![]() | Annu. Rev. Astron. Astrophys. 1988. 26:
631-86 Copyright © 1988 by Annual Reviews. All rights reserved |
The spatial distribution of rich clusters of galaxies and the clustering properties of clusters have been the subject of considerable interest over the past two decades, with a wide range of claims as to the nature and properties of such clustering. Since rich clusters can be used rather efficiently in surveying the structure in large volumes of space, they have recently become an important tool in tracing the large-scale structure of the Universe.
The Abell (1958) catalog of rich clusters has been analyzed by many investigators (e.g. Abell 1958, 1961, Hauser & Peebles 1973, Rood 1976, Bahcall & Soneira 1983, 1984, Klypin & Kopylov 1983, Bahcall et al. 1986, Shvartsman 1988, Kalinkov et al. 1985, Batuski & Burns 1985a, Tully 1986, and references therein) using different techniques in an attempt to determine the spatial distributions of rich clusters. Abell (1958, 1961) found that the surface distribution of the clusters in his statistical sample (see Section 2) was highly nonrandom and reported evidence suggestiug the existence of superclusters; Bogart & Wagoner (1973), Hauser & Peebles (1973), and Rood (1976) (see also references therein) also found, using nearest-neighbor distributions and/or angular correlation functions, strong evidence for superclustering among the Abell clusters. The studies dealt primarily with the surface distribution of clusters and, in some cases, used approximate estimates for cluster redshifts. More recently, Bahcall & Soneira (1983, 1984) and, independently, Klypin & Kopylov (1983) used redshift measurements of complete samples of clusters to determine directly the spatial distribution of rich clusters. The results, discussed in more detail below, indicate that rich clusters of galaxies cluster very strongly in space, forming clusters of clusters of galaxies, or superclusters (see also Section 6). The clustering strength of clusters was observed to be much higher than the clustering strength of galaxies. The clustering or correlation scale for rich clusters was found to be about five times larger than the correlation scale of galaxies. Similar investigations followed therefore after (see below), all yielding consistent results. Since these results provide strong constraints on models for the formation and evolution of galaxies and structure, I review in this section the findings of recent investigations of the clustering of clusters, using various catalogs and methods.
The correlation function
(Limber 1953,
Peebles 1980a)
is one of the
best statistical tools to measure quantitatively the clustering of
objects in a sample, yielding both clustering strength and extent. The
joint probability
dP() of finding two
objects in a sample separated
by an angle
and within
solid angles
d
1 and
d
2 is
written as
![]() | (1) |
where w() is the
two-point angular correlation function and N is the
surface number density of objects in the sample. The two-point angular
correlation function thus describes, as a function of angular scale,
the net projected pair clustering of objects on the sky above that
expected from random distribution.
Similarly, the spatial correlation function
(r) is
defined by the
joint probability dP(r) of finding two objects separated
by a distance
r and within volume elements dV1 and
dV2, such that
![]() | (2) |
where n is the space density of objects in the sample. The correlations are therefore zero for a random distribution of points and are positive for a clumped distribution on the relevant clumping scale.
3.2.1 ABELL CLUSTERS The two-point
spatial correlation function of clusters,
cc(r), was determined by
Bahcall & Soneira
(1983;
hereinafter BS83) using
Abell's (1958)
statistical sample of rich clusters of galaxies of distance class
D
4 (z
0.1), with redshifts
for all clusters reported by
Hoessel et al. (1980).
(For properties of the Abell catalog, see
Section 2.) This sample includes all 104
Abell clusters at D
4 that
are of richness class R
1 and are located at high Galactic latitude
(|b|
30°). A
summary of the sample properties and its division
into distance and richness classes, as well as into hemispheres, is
presented in Table 1 and BS83. Also
listed in Table 1 and BS83 are
properties of the much larger and deeper D = 5 + 6 statistical
sample (z
0.2)
that includes 1547 clusters. While only a small fraction of the
redshifts are measured for this sample, it was 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 frequency distribution F(r) for all pairs of clusters with
separation r in the sample was determined. In order to minimize the
influence of selection effects on the determination of
(r), a set of
1000 random catalogs was constructed, each containing 104 clusters
randomly distributed within the angular boundaries of the survey
region but with the same selection functions in both redshift,
n(z),
and latitude, P(b), as the Abell redshift sample. The
frequency
distribution of cluster pairs was determined in both the real and
random catalogs, and the results were then compared. This procedure
ensures that the selection effects and boundary conditions will affect
the data and random catalogs in the same manner.
