5 Analysis of the void catalogues

5.1 Automated comparison of the void catalogues

The generation of a large number of void catalogues by varying the sample, the search method, or the void selection criterion has evoked the necessity for an easy and objective way to compare the results. AVSAS offers such a possibility, applicable also to the comparison with voids from previous investigations.

The comparison of two void catalogues is based on the comparison of the positions and dimensions of the CS of the voids. The criterion for coincidence of two voids from different catalogues is $\Delta r < \mathrm{max}\,(d_1,\ d_2)$, where $\Delta r$ is the distance between the centres of the compared CS, and $d_1,\ d_2$ are their radii. This criterion must be satisfied for at least one pair of coinciding CS. A rigorous criterion for void coincidence may be chosen by setting $\Delta r =$ 0. The results from the comparison are output as cross-identification tables with the coinciding voids, and as detailed lists of the coinciding CS with the differences in their positions and dimensions.

We have applied both the rigorous and loose criteria to compare the void catalogues listed in Table 2. Table 5 contains the results of the comparison of the two catalogues AR/Lp and A/Lp given in Tables 3 and 4, respectively. The numbers in Table 5 marked by an asterisk correspond to cases satisfying the rigorous criterion $\Delta r =$ 0. As it is seen in Table 5 most of the voids of $R \geq 1$ clusters (90%) can be identified with voids of $R \geq 0$ clusters.

  

Table 5: Identification of the voids of $R \geq 1$ A/ACO clusters (catalogue AR/Lp) with voids of $R \geq 0$ clusters (catalogue A/Lp)

We have compared also our void catalogues with voids known from previous investigations. To do this we have composed a compilation of voids in the NGH, hereafter referred to as VC, containing 39 voids of clusters of galaxies from the following sources: (1) Bahcall & Soneira ([1982b]) - one void of $R \geq 1$ Abell clusters; (2) Batuski & Burns ([1985]) - 22 voids of Abell clusters; (3) Einasto et al. ([1994]) - 16 voids of A/ACO clusters. We shall use the abbreviations BS, BB, and E for sources 1-3, respectively.

The compilation VC occupies a volume considerably smaller than the volume occupied by our void catalogues: the void centres reach a limiting distance of about 250 h^-1 Mpc. The dimensions of the voids of clusters from VC correspond roughly to our criterion for void selection D_min = 50 h^-1 Mpc.

Before the comparison, the voids from VC with non-spherical shapes have been reduced to the largest empty spheres which can be nested in them, with the exception of the huge BS void which, because of its extended form, has been approximated with 21 non-overlapping equal spheres with diameters of 60 h^-1 Mpc.

The results of the comparison of the void catalogues AR/Lp and A/Lp (Tables 3 and 4) with the voids from the compilation VC are given as identification lists in Table 6, where the first column contains the serial numbers (from Tables 3 and 4) of the voids identified in VC, and the second column contains the identifications from VC, presented with their original numbers from the corresponding literature sources, preceded by the source abbreviation. The comparison shows that about 70% of the voids of clusters from VC can be identified in at least one of the two void catalogues. On the other hand, 35-40% of the voids in these catalogues, most of them in the nearer subvolume V1, are identified in VC. The rest > 60% of the voids lie predominantly beyond subvolume V1, i.e. in regions not covered by the compilation VC. We can conclude that about 20% of the voids detected by us in volume V1 and 90% of the voids outside volume V1 are newly discovered voids or void candidates. The reality of the latter needs to be confirmed by future complete redshift surveys.

5.2 Kolmogorov-Smirnov test

We have applied the two-sample Kolmogorov-Smirnov test in an attempt to estimate the statistical significance of the large voids in the distribution of clusters of galaxies. A similar approach for the voids of galaxies has been used by Ryden & Turner ([1984]).

Since the statistic we use for comparison of the observed distribution of clusters with random distributions - the values of the d-field - is very sensitive to the sample completeness, we have restricted the test to appropriate subsamples of the samples of $R \geq 1$ A/ACO clusters.

