A study of the large voids in the spatial distribution of galaxy clusters in the Northern Galactic Hemisphere

Transcription

1 ASTRONOMY & ASTROPHYSICS JUNE I 2, PAGE 323 SUPPLEMENT SERIES Astrn. Astrphys. Suppl. Ser. 144, (2) A study f the large vids in the spatial distributin f galaxy clusters in the Nrthern Galactic Hemisphere K.Y. Stavrev Institute f Astrnmy, Bulgarian Academy f Sciences, 72 Tsarigradsk Shsse Blvd., BG-1784 Sfia, Bulgaria kstavrev@skyarchive.rg Received July 22, 1999; accepted March 23, 2 Abstract. We present a mapping f the large vids (D 5 h 1 Mpc) in the spatial distributin f clusters f galaxies in the Nrthern Galactic Hemisphere in a vlume defined by galactic latitude b +3 and redshift z.14. Rich clusters with spectrscpically measured r phtmetrically estimated redshifts are used as tracers f the large-scale structure f the Universe. An autmated vid-search and analysis prcedure which identifies a vid as a system f intersecting empty spheres is applied, allwing fr a descriptin f the vid dimensins, vlumes and shapes mre cmplete than in previus investigatins. A number f vid catalgues crrespnding t different tracer types, different surces f redshift data, and different versins f the search methd have been generated and analysed. Visualizatins f the 2-D and 3-D distributins f vids are presented. The estimated mean dimensins f the vids f R 1andR Abell/ACO clusters are D e = 15. ± 5.6 h 1 Mpc and D e = 87.2 ± 4.1 h 1 Mpc, respectively, where D e is the equivalent vid diameter. We have identified the pr clusters, grups, and galaxies ppulating the vids f rich clusters and cnstructed radial density prfiles f the vids, which shw the presence f a vid hierarchy. Key wrds: csmlgy: bservatins large-scale structure f Universe galaxies: clusters: general methds: data analysis 1. Intrductin The vids in the spatial distributin f galaxies and clusters f galaxies, discvered mre than tw decades ag (Tifft & Gregry 1976; Gregry & Thmpsn 1978; Jõeveer & Einast 1978; Kirshner et al. 1981; Bahcall & Sneira 1982a, 1982b), are a dminant feature f the largescale structure (LSS) f the Universe. The bservatinal Send ffprint requests t: K.Y.Stavrev studies f their prperties (gemetrical parameters, cntent, envirnment) are f great imprtance fr testing basic assumptins in the cntemprary theretical mdels and scenaris f the frmatin and evlutin f the LSS. One majr cmplicatin in vid studies arises frm the difficulty t btain spatial distributins f the tracers f the LSS with spectrscpically measured redshifts which are fairly cmplete in vlumes bth wide and deep. Frm this pint f view the use f rich clusters f galaxies as tracers allws t study the largest spatial vlumes. Wide-angle studies f the vids in the distributin f clusters f galaxies cntaining lists (catalgues) f the identified vids have been carried ut s far by Batuski & Burns (1985), Tully (1986), Stavrev (199a, 199b), and Einast et al. (1994). All f them are based n Abell/ACO clusters (Abell 1958; Abell et al. 1989) as tracers. Since then, new redshift data fr the clusters have been accumulated. An imprtant step in this directin is the MX Nrthern Abell Cluster Redshift Survey (Slinglend et al. 1998), nt nly because f the increased number f clusters with measured redshifts, but als because f the reliable determinatin f the cluster redshifts excluding t a great extent the fregrund/backgrund cntaminatin. The newly accumulated redshift data allws nw fr a mre cmplete and crrect mapping f the vids in the distributin f galaxy clusters. Recent bservatinal studies cnfirm a hierarchical structure f the vids, suggested frm theretical cnsideratins fr the vid evlutin by Dubinski et al. (1993), and van de Weygaert & van Kampen (1993). Accrding t Einast et al. (1994) the vids determined by rich clusters f galaxies are divided int smaller vids by filaments f very pr clusters and galaxies. Lindner et al. (1995, 1996) find a vid hierarchy in the Nrthern Lcal Supervid (Einast et al. 1983): vids defined by pr (Zwicky) clusters are divided int smaller vids by latetype galaxies. These vids cntain fainter galaxies which define still smaller vids. The presence f the vid hierarchy makes it reasnable t cmbine the bservatins f