The spatial correlation function was determined from the relation
![]() | (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 1000 random catalogs). An
ensemble average random frequency is used in order that
(r) 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) = 1]; (b) full selection function in
latitude; (c) Northern
and Southern Hemispheres treated separately; and (d) high- and
low-latitude zones (|b| > 50° and
|b|
50°) treated
separately, each
with its observed n(z) [and P(b)] selection
function.
The resulting correlation function is presented in
Figure 3. Strong
spatial correlations are observed at separations
25h-1 Mpc. Weaker
correlations are observed to larger separations of at least
~ 50h-1 Mpc, and possibly
~ 100h-1 Mpc, where
cc ~ 0.1;
beyond 150 h-1 Mpc, no
statistically significant correlations are observed in the present
sample.
The correlation function of Figure 3 can be well
approximated by a single power-law relation of the form
cc(r)
= 300 r-1.8 for
5
r
150h-1
Mpc. The function is smooth, with little scatter at
r
50h-1 Mpc. At
r > 50h-1 Mpc, the scatter and uncertainties
increase, but weak
correlations of order 0.2 are still detected at these very large
separations. When corrected for velocity broadening among clusters,
the intrinsic rich (R
1)
cluster correlation function was determined by
Bahcall & Soneira to be (BS83)
![]() |
Figure 3. (Top) The spatial
correlation function of the D
|
![]() | (4) |
In comparison, the correlation function of galaxies is given by (Groth & Peebles 1977, Davis & Peebles 1983)
![]() | (5) |
The rich cluster correlation function has the same shape and slope
as those of the galaxy correlation function, but it is considerably
stronger at any given scale (by a factor of ~ 18) than the correlation
function of galaxies. The cluster correlations also extend to greater
separations than the scales observed in the galaxy correlations. The
cluster correlation scale length, i.e. the scale at which the
correlation function is unity, is
r0
26h-1 Mpc (Equation 4), as
compared with
r0
5h-1 Mpc for galaxies. The extent of the rich
cluster correlation function beyond the reported
~ 15h-1 Mpc break in the galaxy correlation function
(Groth & Peebles
1977)
suggests the existence of large-scale structure in the Universe (~
15h-1
Mpc). While
the reason for the strong increase of correlation strength and scale
from galaxies to clusters is still a theoretical challenge, some
possible explanations are discussed in
Sections 5 and 9.
The cluster
correlation function determined above places constraints on models for
the formation of galaxies and structure (see
Section 9).
In order to ensure that the spatial correlation function is not due
to some special peculiarities in the nearby D
4 sample. Bahcall &
Soneira (BS83) carried out several tests that are discussed below.
First, the angular correlation function of the much larger and
deeper D = 5 + 6 sample (1547 R
1 clusters to
z
0.2) was
determined and compared with that expected from the spatial
correlation function above (Equation 4), as well as from the expected
scaling law
(Peebles 1980a)
of the D
4 angular correlation
function. The angular correlation functions of the nearby D
4 and
distant D = 5 + 6 samples are determined to be (BS83)
![]() | (6) |
![]() | (7) |
The angular correlations scale as expected from the scaling law
applied to their respective distances. A comparison of the scaled
functions is shown in figure 4. If the
correlations were mainly due to
patchy obscuration or other omissions by Abell, the (observed) scaling
would not be expected. The scaling agreement indicates that any
possible projection biases in the catalog (e.g.
Sutherland 1988)
are rather small and do not significantly affect the correlation results
(see Bahcall 1988b,
Dekel 1988).
The reduced correlation scale
suggested by Sutherland may result from overcorrecting the actual
correlation power on large scales. A comparison of the D = 5 + 6
angular function with that expected from the spatial correlation
function of Equation (4), when integrated over the relevant redshift
distribution, is shown in Figure 5. The
agreement between the D 4
and D = 5 + 6 functions is excellent. This agreement indicates that
the D
4 redshift sample
is a fair sample of the much larger sample,
and that the observed correlations represent real correlations of
clusters in space. The scaling law of the angular functions was also
studied by
Hauser & Peebles
(1973),
who reached similar conclusions
with regard to the reality of the intrinsic correlations.