The observed sample used in the test is a subsample from AR/L limited to volume V1 ($b \geq$ +40$^{\circ }$, $z \leq$ 0.09) with 128 clusters. It is compared to a random sample generated with the same number of objects in the same volume. The number of points (grid nodes) of the d-field in volume V1 is 14736.

For each run of the test the maximum value of the test statistic D has been compared with the critical value $D_{\alpha}$ for three values of the significance level $\alpha$: 0.1, 0.05, and 0.01. The result for a series of 50 generations of random samples is: the null hypothesis (observed sample and random sample are from the same population) is rejected for all significance levels.

  

Table 6: Identification of the voids of $R \geq 1$ and $R \geq 0$ A/ACO clusters (catalogues AR/Lp and A/Lp) with voids known from previous studies

The relative frequency distributions of the values of the d-field (the nearest-neighbour distances) of the observed and random samples for one run of the Kolmogorov-Smirnov test are compared in Fig. 11. The excess of large voids in the observed distribution in comparison with the random distribution is well manifested in this figure.

Figure 11: Comparison of the relative frequency distributions of the distance field values for the observed sample and a random sample generated in one run of the Kolmogorov-Smirnov test. A spline fit is used for the random distribution

One could argue that the definite result of the Kolmogorov-Smirnov test is due to the incompleteness of the observed sample. Therefore, we have repeated the test with a more "clean'' subsample limited by $b \geq$ +50$^{\circ }$ and 0.0133 $\leq z \leq$ 0.08, with 76 clusters of galaxies and 6712 grid nodes. As a result from 50 generations of random samples the null hypothesis has been rejected in this case, too.

We conclude that if our samples are not seriously affected by incompleteness, the observed voids in the distribution of $R \geq 1$ A/ACO clusters are not likely to be random fluctuations. This conclusion is in disagreement with the results of Otto et al. ([1986]) who have suggested a mechanism of void formation based on the interpretation of rich clusters as rare events in the primordial Gaussian density fluctuations. Our result could possibly indicate that the distribution of rich clusters might be described by a statistic more complicated than a simple Gaussian, as suggested by Plionis et al. ([1992]).

5.3 Statistical analysis of the void catalogues

The generated large number of void catalogues (Sect. 4) allows for various statistical investigations of the voids, and first of all, for the determination of the mean void characteristics describing the void dimension, volume, and shape.

We have first calculated for all the 16 catalogues generated with the medium-compact grouping option (see Table 2) the mean and median values for the diameters of the largest CS and the equivalent void diameters, the volumes of the largest CS and the total void volumes, as well as the void sphericities, separately for volumes V1 and V2.

The difference between the mean dimensions for volumes V1 and V2 is very large - about 30-35% for all types of tracers with measured redshifts -, due mainly to the incompleteness in volume V2, and probably partly to the real presence of larger voids in this much larger volume. This difference reduces to 10-15% when A/ACO clusters with estimated redshifts are used in addition to the clusters with measured redshifts (catalogues Nos. 2, 4, 6, 8, and 10, 12, 14, 16). It can be explained as a combined effect of large errors in the estimated cluster distances and the real presence of larger voids in V2 compared with V1.

The comparison of the mean dimensions for different search methods (p or $\nu$) shows that the $\nu$-method gives systematically higher values. Contrary to this, the voids based on different sources of redshift data (L or N) do not show significant systematic differences in their mean dimensions.