2 324 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters different types f tracers f the LSS frm rich clusters f galaxies dwn t pr clusters, grups, and single galaxies. The usefulness f this apprach, hwever, is limited by the incmpleteness f the samples f pr clusters, grups and galaxies with measured redshifts in vlumes in which the rich clusters f galaxies frm cmplete samples. The chice f apprpriate numerical techniques fr vid search and analysis is an imprtant aspect f the vid studies. The methd f the largest empty sphere, nested in the vid, has been mst ften used in searches fr vids in the distributin f clusters. It has been first applied by Tully (1986), and later by Einast et al. (1989) and Stavrev (199a, 199b). Tifft & Gregry (1988) and Stavrev (1991) have prpsed an imprvement f this methd thrugh apprximatin f a vid by a system f intersecting spheres (spherids). Such an apprach allws fr a mre cmplete descriptin f the vid dimensins, vlumes and shapes. An applicatin f this versin f the methd is given in Stavrev (1998). Recently similar multisphere techniques were develped by El-Ad et al. (1996), as well as by Aiki & Mähönen (1998), and applied t the study f vids in the distributin f IRAS galaxies (El- Ad et al. 1997) and suthern sky galaxies (El-Ad & Piran 1997). The gals f this paper are (1) t map as cmpletely as pssible the large vids in the distributin f clusters f galaxies in the Nrthern Galactic Hemisphere ut t medium-large distances ( 4 h 1 Mpc) cmbining redshift data frm many surces, and using estimated redshifts, (2) t define vid parameters n the basis f a mre elabrate prcedure fr vid search and analysis, (3) t btain estimates f the mean characteristics f vids frm a statistical analysis f hmgeneus samples f vids, and (4) t identify the vid ppulatin and study its distributin in the light f the cncept f a vid hierarchy. T achieve these gals we have develped and applied an autmated system fr vid search and analysis. The bservatinal data used in this study are described in Sect. 2. The Autmated Vid Search and Analysis System is presented in Sect. 3, including the vid-search algrithm and the vid parameterizatin. Sme functins f the system are explained als in Sects. 5 and 6. Sectin 4 presents the generated vid catalgues, and Sect. 5 cntains the results frm their analysis, including an autmated cmparisn f the catalgues btained with different tracers, a cmparisn with vids knwn frm previus investigatins, and a statistical analysis. The vid ppulatin and vid shell are discussed in Sect. 6. Finally, Sect. 7 gives a summary f the results. 2. Cmpsitin f the bservatinal samples 2.1. Cmbining redshift data frm different surces The redshift data used in the present study which is limited t the Nrthern Galactic Hemisphere (NGH) have been cmpiled frm the fllwing surces: (1) A Cmpilatin f Clusters f Galaxies with Published Redshifts, versin 1996., by V.S. Lebedev and I.A. Lebedeva (a descriptin f an earlier versin is given in Lebedev & Lebedeva 1991), with redshifts fr 1 5 clusters and grups (with at least 2 member galaxies) in the NGH, 859 f which are Abell/ACO (A/ACO) clusters. (2) The NASA/IPAC Extragalactic Database (NED), versin 1997 (see descriptin in Madre et al. 1992), with redshifts fr 968 clusters and 19 grups in the NGH, 759 f which are A/ACO clusters. (3) A Catalgue f Measured Redshifts f Abell Clusters f Galaxies, versin 1997 (Andernach & Tag 1998; Andernach 1997), with 944 A/ACO clusters in the NGH. (4) The MX Nrthern Abell Cluster Redshift Survey (Slinglend et al. 1998; Batuski et al. 1998), with redshifts fr 62 clusters and 81 galaxies in the NGH. (5) The Center fr Astrphysics (CfA) Redshift Catalgue (ZCAT), versin March 3, 1997 (see descriptin in Huchra et al. 1992), cntaining redshifts fr galaxies in the NGH. Fr cnvenience, we shall use the fllwing abbreviatins fr surces 1-5, respectively: LEB, NED, AND, MXS, CfA. As can be seen, we use fur cmpilatins, based n hundreds f riginal surces, and ne new redshift survey, nt yet included in any cmpilatin at the time f the present study. They reflect the status f the available redshift data fr the clusters, grups, and galaxies in the beginning f The numbers f the A/ACO clusters given abve include the supplementary suthern clusters (ACO S) as well. Let us nte that because f the limitatin t the NGH (galactic latitude b ) the great majrity (> 9%) f the cmpiled A/ACO clusters are nrthern Abell clusters (Abell 1958). Besides A/ACO clusters the cmpilatins LEB and NED cntain als Zwicky clusters (Zwicky et al. 1961), ther types f clusters like pr clusters (Mrgan et al. 1975; Albert et al. 1977), clusters frm the Shane-Wirtanen cunts (Shectman 1985), distant clusters (Gunn et al. 1986), clusters frm the Einstein Observatry Medium Sensitivity Survey (Giia et al. 199) and the ROSAT all-sky survey (Vges 1992). The grups f galaxies in LEB and NED are mainly nearby grups (Huchra & Geller 1982), cmpact grups (Hicksn 1982), grups in the tw CfA redshift surveys (Geller & Huchra 1983; Ramella et al. 1989), and grups in individual clusters and superclusters (e.g. Ciardull et al. 1983; Hpp & Materne 1985). N indicatin which bject is a grup is given in LEB. We have fund frm the riginal data surces referenced in LEB that the number f grups in this cmpilatin (in the NGH) is < 1.

3 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 325 The preliminary examinatin f the spatial distributin f the different tracers frm data surces 1-5 shws that nly the A/ACO clusters can frm cmplete enugh samples in large vlumes suitable fr large vid studies. The cmpleteness limits fr all ther tracers (nn- A/ACO clusters, grups, galaxies) d nt g beynd z.3.4. Therefre, we use in this study fr vid identificatin nly distributins f A/ACO clusters. A limited applicatin f the ther tracers has been dne nly fr investigating the substructure f vids in the distributin f rich A/ACO clusters (Sect. 6). (Prcessing f samples cntaining galaxies, as well as grups and pr clusters, in smaller and nearer spatial vlumes in which the cmpleteness f these bjects is higher will be dne in a separate study.) The cmpilatins LEB, NED, and AND are maintained independently by their authrs. Furthermre, the cmpilatin versins we use refer t smewhat different times f last update. Therefre, they d nt verlap cmpletely and each ne f them cntains bjects which are nt present in the ther cmpilatins. It fllws that by merging them we can increase the ttal number f bjects with spectrscpically measured redshifts. We have first cmbined the data frm LEB, NED, and MXS. The data frm the new unpublished versin f AND, nly fr A/ACO clusters missing in the ther three surces, became available t us and was added later as a final enlargement f the data. After a preliminary check f the data in LEB and NED we have remved frm them several dzens f bjects which are predminantly subcmpnents f clusters with the same crdinates and redshift as the main cmpnent, r bjects which are errneusly included in the cmpilatins as spectrscpically bserved, while, in fact, their redshifts are phtmetric estimates (Andernach 1997). Befre cmbining the data, the redshifts, when helicentric (z h ), have been transfrmed t galactcentric (z g ) fllwing z g = z h + 3 sin l cs b, c where l and b are the galactic crdinates, and c is the speed f light. The distance r t the bjects has been derived frm the Hubble law assuming a Hubble cnstant H = 1 km s 1 Mpc 1. The Cartesian crdinates x, y, z f the bjects have been calculated frm their spherical crdinates r, l, b with the transfrmatin x = r cs l cs b, y = r sin l cs b, z = r sin b in a crdinate system centred in the Galaxy with axes x and y directed t l =, b = and l =9, b =,respectively, and axis z directed t the Nrth Galactic Ple (NGP). In rder t be able t cnstruct the 2-D distributin f bjects n the sky we have calculated als their z NED z LEB Fig. 1. Cmparisn f the redshifts f clusters and grups in the cmpilatin f Lebedev and Lebedeva (z LEB) with the redshifts in NED (z NED) fr galactic latitude b rectangular crdinates x L, y L in an azimuthal equal-area prjectin (Lambert prjectin) frm ( A =2sin 45 b ), 2 x L = A cs l, y L = A sin l, where b, fr a crdinate system centred n the NGP and axes x and y directed respectively t l =, b = and l = 27, b =. The cmbinatin f the data frm the different surces is cmplicated by the presence f discrepant redshifts fr a part f the bjects, as well as by the necessity t crss-identify crrectly (autmatically) the identical bjects whse designatins, crdinates, redshifts, etc. may nt exactly cincide in the different surces. The cmparisn f the LEB and NED cmpilatins gives 8 cinciding bjects whse redshifts are cmpared in Fig. 1 (fr z.4). As it is seen, mst f the clusters and grups are situated alng the line z LEB = z NED. Hwever, fr several dzens f bjects ( 1%) the differences z = z LEB z NED are significant. We have fund that in many cases the large z are an effect f the backgrund cntaminatin which splits clusters int tw r mre cmpnents at different distances alng the line-fsight. The MX Survey cntains the mst reliable data fr the cluster redshifts in cmparisn with the ther three surces f data. This is due t the large mean number N z = 9 f cluster members with measured redshifts per cluster, allwing fr the eliminatin f the backgrund cntaminatin. At the same time, nly less than half f the clusters in LEB, NED and AND have N z 3. We can use MXS as a cntrl sample t evaluate the quality f the redshift data in LEB and NED. The cmparisns LEB MXS and NED MXS fr 44 and 42 cmmn