![]() |
Figure 4. (a) The angular
correlation function of the D
|
Second, the angular correlation function was compared with the pure redshift (i.e. line-of-sight) correlations of the clusters. If the correlations were mostly due to patchy obscuration on the sky or other similar biases, no extensive redshift correlations would be expected. It is observed (BS83) that the projected and redshift correlations are consistent with each other, further strengthening the reality of the correlations.
Third, the angular correlation function of the D = 5 + 6 sample was determined in different regions of the sky, yielding consistent results within the uncertainties (see Figure 6).
These tests, and those listed in Section 3.3 below, suggest that the observed cluster correlation function is mostly due to physical clustering of rich clusters of galaxies that extends to large scales.
![]() |
Figure 5. The curves represent the angular
correlation function
expected from an intrinsic cluster spatial function given by
|
Since the correlation strength appears to increase from galaxies to
clusters, Bahcall & Soneira also investigated whether a similar trend
is observed between the correlation function of poor and rich
clusters. The angular correlation functions of different richness
classes (R = 1 and R
2) were determined for the
large D = 5 + 6 sample (1125 R = 1 clusters, 422 R
2 clusters). The amplitude of the
correlation function was 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 = 1) clusters. The
results are shown in Figure 7. Both richness
classes exhibit the same
power-law shape correlation function as observed in the total sample;
they satisfy
![]() | (8) |
![]() |
Figure 7. The angular correlation function
of richness 1 and richness
|
![]() | (9) |
The implied spatial correlation can then be represented by
![]() | (10) |
![]() | (11) |
The amplitude of the total (R
1) correlation function is
dominated
by the lower amplitude of the poorer, but more numerous, R = 1
clusters. Figure 8 shows the dependence of the
correlation function
on the richness of the system, from single galaxies to poor and rich
clusters, as suggested by BS83 (see also Section 3.5
for a more
updated richness dependence). The correlations become stronger with
increasing richness (or luminosity) of the system, suggesting that the
correlation function depends upon richness. The galaxy-cluster
cross-correlation function
(Seldner & Peebles
1977;
see, however,
Efstathiou 1988
(also Section 3.2.4)] is consistent with the cluster
correlations and the trend observed above
(Figure 8). Recent
observations of clusters of different types and richnesses (see
summary below) yield results that are consistent with the richness
trend suggested by BS83 (Section 3.5).
![]() |
Figure 8. The dependence on richness of the
two-point spatial
correlation function suggested by BS83. The spatial correlation
function at r = 5h-1 Mpc is plotted as a
function of richness for the
galaxy-galaxy, galaxy-cluster, and cluster-cluster pairs (R
|
Klypin & Kopylov
(1983)
investigated the spatial correlation
function of a nearby sample of Abell clusters similar to the one
described above, supplementing available redshift data with their own
observations. Their sample includes 158 Abell clusters of all richness
classes [R 0
(i.e. including the somewhat incomplete class of R = 0
clusters] in distance group D
4 and located at
|b|
30°. Their
results are consistent with those of BS83. They find
cc(r) = (r/25)-1.6
for their observed range of
r
50h-1 Mpc. The approximately 10%
difference in slope is within the
1
uncertainty of the slope
determination estimated by BS83 (10%).
The earlier work of Hauser & Peebles (1973) used power-spectrum analysis and angular correlations to investigate the distribution of clusters in the Abell catalog. They also find evidence for strong superclustering of clusters and show that the degree and angular scale of the apparent superclustering varies with distance in the manner expected if the clustering is intrinsic to the spatial distribution rather than a consequence of patchy local obscuration.
Additional investigations of the spatial distribution of rich clusters of galaxies in the Abell catalog include those by Kalinkov et al. (1985), Batuski & Burns (1985a), Postman et al. (1986), Shvartsman (1988), and Szalay et al. (1988). These studies investigate different subsamples of the catalog, to different distances, regions, and/or richnesses, as well as apply different techniques and/or corrections. All the investigations yield consistent results with those described above, as summarized below.
