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Proceedings: Evolution of Large Scale Structure -- Garching, August 1998
CLUSTERS AS TRACERS OF LARGE--SCALE
STRUCTURE
Marc Postman, Space Telescope Science Institute
Baltimore, Maryland 21218 U.S.A.
ABSTRACT. By virtue of their high galaxy space densities and their large
spatial separations,clusters are e#cient and accurate tracers of the large--scale
density and velocity fields. Substantial progress has been made over the past
decade in the construction of homogeneous, objectively derived cluster cata­
logs and in characterizing the spatial distribution of clusters. Consequently,
the constraints on viable models for the growth of structure have been refined.
A review of the status of cluster--based observations of large--scale structure is
presented here, including discussions of the second and higher order moments,
the dependenceof clustering on richness (mass), recent and new measurements
of bulk flows, and a new constraint on the cluster mass function in the range
0.7 <
# z <
# 1.
1 Introduction
The study of the evolution of
large--scale structure is fundamental
to cosmology. When observations of
the galaxy and cluster spatial distri­
butions are combined with comple­
mentary spectroscopic and morpho­
logical information, one can probe
the abundance and form of dark
matter, the mean baryon and mat­
ter densities, the turnover scale in
the perturbation power spectrum,
and the formation epoch of galax­
ies and clusters. Clusters of galaxies
are particularly well suited to these
studies and, over the past decade,
there have been substantial break­
throughs in and challenging ques­
tions raised about our understand­
ing of large--scale structure from red­
shift and peculiar velocity surveys of
clusters. In this review, I summa­
rize the current constraints on the
two--point cluster--cluster correlation
function (and its dependenceon clus­
ter richness), the cluster power spec­
trum, the very large--scale distribu­
tion of clusters, bulk flow measure­
ments, and the evolution of the clus­
ter mass function. I also emphasize
the importance of characterizing how
ones definition of a cluster ultimately
a#ects the interpretation of obser­
vational and N­body simulation re­
sults. I conclude by highlighting sev­
eral exciting cluster research pro­
grams now (or soon to be) under­
way which will provide new levels of
accuracy in defining the relationship
between the large--scale distribution
and properties of clusters and the un­
derlying astrophysics responsible for
their formation and evolution.
2 The Case for Clusters As
Tracers of LSS
There are several powerful argu­
ments in support of using clusters
to trace large--scale structure in the
universe. Clusters lie 10 - 30 times
farther apart than galaxies, with in­
tercluster separations in the range
30 - 100/h Mpc. This fact immedi­
ately makes them promising candi­
dates for tracing large--scale struc­
ture because a relatively small sam­
ple can be used to probe great dis­
tances. Clusters are probably ``close''
to their initial formation positions
(±10(vpec/1000 km/s)(to/10 Gyr) h -1
Mpc) because the typical peculiar ve­
locity of an individual cluster is 500
km/s or less (Watkins 1997). Fur­
thermore, clustering of clusters is sig­
nificantly amplified with respect to
that for galaxies. This is, in part, due
to the fact that clusters are located
at highest peaks in matter distribu­
tion (# CL /#c >
# 100). This enables
one to study features of the distri­
bution that might otherwise be too
weak to measure from galaxy redshift
surveys (e.g, correlations on scales
in excess of 30h -1 Mpc) and fea­
tures which are dependent on the
higher moments of the galaxy dis­
tribution. High order moments, for
1

2 Marc Postman
example, are highly sensitive to the
details of most biasing prescriptions.
In addition, the dark matter on typ­
ical intercluster distance scales is
likely still in the quasi­linear (or
only mildly non­linear) regime and,
thus, it is easier to associate present
epoch constraints on the shape of
the perturbation spectrum with its
primordial counterpart. N­body sim­
ulations show that clusters are un­
biased tracers of the underlying ve­
locity field (Strauss et al. (1995);
Gramann et al. (1995)) and provides
further theoretical justification for
the extensive observational e#orts to
measure large--scale flows (see Sec­
tion 7). The observations have been
motivated by the fact that rela­
tive redshift­independent distances
to clusters can be measured to accu­
racies <
# 5% owing to the large num­
ber of galaxies in each system. Clus­
ters are also the exclusive sites for
application of a number of secondary
distance indicators (e.g, BCGs, SZ
E#ect) and large elliptical popula­
tions (allowing accurate fundamental
plane measurements). Lastly, clus­
ters can be easily detected out to z #
1 (and beyond, if NIR data are avail­
able) owing to their high density con­
trast, red elliptical population, and
bright member galaxies. This enables
important constraints to be made on
the evolution of large--scale structure
and# o .