To study better the dependence of the mean void dimensions on the completeness of the samples of objects used we have constructed plots of the type diameter - distance and diameter - galactic latitude for all void catalogues using the diameters and positions of the centres of the constituent void spheres. Such a plot for catalogue AR/Lp concerning the voids of $R \geq 1$ A/ACO clusters in volume V1 is presented in Fig. 12 which shows a well defined dependence of the diameters of the CS in volume V1 on the galactic latitude as an apparent presence of larger voids in the latitude zone $+40^{\circ }\,-\,+50^{\circ }$. This effect is most probably due to observational selection caused by the galactic obscuration. A similar dependence on the galactic latitude has been found for the other void catalogues, too. Therefore, we conclude that better estimates of the mean void characteristics will be obtained for a subvolume V1A of volume V1 with $b \geq +50^{\circ }$, in spite of the substantial decrease of the number of voids used for statistics.

A similar check of the dependence of the CS diameters on the distance has shown that the dimensions of the voids in volume V1 are not significantly affected by the observational selection due to the distance.

As seen from Table 2 in Sect. 4, we have generated four void catalogues for each of the four tracer types (AR, ARE, A, AE), corresponding to the four combinations of redshift data source and search method: Lp, L $\mathrm{\nu}$, Np, N $\mathrm{\nu}$. Since we do not give priority to any of these combinations, we use all of them to obtain estimates of the mean void characteristics, assuming the corresponding four void catalogues to be independent realizations of the void distribution. Such an averaging over the source of data and search method increases the number of voids used for statistics, decreasing in this way the statistical errors.

Figure 12: Diameter of constituent sphere versus galactic latitude for the voids of $R \geq 1$ A/ACO clusters (catalogue AR/Lp) in volume V1. The line is a least squares fit

The results of the estimation of the mean void characteristics - dimension, volume and shape (sphericity) - are presented in Table 7 with the following columns: Col. (1) - serial numbers of the void catalogues used for the calculation of the mean characteristics (see Table 2); Col. (2) - tracer type (see Table 2); Col. (3) - sample volume: V1A defined by $b \geq +50^{\circ }$ and $z \leq$ 0.09, V2A defined by $b \geq +50^{\circ }$ and $z \leq$ 0.14; Col. (4) - number of voids (sum of the number of voids in the catalogues listed in Col. (1)); Col. (5) - dimension type: D is the diameter of the largest constituent sphere of the void, D_e is the equivalent void diameter (see Sect. 3.2); Col. (6) - mean value of the dimension; Col. (7) - error of the mean for significance level $\alpha =$ 0.05 (95% confidence interval): $\epsilon = t_{\alpha_n}\sigma_{n}/\sqrt{n}$, where $t_{\alpha_n}$ is Student's distribution, $\sigma_n$ is the standard deviation, and n is the number of voids; Col. (8) - standard deviation; Col. (9) - median; Col. (10) - volume type: V is the volume of the largest CS of the void, V_T is the total void volume; Cols. (11)-(14) - same as Cols. (6)-(9), but for the void volume; Cols. (15)-(18) - same as Cols. (7)-(9), but for the void sphericity.

  

Table 7: Mean void characteristics

Figure 13: Number versus diameter (equivalent and of the largest empty sphere) for voids of $R \geq 1$ and $R \geq 0$ A/ACO clusters a) with measured redshifts (volume V1A), and b) with measured or estimated redshifts (volume V2A)

Histograms of the void diameters (D and D_e) for the four samples of voids used to calculate the mean void characteristics (see Table 7) are given in Fig. 13.

We consider as most representative the estimates of the mean void characteristics in Table 7 for the voids of $R \geq 1$ A/ACO clusters in sample volume V1A, since they are based on most complete samples of the tracing objects. The other estimates in Table 7 are affected to a larger extent by sample incompleteness, and/or by large errors of the estimated redshifts. It is seen that for tracer types ARE and AE (i.e. samples of clusters with measured or estimated redshifts) the estimated dimensions in volume V2A are 10-15% higher than in V1A.

