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Superclustering and the large-scale structure of the universe
Adv. Space Res. Vol.3, No.10-12, pp.367-378, 1984 Printed in Great Britain.
0273-1177/84 $0.00 + .50 Copyright © COSPAR
All rights reserved.
S U P E R C L U S T E R I N G A N D THE L A R G E - S C A L E S T R U C T U R E OF THE U N I V E R S E Neta A. Bahcall Space Telescope Science Institute, Baltimore, MD, U.S.A.
ABSTNICT The spatial distribution of a complete redshift sample of nearby rich clusters of galaxies is studied. Strong superclustering, extending to very large scales (~ 100h-IMpc), is observed in the cluster distribution. In particular, we determined and discuss below the following studies: I.
II.
III.
IV.
The spatial correlation function of rich clusters of galaxies. The construction of a complete catalog of superclusters and a study of their growth (percolation). The finding of large-scale superclusters surrounding the giant galaxy void in Bootes. The finding of a ~ 300 l~c void of rich clusters of galaxies.
All the above studies provide evidence for the existence of large-scale structure in the universe, and reveal some of its characteristics. These findings are of importance to models of the formation of galaxies and structure in the universe.
KEYWORDS
Clusters of galaxies;
superclusters;
correlation function; voids;
large-scale structure.
THE S ~ P L E The sample used is Abell's (1958) statistical sample of rich clusters of galaxies of distance class D < 4 (z ~ 0.1), with redshifts for all but one of these clusters reported by Hoessel, Gunn, and Thuan (1980). This sample includes all Abell clusters at D $ 4 that are of richness class R ~ 1 and are located at high galactic latitude (as specified in Abell's Table 1 plus the requirement of [bl ~ 30o). A total of 104 clusters belong in this sample. The area subtended by the above statistical sample is 4.26 steradians (2.64 in the north and 1.62 in the south). A summary of the sample properties and its division into distance and richness classes as well as into hemispheres is presented in Bahcall & Soneira 1983a (hereafter BS 1983a). Also listed in the above reference are properties of the much larger and deeper D = 5 + 6 statistical sample ([b] ~30°), that includes 1547 clusters; while only a small fraction of the redshifts are measured for this sample, it is used, because of its much larger number of clusters, in various comparison tests to strengthen and confirm the results obtained from the D ~ 4 sample. The distances to the clusters are calculated from the standard relation 1961):
(e.g., Sandage,
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The spatial density of clusters in the sample as a function of redshift is given in BS 1983a. As expected, the mean density remains approximately constant for small redshifts and then, due to the limited precision in estimating distances from ml0 magnitudes, the density falls off, exhibiting a low-density tail to z = 0.14. This observed variation of the Abell selection function with distance (i.e., n(z) in Fig. 3 of BS 1983a) must be taken into account in any analysis requiring a distance-limited sample. We do so in our determination of the spatial correlation function by comparing the data with random catalogs constructed to have the same density selection function n(z) and boundaries as those of the real sample. We also use the observed n(z) in our construction of the supercluster catalog (see below). Selection effects that depend on position in the sky (e.g., due to absorption in the Galaxy or confusion with galactic stars) are also of importance in determining the correlation function or other distributions of clusters. We determined as a function of galactic latitude and longitude the ratio of observed number of clusters to that expected if the distribution on the sky was uniform. The results for both D < 4 and D = 5 + 6, presented in BS 1983a, show some decrease of cluster density with latitude. This dependence is corrected for in the determination of the cluster correlation function by including the same selection effect, P(b), in the random catalogs (see below).
THE SPATIAL CORRELATION FUNCTION OF RICH CLUSTERS OF GALAXIES The availability of a complete redshift sample of rich clusters of galaxies makes it possible, for the first time, to calculate relatively accurately cluster distances and separations and, in turn, to determine directly the spatial correlation function of rich clusters of galaxies. We do so in the present section. A more detailed description is given in BS 1983a. The joint probability, dP, of finding two clusters separated by a distance r and within volume elements dV I and dV 2 is written as: dP = n(R~)n(R2)[1
+ ~(r)]dVxdV2 ,
(2)
where ~ (r) is the two-point spatial correlation function and n(R) is the space density of clusters at the distance R from the Sun. (The space density in the present sample varies with the distance R due to the sample selection function discussed in the previous section.) The frequency distribution F(r) of all pairs of clusters with separation r in the present sample was determined. Cluster separations were calculated using the redshift d~stances (eq. (I)) and the angular separation on the sky (i.e., r = (R~+ R~ - 2RiR2cos 0)~); peculiar velocities were assumed to be small. In order to minimlze the influence of selection effects on the determination of ~ (r), we constructed a set of i000 random catalogs, each containing 104 clusters randomly distributed within the angular boundaries of the survey region, but with the same selection functions in both redshift and angular position as the Abell redshift sample (see the previous section). The frequency distribution of cluster pairs was determined in both the real and random catalogs, and the results were compared. This procedure ensures that the selection effects and boundary conditions will affect the data and random catalogs in the same manner, thus minimizing most of these effects. The spatial correlation function was determined from the relation ~(r) = F(r)/Fn(r)
-
1 ,
(3)
where F(r) is the observed frequency of pairs in the Abell sample and FR(r) is the corresponding frequency of random pairs (as determined by the ensemble average frequency of the i000 random catalogs). An ensemble average random frequency is used in order that (r) will not be affected by the fluctuations present in any particular realization of a single random sample. The correlation function was evaluated for various cases including: (a) no selection function in latitude (i.e., P(b) = i); (b) full selection function in latitude (discussed above); (c) northern and southern hemispheres treated separately; and (d) high and low latitude zones (Ibt > 50 ° and ]bl ~ 5 0 ° ) treated separately, each with its observed n(z) (and P(b)) selection function. The resulting correlation function is presented in Fig. i. Strong spatial correlations are observed a T separations ~ 25h-iMpc. Weaker correlations are observed to large separations of ~ 100h--~pc, where ~ 0.I; beyond 150h-iMpc, no statistically significant correlations are observed in the present sample. The results for the correlation function do not differ greatly between the two extreme cases of zero and full correction to the latitude selection function P(b), implying that our main conclusions are not strongly influenced by the latitude selection. In addition, the overall results obtained using a two-zone random
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a) The spatial correlation function of the D ~ 4 sample. (Crosses refer to no correction for latitude selection function; dots refer to the full correction of P(b)). The solid line is the best-fit 1.8 power-law to the data. The dashed line is the galaxy-galaxy correlation function of Peebles and co-workers. b)
Same as a) but plotted in larger bins at large separations.