Kalinkov et al. (1985)
find a spatial correlation function for rich
(R 1) Abell clusters,
using new redshift estimator calibrations and
richness corrections, of
cc(r) = (r/22.4)-1.9
for r
80h-1 Mpc.
Batuski & Burns
(1985a)
determined the spatial correlation function
for Abell clusters of all richness groups (R
0) to
z
0.085. Their
sample includes 226 clusters. (The higher spatial density of this
sample as compared with the R
1 sample is due to the
inclusion of the R = 0 clusters.) For this sample they find
ccR
0(r) =
65r-1.5 for
r
150h-1 Mpc. The somewhat shallower
slope, while within 2
of the
1.8 slope, may be partially due to the use of some estimated rather than
measured redshifts, which reduces the correlations on small scales and
flattens the slope (see BS83). When approximated as a -1.8 power-law
slope, the function is
ccR
0(r)
200r-1.8
(r/19)-1.8. This correlation
function is one order of magnitude stronger than the galaxy
correlations and about 50% lower than BS83 correlation function for
R
1 clusters. The
somewhat reduced correlation strength is consistent
with the richness dependence suggested by BS83 and
Bahcall & Burgett
(1986)
(Section 3.5).
Postman et al. (1986)
reanalyzed the D 4
sample used by Bahcall &
Soneira, as well as a sample of 152 Abell clusters to z
0.1. Their
results are consistent with the BS83 correlation functions.
Shvartsman (1988)
and
Kopylov et al. (1987)
used the 6-m USSR telescope to measure redshifts of all very rich (R
2) Abell clusters to
z
0.23,
located at b > 60°. They calculated the spatial
correlation function of this deep sample of very rich clusters, which
includes 50 clusters in the redshift range
0.10
z
0.23. They find
ccR
2(r) =
(r/40)-1.5 ± 0.5 for the range
5
r
50h-1 Mpc, consistent with the
BS83 correlations of very rich (R
2) clusters and with the suggested
increase of correlation strength (and length) with richness. The
correlation scale for the R
2 clusters is
~ 40h-1 Mpc, while the
correlation scale for the R
1 clusters is ~
25h-1 Mpc. The above
authors also report weak but positive correlations at much larger
separations:
(100-150
h-1 Mpc) = 0.47 ± 0.14. A similar result is
suggested by
Batuski et al. (1988).
This is comparable to the supercluster correlation results of
Bahcall & Burgett
(1986)
(Section 3.4), who detect similar marginal
(3
) correlations. Systematic
effects, however, which may be important on these scales, are
difficult to assess.
Huchra (1988)
used a deep redshift sample (z
0.2) of Abell
clusters complete over a small region of the northern sky. He finds
cc(r) ~ (r/20)-1.8 for
R
0 clusters,
consistent with the results discussed above.
The new Southern Hemisphere catalog of rich clusters (Abell et al. 1988; see Section 2) can also be analyzed for structure. Bahcall et al. (1988b) have recently analyzed the distribution of clusters in this catalog. The results suggest that the correlation function of clusters in the southern sky is consistent with the results presented above for northern clusters.
3.2.2 SHECTMAN CLUSTERS Shectman (1985) used the Shane-Wirtanen counts to identify clusters of galaxies by finding local density maxima above a threshold value, after slightly smoothing the data to reduce the effect of the sampling grid. A total of 646 clusters of galaxies were identified using the specified selection algorithm (Section 2).
The radial velocity distribution of these clusters is similar to the
radial velocity distribution of Abell clusters of distance class
D 4
as determined by Shectman from comparisons of velocity data for a
complete sample of 112 clusters. The space density of the Shectman
clusters is therefore ~ 6 times greater than the space density of the
104 R
1, D
4 Abell cluster sample.
The angular two-point correlation function of the Shectman clusters
at |b| 50° (a
sample of 488 clusters in total) was determined by
Shectman (1985).
