Analysis of the Spatial Distribution of Galaxies by Multiscale Methods
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Analysis of the Spatial Distribution of Galaxies by Multiscale Methods J-L. Starck DAPNIA/SEDI-SAP, Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France Email: [email protected]
V. J. Mart´ınez Observatori Astron`omic de la Universitat de Val`encia, Edifici d’Instituts de Paterna, Apartat de Correus 22085, 46071 Val`encia, Spain Email: [email protected]
D. L. Donoho Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA 94305, USA Email: [email protected]
O. Levi Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA 94305, USA Email: [email protected]
P. Querre DAPNIA/SEDI-SAP, Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France Email: [email protected]
E. Saar Department of Cosmology, Tartu Observatory, Toravere 61602, Estonia Email: [email protected] Received 17 June 2004; Revised 17 February 2005 Galaxies are arranged in interconnected walls and filaments forming a cosmic web encompassing huge, nearly empty, regions between the structures. Many statistical methods have been proposed in the past in order to describe the galaxy distribution and discriminate the different cosmological models. We present in this paper multiscale geometric transforms sensitive to clusters, sheets, and walls: the 3D isotropic undecimated wavelet transform, the 3D ridgelet transform, and the 3D beamlet transform. We show that statistical properties of transform coefficients measure in a coherent and statistically reliable way, the degree of clustering, filamentarity, sheetedness, and voidedness of a data set. Keywords and phrases: galaxy distribution, large-scale structures, wavelet, ridgelet, beamlet, multiscale methods.
1.
INTRODUCTION
Galaxies are not uniformly distributed throughout the universe. Voids, filaments, clusters, and walls of galaxies can be observed, and their distribution constrains our cosmological theories. Therefore we need reliable statistical methods to compare the observed galaxy distribution with theoretical models and cosmological simulations. The standard approach for testing models is to define a point process which can be characterized by statistical
descriptors. This could be the distribution of galaxies of a specific type in deep redshift surveys of galaxies (or of clusters of galaxies).1 In order to compare models of structure formation, the different distribution of dark matter particles 1 Making 3D maps of galaxies requires knowing how far away each galaxy is from Earth. One way to get this distance is to use Hubble’s law for the expansion of the universe and to measure the shift, called redshift, to redder colors of spectral features in the galaxy spectrum. The greater the redshift, the larger the velocity, and, by Hubble’s law, the larger the distance.
2456 in N-body simulations could be analyzed as well, with the same statistics. The two-point correlation function ξ(r) has been the primary tool for quantifying large-scale cosmic structure [1]. Assuming that the galaxy distribution in the Universe is a realization of a sta
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