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catalog (high and low latitude zones) show no significant difference from those obtained usin~ the entire sample. The correlation function in Fig. 1 can be well approximated by a single power-law relation of the form ~{r) = 300(rh)
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where r is in Mpc. The correlation function is smooth, with little scatter at r ~50h-IMpc. At r >50h-iMpc the scatter and uncertainties increase, but weak correlations of order 0.2 are still detected to these very large separations. These results, as expected, are consistent with the angular and redshift correlations observed at large angular and radial separations (BS 1983a). If real, the weak correlations at ~50-150h-iMpc should also be found in future, larger (deeper) redshift samples of rich clusters. In the meantime, the agreement between the above derived correlation function and the angular correlation function of the much larger D = 5 + 6 sample (which includes 1547 clusters), both of which show correlation to projected separations of at l e a s t ~ 100h-lMpc, suggests that the present results may indeed be representative of rich clusters in general. At 1.5
(5)
For f ~ 5 % (corresponding to galaxies within one Abell radius; Bahcall, 1979), the contribution to the galaxy correlation function from rich clusters, at any separation, is of the order of ~cc/400 ( ~ $ ~ / 2 5 ) . Thus, beyond the reported break in ~ , e.g., at r ~20h-iMpc, a lower limit to the expected galaxy-galaxy correlation is: ~gg(2~ Mpc) ~ 0 . 0 0 5 . Since galaxy-cluster cross-correlations indicate that a galaxy excess @fists well beyond an Abell radius of a cluster, the fraction f is most likely larger than 5% and the galaxy correlation limit becomes higher (for f = 10%, ~ gg(20h -I) ~0.02). The correlation function of clusters of different richness classes (R = 1 and R ~ 2) were determined separately in order to test for possible dependance on richness. The large sample of 1547 D = 5 + 6 clusters was used for that purpose, and the angular correlation functions of its 1125 R=I and 422 R ~ 2 clusters were determined separately. The amplitude of the angular correlation function is found to be strongly dependent on cluster richness, with richer clusters (R ~ 2) showing stronger correlations by a factor of ~ 3 as compared with the poorer (R = I) clusters (Fig. 2). (The D < 4 sample contains too few R ~ 2 clusters to yield statistically significant comparisons.) Both richness classes exhibit the same power-law shape correlation function as observed in the total sample. (See also BS 1983a.) In Fig. (2b) we show the dependence of the correlation function on the richness of the system, from single galaxies to poor and rich clusters. It is apparent that the correlations become stronger with increasing richness (or luminosity) of the system. This implies that the chance, above random, of finding a rich system next to another rich system is considerably higher than that of finding two neighboring poorer systems.
Superclustering
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a) The angular correlation function of richness 1 clusters (dots; 1125 clusters) and richness ~ 2 clusters (circles; 422 clusters) in the D = 5 + 6 sample. b) The dependence on richness of the two-p~int spatial correlation function. The spatial correlation function at r = 5h-~Mpc is plotted as a function of richness (roughly proportional to luminosity) for the galaxy-galaxy, galaxycluster, and cluster-cluster pairs (R ~ 1 as well as R = 1 and R ~ 2 independently; the/$atter are shown by open circles). Richness represents < n > = (nln 2) l z , where the subscripts 1 and 2 refer to the two members of t h e pair, n i is the mean Abell galaxy richness count for the appropriate clusters, and n I = 1 is used for a galaxy. (The appropriate galaxy-cluster cross-correlation is from Peebles, 1980; it excludes the term due to internal galaxy correlations within the cluster.) The dashed line, $ ( n ) ~ n 0"8, is presented in order to guide the eye in the approximate richness dependence of the correlation.
372
N.A. Bahcall
In order to ensure that the above derived spatial correlation function is not due to some special p e c u l i a r i t i e s in the nearby D < 4 sample, the an~ular c o r r e l a t i o n function of the m u c h larger and deeper D = 5 + 6 sample (1547 clusters) was determined and compared with that expected from the above spatial function. The a~reement b e t w e e n the D ~ 4 and D = 5 + 6 functions is excellent (see Fig. 12 of BS 1983a), indicating that the derived spatial function indeed represents well the overall cluster d i s t r i b u t i o n in the entire catalog. A d d i t i o n a l checks are presented in BS 1983a.
Tha m a i n conclusions reached from the above study are summarized below.
i. Strong spatial correlations of rich Abell clusters are observed in both the D < 4 redshift sample and the larger and deeper D = 5 ÷ 6 sample. Cluster correlations are detected to large separations of r £ 150h-iMpc. The correlations are consistent in the angular, redshift, and sDatial c o r r e l a t i o n functions of these samples (see also BS 1983a), with good agreement b e t w e e n the results obtained from the D < 4 and D = 5 + 6 samples. Various selection effects have been removed by comparing with a p p r o p r i a t e l y synthesized random catalogs. 2. The spatial correlation function of rich (R ~ i) clusters is observed to fit a powerlaw relation of the form ~ (r) = 300(rh) -1.8 for 5 ~ r ~ 150h-IMpc (r in Mpc). (When corrected for v e l o c i t y broadening, the intrinsic function becomes ~ (r) = 360(rh) -1"8, w h i c h best fits the observed angular and redshift c o r r e l a t i o n functions.) This correlation function has the same shape as the galaxy c o r r e l a t i o n function, but has an amplitude larger by a factor of ~ 18, and extends to greater distances than those observed for the galaxy c o r r e l a t i o n function. The cluster c o r r e l a t i o n function is unity at r ~ 2 5 h - i M p c , as compared with r 2 5h-iMpc in the galaxy c o r r e l a t i o n function. The extent of the rich cluster correlations beyond the reported 15h-iMpc break in the galaxy correlation function yields a lower limit to the galaxy correlation at r ~ 15h-lMpc. 3. Integrating the intrinsic spatial c o r r e l a t i o n g (r) = 360(rh) -1"8 ( r < 150h-iMpc) yields good agreement w i t h the observed angular c o r r e l a t i o n functions of both the D 4 4 and D = 5 + 6 samples. This agreement indicates that the D < 4 redshift sample is a fair sample of the much larger D = 5 + 6 sample, and that the observed correlations represent real spatial correlations. 4. The correlation function appears to depend strongly on cluster richness, w i t h rich clusters (R ~ 2) showing stronger correlations by a factor of ~ 3 as compared with the poorer (R = i) clusters (both are consistent w i t h an r -1"8 power law). This result, combined w i t h the lower c o r r e l a t i o n amplitude of individual galaxies, indicates that p r o g r e s s i v e l y stronger correlations exist, at a given separation, for richer galaxy systems.
A CATALOG OF SUPERCLUSTERS Introduction The strong c o r r e l a t i o n function among clusters of galaxies discussed in the p r e c e d i n g section arises from the strong tendency of clusters to cluster themselves, i.e., to form superclusters. In order to i n v e s t i g a t e the properties of the large-scale clustering of clusters, a complete, well defined catalog of superclusters - defined as clusters of clusters of galaxies - is required. We have recently c o n s t r u c t e d such a catalog (Bahcall and Soneira, 1983b, h e r e a f t e r BS 1983b) using the complete redshift sample of Abell clusters (discussed above) and an objective selection criterion of a spatial density enhancement. We report on these results in the present section.