4 326 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters bjects, respectively, shw large redshift discrepancies fr abut 1% f the cmpared clusters in bth cases. We cnclude that (1) the cmpilatins LEB and NED cntain similar amunts f large redshift errrs, and (2) such errrs strngly affect nly a cmparatively small part f the clusters. Therefre, we accept that bth cmpilatins are equally suitable fr studies f the large ( 5 h 1 Mpc) vids in the spatial distributin f clusters f galaxies. The cmbinatin f the data frm LEB, NED, and MXS has been dne by assigning different pririties t them. If an bject is present in mre than ne surce it btains the redshift frm that surce with the higher pririty. We give the highest pririty t MXS, and lwer, but equal pririties t LEB and NED. Because f this equality we merge the data int tw parallel cmpilatins: in the first ne the bjects which are bth in LEB and NED have the redshifts frm LEB (cmpilatin CL), and in the secnd ne the redshifts frm NED (cmpilatin CN). After adding the data frm AND, the tw versins f the final cmpilatin cntain bjects each, 1 14 f which are A/ACO clusters ( 45% f all A/ACO clusters in the NGH). The different data surces cntribute t cmpilatins CL and CN as fllws: MXS 62 A/ACO clusters bth in CL and CN, LEB bjects (812 A/ACO clusters) in CL and 679 (15) in CN, NED 267 (37) in CL and 1 24 (699) in CN, AND 13 A/ACO clusters bth in CL and CN. The cmbinatin f several surces has led t a 25% increase in cmparisn with the largest cmpilatin LEB (1 5 clusters and grups). Fr the A/ACO clusters alne, the increase is abut 1% cmpared t AND (944 clusters). Cnsequently, ur cmpilatins ffer the pssibility f a mre cmplete mapping f the large vids in the spatial distributin f cncentratins f galaxies in the NGH. Besides the tw cmpilatins CL and CN we have cmpsed als a cmpilatin f galaxies, hereafter referred t as CG, by merging the data frm CfA (ZCAT) and MXS. CG cntains galaxies with b.itisusedin Sect. 6 t examine hw galaxies ppulate the vids in the cluster distributin. In rder t define a vlume suitable fr vid investigatins we have first studied the variatins f the surface and spatial number densities with decreasing galactic latitude and with increasing redshift, respectively, fr the A/ACO clusters frm the cmpilatins CL and CN. The surface number density as a functin f galactic latitude fr 5 bins in b is shwn in Fig. 2 fr samples frm CL separately fr all A/ACO clusters (richness class R, including the ACO S clusters) and fr the R 1 clusters. It is well seen that the effect f galactic bscuratin is nt strng dwn t b =+4. The spatial number density as a functin f redshift (Fig. 3) is cmputed fr cncentric shells f thickness z =.1. Only bjects with b +3 have been cnsidered in rder t reduce the effect f galactic bscuratin. Figure 3a shws the variatin f the number density with NUMBER DENSITY [1-2 deg -2 ] A/ACO clusters R>1 R> GALACTIC LATITUDE Fig. 2. Surface number density as a functin f galactic latitude fr A/ACO clusters f richness classes R 1andR z fr the R 1andR A/ACO clusters frm CL. As can be seen, the spatial density f the R 1 clusters remains cmparatively high up t z.9. In this range the average density is in gd agreement with the estimate by Bahcall & Cen (1993) n =61 6 h 3 Mpc 3 fr the mean spatial density f the R 1 A/ACO clusters. In the redshift range.9.14 the density f the R 1 clusters with measured redshifts drps by mre than 5%, remaining, hwever, cmparatively cnstant in this range. Fr z>.14 the density is belw 25% f the mean estimate f Bahcall & Cen. Figure 3a shws that the spatial distributin f the R A/ACO clusters is affected by the bservatinal selectin even in the redshift range z <.9. This is expected since the R = clusters represent an incmplete class f bjects in Abell s catalgue. While the density f the R clusters fr z =.4.9 agrees well with the estimate by Bahcall & Cen (1993) n = h 3 Mpc 3 fr this class f bjects, in the nearest vlume fr z<.4 the density agrees better with the mre recent estimate n =261 6 h 3 Mpc 3 f Einast et al. (1997). Fr z >.9 the relative decrease f the density is strnger than fr the R 1 clusters. We cnclude frm the examinatin f Figs. 2 and 3a that (1) bth the R 1andR A/ACO clusters shw a weak effect f the galactic bscuratin fr b +4, and (2) the sample f R 1 A/ACO clusters seems t be cmplete up t z.9 while the sample f R clusters is incmplete even in this clser range Clusters with estimated redshifts One pssibility t cmpensate fr the grwing incmpleteness with distance f the clusters is the usage f estimated distances t the clusters with unmeasured redshifts. A disadvantage f this apprach are the large errrs (2 3%) f the estimated redshifts (Peacck & West 1992). The unfavurable effect frm the large errrs may be reduced by keeping small the fractin f clusters with estimated

