmological projects may change in the future our view of the universe.Comment: 10 pages, ASP style (asp2006.sty), invited talk, to be published in the proceedings of the conference "Cosmology across Cultures" (held at Granada, Spain, on 2008, September 8th to 12th), J. A. Belmonte, F. Prada, J. A. Rubino Martin, & A. Alberdi, Eds., ASP, S. Francisco. Comments are welcom

Searching for the scale of homogeneity

We introduce a statistical quantity, known as the $K$ function, related to the integral of the two--point correlation function. It gives us straightforward information about the scale where clustering dominates and the scale at which homogeneity is reached. We evaluate the correlation dimension, $D_2$, as the local slope of the log--log plot of the $K$ function. We apply this statistic to several stochastic point fields, to three numerical simulations describing the distribution of clusters and finally to real galaxy redshift surveys. Four different galaxy catalogues have been analysed using this technique: the Center for Astrophysics I, the Perseus--Pisces redshift surveys (these two lying in our local neighbourhood), the Stromlo--APM and the 1.2 Jy {\it IRAS} redshift surveys (these two encompassing a larger volume). In all cases, this cumulant quantity shows the fingerprint of the transition to homogeneity. The reliability of the estimates is clearly demonstrated by the results from controllable point sets, such as the segment Cox processes. In the cluster distribution models, as well as in the real galaxy catalogues, we never see long plateaus when plotting $D_2$ as a function of the scale, leaving no hope for unbounded fractal distributions.Comment: 9 pages, 11 figures, MNRAS, in press; minor revision and added reference