M e t h o d of Selection The following selection procedure is adopted. For each cluster in the sample a sphere of m a x i m u m radius rm, centered on the cluster, is determined. The radius is chosen so that the density of clusters w i t h i n rm, n(r ~ r m) = N( ~ rm)/ 4~3~-- r~, satisfles: n ( r @ rm) ~ f n o ( R )
(6)
Here N ( < r m) is the number of sample clusters w h o s e centers fall w i t h i n the sphere of radium rm, no(R) is the m e a n space density of clusters at the cluster distance R(z) from us, and f is the density enhancement factor. The radius rm is the m a x i m u m radius around the given cluster at w h i c h the cluster density exceeds the m e a n level by a factor of f. The density of clusters in the sample, especially at high z, is observed to depend on the distance R(z) from the Galaxy (discussed above and BS 1983a), since the sample is
Superclustering
and L a r g e - S c a l e
Structure
373
m a g n i t u d e - l i m i t e d (with some precision) rather than r e d s h i f t - l i m i t e d (as defined in the Abell catalog); therefore, for any given value of f, e q u a t i o n (6) is calculated using the sample's actual v a l u e of n o (R) for each cluster at its a p p r o p r i a t e distance R. For each of the 104 clusters in the sample, a sphere of radius rm is determined, r e s u l t i n g in a series of 104 spheres of v a r i o u s sizes. The spheres typically fall into two categories: i) single, isolated spheres, and 2) distinct, n o n - o v e r l a p p i n g clumps of spheres. The exact d i v i s i o n into these categories depends on the assumed value of f. The outside b o u n d a r y of each clump of spheres a p p r o x i m a t e s an i s o - d e n s i t y e n h a n c e m e n t contour w i t h an average volume density of clusters w i t h i n the b o u n d a r y of fno(R). Each of the distinct clumps of spheres at a given f is i d e n t i f i e d as a separate group of clusters, i.e., a supercluster. Any cluster lying outside the clump boundaries, i.e., all the isolated spheres, are considered single clusters for that v a l u e of f. The sphere size for any value of f depends on the density of n e i g h b o r i n g clusters. We have also carried out a somewhat different analysis using a constant sphere size. Both of these m e t h o d s are similar to the p e r c o l a t i o n - a n a l y s i s technique (e.g., Zeldovich e t al., 1982), w h e r e the p e r c o l a t i o n - s i z e p a r a m e t e r is rm (or, approximately, c~g f-i/3). The groups and singles s e l e c t e d as described above, as well as the b o u n d a r i e s and richnesses of the groups (i.e., number of clusters per group) depend, by definition, on the v a l u e adopted for the d e n s i t y enhancement, f (or, equivalently, the size of the selecting sphere). Low values of f will tend to select groups of low density and h i g h separation, some of w h i c h may not be real p h y s i c a l associations. Higher values of f w i l l select tighter, m o r e compact groups. Many of these tight groups constitute the dense cores of larger groups selected with lower f values (see below). The group b o u n d a r i e s clearly do not define strict p h y s i c a l limits to the s u p e r c l u s t e r s but rather define volumes of various levels of overdensities. In order to q u a n t i t a t i v e l y study how the selection of superclusters is assumed v a l u e of f, we p e r f o r m e d the analysis for various values of f, to 400. The typical m a x i m u m radii r m around single clusters are 17.5, and 5.1 h-IMpc respectively, for f = i0, 20, 40, i00, 200, and 400 (at h-iMpc). Due to the p r o c e d u r e by w h i c h connected (i.e., "percolated") the m a x i m u m s e p a r a t i o n b e t w e e n any b i n a r y cluster is 2r m.
a f f e c t e d by the ranging from f = i0 13.8, 10.9, 8.1, 6.4 a distance of R = 200 spheres are selected,
The S u p e r c l u s t e r s The results for f = 20, 40, i00, 200, and 400 are summarized in Table 1 (for the complete sample of 104 R ~ I clusters). In this table we present a catalog of superclusters, complete to z ~ 0.08, for each of the various values of f. The m e a n p o s i t i o n and redshift of the s u p e r c l u s t e r s as d e t e r m i n e d from the m e m b e r s in the f = 20 catalog are also listed. These values change somewhat as f increases and the groups b r e a k - u p into m o r e compact subgroups (or vice versa). The total number of s u p e r c l u s t e r s decreases from 16 at f = 20 to 7 at f = 400. This factor of two d e c r e a s e is small c o n s i d e r i n g the factor of twenty i n c r e a s e in the overdensity. (Random catalogs yield a factor of 14 decrease in the number of r a n d o m "superclusters"). The m u l t i p l i c i t y , or n u m b e r of clusters per supercluster, Ncl/sc, varies from 2 to 15 for the f = 20 superclusters, and reduces to a value of 2 to 3 clusters per f = 400 supercluster. The s u p e r c l u s t e r s contain a large fraction of all clusters; this fraction, Fcl(SC), is 54% at f = 20 and reduces to 16% at f = 400 (see also Fig. 4 below). The fractional volume of space o c c u p i e d by the s u p e r c l u s t e r s is, however, small; it varies from ~ 3% at f = 20 to 0.04% at f = 400. The f = 400 s u p e r c l u s t e r s have, as expected, the smallest cluster s e p a r a t i o n s observed; all cluster pairs in these s u p e r c l u s t e r s are separated b y £ 13h-1Mpc. These h i g h density s u p e r c l u s t e r s are either identical to, or constitute the densest part (or parts) of the large s u p e r c l u s t e r s selected with the lower f values. The effect of increasing f on the b r e a k u p of large groups into dense, compact subgroups, and e v e n t u a l l y into cores of tight b i n a r i e s and triplets is apparent in Table i. The linear size of the largest observed s u p e r c l u s t e r s increases rapidly from 13h-iMpc for the h i g h e s t density clumps (f = 40) to over ]00h-iMpc at f = 20. This increase is considerably steeper than o b s e r v e d in the random catalogs and indicates the stronger connections among clusters in the real data. A c o m p a r i s o n of the m e a n separation of all cluster pairs in s u p e r c l u s t e r s as seen both p r o j e c t e d on the sky and in the redshift d i r e c t i o n appears to suggest a v e l o c i t y b r o a d e n i n g component s u p e r i m p o s e d on the Hubble flow (see BS 1983a, 1983b) A m a p of all s u p e r c l u s t e r s s e l e c t e d at f = 20, 40, i00, 200 and 400 is p r e s e n t e d in Fig. 3 for both the n o r t h e r n and southern g a l a c t i c hemispheres. The density e n h a n c e m e n t s of the s u p e r c l u s t e r s for f = 20 to 400 are indicated by the density contours. The s u p e r c l u s t e r numbers refer to those listed in Table i. Some specific s u p e r c l u s t e r s are noteworthy. S u p e r c l u s t e r BS8 is a rich Ursa M a j o r supercluster. BSI0 is the Coma s u p e r c l u s t e r (which contains A 1 6 5 6 and A1367 as its R $ 1
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376
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members). BS12B is a dense Corona Borealis supercluster (that may extend to 12A and 12C, as shown, and possibly even further; located in the region of highest density of rich clusters in the sample). BSI5 is the Hercules supercluster, which, from galaxy redshift surveys (Chincarini, Rood, and Thompson, 1981) as well as from our extended R ~ 0 cluster sample distribution (BS 1983b), appears to extend at least ~ 50h-iMpc north to connect with BSI6. It is apparent from the maps that a large concentration of superclusters (i.e., BSI2 Lhrough 16) is present in one quadrant of the northern hemisphere. The large galaxy void in Bootes (Kirshner et al., 1981) is located close-by, and is apparently related to this high concentration of rich galaxy clusters and superclusters (Bahcall and Soneira, 1982). Superclusters were selected also from a larger sample of 175 clusters that included all R = 0 clusters with measured redshifts (D ~ 4). The superclusters identified in this R ~ 0 sample are generally consistent with, and strengthen those found in the smaller R ~ i sample (see BS 1983b).
