Annu. Rev. Astron. Astrophys. 1988. 26: 245-294 Copyright © 1988 by Annual Reviews. All rights reserved

3.1.4. POINT-SMOOTHING ANALYSIS

Gott et al. (80) [also see (80a, 84a, 211), J.R. Gott III & A.L. Melott (preprint, 1986), D. Weinberg & A.L. Melott (preprint, 1986), and A.L. Melott, D. Weinberg & J.R. Gott III (preprint, 1987)] introduce at least three new ingredients into the analysis of three-dimensional large-scale structure: point smoothing, high-contrast mapping, and a topological genus parameter to distinguish between different models. The analysis is done with data from the CfA redshift survey complete to apparent magnitude m_p ~ 14.5. A "cosmic cube" with an effective edge length of ~ 140 Mpc was selected for which the census of galaxies with absolute magnitude M_p - 20.5 is complete. Each of the 153 galaxies in the cosmic cube is given a unit mass that is smoothed into a density distribution over a volume with an effective smoothing length R_es 19 Mpc selected to be sufficiently large so that distorting effects of motions of galaxies [ef. (100b) and Sections 2.1.3 and 2.1.4] are "small." At each location in the cosmic cube, the total local density is then calculated by summing the contributions from the smoothed individual galaxies. The known density distribution in the cosmic cube is then used to construct a high-contrast three-dimensional map that consists of "white" and "black" regions corresponding to local densities larger or smaller than the medium value, respectively. Thus, half of the volume of the cosmic cube is "white" and half is "black." If theoretical considerations are neglected, then the resulting topology of the cosmic cube is surprising. Instead of white polka dots (individual superclusters) on a black background [model (a) in Section 1.1] or black polka dots (individual voids) on a white background (model (b) in Section 1.1], the white and black regions were found to be connected, equivalent, and completely interlocking, a connected white structure intertwined with a connected black structure (one supercluster, one void), a sponge-like topology (model (c) in Section 1.1].

Gott and coworkers point out that for a large class of cosmological models [cf. Oort (128, pp. 418-25) and Section 3.2.2], sponge-like topologies are physically consistent with having equivalent high- and low-density regions in the initial fluctuations. The topology does not change so long as the fluctuations are in the linear regime. The topology of the two-dimensional surface specified by the median density separating the black and white regions is characterized at a particular location by the Gaussian curvature K = 1/a₁a₂, where a₁ and a₂ are the principal radii of curvature at that point. For a large class of surfaces, the integral of the Gaussian curvature over the surface is given by the Gauss-Bonnet theorem, I = KdA = 4 (1 - g), where g is the genus of the surface. Loosely speaking, g is the number of holes contained in the surface: for a sphere, g = 0; for a donut, g = 1; for a sphere with N_h handles, g = N_h; and for two spheres, g = - 1. In applications, Gott and collaborators calculate (a) the genus of the median density surface and other contours of the smoothed density distribution and (b) the mean genus per unit volume. Results derived for the galaxy distribution in the cosmic cube of the CfA sample are compared with corresponding results for models derived from numerical simulations of galaxy clustering (cf. Section 3.2.2).