Mathematics – Probability
Scientific paper
Jan 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995phdt.........7s&link_type=abstract
Thesis (PH.D.)--THE JOHNS HOPKINS UNIVERSITY, 1995.Source: Dissertation Abstracts International, Volume: 56-03, Section: B, page
Mathematics
Probability
1
Scientific paper
We lay the groundwork for subsequent analyses of the higher order statistics of the distributions in galaxy catalogs, applying generating functions (GF). The GF of the distribution is connected to the continuum limit characteristic function. The factorial moments correspond to continuum limit the moments. The GF of the distribution is related to the GF of the hierarchal structure constants, the moments and factorial moments, and the combinants. We investigate N-point and cumulative distributions as well. We present a formalism describing a continuum limit non-Gaussian biasing scheme. GFs are used to describe how cluster correlations arise from the galaxy correlations. We find that the shape of the cluster correlation function is identical to the galaxy correlation function and we calculate the amplification. The results agree with observations: the biased density field has amplifications similar to rich clusters, and the galaxy -'cluster' cross-correlation satisfies the observed geometric mean law. Under the hierarchy we accurately evaluate the cell averages of correlations, used in moment methods. We extract the (factorial) moment correlators from the Lick, IRAS, and the APM catalogs. The resulting correlators are all proportional to the two point correlation function, substantially amplified, with an amplification nearly exponential with the total rank of the correlators. This confirms the validity of the hierarchical clustering assumption on the dynamic range examined, corresponding to 0.5 Mpc -50 Mpc in three dimensional space, while the Kirkwood superposition with loop terms is rejected. We obtain values of q N, the structure constants of the gravitational hierarchy, up to N = 10, considerably beyond previous attempts. The high quality of the APM catalog enabled us to disentangle the various contributions from the power spectrum, small scale nonlinear clustering, and combinatorial effects, which all affect the amplification of the correlators. These effects appear almost identically in correlations of clusters. Using this statistic an extremely simple gravitational hierarchy seems to emerge from the data providing useful clues to solve the BBKGY equations describing the gravitating statistical system. We investigate in detail the various sampling effects arising when reconstructing the distribution of galaxies. We use simple probability theory approach, Bayesian classifier theory, and Bayesian transition probabilities.
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