Tests and Comparisons with Random Cataloss In order to evaluate the statistical significance of the superclusters in Table 1, we used i00 random catalogs that were generated with the same selection functions as the real sample (see BS 1983a) and analyzed with an identical procedure to that used on the real clusters. The number of superclusters as a function of richness ncl/s c (i.e., number of cluster members per supercluster) was determined for both the observed and the average of the i00 random catalogs. For all the values of f which were studied, the observed catalog yields a larger number of superclusters with high richnesses than do the random catalogs. The fraction of rich clusters that belong in the superclusters, Fcl(SC), is presented as a function of f in Fig. 4 for both the observed and random catalogs. As expected, Fcl(SC) decreases smoothly with f in both the observed and random catalogs. However, a larger fraction of clusters are supercluster members in the real data than in the random catalogs for f ~ I0. In addition, the decrease in Fcl(SC) with f is considerably less steep in the real data than in the random catalogs. The observed fraction of rich (R $ I) clusters that are in superclusters can be approximated by Fcl(SC) 2 1.8 f-0.4 (for f >>i). The fractional volume of space occupied by the superclusters is very small ( < 10% for f £ i0), and decreases rapidly with increasing density enhancement.
Superclustering
and Large-Scale
Structure
377
The superclusters selected by using a constant-size sphere around each cluster are found to be similar to those identified using a density enhancement criterion (see BS 1983b). By repeating the selection procedure using a somewhat different redshift catalog than that of HGT (i.e., those of Sarazin, Rood and Struble, 1982; Fetisova, 1981), we conclude that the main features of the superclusters identified in this paper are insensitive to the precise determination of the cluster redshifts within the typical expected uncertainties.
LARGE-SCALE
SUPERCLUSTERS
SURROUNDING
THE GIANT GALAXY VOID IN BOOTES
The overdensity of galaxies observed by Kirshner et al. (1981) on both redshift sides of the z ~ 0.04-0.06 galaxy void in Bootes is found to coincide in redshift space with similar overdensities of clusters and superclusters of galaxies (see Bahcall and Soneira, 1982a). The main contributors to these overdensities are superclusters BSI2 and BS15-16 from our catalog (Fig. 3). These dense, large-scale superclusters appear to surround the giant galaxy void, and extend to scales of ~ lOOh-IMpc in the tail of their galaxy distribution. Previous observational evidence, together with these results suggest that galaxy voids may generally be associated with surrounding galaxy excesses; moreover, the bigger the void, the stronger is the related excess (see Bahcall & Soneira 1982a for more details).
A
~ 300 Mpc VOID OF RICH CLUSTERS OF GALAXIES
A hugh void of cataloged nearby rich clusters of galaxies is observed in the complete D ( 4 Abell sample discussed above (see Bahcall & Soneira 1982b). The void is in the approximate redshift range of z ~ 0.03-0.08, and it extends ~ I00 ° across the sky (i.e., ~ 300h-iMpc). Its projected area is completely devoid of nearby - but not distant - rich clusters (R > I). The void does not appear to be caused by absorption in the Galaxy. If this apparent void in nearby rich clusters is real, it subtends avolume of more than 106h-3Mpc 3. All the findings described above appear to indicate the existence of structure in the universe on scales much larger ( ~ 100h-IMpc) than previously seen or expected. This provides strong limitations to models of the formation of galaxies, clusters and the largescale structure of the universe.
REFERENCES Abell, G.O., 1958, Ap. J. Suppl. 3, 211. Bahcall, N.A., 1979, Ap. J. 232, 689. Bahcall, N.A., and Soneira, R.M., 1982a, Ap. J. Letters 258, LI7. Bahcall, N.A., and Soneira, R.M., 1982b, Ap. J. 262, 419. Bahcall, N.A., and Soneira, R.M., 1983a, Ap. J. 270~ 20. Bahcall, N.A., and Soneira, R.M., 1983b, Ap. J. 277, 0. Chincarini, G., Rood, H.J., and Thompson, L.A., 1981, Ap. J. Letters 249, L47. Fetisova, T.S., 1981, Astron. Zh. 58, 1137. Groth, E.J., and Peebles, P.J.E., 1977, Ap. J. 217, 385. Hoessel, J.G., Gunn, J.E., and Thuan, T.X., 1980, Ap. J. 241, 486. Kirshner, R.P., Oemler, A. Jr., Schechter, P.L., and Shectman, S.A., 1981, Ap. J. Letters 248, L57. Peebles, P.J.E., 1974, Ap. J. Suppl. 28, 37. Peebles, P.J.E., 1980, in Proceedings of Les Houches Summer School, session XXXII, Physical Cosmology. Sandage, A., 1961, Ap. J. 133, 355. Sarazin, C.L., Rood, H.J., and Struble, M.F., 1982, Astron. Ap. Letters 108, L7. Soneira, R.M., and Peebles, P.J.E., 1978, A.J. 833, 845. Zeldovich, Ya. B., Einasto, J., and Shandarin, S.F., 1982, Nature, 300, 407.
0273-1177/84 $0.00 + .50 Copyright © COSPAR
All rights reserved.
S U P E R C L U S T E R I N G A N D THE L A R G E - S C A L E S T R U C T U R E OF THE U N I V E R S E Neta A. Bahcall Space Telescope Science Institute, Baltimore, MD, U.S.A.
ABSTNICT The spatial distribution of a complete redshift sample of nearby rich clusters of galaxies is studied. Strong superclustering, extending to very large scales (~ 100h-IMpc), is observed in the cluster distribution. In particular, we determined and discuss below the following studies: I.
II.
III.
IV.
The spatial correlation function of rich clusters of galaxies. The construction of a complete catalog of superclusters and a study of their growth (percolation). The finding of large-scale superclusters surrounding the giant galaxy void in Bootes. The finding of a ~ 300 l~c void of rich clusters of galaxies.
All the above studies provide evidence for the existence of large-scale structure in the universe, and reveal some of its characteristics. These findings are of importance to models of the formation of galaxies and structure in the universe.
KEYWORDS
Clusters of galaxies;
superclusters;
correlation function; voids;
large-scale structure.