The implied spatial correlation function is
cc(r)
180r-1.8
(r/18)-1.8. This correlation function is about 10
times larger
than the galaxy correlation (Equation 5) and is about a factor of 2
lower than the rich (R
1) cluster correlations (Equation 4). Since
the space density of the Shectman clusters is ~ 6 times higher than
the density of the R
1
clusters, and thus the identifications of the
former are with poorer clusters, the results of the Shectman cluster
correlations are consistent both with those of the Abell clusters and
with the trend suggested by Bahcall & Soneira of increased correlation
strength with cluster richness
(Section 3.5). Recently, S. Shectman
(private communication, 1988) determined the redshifts of all clusters
in the sample, enabling the direct determination of the cluster
spatial correlation function. The spatial function was observed to be
in agreement with the implied spatial correlation function discussed
above.
3.2.3 ZWICKY CLUSTERS The angular distribution of clusters in the Zwicky (1961-1968) catalog was analyzed by Postman et al. (1986). The cluster selection algorithm in the Zwicky catalog differs markedly from the cluster selection definition of Abell (Section 2). Abell's definition of a cluster relates to the cluster intrinsic properties (i.e. the number of galaxies within a given linear scale and a given absolute magnitude range) and thus is independent of redshift (except for standard selection biases). Zwicky's clusters are defined relative to the mean density of the field, with varying cluster sizes and contours, and all galaxies down to the plate limit are considered. Therefore, the cluster selection is by definition strongly dependent on redshift. A direct comparison between the correlation functions of Zwicky and Abell clusters is therefore not straightforward. However, an uncorrected comparison will test to some extent the universality of the cluster correlation function, with its suggested dependence on richness, as well as further test the sensitivity of the correlation function to the cluster identification procedure.
It is found that in the distance range where Abell and Zwicky
identify clusters of comparable overdensity (1173 Distant Zwicky
clusters with
z 0.1 - 0.14), the
correlation functions of the Abell
and Zwicky clusters are indeed the same in the scale range studied (
r
60h-1 Mpc). The angular correlation functions of the
two nearer samples of the Zwicky clusters (377 Near clusters and 680
Medium-Distant clusters) are observed to be weaker (when scaled to the
same depth as the D
4
Abell sample) than the rich (R
1) Abell
clusters. Since these nearer Zwicky clusters are by definition much
poorer clusters, with a considerably higher space density than the
R
1 Abell clusters, they
are expected to have a weaker correlation
strength (BS83; see also Section 3.5).
A comparison of the cluster correlation functions determined by the various investigators discussed above using different catalogs and samples is summarized in Figures 9a and 9b. A general agreement is observed among all the results. The consistency of the correlation functions determined from different catalogs, cluster selection criteria, redshift and richness ranges, and by diferent investigators strongly supports the reality and universality of the cluster correlations described in this section.
![]() |
Figure 9a. A composite of the spatial
cluster correlation function
determined by different investigators from different cluster samples
[Abell clusters to different depths (z
|
![]() |
Figure 9b. A composite of the angular cluster correlation function determined from different catalogs and samples (indicated by the different symbols; see Section 3). All results are scaled to the D = 5 + 6 distance. The BS83 correlation function for the D = 5 + 6 clusters (Figure 4) is indicated by the solid line. The consistency among the different samples, as well as the dependence of the correlation strength on richness (Sections 3.2, 3.5), are apparent. |
3.2.4 GALAXY-CLUSTER CROSS-CORRELATIONS
The angular cross-correlation between the galaxy distribution in the
Shane-Wirtanen galaxy counts and the positions of rich Abell clusters
was studied by
Seldner & Peebles
(1977)
and more recently by
Efstathiou (1988).
This cross-correlation function,
wgc(),
measures
the excess probability, over random, of finding a galaxy within a
given separation from a cluster (i.e. it describes the enhanced
density of galaxies around a cluster).
Seldner & Peebles
(1977)
find that the angular function
wgc()
scales with cluster distance-class D as expected from the galaxy
luminosity function. The
wgc(
)
estimates are reasonably well fitted
by a two-power-law model for the spatial function
gc(r)
(Peebles 1980b):
![]() | (12) |
The enhancement of Lick counts around cluster centers is traced to r ~ 40h-1 Mpc before it is lost in the noise.
