Analysis of Large-Scale Matter Distribution with the Minimal Spanning Tree Technique

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The application of the Minimal Spanning Three technique to the description of large scale object distribution in observed and simulated catalogues is demonstrated. We show that it can be roughly described as a system of high density filaments half of which is accumulated by wall-like condensations. 1 Minimal Spanning Tree The MST is a construct from graph theory and is an unique network associated with a given point sample and connects all points of the sample to a tree in a special and unique manner which minimizes the full length of the tree. Cosmological implications of this technique were firstly discussed in Barrow, Bhavsar, & Sonoda (1985) and van de Weygaert (1991). Basically, the MST for a galaxy distribution contains within it all ‘friendsof-friends’ cluster catalogues for all linking lengths; if the MST is separated for a given linking length, one then has the ‘friends-of-friends’ cluster catalogue for that linking length. The probability distribution function of MST edge lengths, (PDF MST), WMST (l), depend on the correlation functions (or cumulants) of all orders. For larger point separations, however, where correlations become small and the cumulants tend to constants, the appearance of a Poisson-like point distribution can be expected. In such a case the PDF MST techniques can characterize the geometry of a point distribution and discriminate 1D, 2D and 3D Poisson-like point distributions. For the 1D and 2D Poissonian distributions of points there are analytical expressions for the PDF MST: WMST (l) = e−l/〈l〉, WMST (l) = 2 l 〈l2〉 − l2 〈l2〉 . When the measured PDFs are found to be similar to these theoretical expressions then it can be expected that the sample under investigation can be successfully constructed as a superposition of a system of 1D filaments and 2D sheets (or walls). ESO Symposia: Mining the Sky, pp. 283–285, 2001. c © Springer-Verlag Berlin Heidelberg 2001 284 A. Doroshkevich and V. Turchaninov 2 Wall-Like and Filamentary Structure Elements In catalogues under consideration the sample of structure elements can be identified with clusters found for a threshold linking length, lthr, and a threshold richness, Nthr. The threshold linking length characterizes actually the threshold overdensity, δthr, above the mean density bounded the cluster as δthr = b−3 = 3 4π〈ngal〉l thr , where 〈ngal〉 is the mean density of points in the sample and b is a dimensionless linking length. The threshold richness,Nthr, allows to select set of clusters with a given richness interval. The wall-like structure elements are usually identified with richer clusters and the threshold richness, Nthr, restricts the fraction of points, fpnt, associated with walls for a threshold overdensity used. 3 Selection Function For observed and simulated mock catalogues the density of galaxies varies with the distance due to the galaxy selection. For many catalogues the selection function, fs, can be well fitted to the expression (Baugh & Efstathiou 1993) dNgal(r) = fsdr ∝ x2 exp(−x1.5)dx, x = r/Rsel, where the typical scale Rsel depends on the considered catalogue. This means that the observed galaxy distribution becomes homogeneous if we will use the artificial radial coordinate, ra(r), instead of the real radial coordinate, r: