7 Summary

We have presented in this paper the so far most complete mapping of the large ($\geq$ 50 h^-1 Mpc) voids of clusters of galaxies in the Northern Galactic Hemisphere for $b \geq$ +30$^{\circ }$ to a limiting distance of 420 h^-1 Mpc. Abell/ACO clusters with spectroscopically measured redshifts, as well as with estimated redshifts, have been used as tracers of the large-scale structure, combining the data from four different redshift sources. The analysis of the spatial number density has shown that the $R \geq$ 1 A/ACO clusters can form complete samples to a limiting distance of $\sim$ 270 h^-1 Mpc, while the samples of $R \geq$ 0 clusters are incomplete even in this nearer distance range.

Eight observational samples have been composed and processed for two types of tracers: (1) $R \geq$ 1 A/ACO clusters, and (2) $R \geq$ 0 A/ACO clusters, varying for each tracer type the priority of the source of redshift data (LEB or NED), and varying, as well, the redshift type (measured redshifts versus measured or estimated redshifts).

The observational samples have been processed with the help of an Automated Void Search and Analysis System (AVSAS). The void-search algorithm in AVSAS is based on an identification of the local maxima in the nearest-neighbour distance field, defined on a regular grid. Two alternative methods, the p-method and the $\nu$-method, are used to identify the local maxima. The first one searches the maxima by direct comparisons of the distance field values, and the second one uses for this purpose the standard deviation of the field values. The number of detected local maximum points depends on the choice of an appropriate value for a parameter p for the first method, respectively $\nu$ for the second method. Next, the search algorithm groups the local maximum points according to their mutual positions on the basis of a choice between three possible options for grouping - compact, medium-compact, and loose. Each group of local maximum points identifies a separate void, which is approximated by a system of overlapping empty spheres. The approximation of a void by more than one sphere (the largest empty sphere) allows for a more complete description of the void properties - dimension, volume and shape.

The tests of the void-search procedure have shown that it can be optimized by choosing such values of the parameters p and $\nu$ which maximize the number of detected voids for a minimum degree of overlap of the neighbouring voids.

Choosing a grid constant k = 10 h^-1 Mpc and a minimum dimension for the large voids of D_min = 50 h^-1 Mpc, we have processed the 8 observational samples with both search methods (p and $\nu$) and with all three grouping options (compact, medium-compact, loose), producing in this way 48 void catalogues. However, in this paper only the 16 catalogues corresponding to the medium-compact grouping option have been used to analyse further the void properties.

The generated void catalogues, compared to similar wide-field studies, contain a larger number of voids out to larger distances, described by more parameters (e.g. equivalent diameter, total volume, sphericity). They can be used as identification lists of the voids of clusters in the Northern Galactic Hemisphere, for studies of individual voids, as well as for statistical studies.

The examination of visualizations of the 3-D distribution of the voids from different void catalogues suggests a void-filled Universe with closely packed and intersecting voids. 2-D cross-sections of the 3-D distribution, however, indicate that the large voids may be separated by large zones of enhanced density of the tracing objects.

The automated comparison of the void catalogues corresponding to tracers of different richness class has shown that most of the voids in the distribution of the $R \geq$ 1 A/ACO clusters can be identified in the distribution of the $R \geq$ 0 clusters.

The comparison of our void catalogues with compilations of voids known from other studies has shown that about 20% of the voids in the nearer and more complete volume V1 ($z \leq$ 0.09) and 90% of the more distant voids are newly discovered voids or void candidates. Their reality must be confirmed by future complete redshift surveys of clusters of galaxies.

The statistical significance of the voids of $R \geq$ 1 A/ACO clusters has been verified by a two-sample Kolmogorov-Smirnov test. The results from the test reject the hypothesis that the large voids of rich clusters are random fluctuations.

We have processed statistically the void catalogues in order to derive mean characteristics (mean and median values) for the void dimensions, volumes, and sphericities. Analyzing the dependence of the diameters of the void constituent spheres on distance and on galactic latitude, we have found that homogeneous samples of voids of $R \geq$ 1 A/ACO clusters not affected significantly by the observational selection should be extracted from a volume limited by $b \geq$ +50$^{\circ }$ and $z \leq$ 0.09. We suggest that our estimates of the equivalent void diameters of the $R \geq$ 1 and $R \geq$ 0 A/ACO clusters $D_\mathrm{e} = 105.0\pm5.6$ h^-1 Mpc, and $D_\mathrm{e} = 87.2\pm4.1$ h^-1 Mpc, respectively, are improved estimates of the dimensions of large voids. We have obtained for the first time mean estimates of the volumes and sphericities of the voids of clusters. The high value of the mean sphericity ($\sim$ 0.9) shows that the voids of clusters have quite spherical shapes.

Jointly processing the void catalogues and samples of objects of different type, we have identified the populations of clusters, groups, and galaxies in the voids of $R \geq$ 1 A/ACO clusters, as well as in the void shells. The estimates of the mean number densities of the void populations show that the voids of $R \geq$ 1 A/ACO clusters are also underdense regions in the distribution of the poorer clusters and groups, and in the distribution of galaxies. We have constructed radial density profiles of the voids of $R \geq$ 1 A/ACO clusters which suggest that they contain smaller voids of poorer clusters and groups, and still smaller voids of galaxies, in agreement with the concept for a void hierarchy.

Acknowledgements

The author is very grateful to David Batuski and his co-authors for providing the data of the MX Northern Abell Cluster Redshift Survey in advance of publication, as well as to Viktor Lebedev and Irina Lebedeva for the use of the updated version of their Compilation of Clusters of Galaxies with Published Redshifts. Thanks are due to Heinz Andernach for providing redshift data for the Abell/ACO clusters. His remarks on the redshifts used in this study were very helpful for improving the data. Thanks are due to Rudolf Dümmler and Gotthard Richter for useful comments on part of the results. The author wishes to thank Milcho Tsvetkov for the use of the computer complex of the Wide-Field Plate Database, as well as Katya Tsvetkova and Evgeni Semkov for help in the preparation of the manuscript. Thanks are due to the anonymous referee for a number of helpful suggestions. This research has made use of the NASA/IPAC extragalactic database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The present study was partly supported by the National Science Fund of Bulgaria under Grant F-650/1996.