THE S ~ P L E The sample used is Abell's (1958) statistical sample of rich clusters of galaxies of distance class D < 4 (z ~ 0.1), with redshifts for all but one of these clusters reported by Hoessel, Gunn, and Thuan (1980). This sample includes all Abell clusters at D $ 4 that are of richness class R ~ 1 and are located at high galactic latitude (as specified in Abell's Table 1 plus the requirement of [bl ~ 30o). A total of 104 clusters belong in this sample. The area subtended by the above statistical sample is 4.26 steradians (2.64 in the north and 1.62 in the south). A summary of the sample properties and its division into distance and richness classes as well as into hemispheres is presented in Bahcall & Soneira 1983a (hereafter BS 1983a). Also listed in the above reference are properties of the much larger and deeper D = 5 + 6 statistical sample ([b] ~30°), that includes 1547 clusters; while only a small fraction of the redshifts are measured for this sample, it is used, because of its much larger number of clusters, in various comparison tests to strengthen and confirm the results obtained from the D ~ 4 sample. The distances to the clusters are calculated from the standard relation 1961):
(e.g., Sandage,
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A tlubble distances
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o f Ho = 1 0 0 h km s - 1 M p c - 1 a n d qo = ½ a r e u s e d t h r o u g h o u t the paper. b y ~ 2% f o r qo = 0 ) . In equation ( 1 ) we a s s u m e t h a t F e c u l i a r v e l o c i t i e s
are small compared to the ~ubble velocity.
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The spatial density of clusters in the sample as a function of redshift is given in BS 1983a. As expected, the mean density remains approximately constant for small redshifts and then, due to the limited precision in estimating distances from ml0 magnitudes, the density falls off, exhibiting a low-density tail to z = 0.14. This observed variation of the Abell selection function with distance (i.e., n(z) in Fig. 3 of BS 1983a) must be taken into account in any analysis requiring a distance-limited sample. We do so in our determination of the spatial correlation function by comparing the data with random catalogs constructed to have the same density selection function n(z) and boundaries as those of the real sample. We also use the observed n(z) in our construction of the supercluster catalog (see below). Selection effects that depend on position in the sky (e.g., due to absorption in the Galaxy or confusion with galactic stars) are also of importance in determining the correlation function or other distributions of clusters. We determined as a function of galactic latitude and longitude the ratio of observed number of clusters to that expected if the distribution on the sky was uniform. The results for both D < 4 and D = 5 + 6, presented in BS 1983a, show some decrease of cluster density with latitude. This dependence is corrected for in the determination of the cluster correlation function by including the same selection effect, P(b), in the random catalogs (see below).
THE SPATIAL CORRELATION FUNCTION OF RICH CLUSTERS OF GALAXIES The availability of a complete redshift sample of rich clusters of galaxies makes it possible, for the first time, to calculate relatively accurately cluster distances and separations and, in turn, to determine directly the spatial correlation function of rich clusters of galaxies. We do so in the present section. A more detailed description is given in BS 1983a. The joint probability, dP, of finding two clusters separated by a distance r and within volume elements dV I and dV 2 is written as: dP = n(R~)n(R2)[1
+ ~(r)]dVxdV2 ,
(2)
where ~ (r) is the two-point spatial correlation function and n(R) is the space density of clusters at the distance R from the Sun. (The space density in the present sample varies with the distance R due to the sample selection function discussed in the previous section.) The frequency distribution F(r) of all pairs of clusters with separation r in the present sample was determined. Cluster separations were calculated using the redshift d~stances (eq. (I)) and the angular separation on the sky (i.e., r = (R~+ R~ - 2RiR2cos 0)~); peculiar velocities were assumed to be small. In order to minimlze the influence of selection effects on the determination of ~ (r), we constructed a set of i000 random catalogs, each containing 104 clusters randomly distributed within the angular boundaries of the survey region, but with the same selection functions in both redshift and angular position as the Abell redshift sample (see the previous section). The frequency distribution of cluster pairs was determined in both the real and random catalogs, and the results were compared. This procedure ensures that the selection effects and boundary conditions will affect the data and random catalogs in the same manner, thus minimizing most of these effects. The spatial correlation function was determined from the relation ~(r) = F(r)/Fn(r)
-
1 ,
(3)
where F(r) is the observed frequency of pairs in the Abell sample and FR(r) is the corresponding frequency of random pairs (as determined by the ensemble average frequency of the i000 random catalogs). An ensemble average random frequency is used in order that (r) will not be affected by the fluctuations present in any particular realization of a single random sample. The correlation function was evaluated for various cases including: (a) no selection function in latitude (i.e., P(b) = i); (b) full selection function in latitude (discussed above); (c) northern and southern hemispheres treated separately; and (d) high and low latitude zones (Ibt > 50 ° and ]bl ~ 5 0 ° ) treated separately, each with its observed n(z) (and P(b)) selection function. The resulting correlation function is presented in Fig. i. Strong spatial correlations are observed a T separations ~ 25h-iMpc. Weaker correlations are observed to large separations of ~ 100h--~pc, where ~ 0.I; beyond 150h-iMpc, no statistically significant correlations are observed in the present sample. The results for the correlation function do not differ greatly between the two extreme cases of zero and full correction to the latitude selection function P(b), implying that our main conclusions are not strongly influenced by the latitude selection. In addition, the overall results obtained using a two-zone random
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a) The spatial correlation function of the D ~ 4 sample. (Crosses refer to no correction for latitude selection function; dots refer to the full correction of P(b)). The solid line is the best-fit 1.8 power-law to the data. The dashed line is the galaxy-galaxy correlation function of Peebles and co-workers. b)
Same as a) but plotted in larger bins at large separations.
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catalog (high and low latitude zones) show no significant difference from those obtained usin~ the entire sample. The correlation function in Fig. 1 can be well approximated by a single power-law relation of the form ~{r) = 300(rh)
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5 ~ r ~ 150h t M p c ,
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where r is in Mpc. The correlation function is smooth, with little scatter at r ~50h-IMpc. At r >50h-iMpc the scatter and uncertainties increase, but weak correlations of order 0.2 are still detected to these very large separations. These results, as expected, are consistent with the angular and redshift correlations observed at large angular and radial separations (BS 1983a). If real, the weak correlations at ~50-150h-iMpc should also be found in future, larger (deeper) redshift samples of rich clusters. In the meantime, the agreement between the above derived correlation function and the angular correlation function of the much larger D = 5 + 6 sample (which includes 1547 clusters), both of which show correlation to projected separations of at l e a s t ~ 100h-lMpc, suggests that the present results may indeed be representative of rich clusters in general. At 1.5
(5)
For f ~ 5 % (corresponding to galaxies within one Abell radius; Bahcall, 1979), the contribution to the galaxy correlation function from rich clusters, at any separation, is of the order of ~cc/400 ( ~ $ ~ / 2 5 ) . Thus, beyond the reported break in ~ , e.g., at r ~20h-iMpc, a lower limit to the expected galaxy-galaxy correlation is: ~gg(2~ Mpc) ~ 0 . 0 0 5 . Since galaxy-cluster cross-correlations indicate that a galaxy excess @fists well beyond an Abell radius of a cluster, the fraction f is most likely larger than 5% and the galaxy correlation limit becomes higher (for f = 10%, ~ gg(20h -I) ~0.02). The correlation function of clusters of different richness classes (R = 1 and R ~ 2) were determined separately in order to test for possible dependance on richness. The large sample of 1547 D = 5 + 6 clusters was used for that purpose, and the angular correlation functions of its 1125 R=I and 422 R ~ 2 clusters were determined separately. The amplitude of the angular correlation function is found to be strongly dependent on cluster richness, with richer clusters (R ~ 2) showing stronger correlations by a factor of ~ 3 as compared with the poorer (R = I) clusters (Fig. 2). (The D < 4 sample contains too few R ~ 2 clusters to yield statistically significant comparisons.) Both richness classes exhibit the same power-law shape correlation function as observed in the total sample. (See also BS 1983a.) In Fig. (2b) we show the dependence of the correlation function on the richness of the system, from single galaxies to poor and rich clusters. It is apparent that the correlations become stronger with increasing richness (or luminosity) of the system. This implies that the chance, above random, of finding a rich system next to another rich system is considerably higher than that of finding two neighboring poorer systems.
