Annu. Rev. Astron. Astrophys. 1988. 26:
245-294
Copyright © 1988 by . 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/*a*_{1}*a*_{2}, where
*a*_{1} and
*a*_{2} 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).