Mathematics – Probability
Scientific paper
Nov 1988
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1988pasp..100.1343h&link_type=abstract
Astronomical Society of the Pacific, Publications (ISSN 0004-6280), vol. 100, Nov. 1988, p. 1343-1350.
Mathematics
Probability
6
Astronomical Models, Cosmology, Fractals, Topology, Universe, Many Body Problem, Normal Density Functions, Random Walk, Statistical Analysis
Scientific paper
It is shown how the genus per unit volume of isodensity surfaces in general nonlinear universes is related to the entire hierarchy of correlation functions. The general relation between the correlation function, the probability distribution of densities at several points, and the probability distributions of density and its derivatives at a point are given. Formulas for the area and genus per unit volume of isodensity surfaces are presented. As an application, after first reviewing the case of Gaussian fields, analytic results are reported for one particular example of a thoroughly nonlinear universe, Mandelbrot's Rayleigh-Levy random-walk fractal. While this fractal bears little resemblance to the real universe of galaxies, it possesses the singular and theoretically interesting property that in it cluster-cluster correlations are identically equal to galaxy-galaxy correlations to all orders.
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