Beyond Genus Statistics: A Unifying Approach to the Morphology of Cosmic Structure

Abstract

The genus statistics of isodensity contours has become a well-established tool in cosmology. In this Letter we place the genus in the wider framework of a complete family of morphological descriptors. These are known as the Minkowski functionals, and we here apply them for the first time to isodensity contours of a continuous random field. By taking two equivalent approaches, one through differential geometry, the other through integral geometry, we derive two complementary formulae suitable for numerically calculating the Minkowski functionals. As an example we apply them to simulated Gaussian random fields and compare the outcome to the analytically known results, demonstrating that both are indeed well suited for numerical evaluation. The code used for calculating all Minkowski functionals is available from the authors.Subject headings: large-scale structure of universe, methods: statistical

Statistics of Smoothed Cosmic Fields in Perturbation Theory. I. Formulation and Useful Formulae in Second‐Order Perturbation Theory

“…1 ¼ 2, ! 2 ¼ , and Schmalzing & Buchert 1997). It turns out that the Minkowski functionals in two and three dimensions are identical to the statistics N 1 , G 2 , and G 3 in each dimension, except for normalization factors.…”

“…Rather recently, more complex statistics of smoothed cosmic fields have become popular in cosmology, such as the genus statistic (Gott, Melott, & Dickinson 1986), density peak statistics (Bardeen et al 1986), area, length, and level-crossing statistics (Ryden 1988a), Minkowski functionals (Schmalzing & Buchert 1997), etc. These statistics provide assuring characterizations of the clustering pattern that cannot be perceived only by the hierarchy of cumulants or by the PDF.…”

Morphometry of spatial patterns

“…The smoothing length determines the scale of interest. Then we construct the excursion sets and investigate their topology and geometry using the Quermaß vectors and the Minkowski functionals [17]. Figure 6 vs. the density threshold u (u is given in units of the mean density within the cluster, N is the number of cluster particles, R m the maximum distance of cluster particles from the center of mass of all cluster particles).…”

Studying topological structure of 21-cm line fluctuations with 3D Minkowski functionals before reionization

“…Negative value of V3, on the other hand, indicates that the excursion set has a multi-connected structure. The calculation method of the MFs is shown in Gleser et al (2006) and Schmalzing & Buchert (1997). When p(x) has a gaussian distribution, the MFs can be written analytically and have symmetric shapes, as shown in Fig.…”

“…The MFs are known to be useful to characterize this kind of topology. In the context of observational cosmology, the MFs have been used to investigate the geometrical feature of galaxy distribution (Gott et al 1986;Schmalzing & Buchert 1997) and to estimate the non-gaussianity appeared in the CMB temperature map (Komatsu et al 2009). …”