Superclustering
40
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a) The angular correlation function of richness 1 clusters (dots; 1125 clusters) and richness ~ 2 clusters (circles; 422 clusters) in the D = 5 + 6 sample. b) The dependence on richness of the two-p~int spatial correlation function. The spatial correlation function at r = 5h-~Mpc is plotted as a function of richness (roughly proportional to luminosity) for the galaxy-galaxy, galaxycluster, and cluster-cluster pairs (R ~ 1 as well as R = 1 and R ~ 2 independently; the/$atter are shown by open circles). Richness represents < n > = (nln 2) l z , where the subscripts 1 and 2 refer to the two members of t h e pair, n i is the mean Abell galaxy richness count for the appropriate clusters, and n I = 1 is used for a galaxy. (The appropriate galaxy-cluster cross-correlation is from Peebles, 1980; it excludes the term due to internal galaxy correlations within the cluster.) The dashed line, $ ( n ) ~ n 0"8, is presented in order to guide the eye in the approximate richness dependence of the correlation.
372
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In order to ensure that the above derived spatial correlation function is not due to some special p e c u l i a r i t i e s in the nearby D < 4 sample, the an~ular c o r r e l a t i o n function of the m u c h larger and deeper D = 5 + 6 sample (1547 clusters) was determined and compared with that expected from the above spatial function. The a~reement b e t w e e n the D ~ 4 and D = 5 + 6 functions is excellent (see Fig. 12 of BS 1983a), indicating that the derived spatial function indeed represents well the overall cluster d i s t r i b u t i o n in the entire catalog. A d d i t i o n a l checks are presented in BS 1983a.
Tha m a i n conclusions reached from the above study are summarized below.
i. Strong spatial correlations of rich Abell clusters are observed in both the D < 4 redshift sample and the larger and deeper D = 5 ÷ 6 sample. Cluster correlations are detected to large separations of r £ 150h-iMpc. The correlations are consistent in the angular, redshift, and sDatial c o r r e l a t i o n functions of these samples (see also BS 1983a), with good agreement b e t w e e n the results obtained from the D < 4 and D = 5 + 6 samples. Various selection effects have been removed by comparing with a p p r o p r i a t e l y synthesized random catalogs. 2. The spatial correlation function of rich (R ~ i) clusters is observed to fit a powerlaw relation of the form ~ (r) = 300(rh) -1.8 for 5 ~ r ~ 150h-IMpc (r in Mpc). (When corrected for v e l o c i t y broadening, the intrinsic function becomes ~ (r) = 360(rh) -1"8, w h i c h best fits the observed angular and redshift c o r r e l a t i o n functions.) This correlation function has the same shape as the galaxy c o r r e l a t i o n function, but has an amplitude larger by a factor of ~ 18, and extends to greater distances than those observed for the galaxy c o r r e l a t i o n function. The cluster c o r r e l a t i o n function is unity at r ~ 2 5 h - i M p c , as compared with r 2 5h-iMpc in the galaxy c o r r e l a t i o n function. The extent of the rich cluster correlations beyond the reported 15h-iMpc break in the galaxy correlation function yields a lower limit to the galaxy correlation at r ~ 15h-lMpc. 3. Integrating the intrinsic spatial c o r r e l a t i o n g (r) = 360(rh) -1"8 ( r < 150h-iMpc) yields good agreement w i t h the observed angular c o r r e l a t i o n functions of both the D 4 4 and D = 5 + 6 samples. This agreement indicates that the D < 4 redshift sample is a fair sample of the much larger D = 5 + 6 sample, and that the observed correlations represent real spatial correlations. 4. The correlation function appears to depend strongly on cluster richness, w i t h rich clusters (R ~ 2) showing stronger correlations by a factor of ~ 3 as compared with the poorer (R = i) clusters (both are consistent w i t h an r -1"8 power law). This result, combined w i t h the lower c o r r e l a t i o n amplitude of individual galaxies, indicates that p r o g r e s s i v e l y stronger correlations exist, at a given separation, for richer galaxy systems.
A CATALOG OF SUPERCLUSTERS Introduction The strong c o r r e l a t i o n function among clusters of galaxies discussed in the p r e c e d i n g section arises from the strong tendency of clusters to cluster themselves, i.e., to form superclusters. In order to i n v e s t i g a t e the properties of the large-scale clustering of clusters, a complete, well defined catalog of superclusters - defined as clusters of clusters of galaxies - is required. We have recently c o n s t r u c t e d such a catalog (Bahcall and Soneira, 1983b, h e r e a f t e r BS 1983b) using the complete redshift sample of Abell clusters (discussed above) and an objective selection criterion of a spatial density enhancement. We report on these results in the present section.
