6. Correlation functions
Correlation functions are very powerful instruments to study the distribution of galaxies and clusters (Totsuji & Kihara 1969, Peebles 1973, 1980). It is established that the two-point space correlation functions of galaxies and clusters are power laws (with many modifications for dependence on the sample, luminosity, morphological type, richness etc.). But there is no agreement about the two-point space correlation function for superclusters of galaxies.
The first determination of was given by Kalinkov & Kuneva (1985). It is shown that for Mpc. Bahcall & Burgett (1986) have used the catalog of superclusters of Bahcall & Soneira (1983) to estimate the correlation function. Assuming the same power index as for galaxies and clusters, Bahcall & Burgett conclude
Moreover, based on the galaxy and cluster correlation functions
they presume the existence of a dimensionless or universal correlation function (such a function is discussed by Szalay & Schramm 1985, also). The universal function is quoted in many papers - e.g. Bahcall & West (1992).
First of all, Fig. 1 in Bahcall & Burgett (1986) is not convincing, because of the scatter of the six points in the range 90-140 Mpc. But the scatter in their Fig. 2 is even larger.
Kalinkov & Kuneva (1986) used another catalog of superclusters with conclusions:
Lebedev & Lebedeva (1988) have computed for the supercluster catalog of Batuski & Burns (1985a); this is a doubtful result because selections in the real catalog are not fully taken into account. Besides, the supercluster catalog is polluted, because of a crude estimation for redshift.
Now it is possible to compute for more superclusters and for various density enhancements.
We use the standard estimate
where DD is the number of supercluster pairs in the examined sample with space separation between and , and DR is the number of pairs in a cross correlation between the observed supercluster catalog and a simulated one. The objects in the mock catalog are in an identical volume and are subject to the same selection effects presented in the observed sample. The sample size is N. So the mock catalog contains just the same number of objects as the real one. We use for the uncertainty of the correlation function
Let us mention that the Hamilton (1993) estimator, namely
leads to the same results when (13) is used.
Various simulated catalogs with different assumptions are generated.
Obviously polynomials of higher order would produce catalogs very close to bootstrap resampling ones. But polynomials of first power, straight lines, operate as a function which smooths at the most possible degree. So the limiting cases, within which the real correlation function must be located, correspond to the bootstrap and polynomials of first power.
All calculated correlation functions are in the range defined by cases (bbb) and (111). However sometimes, when cumulative distributions are badly fitted with straight lines, the estimates according to (111) differ substantially from all other estimates.
In general, the first distance bin contains a biased estimate of the correlation function, since in the real samples there are no separations smaller than the diameters of the superclusters, while in the mock catalogs these separations could be smaller. In a more refined procedure for estimation of any correlation function, a cut-off parameter, or a softening distance, must be introduced.
Each correlation function is determined for 1000 "random" catalogs and in the examined cases all (i) - (v) various estimates do not differ substantially.
Fig. 14 presents for samples 3(N+S), and it is zero in the entire range of separations to 600 Mpc. One should note that, depending on bin, the first point of the smallest one is a biased estimate of the correlation function.
The same situation exists for samples 3N30 (Fig. 15). But for sample 3S30 (Fig. 16), there are indications for anticorrelation between 200 and 400 Mpc at all density enhancements. This might be due to uncertain redshift estimates.
The application of other estimators, including Landy & Szalay (1993), does not change the results (Kalinkov et al. 1997). Consequently the 2-point space correlation function of the superclusters is zero in the range .
We have attempted to estimate the correlation function as a function of the separation parallel and perpendicular to the line of sight, namely , e. g. Fisher et al. (1994). All attempts failed - is zero along as well as transverse to the line of sight.
This result that the correlation function is zero means there are no distortions in redshift space. Therefore, the correlation function in real space is zero too.
Up to now, we have been examining the autocorrelation function. Here some results on the crosscorrelation function will be given. All estimates are obtained with the bootstrap resampling technique.
Our calculation of the crosscorrelation function for superclusters and clusters show no systematic difference between north and south caps. Therefore, only the crosscorrelation function for merged samples worth examining.
The crosscorrelation function for superclusters and member clusters is given in Fig. 17, while Fig. 18 presents the crosscorrelation function with all A- and ACO-clusters. In fact is given. Of course only clusters located in the same volume defined for the superclusters are treated.
According to Fig. 17, the amplitude and the correlation radius of increase when the density contrast increases. Fig. 18 establishes that "field" clusters of galaxies which do not belong to any supercluster are not correlated with the superclusters. Naturally, the corresponding amplitudes for member clusters are higher than for all clusters.
Crosscorrelation functions were determined for clusters and galaxies (Seldner & Peebles 1977; more details in Peebles 1980) and Stevenson et al. (1985). Lilje & Efstathiou (1988) proposed another model for the crosscorrelation function. However the supercluster-cluster crosscorrelation function cannot be fitted with the models applicable to the cluster-galaxy crosscorrelation function.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998