The first term of the galaxy-cluster cross-correlation (Equation 12) represents the "standard" internal density profile of galaxies in a cluster (which generally has the shape of a bounded isothermal sphere; e.g. Bahcall 1977). The more slowly varying part of the cross-correlation function found at larger scales and represented by the r-1.7 part of Equation 12 is produced by the clustering of clusters, as discussed in the previous subsections (i.e. galaxies from one cluster provide excess concentration of galaxies near a neighboring "correlated" cluster). The above cross-correlation is consistent with the cluster-cluster correlation function discussed in Section 3.2.1 (Equation 4). It is expected that the cross-correlation term will be a geometrical mean of the correlation functions of the galaxies and clusters. Thus, it is expected that
![]() | (13) |
Using the galaxy and cluster correlation functions discussed in
Section 3.2.1, i.e.
gg
20r-1.8 and
cc
360r-1.8,
the expected cross-correlation term is
gc
85r-1.8. This
compares remarkably well with the second term of Equation 12,
gc(r
/ 12.5)-1.7
73r-1.7. This result
implies that the cluster correlation function is stronger by a factor
of about 16 than the galaxy correlations, and that it extends to
scales of at least 40h-1 Mpc, as is observed directly.
Recently, however, Efstathiou (1988) re-analyzed the galaxy-cluster cross-correlations using only the subsample of clusters for which redshift measurements are available, finding a somewhat weaker and less extended galaxy-cluster cross-correlation function. A more complete redshift sample of clusters may be needed before a galaxy-cluster cross-correlation function can be established with greater precision.
3.3. Supporting Evidence for the Cluster Correlation Function
I summarize below several observations that support the physical reality of the cluster correlation function discussed above.
The evidence listed above supports the reality of the cluster correlation function and suggests that it is unlikely that the correlations are mainly a result of catalog biases or omissions. A determination of the cluster correlation function from catalogs with automated selection procedures will improve the accuracy of the intrinsic cluster correlations, especially at large separations where the correlations are rather weak.
3.4. Supercluster Correlations
Bahcall & Burgett
(1986)
carried the study of rich galaxy clusters one
step further by studying the spatial distribution of
superclusters. The sample used was the
Bahcall & Soneira
(1984)
complete catalog of superclusters to
z 0.08, where
superclusters are
defined as groups of rich clusters and identified by a spatial density
enhancement of clusters. All volumes of space with a spatial density
of clusters f times larger than the mean cluster density are
identified in the above catalog as superclusters for a specified value
of f (Section 6.1). The
supercluster selection process was repeated
for various overdensity values f, from f = 10 to f
= 400, yielding
specific supercluster catalogs for each f value. A total of 16
superclusters are cataloged for R
1 and f = 20, and 26
superclusters
for R
0 and f = 20.
The spatial correlation among the superclusters was determined by
Bahcall & Burgett
(1986)
for samples of different richness and
overdensity. Because of the large size of the superclusters
themselves, no meaningful correlations are expected at small separations
(
50h-1 Mpc). In addition, no detectable
correlations are expected at very large separations
(> 200h-1 Mpc), since this scale is
comparable to the limits of the sample. Any observable correlations
are therefore expected only in a separation "window" around
~ 100h-1 Mpc.
The results, presented in Figure 10, reveal
correlations among superclusters on a very large scale:
~ 100 - 150h-1 Mpc. Because of the
small size of the supercluster sample, the statistical uncertainty is
appreciable; the observed effect is at the
3 level (as determined by
comparisons with numerical simulations of random catalogs). In
addition, all the samples with different overdensities and cluster
richnesses show a similar effect at a similar scale length. The
results imply the existence of very large-scale structures with scales
of ~ 100 - 150h-1 Mpc.
Similar results have been recently obtained by
Kopylov et al. (1987)
by studying correlations of very rich clusters to
z 0.2
(Section 3.2.1). They report
cc(100-150
h-1 Mpc) = 0.47 ± 0.14.
Tully's (1986,
1987b)
observations of very large-scale structures in the cluster
distribution, up to
~ 300h-1 Mpc, may also reflect the above
observed tendency of superclusters to cluster.
![]() |
Figure 10. The spatial correlation of
superclusters for the R
|
Figure 10 shows that the supercluster correlation strength is stronger than that of the rich-cluster correlations by a factor of approximately 4. It is approximately two orders of magnitude stronger than the galaxy correlation amplitude. While this enhancement is observed in the ~ 100 - 150h-1 Mpc range, it is possible that the supercluster correlation function also follows an r-1.8 law. If the correlations follow an r-1.8 law, then the function would satisfy the relation
![]() | (14) |
The implied correlation scale of superclusters would be
60h-1 Mpc, as
compared with 5h-1 Mpc for the correlation scale of
galaxies
(Groth & Peebles
1977)
and 25h-1 Mpc for rich (R
1) clusters (BS83). This
apparent increase in correlation strength is consistent with the
earlier prediction of BS83 of increased correlations with richness
(luminosity) of the system.