5 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 327 NUMBER DENSITY [1-6 h 3 Mpc -3 ] a A/ACO clusters, b> REDSHIFT R>1 R> NUMBER DENSITY [1-6 h 3 Mpc -3 ] b A/ACO clusters, R>1, b> REDSHIFT measured z measured r estimated z Fig. 3. Spatial number density as a functin f redshift fr: a) A/ACO clusters f richness classes R 1andR, b) R 1 A/ACO clusters with measured redshifts, cmpared with that fr the R 1 A/ACO clusters with measured r estimated redshifts redshifts when cmbining them with clusters with measured redshifts. T estimate the distances t the A/ACO clusters we use the calibratin equatin (Kalinkv et al. 1985) lg z = m 1, ±829 ±5 btained fr a sample f 529 Abell clusters with measured redshifts, where m 1 is the magnitude f the 1th brightest cluster member frm Abell (1958). We have applied this calibratin fr A/ACO clusters in the NGH (b ) fr which n measured redshifts are available. The clusters with phtmetrically estimated redshifts have been mixed with the clusters with measured redshifts frm cmpilatins CL and CN. It is seen frm Fig. 3b, where the spatial number density f the R 1 A/ACO clusters with measured r estimated redshifts is cmpared with that f the same type f clusters with measured redshifts (frm CL), that after the additin f the clusters with estimated z a deep minimum in the number density distributin is frmed at z =.1.11, fllwed by a strng enhancement fr z = The latter feature is prbably due t systematic effects in Abell s catalgue (Abell 1958). A similar density enhancement in the distance range 3 45 h 1 Mpc has been fund by Scaramella et al. (1991) fr a sample f Abell clusters, abut 45% f which had estimated redshifts, and has been explained as an effect f a systematic verestimate f m 1 fr the distant clusters in Abell s catalgue. Hwever, in spite f the large errrs f the estimated distances, the reality f the maximum in this range cannt be definitely ruled ut. It is interesting that the density enhancement in the range cincides well with the third f the peaks in the number f galaxies twards the NGP discvered in the ne-dimensinal deep survey f Bradhurst et al. (199). If the density distributin is cut at z =.14 the crrespnding selectin functin fr the R 1 A/ACO clusters with measured r estimated redshifts becmes flat at adensitylevelf61 6 h 3 Mpc 3, in gd agreement Table 1. Cluster samples fr b +2 and z.16 with the Bahcall & Cen estimate. Therefre, we cnclude that this redshift limit is apprpriate fr the samples cntaining clusters with measured r estimated redshifts. Let us nte that if the deep minimum at z =.1.11 in Fig. 3b is real then the sample f R 1 A/ACO clusters with measured redshifts may be cmplete t z.12, i.e. nearly 1 h 1 Mpc mre than the adpted cmpleteness limit z = The cluster samples Taking int accunt the analysis f the cmpleteness f the cmpilatins CL and CN, we define a vlume V 2 in the NGH with b +3 and z.14 in which the R 1 A/ACO clusters with measured r estimated redshifts frm a cmplete sample and delimit in it a subvlume V 1 with b +4 and z.9 in which the sample f R 1 A/ACO clusters with measured redshifts is cmplete. T reduce the bundary effects we intrduce additinally an enlargement f vlume V 2dwntb =+2 and up t z =.16. While the vid search is limited t vlume V 2 (i.e. vid centres must lie in V 2), the bjects utside f vlume V 2 are used nly t cnstrain the vids which intersect the vlume bundaries.

6 328 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 9 9 A/ACO clusters R>1 Sample AR/L n=415 A/ACO clusters,r>1 measured r estimated z Sample ARE/L n=813 l= 18 l= 18 a b= b b= A/ACO clusters R> Sample A/L n=79 A/ACO clusters,r> measured r estimated z Sample AE/L n=1455 l= 18 l= 18 c b= d b= Fig. 4. Surface distributins in galactic crdinates fr samples f A/ACO clusters with b +2 and z.16 (see Table 1) in Lambert equal-area prjectin centred n the NGP: a) R 1 clusters with measured redshifts, b) R 1 clusters with measured r estimated redshifts, c and d) same as a and b) but fr the R clusters The samples used in the present study t search fr vids are frmed by extracting frm the cmpilatins CL and CN all A/ACO clusters with b +2 and z.16 separately fr (1) the R 1 clusters, and (2) all (R ) A/ACO clusters, and fr each f these tw tracer types separately fr the clusters (1) with measured redshifts, and (2) with measured r estimated redshifts. Thus, a ttal f 8 samples are cmpsed. Table 1 cntains the adpted sample designatins and the number f bjects in each sample. Letter L r N in the sample designatin means extractin frm CL r CN, respectively, and E means that the sample cntains clusters with estimated redshifts. Fr z.14 the fractin f clusters with estimated redshifts is abut 1/3 f the ttal number (33% fr the R 1 clusters and 36% fr the R clusters). Hwever, fr z>.12 the number f clusters with estimated redshifts is already larger than that f the measured clusters (see Fig. 3b). There are almst n clusters with nly estimated redshifts in the clseby subvlume V 1. Figures 4a d and 5a d shw 2-D and 3-D visualizatins, respectively, f the samples frm Table 1 (nly the samples extracted frm CL are shwn). The 2-D distributins are Lambert equal-area prjectins, centred n the NGP, while the 3-D visualizatins are perspective graphs. The 2-D distributins (Fig. 4) shw that the NGH is cvered satisfactrily by the bservatins with the exceptin f the sparsely ppulated regin at lw galactic latitudes arund l = affected strngly by the galactic bscuratin (Abell 1958; Bahcall & Sneira 1982b). The inclusin f clusters with estimated redshifts (Figs. 5b and d) increases the cmpleteness but sme stratificatin in

7 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters z (h -1 Mpc) z (h -1 Mpc) -4 x (h -1 Mpc) 4-4 A/ACO clusters, R>1, Sample AR/L (n=415) y (h -1 Mpc) 4 a -4 x (h -1 Mpc) 4-4 y (h -1 Mpc) A/ACO clusters, R>1, measured r estimated z, Sample ARE/L (n=813) 4 b z (h -1 Mpc) z (h -1 Mpc) -4 A/ACO clusters, R>, Sample A/L (n=79) x (h -1 Mpc) 4-4 y (h -1 Mpc) 4 c -4 x (h -1 Mpc) 4-4 y (h -1 Mpc) A/ACO clusters, R>, measured r estimated z, Sample AE/L (n=1455) 4 d Fig. 5. Spatial distributins fr samples f A/ACO clusters with b +2 and z.16 (see Table 1) shwn as perspective graphs with the crdinate system centred n the Galaxy, axes x and y directed twards l =, b =,andl =9, b =, respectively, axis z directed twards the NGP, and lcatin f the view pint at b +18, l 32 : a) R 1 clusters with measured redshifts, b) R 1 clusters with measured r estimated redshifts, c and d) same as a and b) but fr the R clusters the spatial distributin appears at larger distances, mst prbably due t the systematic verestimate f m 1 in Abell s catalgue. 3. Autmated vid search and analysis In rder t facilitate the quantitative studies f vids we have develped an Autmated Vid Search and Analysis System (AVSAS). Applicatins f earlier versins f this system are given in Stavrev (199a, 199b, 199c, 1991, 1998). AVSAS is a prgramme package f abut 5 mdules executing the fllwing functins: (1) preliminary data analysis hmgenizatin, cmparisn and cmbinatin f data frm different surces; (2) lcal analysis f galaxy and cluster samples cnstructin and visualizatin f the nearest-neighbur distance field, search fr and parameterizatin f vids (generatin f vid catalgues), visualizatin f the 2-D and 3-D distributins f vids, cmparisn f vid catalgues, identificatin f the vid ppulatin and the shell ppulatin, cnstructin f vid prfiles; (3) statistical analysis statistical cmparisn f bserved samples with randm samples, statistical analysis f vid catalgues The vid-search algrithm The vid-search algrithm in AVSAS is based n a definitin f the vids as space regins cmpletely devid f a certain type f bjects (clusters, grups, galaxies). T cnstruct the search algrithm we intrduce the cncept f a distance field (Stavrev 199a). Such an apprach has been applied als by Frisch et al. (1995), Lindner et al. (1995), and recently by Aiki & Mähönen (1998). Let S be a sample f n bjects A i, i = 1,..., n, with Cartesian crdinates x i, y i, z i in a 3-D crdinate system as described in Sect We cnstruct in the spatial vlume V S f sample S a cubic grid G, riented alng the axes x, y, z, with grid cnstant k [h 1 Mpc] and grid ndes g j with crdinates x j, y j, z j,wherej = 1,..., m. Fr each nde the distance t its nearest neighburing bject is cmputed frm d j =min{[(x i x j ) 2 +(y i y j ) 2 +(z i z j ) 2 ] 1/2 }, i where i = 1,..., n. The set f distances d j in the ndes g j frms a 3-D discrete scalar field d (x, y, z), which we shall call nearest-neighbur distance field, rd-field fr brevity. The lcal maxima f the d-field indicate regins devid f bjects, while the lcal minima indicate regins ppulated by bjects. Thus, the task f identificatin f vids is

