4 Generation of void catalogues

We have applied AVSAS over the 8 observational samples described in Sect. 2 (see Table 1) which occupy the same volume. The generated void catalogues contain the voids whose CS have centres lying in volume V2.

All catalogues are generated with the same grid constant k = 10 $h^{-1}\,\mathrm{Mpc}$ (see discussion in Sect. 3.1), producing 77603 grid nodes in volume V2, on which the d-field is calculated. The semi-cube containing the conical volume V2 has dimensions $85k \times 85k \times 43k$ along the $x,\ y,\ z$ axes, respectively.

  

Table 2: Void catalogues: number of local maximum points and voids in volumes V1 and V2

The void selection criterion with which a void catalogue is generated is a combination of five conditions for (1) the minimum void diameter D_min, (2) the search method p or $\nu$, (3) the value of the search parameter p or $\nu$, (4) the option for the grouping of the LM points (compact, medium-compact, loose), and (5) the treatment of the peripheral LM points (acceptance or rejection).

Since we are interested in large voids we have chosen D_min = 50 $h^{-1}\,\mathrm{Mpc}$ (see discussion in Sect. 3.1). This dimension is comparable with the characteristic size of the superclusters of galaxies.

We have applied for each sample both search methods, setting p = p_m and $\nu = \nu_\mathrm{m}$ to optimize the search. The values of p_m or $\nu_\mathrm{m}$ for each generated catalogue are given in Table 2.

All three options for the grouping of the LM points have been used for each sample and each method to generate void catalogues. Thus, 6 catalogues - 3 for the p-method and 3 for the $\nu$-method - have been produced for each sample. However, for further void analysis in this paper we have chosen only the voids generated by the medium-compact grouping of the LM points assuming that the compact grouping is more suitable for the study of void substructures, while the loose grouping is suitable for the study of void superstructures (see Sect. 3.1).

  

Table 3: Voids of $R \geq$ 1 A/ACO clusters (sample AR/L, p-method)

All void catalogues have been generated rejecting the peripheral LM points.

The total number of generated void catalogues is 48. The numbers of voids and LM points in each catalogue are given in Table 2. Hereafter, we shall designate the void catalogues for the medium-compact option by the corresponding sample designation (Table 2, Col. 2) supplemented by the suffix p or $\nu$ for the method applied (Table 2, Col. 3), or we shall simply use the number given in the first column of Table 2.

Two of the void catalogues listed in Table 2 corresponding to samples AR/L and A/L are presented in Tables 3 and 4. They are generated with the p-method and the medium-compact grouping of the LM points. Because of the large number of void parameters and void CS the catalogues are given in a concise form, with one or two lines of data for each void: the first line contains (after Col. 2) the parameters of the largest CS of the void, and the second line contains the parameters of the whole void. Only one line appears for voids consisting only of one CS. If the void centre coincides with the centre of the largest CS, the positional parameters are given only in the first line. Voids are ordered by increasing right ascension of the centre of the largest CS.

  

Table 4: Voids of $R \geq$ 0 A/ACO clusters (sample A/L, p-method)

Tables 3 and 4 have the following contents: Col. (1) - void serial number; Col. (2) - number of CS of the void; Cols. (3) and (4) - equatorial coordinates $\alpha,\ \delta$ for B1950.0 of the centre of the largest CS of the void (line 1), and $\alpha_\mathrm{c},\ \delta_\mathrm{c}$ of the centroid of all CS (line 2); Cols. (5) and (6) - galactic coordinates $l,\ b$ of the centre of the largest CS (line 1), and $l_\mathrm{c},\ b_\mathrm{c}$ of the centroid of all CS (line 2); Cols. (7)-(9) - Cartesian coordinates $x,\ y,\ z$ of the centre of the largest CS (line 1), and $x_\mathrm{c},\ y_\mathrm{c},\ z_\mathrm{c}$ of the centroid of all CS (line 2); Col. (10) - distance r to the centre of the largest CS (line 1), and r_c to the centroid of all CS (line 2); Col. (11) - position of the centre of the largest CS (line 1) and of the centroid of all CS (line 2) with respect to volumes V1 and V2: "1'' - in volume V1, "2'' - in volume V2, outside volume V1; Col. (12) - diameter D of the largest CS (line 1), and equivalent void diameter D_e (line 2); Cols. (13)-(15) - void dimensions along axes $x,\ y,\ z$; Col. (16) - largest void dimension D_max; Col. (17) - volume V of the largest CS (line 1), and total volume of the void V_T(line 2); Col. (18) - sphericity s.

  

Table 4: continued

The void catalogues listed in Table 2, two of which are given in Tables 3 and 4, represent the currently most complete mapping of the large voids in the distribution of galaxy clusters in the NGH to a limiting distance of 420 h^-1 Mpc. Compared to similar wide-angle studies (Batuski & Burns [1985]; Tully [1986]; Einasto et al. [1994]) they contain larger numbers of voids to larger distances. They can be used as identification lists of the voids in the NGH, in studies of individual voids, as well as for statistical investigations of the void properties.

The spatial and surface distributions of the voids in Tables 3 and 4 are presented in Figs. 8-10 by different types of visualizations allowing for a visual examination and comparison of the void catalogues. The spatial distributions (Fig. 8) can be rotated and examined from an arbitrary view point on a computer screen. The cross-sections of the 3-D void distribution (Fig. 9) are shown jointly with the distribution of the corresponding tracers in slices of a thickness equal to the grid constant k = 10 h^-1 Mpc.

The 3-D distribution of the voids in Fig. 8 (left panel) suggests a void-filled Universe with closely packed and intersecting voids. However, the cross-sections of the 3-D distribution in Fig. 9 show that the large voids may be separated by large zones of enhanced density of the tracing objects. Such a zone is best outlined in the y = 0 Mpc cross-section of the distribution of the $R \geq$ 1 A/ACO clusters for z = 200-250 h^-1 Mpc (Fig. 9a), but it can be identified in the other cross-sections, as well as in the adjacent cuts for $y \neq$ 0 and $x \neq$ 0, although in the direction along the y axis (Fig. 9, right panel) it is smaller and not well outlined. This feature is probably due to the presence of a large orthogonal structure at this distance similar to the Great Wall (Tully [1986], [1987]).

Figure 8: Spatial distribution of the voids (left panel) and of the void and constituent sphere centres (right panel) from the catalogues in Tables 3 and 4: a and b) voids of $R \geq 1$ A/ACO clusters (catalogue AR/Lp), c and d) voids of $R \geq 0$ A/ACO clusters (catalogue A/Lp). Coordinate system and view point are the same as in Fig. 5

Figure 9: Central cross-sections in the x-z plane (left panel) and y-z plane (right panel) of the joint spatial distribution of voids and tracers: a-d) same void catalogues as in Fig. 8. Voids are marked with their serial numbers from Tables 3 and 4

Figure 10: Surface distribution of the void and constituent sphere centres in a Lambert equal-area projection centred on the NGP: a-b) same void catalogues as in Fig. 8