7. Reliability of the results
We use superclusters that have member clusters with estimated redshift. This could be regarded as a serious objection against our results. That is why some tests are presented in this section to verify the influence of the redshift estimates on our conclusions.
First we note that a naive assumption that estimated redshifts would "smear" any weak correlation (if any) is not correct. Suppose we estimate cluster redshift with one estimator only - the magnitude of the tenth rank galaxy. It is known that the distribution of is clumpy, due to systematic effects (cf. Rowan-Robinson 1972). Then redshift estimates would be clumpy also, and preferred redshifts would appear. In a sense, the distances to clusters will be discrete and not continuous. Therefore the correlation would grow. This effect would be attenuated if there are several carefully selected estimators as in the our case.
We need to obtain the correlation function for samples containing only those superclusters having measured redshift for all cluster members. The sample sizes are not large and natural uncertainties would be larger. Secondly, if the KK catalog is not substantially different from other catalogs, we would expect the same correlation function for superclusters found according to quite different searching criteria. Finally a dependence of on distance R could be tested when samples of near and distant superclusters are examined.
7.1. Correlation function for superclusters with measured redshift
We have created some samples of superclusters with measured redshift for all cluster members. The new samples are denoted with zz instead of z. The results for samples 3N30zz.10 (n = 40), 3S30zz.10 (n = 28) and 3(N S)zz.10 for the bbb method are given in Fig. 19. Again 1000 random catalogs of superclusters are used. Again, the correlation function is zero, which is very strong evidence that results presented in Figs. 14-16 are consistent. The consistency is manifested for the other samples and for different density enhancements.
Now let us look at a sample, denoted as 3N30zc which together with 3N30zz constitute the sample 3N30. The sample 3N30zc includes samples 3N30z and 3N30c - containing superclusters with cluster members without measured redshift. In the light of previous results, it is not surprising that for 3N30zc.10 as well as for 3S30zc.10.
7.2. Correlation function for other catalogs
The correlation functions for three other catalogs are presented in Fig. 20:
Again the bbb method is used and 1000 simulated catalogs are generated for each case.
For all catalogs . Therefore the zero correlation is an intrinsic property of all superclusters found, independent of the searching procedure.
7.3. Correlation function for near and distant superclusters
Let us name near superclusters those having . Then distant superclusters have . Results for samples 3N30.10, 3S30.10 and 3N30z.10, containing respectively 44, 82 and 33 near, and 82, 49 and 31 distant superclusters are given in Fig. 21.
Obviously the incompleteness in the basic cluster catalog does not substantially affect the conclusions.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998