8 33 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters reduced t the task f determinatin f the lcal maxima f a 3-D field. The lcal maxima (LM) f the d-field crrespnd in real space t the centres f the largest empty spheres embedded in the vids. A better apprximatin f a vid (its psitin, dimensin, shape) can be achieved by a system f crssing (verlapping) empty spheres, i.e. by a grup f grid ndes defining a lcal enhancement f the d-field (Stavrev 1991; see als Stavrev 199c). Similar methds fr vid search have been develped by El-Ad et al. (1996) and Aiki & Mähönen (1998).The apprximatin f a vid by mre than ne sphere is shwn schematically in Fig. 6. T determine the LM we intrduce first a threshld value d min crrespnding t the radius f the smallest vids which are t be identified, i.e. the search fr the LM is restricted ver the pints f the d-field with d d min. The neighburhd f a pint f the d-field with value d i in which a LM is searched is a sphere with radius r i just equal t d i. This means that the search prcedure uses a variable neighburhd which depends n the lcal prperties f the distributin f bjects: the neighburhd is larger in the mre sparsely ppulated regins and smaller in the denser regins. This feature f the prcedure makes it suitable fr prcessing spatial distributins f incmplete samples f bjects when the number density varies strngly depending n the distance r the galactic latitude. AVSAS ffers tw different methds fr determining the LM: (1) by direct cmparisn f the d-field values in the neighburhd f the d-field pints, and (2) by a criterin νσ fr the standard deviatin f the d-field values in the neighburhd f the d-field pints. Let di d min be the value f the d-field in the running nde gi which has neighburhd Vi with radius di, where i = 1,..., m 1, m 1 m (m 1 is the number f ndes with d d min ). Let Vi cntain N ndes with values d j, j = 1,..., N. We define di as a lcal maximum pint (i.e. a pint in the regin f a lcal maximum f the d-field) in if fr all j V i di d j 1+p, where p. If the parameter p = the abve cnditin reduces t di d j fr all j. Inthiscasedi is an abslute maximum in Vi and crrespnds t the largest empty sphere embedded in the vid. If p>thealgrithm identifies di as a lcal maximum pint nt nly when it is an abslute maximum in Vi but als when it is smaller by less than p 1% than the maximum value f the d-field in Vi. In this case the algrithm describes the LM f the d-field by a larger number f pints than fr p =. Thus, a vid is identified (in the next stages f the prcedure) as a grup f LM pints, i.e. the vid is apprximated by a system f crssing empty spheres. Distance field crss-sectin A 3 d 3 d 1 d2 A 1 A 2 largest cnstituent sphere k cnstituent sphere lcal maximum pint nearest neighbur Fig. 6. A schematic representatin f the multi-sphere methd fr the vid search. A crss-sectin f the distance field defined n a grid with cnstant k is shwn with three lcal maximum pints d 1, d 2, d 3 and their nearest neighburs A 1, A 2, A 3, defining three cnstituent spheres f a vid, ne f which is the largest empty sphere We shall call hereafter the empty spheres cnstituting a vid cnstituent spheres (CS). Fr cnvenience, the methd fr the determinatin f the LM pints by direct cmparisnusing the parameter p will be called p-methd. The secnd methd ptinally used in AVSAS fr the determinatin f the LM pints cnsists f the fllwing. First, we calculate the mean f the d-field in the neighburhd Vi f the running nde gi d i = 1 N N j=1 d j and the standard deviatin σi 1 N = dj 2 1 N N 1 N j=1 d j j=1 2 Then, di is a LM pint in Vi if di d i + νσ i, where ν. Chsing higher r lwer values fr the parameter ν we can tune the search algrithm t select mre r less statistically significant vids. We shall call this methd ν-methd. Applying any f the tw described methds n all d i d min, the search prcedure prduces a set f LM pints (r CS). Part f them lie at the bundaries f the examined space vlume. These peripheral LM pints can ptinally be remved frm further analysis if it is judged that they are f lw significance because f bundary effects r/and sample incmpleteness. In the next stage, the search algrithm grups the LM pints n the basis f criteria fr their mutual psitins. 1/2.

