Minkowski functionals of Abell/ACO clusters

Abstract

We determine the Minkowski functionals for a sample of Abell/ACO clusters, 401 with measured and 16 with estimated redshifts. The four Minkowski functionals (including the void probability function and the mean genus) deliver a global description of the spatial distribution of clusters on scales from 10 to 60h −1 Mpc with a clear geometric interpretation. Comparisons with mock catalogues of N-body simulations using different variants of the CDM model demonstrate the discriminative power of the description. The standard CDM model and the model with tilted perturbation spectrum cannot generate the Minkowski functionals of the cluster data, while a model with a cosmological constant and a model with breaking of the scale invariance of perturbations (BSI) yield compatible results.

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“…F(r) is equal to the volume fraction occupied by the set of all points in D that are closer than r to a point of the process. Hence, F(r) coincides with the volume density of the Ðrst Minkowski functional (Mecke et al 1994 ;Kerscher et al 1997), and is related to the void probability function via P 0 (r) F(r) \ 1 [ P 0 (r). For a homogeneous Poisson process, we have…”

Morphology of the local volume

“…Minkowski functionals provide a unifying framework for the topology and the geometrical quantities like volume, surface area, and integrated mean curvature. They have been developed for the morphological characterisation of the large scale distribution of galaxies and galaxy clusters by Mecke et al (1994) and have successfully been used in cosmology (Kerscher et al 1997;Kerscher 2000), porous and disordered media, dewetting phenomena, fluid and magneto hydrodynamics (see Mecke 2000, for a review). In order to quantify the spatial distribution of the galaxies we decorate the galaxies with balls of varying radii (see Fig.…”

“…The code used for the calculations of the Minkowski functionals is an updated version of the code developed by Kerscher et al (1997), based on the methods outlined in Mecke et al (1994). The code is made available to the general public via http:// www.math.lmu.de~kerscher/software/ Below we present the boundary corrections used by Schmalzing et al (1996).…”

The cluster distribution as a test of dark matter models - IV. Topology and geometry

“…Rhoads, Gott & Postman (1994) have used the more standard 'genus' algorithm to study the topological properties of contour surfaces constructed from the cluster distribution, including the different choice of ν we described above. On the other hand, Kerscher et al (1997) have used a different mathematical approach, based on the so-called Minkowski functionals, which incorporates as one of the descriptors a quantity analogous to the genus; for a further development of these ideas, see Schmalzing & Buchert (1997). These analyses are to some extent similar in spirit to that which we present here, but there are significant differences in both philosophy and implementation.…”