3 Identifying Clusters
There are three basic ways to ``de­
fine'' what one means by a cluster of
galaxies. They are:
The Physical Definition (at z #
0): Gravitationally bound, virialized
system of dark matter, gas, and
galaxies with a mass of at least #
10 14 M# inside a region # 1/h Mpc
in radius.
The N­body Definition: Peaks in a
dark­matter dominated density field,
perhaps satisfying additional con­
straints such as a minimum veloc­
ity dispersion, #, and/or minimum
value of ##. The peaks can be lo­
cated via Friends­of­Friends (FoF)
or Local Overdensity (LO; nearest
n particles) algorithms. The loca­
tions in real space are known pre­
cisely but the precise relationship be­
tween the dark and luminous matter
distributions is poorly understood at
present.
The Observational Definition(s):
These can vary substantially de­
pending on number of available
positional coordinates per galaxy
(2D/3D) and on the wavelength of
the survey (opt, IR, x­ray). Mod­
ern optical/IR searches identify den­
sity enhancements using FoF, LO, or
matched filter algorithms in either
2 or 3 dimensions. X­ray searches
look for sources with extended emis­
sion and/or cross­correlate all x­
ray sources with optically­selected
galaxy catalogs. Complications in­
clude projection e#ects (2D), red­
shift distortions (3D), sensitivity
variations across the survey, and
for the Abell (1958), Abell, Corwin,
Olowin (ACO; 1989), and Zwicky
et al. (1968) catalogs, some degree of
human error.
Because the clustering properties
of clusters will depend on how they
were selected (e.g, Eke et al. 1996),
any comparisons between observa­
tional datasets or between observa­
tions and simulations must quantify
and correct for biases introduced by
the selection process. This task is
tractable for samples derived using
well­understood and quantifiable se­
lection criteria.
3.1 Cluster Catalogs
There have been at least 14 orig­
inal cluster catalogs constructed to
date and Table 1 summarizes their
basic parameters. The impact of high
speed, memory­rich computers and
digital data is readily apparent: sub­
sequent to 1992, cluster detection
relies exclusively on the objective
application of algorithms with ac­
curately quantifiable selection func­
tions (``Bulleted'' catalogs). None the
less, results from the visually derived
Abell and ACO catalogs are still
widely cited because they have been
the only all sky cluster surveys avail­
able to date. The Zwicky cluster cat­
alog, by contrast, is only infrequently
used largely because their cluster
finding procedure, visual identifica­
tion of global isodensity contours
twice as high as the mean contour,
is fraught with many pitfalls includ­
ing enhanced sensitivity to plate­to­
plate photometric zeropoint varia­
tions. The cluster identification pre­
Evolution of Large Scale Structure / Garching August 1998

Clusters as Tracers of Large--Scale Structure 3
scription developed by Abell is a bit
more robust and, indeed, the auto­
mated APM catalog is based upon a
modified version of this approach.
Postman et al. (1986) demon­
strated that the angular correlation
functions of Abell and Zwicky clus­
ters agree when appropriate sub­
samples of each catalog are chosen
(corresponding to the spatial regime
where the di#erent algorithms iden­
tify similar types of clusters). With­
out such careful comparison, the
#cc (#) from Abell and Zwicky clus­
ters di#er significantly. A similar
level of discrepancy has existed be­
tween the spatial correlation func­
tions for the Abell/ACO clusters and
that for the APM clusters (e.g, Dal­
ton et al. 1992; Postman et al. 1992).
These di#erences are, in large part,
due to di#erences in the respec­
tive selection criteria which result
in di#erent minimum richness lim­
its in the catalogs (Bahcall & West
1992). To a lesser degree, di#erences
in the large--scale structure statis­
tics for the Abell/ACO and APM
catalogs are due to projection ef­
fects which appear to be strongest
in the original Abell catalog (Suther­
land 1988; Soltan 1988; Efstathiou
et al. 1992). A cross­correlation of
the ACO and APM catalogs in a sec­
tor of sky covered by both reveals
that about 75% of the ACO clus­
ters with m 10 # 16.5 (corresponding
to z <
# 0.1) have an APM counter­
part. The percentage drops to 50%
for m 10 < 18. Conversely, about 70%
of the APM clusters with mX # 16.9
have ACO counterparts; 50% match
when mX # 17.8. Some of the failed
matches are due to di#erences in the
respective depths of the two catalogs
and most of the remainder are due to
di#erences in richness cuts imposed
by the di#erent detection schemes.