9 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 331 Each defined grup f LM pints cnstitutes a separate vid in the spatial distributin f the sampled bjects. The prcess f gruping f the LM pints is cmplicated by the perclatin f the neighburing vids, especially when samples f rare tracers f the LSS, such as the rich clusters f galaxies, are used. Often the perclatin is strengthened by the sample incmpleteness. Because f the uncertainty in delimiting neighburing vids the search algrithm allws fr a certain degree f verlap amng them. It ffers a chice between three ptins fr gruping the LM pints: (1) cmpact gruping, (2) medium-cmpact gruping, and (3) lse gruping. Accrding t the first ptin tw LM pints affiliate t the same grup if the distance between their psitins r = k, wherek is the grid cnstant. This ptin leads t cmpact grups f a cmparatively small number f LM pints, and cnsequently a large number f vids with a high degree f perclatin f neighburing vids. The criterin fr the medium-cmpact gruping is r 3 k (i.e. a distance equal t the diagnal f the elementary grid cube). This ptin prduces vids with mre cmplicated cnfiguratins f CS and less verlap with neighburing vids than the first ptin. The criterin fr the third ptin lse gruping, unlike the first tw ptins, des nt depend n k: tw LM pints with values d 1 and d 2 are gruped tgether if r <max(d 1,d 2 ). This criterin assures that the center f at least ne f the tw CS lies within the ther CS. The vids prduced by lse gruping f LM pints may have cmplicated (irregular) shapes and a large number f CS, i.e. large sizes. Hwever, the verlap f neighburing vids is less (relative t vid size) cmpared with the first tw ptins. The vids identified with ptin 1 can be cnsidered as substructures f the vids identified with ptins 2 and 3, and vice versa the vids identified with ptin 3 can be cnsidered as unificatins f the vids identified with ptins 1 and 2. The chice f the gruping ptin frm 1 t 3 acts parallel t the chice f a higher value f the parameter p and a lwer value f the parameter ν, i.e. twards larger vids with mre cmplicated shapes. As is seen frm the abve the vid-search algrithm needs chices f suitable values fr the fllwing free parameters: k, d min, p/ν, and the gruping ptin. The questin arises: given the ther values, what values f p and ν, respectively, are mst suitable fr the vid search? T answer this questin a number f tests f the p- methd and the ν-methd have been carried ut ver tw test samples (AR/L and AR/N, limited t vlume V 2, see Sect. 2.3) using tw values fr k (1 and 2 h 1 Mpc), tw values fr the minimum vid diameter D min =2d min (5 and 8 h 1 Mpc), and tw gruping ptins (mediumcmpact and lse gruping). Then, fr a certain cmbinatin f k, D min, and the gruping ptin we check hw the number f identified vids and CS changes as a functin f p r ν. The values f D min have been chsen t be rughly equal t r larger than the mean separatin f clusters which is 53 h 1 Mpc and 42 h 1 Mpc fr the R 1 and R Abell clusters, respectively (Batuski et al. 1991). With these D min the values f k are chsen t be sufficiently small, s that the LM crrespnding t the smallest identified vids can be determined frm statistics ver at least several dzens f d-field pints. The nly cmbinatin fr which this requirement is nt fulfilled is the ne with D min =5h 1 Mpc, k =2h 1 Mpc. The chice f a sufficiently small value f k is als f imprtance fr the accuracy f the determined vid parameters. Frm elementary cnsideratins fllws that the errr f the psitins f the CS centres due t the discreteness f the d-field can be estimated as 3 4 k.fr the tw chsen values f k (2 and 1 h 1 Mpc) the errr is 8.66 and 4.33 h 1 Mpc, respectively. In the latter case the errr is cmparable t the diameter f Abell clusters (3 h 1 Mpc), and it is less than 1% f the diameter f the smallest identified vids (5 h 1 Mpc). Let us als nte that the backgrund cntaminatin, the large peculiar velcities f the cluster members, and the use f estimated redshifts may lead t psitinal errrs larger than bth chsen values fr k. The results frm the tests are shwn in Fig. 7. Meaningful values f p and ν lie in the ranges p <.2 and ν > 1.. Outside these ranges the search prcedure prduces a t large number f LM pints and this causes difficulties in delimiting neighburing vids because f increased verlap amng them. Fr the case f the medium-cmpact gruping (Fig. 7, left panel) the number f vids increases cntinuusly with increasing p and decreasing ν, while fr the lse gruping (Fig. 7, right panel) the curves shw well defined maxima in the number f vids. Fr values f p andν the search algrithm identifies nly the mst significant vids deep, spherical and well islated regins, apprximated by single empty spheres. The increase f p, respectively the decrease f ν, relaxes the criterin fr the selectin f LM pints, hence the number f vids grws. At the same time, vids enlarge and start t verlap mre due t the increased number f CS. This causes the merging f neighburing vids and cnsequently a decrease f the number f vids. In the case f the lse gruping f the LM pints, fr certain values f the parameters p and ν the prcess f merging vertakes the increase f the number f LM pints, hence the number f vids decreases. If p is left t grw very large r ν t becme very small the number f vids tends t 1: a single, cntinuus, netwrk structure is frmed. Fr the medium-cmpact ptin the prcess f merging is nt strng enugh t stp the grwth f the number f vids with the grwth f p, respectively with the decrease f ν. (This is true fr the case f cmpact gruping f the LM pints, t.)

10 332 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters NUMBER OF VOIDS Medium-cmpact gruping f LM pints Test sample AR/L k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 Test sample AR/N k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 a NUMBER OF VOIDS Lse gruping f LM pints Test sample AR/L k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 Test sample AR/N k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 b p p NUMBER OF VOIDS Medium-cmpact gruping f LM pints Test sample AR/L k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 Test sample AR/N k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 c NUMBER OF VOIDS Lse gruping f LM pints Test sample AR/L k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 Test sample AR/N k=1, D>5 h -1 Mpc k=2, D>5 k=1, D>8 k=2, D>8 d ν ν Fig. 7. Number f vids as functins f the parameters p a and b) and ν c and d) fr medium-cmpact (left panel) and lse (right panel) gruping f the lcal maximum pints, fr tw test samples (AR/L and AR/N) and fur cmbinatins f the grid cnstant k and the minimum vid dimensin D min The test curves fr the p-methd, lse gruping (Fig. 7b) shw maxima apprximately fr the same value p m = The ν-methd, hwever, shws tw distinct maxima depending n the value f k: ν m =1.8 fr k =1h 1 Mpc and ν m = frk =2h 1 Mpc. This strnger dependence f the ν-methd n k in cmparisn with the p-methd is well utlined in Fig. 7c. On the ther hand, it is seen frm Figs. 7a d that the ν-methd is less dependent n D min than the p-methd. Figure 7b shws that fr the p-methd the cmbinatin k =2h 1 Mpc, D min =5h 1 Mpc leads t a substantially higher number f vids than all ther cmbinatins. This case differs frm the ther three cmbinatins by its very lw rati D min /k = 2.5. As a result part f the LM pints are defined in neighburhds Vi cntaining a very small number f grid pints. This prbably has the effect f relaxing the criterin fr the selectin f vids, hence increasing the number f vids. The ν-methd is less sensitive t this effect. We cnclude that values f D min /k lwer than 4 5 shuld be avided, especially when the p- methd is used. If we exclude this peculiar case and cmpare the number f vids fr all ther cases at p m (Fig. 7b) with the number f vids at ν m (Fig. 7d) we see that fr bth methds the number f vids is in the same range f abut 2 3 depending slightly n the cmbinatin (k, D min ), i.e. the agreement between the tw methds is gd. In rder t check hw p m and ν m depend n the number density we have used test samples with grwing numbers f bjects in the same vlume. These tests shw that p m decreases slightly with increasing number density, while ν m is almst independent f it. The reasn fr this difference between the tw methds is that the ν- methd is mre efficient than the p-methd at rejecting the smaller vids (near D min ) in the denser regins as insignificant fluctuatins f the d-field. Fr the p-methd the increase f the sample density leads t an increase f the number and the verlap f the CS, hence the number f vids decreases. This effect is cmpensated by a smaller value f p m which tightens the criterin fr selectin f LM pints. The tests f the vid-search prcedure suggest that it can be ptimized by chsing values f p = p m and ν = ν m. That leads t the detectin f the maximum number f vids at a lw degree f verlapping f the neighburing vids in the case f the lse gruping f the LM pints. These values f p and ν can be applied as well in the ther tw ptins cmpact and mediumcmpact gruping, in spite f the fact that the number f detected vids is nt maximized, but fr the sake f