M e t h o d of Selection The following selection procedure is adopted. For each cluster in the sample a sphere of m a x i m u m radius rm, centered on the cluster, is determined. The radius is chosen so that the density of clusters w i t h i n rm, n(r ~ r m) = N( ~ rm)/ 4~3~-- r~, satisfles: n ( r @ rm) ~ f n o ( R )
(6)
Here N ( < r m) is the number of sample clusters w h o s e centers fall w i t h i n the sphere of radium rm, no(R) is the m e a n space density of clusters at the cluster distance R(z) from us, and f is the density enhancement factor. The radius rm is the m a x i m u m radius around the given cluster at w h i c h the cluster density exceeds the m e a n level by a factor of f. The density of clusters in the sample, especially at high z, is observed to depend on the distance R(z) from the Galaxy (discussed above and BS 1983a), since the sample is
Superclustering
and L a r g e - S c a l e
Structure
373
m a g n i t u d e - l i m i t e d (with some precision) rather than r e d s h i f t - l i m i t e d (as defined in the Abell catalog); therefore, for any given value of f, e q u a t i o n (6) is calculated using the sample's actual v a l u e of n o (R) for each cluster at its a p p r o p r i a t e distance R. For each of the 104 clusters in the sample, a sphere of radius rm is determined, r e s u l t i n g in a series of 104 spheres of v a r i o u s sizes. The spheres typically fall into two categories: i) single, isolated spheres, and 2) distinct, n o n - o v e r l a p p i n g clumps of spheres. The exact d i v i s i o n into these categories depends on the assumed value of f. The outside b o u n d a r y of each clump of spheres a p p r o x i m a t e s an i s o - d e n s i t y e n h a n c e m e n t contour w i t h an average volume density of clusters w i t h i n the b o u n d a r y of fno(R). Each of the distinct clumps of spheres at a given f is i d e n t i f i e d as a separate group of clusters, i.e., a supercluster. Any cluster lying outside the clump boundaries, i.e., all the isolated spheres, are considered single clusters for that v a l u e of f. The sphere size for any value of f depends on the density of n e i g h b o r i n g clusters. We have also carried out a somewhat different analysis using a constant sphere size. Both of these m e t h o d s are similar to the p e r c o l a t i o n - a n a l y s i s technique (e.g., Zeldovich e t al., 1982), w h e r e the p e r c o l a t i o n - s i z e p a r a m e t e r is rm (or, approximately, c~g f-i/3). The groups and singles s e l e c t e d as described above, as well as the b o u n d a r i e s and richnesses of the groups (i.e., number of clusters per group) depend, by definition, on the v a l u e adopted for the d e n s i t y enhancement, f (or, equivalently, the size of the selecting sphere). Low values of f will tend to select groups of low density and h i g h separation, some of w h i c h may not be real p h y s i c a l associations. Higher values of f w i l l select tighter, m o r e compact groups. Many of these tight groups constitute the dense cores of larger groups selected with lower f values (see below). The group b o u n d a r i e s clearly do not define strict p h y s i c a l limits to the s u p e r c l u s t e r s but rather define volumes of various levels of overdensities. In order to q u a n t i t a t i v e l y study how the selection of superclusters is assumed v a l u e of f, we p e r f o r m e d the analysis for various values of f, to 400. The typical m a x i m u m radii r m around single clusters are 17.5, and 5.1 h-IMpc respectively, for f = i0, 20, 40, i00, 200, and 400 (at h-iMpc). Due to the p r o c e d u r e by w h i c h connected (i.e., "percolated") the m a x i m u m s e p a r a t i o n b e t w e e n any b i n a r y cluster is 2r m.
a f f e c t e d by the ranging from f = i0 13.8, 10.9, 8.1, 6.4 a distance of R = 200 spheres are selected,
The S u p e r c l u s t e r s The results for f = 20, 40, i00, 200, and 400 are summarized in Table 1 (for the complete sample of 104 R ~ I clusters). In this table we present a catalog of superclusters, complete to z ~ 0.08, for each of the various values of f. The m e a n p o s i t i o n and redshift of the s u p e r c l u s t e r s as d e t e r m i n e d from the m e m b e r s in the f = 20 catalog are also listed. These values change somewhat as f increases and the groups b r e a k - u p into m o r e compact subgroups (or vice versa). The total number of s u p e r c l u s t e r s decreases from 16 at f = 20 to 7 at f = 400. This factor of two d e c r e a s e is small c o n s i d e r i n g the factor of twenty i n c r e a s e in the overdensity. (Random catalogs yield a factor of 14 decrease in the number of r a n d o m "superclusters"). The m u l t i p l i c i t y , or n u m b e r of clusters per supercluster, Ncl/sc, varies from 2 to 15 for the f = 20 superclusters, and reduces to a value of 2 to 3 clusters per f = 400 supercluster. The s u p e r c l u s t e r s contain a large fraction of all clusters; this fraction, Fcl(SC), is 54% at f = 20 and reduces to 16% at f = 400 (see also Fig. 4 below). The fractional volume of space o c c u p i e d by the s u p e r c l u s t e r s is, however, small; it varies from ~ 3% at f = 20 to 0.04% at f = 400. The f = 400 s u p e r c l u s t e r s have, as expected, the smallest cluster s e p a r a t i o n s observed; all cluster pairs in these s u p e r c l u s t e r s are separated b y £ 13h-1Mpc. These h i g h density s u p e r c l u s t e r s are either identical to, or constitute the densest part (or parts) of the large s u p e r c l u s t e r s selected with the lower f values. The effect of increasing f on the b r e a k u p of large groups into dense, compact subgroups, and e v e n t u a l l y into cores of tight b i n a r i e s and triplets is apparent in Table i. The linear size of the largest observed s u p e r c l u s t e r s increases rapidly from 13h-iMpc for the h i g h e s t density clumps (f = 40) to over ]00h-iMpc at f = 20. This increase is considerably steeper than o b s e r v e d in the random catalogs and indicates the stronger connections among clusters in the real data. A c o m p a r i s o n of the m e a n separation of all cluster pairs in s u p e r c l u s t e r s as seen both p r o j e c t e d on the sky and in the redshift d i r e c t i o n appears to suggest a v e l o c i t y b r o a d e n i n g component s u p e r i m p o s e d on the Hubble flow (see BS 1983a, 1983b) A m a p of all s u p e r c l u s t e r s s e l e c t e d at f = 20, 40, i00, 200 and 400 is p r e s e n t e d in Fig. 3 for both the n o r t h e r n and southern g a l a c t i c hemispheres. The density e n h a n c e m e n t s of the s u p e r c l u s t e r s for f = 20 to 400 are indicated by the density contours. The s u p e r c l u s t e r numbers refer to those listed in Table i. Some specific s u p e r c l u s t e r s are noteworthy. S u p e r c l u s t e r BS8 is a rich Ursa M a j o r supercluster. BSI0 is the Coma s u p e r c l u s t e r (which contains A 1 6 5 6 and A1367 as its R $ 1
N.A. Bahcall
374
Table 1 Supercluster Catalog B5
a.~ (19~0)
=
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0.0682
500.514
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81
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0.0992
1084.1399
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0,0'782
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134~+Z?14a
0.069~
1"/'73,1809 "[7"/5.1793 179,5,1831
1929,2022 1213
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0-0710a
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5uperclu,~ter catalog /or t220 through |=400 [or thn D:;4,R~,I complete redahltt =ample; (uaix~ HGT redshffL=, except z(A1:]18)=0.0566; =ee text). Co],xrnn heacllna~. 1) [IS supercluater number;, 2-3) mean =uperclu~er pc~|tlmz and redaEJ~t for [~-20 (average of ~ [=20 member~, except where otherw~e indicated); 4--8) a Iht of the Abell clu=ter member= (R;=l) in the :luperclu_ltera for each of the overden=itie=/220 to 4C0. a) Mean poslLton and re~hJR~ of the f ;~a,O member=. b) Mean pc=iUon and red.~zUt~ o! the f;zlO0 member=. c) hzc]mied in the f=-lCO =upercJu=ter= ff z(A2152) = G.0383 i~ u.~ed (a= mea=ured from 22 galax7 red~hlft~; SP~).
Superclustering
and L a r g e - S c a l e
50"
Structure
375
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Fig.
3.
