5. Voids among superclusters
Many attempts have been undertaken to define and to characterize voids in the distribution of galaxies and clusters, but there were not sufficient statistics for superclusters. The KK catalog gives an opportunity to study the voids among superclusters.
Let us consider the probability that a disk of area A randomly placed on the 2D distribution of superclusters contains no superclusters. According to White (1979), the 2D void probability function (VPF), namely , where is the 2D mean density, depends on the correlation functions of all orders. It is easy to show that for a Poisson distribution of points
We have applied the VPF to some supercluster samples (Fig. 12). Continuous lines refer to the theoretical VPF, while squares are the observed VPF with Poisson error. Each square is a result of 10000 simulations of the supercluster catalogs obtained by the bootstrap resampling technique. Of course, all randomly distributed disks intersecting the edge were rejected.
Clearly there are no significant voids in the 2D distribution of the superclusters. Having in mind that the superclusters are not points, one would expect even better agreement with a Poisson distribution.
Therefore, the 2D distribution of the voids among superclusters is Poissonian.
Voids in the 3D case, if any, should be named supervoids. The space distribution of the voids essentially depends on a decrease of the mean density D of superclusters with distance. The corresponding VDF is
where is a sphere of radius r which does not contain any supercluster, randomly placed among the real superclusters.
where the coefficients are from Table 6. Note that the VDF depends on the distance and the radius of the empty spheres.
The observed VDF is derived again from random supercluster catalogs but without bootstrap resampling technique. Each random supercluster is simulated with distance according to linear regressions from Table 6 and random galactic coordinates.
But a comparison between observed and theoretical VPF is not so simple as the 2D case since there is an additional parameter, namely the radius r.
Some results are presented in Fig. 13 for samples 3N40 and 3S40. The curves refer to theoretical, and squares to observed VPFs. Radii and 100 Mpc are given. Each square is the result of 10000 random supercluster catalogs.
In general, the agreement between theory and observation is very good. Squares in the first bin are biased estimates, since the volume where random supercluster centers are generated is smaller in order to avoid edge-on effects.
Fig. 13 shows that there is no definitive excess of observed over theoretical VPF. All samples of Table 1, having large size, say , lead to the same result. Hence supervoids among superclusters in real space do not exist with volume in the range Mpc.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998