11 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 333 keeping a lw degree f verlapping f the neighburing vids, hence better delimited vids Determinatin f the vid parameters After vids are identified, AVSAS analyses their prperties and defines the fllwing parameters: (1) Cartesian, equatrial and galactic crdinates f the CS centres and f the vid centre. The latter is defined as the centre f the largest CS f the vid, and alternatively, as the centrid f all CS f the vid. The centrid is determined as the centre f gravity f the system f CS. If a vid is cmpsed f n cs > 1 CS, each ne with vlume V i, i = 1,..., n cs,andweight n cs w i = V i / V j, j=1 then the Cartesian crdinates f the centrid f CS are cmputed frm n cs n cs n cs x c = w i x i, y c = w i y i, z c = w i z i, i=1 i=1 where x i, y i,z i are the Cartesian crdinates f the i-th CS. Fr cnvenience, the centrids are slightly shifted in rder t cincide with the nearest grid nde. The Cartesian crdinates x L,y L in a Lambert equalarea prjectin f the CS centres and f the centrid are cmputed, t, as described in Sect (2) Distances are cmputed t the centres f all CS, as well as t the CS centrid frm the Cartesian crdinates: r i = x 2 i + y2 i + z2 i [h 1 Mpc], and r c = x 2 c + y2 c + z2 c [h 1 Mpc]. (3) Dimensins: (a) Diameters f the CS are cmputed directly frm the values f the d-field as D i =2d i [h 1 Mpc]. (b) Vid dimensins alng x, y, z axes are determined frm the prjected distances between the centres f thse tw CS f the vid, which are the farthest apart alng each axis, plus their radii. Thus, fr the x axis D x = x + d A + d B [h 1 Mpc]. (c) The maximum vid dimensin is determined frm the distance between the centres f the tw mst widely separated CS f the vid plus the radii f these tw CS: D max = max + d A + d B [h 1 Mpc]. (d) The equivalent vid diameter is the diameter f the sphere whse vlume is equal t the ttal vlume V T f the vid (see belw): ( ) 1/3 6 D e = π V T [h 1 Mpc]. i=1 (4) Sphericity s. This parameter is intrduced fr a characterizatin f the vid shape. It is defined as s = D/D max, where D is the diameter f the largest CS f the vid. (5) Vlume: (a) The vlume f each CS is cmputed frm V i = 4 3 πd3 i [h 3 Mpc 3 ]. (b) The ttal vid vlume V T is cmputed numerically because f the cmplicated vid shape (especially when the vid is cmpsed f a large number f CS), by cunting the number f elementary grid cubes n g in the vid (strictly speaking, the cubes whse centres are inside the vid). Then V T = n g k 3 [h 3 Mpc 3 ], where k is the grid cnstant. (6) The bjects surrunding the vid are defined as the nearest neighburs t the centres f the CS f the vid. (See Sect. 6 fr a mre cmplete identificatin f the bjects surrunding a vid.) (7) The neighburing vids f the vid are determined frm a cnditin fr verlapping f at least ne CS f the vid with at least ne CS f the neighburing vid. If d 1 and d 2 are the radii f tw CS belnging t tw different vids, they verlap if r <d 1 + d 2,where r is the distance between the centres f the tw CS. (Nte the difference between this criterin and the criterin fr lse gruping f the LM pints in Sect. 3.1.) 4. Generatin f vid catalgues We have applied AVSAS ver the 8 bservatinal samples described in Sect. 2 (see Table 1) which ccupy the same vlume. The generated vid catalgues cntain the vids whse CS have centres lying in vlume V 2. All catalgues are generated with the same grid cnstant k =1h 1 Mpc (see discussin in Sect. 3.1), prducing grid ndes in vlume V 2, n which the d-field is calculated. The semi-cube cntaining the cnical vlume V 2 has dimensins 85k 85k 43k alng the x, y, z axes, respectively. The vid selectin criterin with which a vid catalgue is generated is a cmbinatin f five cnditins fr (1) the minimum vid diameter D min, (2) the search methd p r ν, (3) the value f the search parameter p r ν, (4) the ptin fr the gruping f the LM pints (cmpact, medium-cmpact, lse), and (5) the treatment f the peripheral LM pints (acceptance r rejectin). Since we are interested in large vidswehavechsen D min =5h 1 Mpc (see discussin in Sect. 3.1). This dimensin is cmparable with the characteristic size f the superclusters f galaxies. We have applied fr each sample bth search methds, setting p = p m and ν = ν m t ptimize the search.

12 334 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters Table 2. Vid catalgues: number f lcal maximum pints and vids in vlumes V 1andV 2 The values f p m r ν m fr each generated catalgue are given in Table 2. All three ptins fr the gruping f the LM pints have been used fr each sample and each methd t generate vid catalgues. Thus, 6 catalgues 3 fr the p-methd and 3 fr the ν-methd have been prduced fr each sample. Hwever, fr further vid analysis in this paper we have chsen nly the vids generated by the medium-cmpact gruping f the LM pints assuming that the cmpact gruping is mre suitable fr the study f vid substructures, while the lse gruping is suitable fr the study f vid superstructures (see Sect. 3.1). All vid catalgues have been generated rejecting the peripheral LM pints. The ttal number f generated vid catalgues is 48. The numbers f vids and LM pints in each catalgue are given in Table 2. Hereafter, we shall designate the vid catalgues fr the medium-cmpact ptin by the crrespnding sample designatin (Table 2, Cl. 2) supplemented by the suffix p r ν fr the methd applied (Table 2, Cl. 3), r we shall simply use the number given in the first clumn f Table 2. Tw f the vid catalgues listed in Table 2 crrespnding t samples AR/L and A/L are presented in Tables 3and 4. They aregenerated with the p-methd and the medium-cmpact gruping f the LM pints. Because f the large number f vid parameters and vid CS the catalgues are given in a cncise frm, with ne r tw lines f data fr each vid: the first line cntains (after Cl. 2) the parameters f the largest CS f the vid, and the secnd line cntains the parameters f the whle vid. Only ne line appears fr vids cnsisting nly f ne CS. If the vid centre cincides with the centre f the largest CS, the psitinal parameters are given nly in the first line. Vids are rdered by increasing right ascensin f the centre f the largest CS. Tables 3 and 4 have the fllwing cntents: Cl. (1) vid serial number; Cl. (2) number f CS f the vid; Cls. (3) and (4) equatrial crdinates α, δ fr B195. f the centre f the largest CS f the vid (line 1), and α c, δ c f the centrid f all CS (line 2); Cls. (5) and (6) galactic crdinates l, b f the centre f the largest CS (line 1), and l c, b c f the centrid f all CS (line 2); Cls. (7) (9) Cartesian crdinates x, y, z f the centre f the largest CS (line 1), and x c, y c, z c f the centrid f all CS (line 2); Cl. (1) distance r t the centre f the largest CS (line 1), and r c t the centrid f all CS (line 2); Cl. (11) psitin f the centre f the largest CS (line 1) and f the centrid f all CS (line 2) with respect t vlumes V 1andV 2: 1 in vlume V 1, 2 in vlume V 2, utside vlume V 1; Cl. (12) diameter D f the largest CS (line 1), and equivalent vid diameter D e (line 2); Cls. (13) (15) vid dimensins alng axes x, y, z; Cl. (16) largest vid dimensin D max ; Cl. (17) vlumev f the largest CS (line 1), and ttal vlume f the vid V T (line 2); Cl. (18) sphericity s. The vid catalgues listed in Table 2, tw f which are given in Tables 3 and 4, represent the currently mst cmplete mapping f the large vids in the distributin f galaxy clusters in the NGH t a limiting distance f 42 h 1 Mpc. Cmpared t similar wide-angle studies (Batuski & Burns 1985; Tully 1986; Einast et al. 1994) they cntain larger numbers f vids t larger distances. They can be used as identificatin lists f the vids in the NGH, in studies f individual vids, as well as fr statistical investigatins f the vid prperties. The spatial and surface distributins f the vids in Tables 3 and 4 are presented in Figs. 8 1 by different types f visualizatins allwing fr a visual examinatin and cmparisn f the vid catalgues. The spatial