P r o j e c t e d e q u a l - a r e a m a p of the n o r t h e r n (a) and s o u t h e r n (b) clusters and s u p e r c l u s t e r s in the R ~ I sample. O u t e r contour is Ibl = 30o; inner contour is the c o m p l e t e n e s s limit of the sample. The g a l a c t i c poles are at the r e s p e c t i v e c e n t e r s of these polar maps. L o n g i t u d e 1 II is 0 ° at the w e s t and i n c r e a s e s clockwise. C l u s t e r s at d i f f e r e n t r e d s h i f t ranges are m a r k e d by different symbols. The d e n s i t y contours r e p r e s e n t the 3 - d i m e n s i o n a l d e n s i t y e n h a n c e m e n t of the s u p e r c l u s t e r s , from f = 20 to f = 400. The s u p e r c l u s t e r s are n u m b e r e d by their BS c a t a l o g n u m b e r s for f = 20 (Table i). At f = I0, s u p e r c l u s t e r s BSI2 + 13 + 15 + 16 c o a l e s c e into a single l a r g e - s c a l e supercluster.
376
N.A. Bahcall
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Fig. 4.
The fraction of all R ~ 1 clusters that are supercluster members as a function of the supercluster density enhancement f. Crosses represent the data, dots represent the ensemble average of i00 random catalogs, l-O- error-bars are shown for the random points. The curves are power-law fits for f >> i.
members). BS12B is a dense Corona Borealis supercluster (that may extend to 12A and 12C, as shown, and possibly even further; located in the region of highest density of rich clusters in the sample). BSI5 is the Hercules supercluster, which, from galaxy redshift surveys (Chincarini, Rood, and Thompson, 1981) as well as from our extended R ~ 0 cluster sample distribution (BS 1983b), appears to extend at least ~ 50h-iMpc north to connect with BSI6. It is apparent from the maps that a large concentration of superclusters (i.e., BSI2 Lhrough 16) is present in one quadrant of the northern hemisphere. The large galaxy void in Bootes (Kirshner et al., 1981) is located close-by, and is apparently related to this high concentration of rich galaxy clusters and superclusters (Bahcall and Soneira, 1982). Superclusters were selected also from a larger sample of 175 clusters that included all R = 0 clusters with measured redshifts (D ~ 4). The superclusters identified in this R ~ 0 sample are generally consistent with, and strengthen those found in the smaller R ~ i sample (see BS 1983b).
Tests and Comparisons with Random Cataloss In order to evaluate the statistical significance of the superclusters in Table 1, we used i00 random catalogs that were generated with the same selection functions as the real sample (see BS 1983a) and analyzed with an identical procedure to that used on the real clusters. The number of superclusters as a function of richness ncl/s c (i.e., number of cluster members per supercluster) was determined for both the observed and the average of the i00 random catalogs. For all the values of f which were studied, the observed catalog yields a larger number of superclusters with high richnesses than do the random catalogs. The fraction of rich clusters that belong in the superclusters, Fcl(SC), is presented as a function of f in Fig. 4 for both the observed and random catalogs. As expected, Fcl(SC) decreases smoothly with f in both the observed and random catalogs. However, a larger fraction of clusters are supercluster members in the real data than in the random catalogs for f ~ I0. In addition, the decrease in Fcl(SC) with f is considerably less steep in the real data than in the random catalogs. The observed fraction of rich (R $ I) clusters that are in superclusters can be approximated by Fcl(SC) 2 1.8 f-0.4 (for f >>i). The fractional volume of space occupied by the superclusters is very small ( < 10% for f £ i0), and decreases rapidly with increasing density enhancement.
Superclustering
and Large-Scale
Structure
377
The superclusters selected by using a constant-size sphere around each cluster are found to be similar to those identified using a density enhancement criterion (see BS 1983b). By repeating the selection procedure using a somewhat different redshift catalog than that of HGT (i.e., those of Sarazin, Rood and Struble, 1982; Fetisova, 1981), we conclude that the main features of the superclusters identified in this paper are insensitive to the precise determination of the cluster redshifts within the typical expected uncertainties.
LARGE-SCALE
SUPERCLUSTERS
SURROUNDING
THE GIANT GALAXY VOID IN BOOTES
The overdensity of galaxies observed by Kirshner et al. (1981) on both redshift sides of the z ~ 0.04-0.06 galaxy void in Bootes is found to coincide in redshift space with similar overdensities of clusters and superclusters of galaxies (see Bahcall and Soneira, 1982a). The main contributors to these overdensities are superclusters BSI2 and BS15-16 from our catalog (Fig. 3). These dense, large-scale superclusters appear to surround the giant galaxy void, and extend to scales of ~ lOOh-IMpc in the tail of their galaxy distribution. Previous observational evidence, together with these results suggest that galaxy voids may generally be associated with surrounding galaxy excesses; moreover, the bigger the void, the stronger is the related excess (see Bahcall & Soneira 1982a for more details).
A
~ 300 Mpc VOID OF RICH CLUSTERS OF GALAXIES
A hugh void of cataloged nearby rich clusters of galaxies is observed in the complete D ( 4 Abell sample discussed above (see Bahcall & Soneira 1982b). The void is in the approximate redshift range of z ~ 0.03-0.08, and it extends ~ I00 ° across the sky (i.e., ~ 300h-iMpc). Its projected area is completely devoid of nearby - but not distant - rich clusters (R > I). The void does not appear to be caused by absorption in the Galaxy. If this apparent void in nearby rich clusters is real, it subtends avolume of more than 106h-3Mpc 3. All the findings described above appear to indicate the existence of structure in the universe on scales much larger ( ~ 100h-IMpc) than previously seen or expected. This provides strong limitations to models of the formation of galaxies, clusters and the largescale structure of the universe.
REFERENCES Abell, G.O., 1958, Ap. J. Suppl. 3, 211. Bahcall, N.A., 1979, Ap. J. 232, 689. Bahcall, N.A., and Soneira, R.M., 1982a, Ap. J. Letters 258, LI7. Bahcall, N.A., and Soneira, R.M., 1982b, Ap. J. 262, 419. Bahcall, N.A., and Soneira, R.M., 1983a, Ap. J. 270~ 20. Bahcall, N.A., and Soneira, R.M., 1983b, Ap. J. 277, 0. Chincarini, G., Rood, H.J., and Thompson, L.A., 1981, Ap. J. Letters 249, L47. Fetisova, T.S., 1981, Astron. Zh. 58, 1137. Groth, E.J., and Peebles, P.J.E., 1977, Ap. J. 217, 385. Hoessel, J.G., Gunn, J.E., and Thuan, T.X., 1980, Ap. J. 241, 486. Kirshner, R.P., Oemler, A. Jr., Schechter, P.L., and Shectman, S.A., 1981, Ap. J. Letters 248, L57. Peebles, P.J.E., 1974, Ap. J. Suppl. 28, 37. Peebles, P.J.E., 1980, in Proceedings of Les Houches Summer School, session XXXII, Physical Cosmology. Sandage, A., 1961, Ap. J. 133, 355. Sarazin, C.L., Rood, H.J., and Struble, M.F., 1982, Astron. Ap. Letters 108, L7. Soneira, R.M., and Peebles, P.J.E., 1978, A.J. 833, 845. Zeldovich, Ya. B., Einasto, J., and Shandarin, S.F., 1982, Nature, 300, 407.