The supercluster correlation amplitude fits well the predicted trend (Section 3.5).
3.5. Richness Dependence of the Correlation Function
As discussed above, the cluster correlation function appears to depend strongly on cluster richness (BS83), with richer clusters showing stronger correlations than poorer clusters. This result, combined with the lower correlation amplitude of individual galaxies, led Bahcall & Soneira to the conclusion that progressively stronger correlations exist, at a given separation, for richer or more luminous galaxy systems (Figure 8, Section 3.2.1). Several recent studies of the correlations of other types and richnesses of clusters, reviewed above [Batuski & Burns (1985a), Shectman (1985), and Postman et al. (1986) for poorer clusters; Kopylov et al. (1987) for richer clusters; Bahcall & Burgett (1986) for superclusters] appear to be consistent with the trend suggested by Bahcall & Soneira and later expanded by Bahcall & Burgett (1986). This dependence of correlation strength on richness is summarized in Figure 11. It can be approximated roughly as follows:
![]() | (15) |
where N is the richness of the system [for galaxies, N = 1; for clusters, N = Abell's richness definition (Section 2)]. L is the luminosity (relative to L* in the Schechter luminosity function), and M is the mass of the system. This relation suggests an average trend in the data and should not be regarded as an exact formula. (Obviously, the relation between N, L, and M is not unique; for a given N, different L's and M's may apply, and vice versa. The difference between the M versus the L slope is due to the higher observed M/L ratios for clusters than for galaxies).
![]() |
Figure 11. The dependence of the
correlation function strength on the
mean richness ( |
The correlation-richness dependence suggests that rich clusters populate the large-scale structures, or superclusters, more abundantly than galaxies do relative to their mean space densities. It also implies that rich clusters are indeed an efficient tracer of large-scale structure in the Universe.
A continuous richness dependence of the correlation strength indicates that no unique correlation function exists for all luminous systems (see, however, Section 4 for a possible universal dimensionless correlation function). It therefore places a new emphasis on the question of what is the underlying mass correlation function in the Universe: Which "richness" or "luminosity" equivalent in Figure 11 does the mass follow? And, specifically, should it follow, as usually assumed, the correlation function of galaxies? (The latter may itself be a continuous function of the luminosity or some other property of galaxies.) Could the mass-correlation amplitude correspond to an extrapolation of Figure 11 (or relation 15) to objects with an even lower richness (or luminosity or mass) than galaxies, and thus with weaker clustering properties than galaxies? This question led to the idea of biased galaxy formation models (e.g. Kaiser 1984, Bardeen et al. 1986), where the mass distribution is assumed to be considerably less clumped than galaxies, and galaxies form in a "biased" way only at higher density peaks of the mass distribution. In this picture, a large fraction of the dark matter in the Universe is not attached to luminous galaxies but rather is floating as smaller dark clouds, more smoothly distributed in space.
Several explanations of the observed increase of correlation with
richness have been suggested, although the phenomenon is still not
fully understood.
Kaiser (1984)
suggested applying the statistics of
rare events. If the density perturbations are described by a random
Gaussian field, and if the regions where clusters form correspond to
densities higher than a given threshold, then the correlation function
of the points above the threshold is amplified over the correlation
function of the underlying point distribution. By filtering out scales
smaller than clusters from the initial power spectrum and selecting
the appropriate threshold, it is possible to match the enhanced
correlations of the Abell clusters if the latter correspond to
3
fluctuations (see, however,
Coles 1986).
The model may have
difficulty, however, in explaining the positive cluster correlations
observed at r
20h-1 Mpc, where the galaxy correlations are
negative or zero.
Several galaxy formation models, such as biased cold dark matter, hybrid scenarios, and cosmic strings, can reproduce a trend of increasing correlation strength from galaxies to clusters; these are discussed in Section 9.