13 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 335 Table 3. Vids f R 1 A/ACO clusters (sample AR/L, p-methd)

14 336 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters Table 4. Vids f R A/ACO clusters (sample A/L, p-methd)

15 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 337 Table 4. cntinued distributins (Fig. 8) can be rtated and examined frm an arbitrary view pint n a cmputer screen. The crsssectins f the 3-D vid distributin (Fig. 9) are shwn jintly with the distributin f the crrespnding tracers in slices f a thickness equal t the grid cnstant k = 1 h 1 Mpc. The 3-D distributin f the vids in Fig. 8 (left panel) suggests a vid-filled Universe with clsely packed and intersecting vids. Hwever, the crss-sectins f the 3-D distributin in Fig. 9 shw that the large vids may be separated by large znes f enhanced density f the tracing bjects. Such a zne is best utlined in the y =Mpc crss-sectin f the distributin f the R 1 A/ACO clusters fr z = 2 25 h 1 Mpc (Fig. 9a), but it can be identified in the ther crss-sectins, as well as in the adjacent cuts fr y andx, althugh in the directin alng the y axis (Fig. 9, right panel) it is smaller and nt well utlined. This feature is prbably due t the presence f a large rthgnal structure at this distance similar t the Great Wall (Tully 1986, 1987). 5. Analysis f the vid catalgues 5.1. Autmated cmparisn f the vid catalgues The generatin f a large number f vid catalgues by varying the sample, the search methd, r the vid selectin criterin has evked the necessity fr an easy and bjective way t cmpare the results. AVSAS ffers such a pssibility, applicable als t the cmparisn with vids frm previus investigatins. The cmparisn f tw vid catalgues is based n the cmparisn f the psitins and dimensins f the CS f the vids. The criterin fr cincidence f tw vids frm different catalgues is r <max (d 1,d 2 ), where r is the distance between the centres f the cmpared CS, and d 1, d 2 are their radii. This criterin must be satisfied fr at least ne pair f cinciding CS. A rigrus criterin fr vid cincidence may be chsen by setting r =. The results frm the cmparisn are utput as crss-identificatin tables with the cinciding vids, and

16 338 K.Y. Stavrev: A study f the large vids in the spatial distributin f galaxy clusters 42 z [h -1 Mpc] -4 x [h -1 Mpc] 4-4 Vids f R>1 A/ACO clusters (sample AR/L, p-methd) y [h -1 Mpc] 4 b vid centre cnstituent sphere centre 42 z [h -1 Mpc] -4 x [h -1 Mpc] 4-4 Vids f R> A/ACO clusters (sample A/L, p-methd) y [h -1 Mpc] 4 d vid centre cnstituent sphere centre Fig. 8. Spatial distributin f the vids (left panel) and f the vid and cnstituent sphere centres (right panel) frm the catalgues in Tables 3 and 4: a and b) vids f R 1 A/ACO clusters (catalgue AR/Lp), c and d) vids f R A/ACOclusters (catalgue A/Lp). Crdinate system and view pint are the same as in Fig. 5 as detailed lists f the cinciding CS with the differences in their psitins and dimensins. We have applied bth the rigrus and lse criteria t cmpare the vid catalgues listed in Table 2. Table 5 cntains the results f the cmparisn f the tw catalgues AR/Lp and A/Lp given in Tables 3 and 4, respectively. The numbers in Table 5 marked by an asterisk crrespnd t cases satisfying the rigrus criterin r =.Asitis seen in Table 5 mst f the vids f R 1 clusters (9%) can be identified with vids f R clusters. We have cmpared als ur vid catalgues with vids knwn frm previus investigatins. T d this we have cmpsed a cmpilatin f vids in the NGH, hereafter referred t as VC, cntaining 39 vids f clusters f galaxies frm the fllwing surces: (1) Bahcall & Sneira (1982b) nevidfr 1 Abell clusters; (2) Batuski & Burns (1985) 22 vids f Abell clusters; (3) Einast et al. (1994) 16 vids f A/ACO clusters. We shall use the abbreviatins BS, BB, and E fr surces 1-3, respectively. The cmpilatin VC ccupies a vlume cnsiderably smaller than the vlume ccupied by ur vid catalgues: the vid centres reach a limiting distance f abut 25 h 1 Mpc. The dimensins f the vids f clusters frm VC crrespnd rughly t ur criterin fr vid selectin D min = 5 h 1 Mpc. Befre the cmparisn, the vids frm VC with nnspherical shapes have been reduced t the largest empty spheres which can be nested in them, with the exceptin f the huge BS vid which, because f its extended frm, has been apprximated with 21 nn-verlapping equal spheres with diameters f 6 h 1 Mpc. The results f the cmparisn f the vid catalgues AR/Lp and A/Lp (Tables 3 and 4) with the vids frm the cmpilatin VC are given as identificatin lists in Table 6, where the first clumn cntains the serial numbers (frm Tables 3 and 4) f the vids identified in VC, and the secnd clumn cntains the identificatins frm VC, presented with their riginal numbers frm the crrespnding literature surces, preceded by the surce abbreviatin. The cmparisn shws that abut 7% f the vids f clusters frm VC can be identified in at least ne f the tw vid catalgues. On the ther hand, 35 4% f the vids in these catalgues, mst f them in the nearer subvlume V 1, are identified in VC. The rest > 6% f the vids lie predminantly beynd subvlume V 1, i.e. in regins nt cvered by the cmpilatin VC. We can cnclude that abut 2% f the vids detected by us in vlume V 1 and 9% f the vids utside vlume V 1 are newly discvered vids r vid candidates. The reality f the latter needs t be cnfirmed by future cmplete redshift surveys.