We consider the equivalent diameter D_e as a more representative estimate of the void dimensions than D, since it takes into account the total void volume. Thus, the best estimate for the mean dimension of the voids of $R \geq 1$ A/ACO clusters is $D_\mathrm{e} = 105.0\pm5.6$ h^-1 Mpc, and for the voids of $R \geq 0$ A/ACO clusters we have $D_\mathrm{e} = 87.2\pm4.1$ h^-1 Mpc (see Table 7). The corresponding total void volumes are $V_\mathrm{T} = 0.667\pm0.102~10^6$ $h^{-3}\ \mathrm{Mpc^3}$ and $V_\mathrm{T} = 0.400\pm0.060~10^6$ $h^{-3}\ \mathrm{Mpc^3}$.

It is seen from the last two columns of Table 7 that the voids have quite spherical shapes independent of the tracer type. This result is in agreement with the qualitative estimate of the shapes of the voids of clusters as "remarkably spherical'' by Batuski & Burns ([1985]). The mean sphericity for the voids of $R \geq 1$ A/ACO clusters is 0.88 $\pm$ 0.03, and for the voids of $R \geq 0$ A/ACO clusters it is 0.91 $\pm$ 0.02, i.e. the deviation from an ideal sphere is about 10%. The estimates of the mean sphericity are influenced to a certain extent by the choice in the search procedure of a comparatively high value D_min = 50 h^-1 Mpc for the minimum void dimension.

Our results for the mean dimensions of the voids of clusters are in good agreement with those of Einasto et al. ([1997]). Using a sample of 1304 A/ACO clusters in both galactic hemispheres to a limiting distance z = 0.12, 1/3 of which with estimated distances, they have obtained for the median diameters of the largest empty spheres estimates of 90 h^-1 Mpc for the voids of $R \geq 0$ A/ACO clusters, 100 h^-1 Mpc for the voids of clusters belonging to superclusters, and 110 h^-1 Mpc for the voids of clusters belonging to rich superclusters. Taking into consideration the tracer type, the sample volume, and the type of dimension, most appropriate for the comparison with their results are our estimates for the median diameters of the largest CS (dimension type D) for tracer types AE and ARE in sample volume V2A. (Note, however, the difference between our tracer type ARE and the corresponding tracer types in Einasto et al. ([1997]) based on the membership of clusters in superclusters.) We have for these cases in Table 7 median diameters, respectively, 92.0 h^-1 Mpc against their value 90 h^-1 Mpc, and 106.0 h^-1 Mpc against their values 100 and 110 h^-1 Mpc. However, if the comparison is made with our best estimates, which, as already explained, correspond to volume V1A, and not to V2A, i.e. with median diameters 80.0 h^-1 Mpc for tracer type A, and 94.0 h^-1 Mpc for tracer type AR (see Table 7) then we see that our estimates are lower by 5-10%. We suppose that one reason for the larger median diameters of Einasto et al. ([1997]) may be an unremoved effect of the galactic obscuration due to the use of voids situated at low galactic latitudes.

It is also interesting to compare our results with theoretical predictions for the size of voids of clusters. Frisch et al. ([1995]) have obtained for the mean diameters of the largest empty spheres of the voids of rich clusters ($R \geq 0$) for three different models the following estimates: $91.3\pm11.6$ h^-1 Mpc for a two-power law model, 82.6 $\pm$ 12.5 h^-1 Mpc for a CDM model with density parameter $\Omega =$ 1, and $90.5\pm14.4$ h^-1 Mpc for a low-density CDM model with $\Omega =$ 0.2. These values have been calculated with the Bahcall & Cen ([1993]) estimate $\bar{n} = 13.5~10^{-6}~h^{3}~\mathrm{Mpc}^{-3}$ for the mean number density of the $R \geq 0$ clusters with which our samples of $R \geq 0$ clusters are in good agreement (see Sect. 2.1). Our estimate $82.3\pm3.1$ h^-1 Mpc in Table 7 (for tracer type A, sample volume V1A, and dimension type D) is in best agreement with the high-density CDM model with $\Omega =$ 1. However, the uncertainties of the model estimates are too large in order to make a definite discrimination between the three models.