The spatial distribution of a
nearly volume limited (z # 0.08),
all--sky sample of # 480 Abell/ACO
clusters is shown in Figure 1. The
space density of southern ACO clus­
ters is about a factor of 2 larger than
that of northern Abell clusters pri­
marily due to the more sensitive II­
IaJ emulsion used in the southern
survey and to the presence of the
Shapley supercluster (z # 0.047) in
the south. The ACO cluster space
density is quite similar to that found
in the APM cluster catalog, about
2.4 â 10 -5 h 3 Mpc -3 .
4 Second Order Cluster Cor­
relations
Second order statistics such as the
two--point spatial correlation func­
tion, #(r), and its Fourier transform,
P (k), have been widely used to con­
strain the clustering properties of
clusters. Although competing struc­
ture formation models can occasion­
ally yield quite similar predictions
for #(r) and P (k), they are none
the less robust measures of clustering
strength which provide basic infor­
mation about the underlying matter
distribution. They are also computa­
tionally straight forward to compute
and substantial work has been dedi­
cated to identifying the optimal esti­
mators for these functions (Landy &
Szalay 1993; Hamilton 1993; Peacock
& Dodds 1994; Landy et al. 1996,
Tegmark et al. 1998).
4.1 The Cluster Auto--
Correlation Function
The cluster--cluster spatial corre­
lation function is often fit to a power­
law of the form
#cc(r) = # r
ro # -#
This appears to be a reasonable
model over 5 # r # 35h -1 Mpc
and results from several independent
cluster catalogs yield 1.8 # # # 2.2.
Determinations of #cc (r) from the
Abell and APM catalogs are shown
in Figure 2. Although the amplitude
of #cc(r) di#ers by factors between 6
to 30 from that for the galaxy auto­
correlation function (with the precise
ratio dependent on the sample com­
positions), the shape of the cluster
and galaxy correlation functions are
very similar. This similarity in shape
is also seen in the respective power
spectra, at least on scales less than
70h -1 Mpc where the signal­to­noise
ratio is high.
The most stringent constraints
that #cc (r) can place on structure
formation models come from ad­
dressing the following questions:
Do ro and # change with the
mass­scale of the systems be­
ing studied?
Are there deviations from a
Evolution of Large Scale Structure / Garching August 1998

4 Marc Postman
Figure 1. The spatial distribution of a nearly volume limited sample of # 480
Abell and ACO clusters (RC # 0). The axes are aligned with the Galactic
coordinate system. The zone of avoidance at |b| = 13 # is shown.
Figure 2. The spatial correlation function for APM and Abell clusters, re­
spectively. The best­fit power laws are shown.
Evolution of Large Scale Structure / Garching August 1998

Clusters as Tracers of Large--Scale Structure 5
Table 1. Catalogs of Clusters of Galaxies
Detection Approximate Number of Number with
Catalog Passband Redshift Range Clusters Spec. Redshifts
Abell (1958) Opt. z <
# 0.3 # 2700 # 1300
Zwicky et al. (1968) Opt. z <
# 0.3 # 9000 . . .
. Shectman (1985) Opt. z <
# 0.3 # 650 . . .
Gunn, Hoessel, Oke (1986) Opt./NIR z <
# 1 # 400 # 50+
Abell, Corwin, Olowin (1989) Opt. z <
# 0.3 # 1350 # 250+
Couch et al. (1991) Opt. z <
# 0.6 # 100 # 20
. Henry et al. (1992) EMSS X­Ray z <
# 0.6 # 95 # 70+
. Lumsden et al. (1992) EDCC Opt. z <
# 0.3 # 700 # 100+
. Dalton et al. (1994a) APM Opt. z <
# 0.3 # 1000 # 360+
. Postman et al. (1996) PDCS Opt./NIR z <
# 1 # 80 # 20+
. Scodeggio, Olsen, et al. (1998) EIS Opt./NIR z <
# 1 # 250 . . .
. Rosati et al. (1998) RDCS X­Ray z <
# 0.8 # 70 # 60+
. Vikhlinin et al. (1998) X­Ray z <
# 0.6 # 200 . . .
. Boehringer et al. (1998) REFLEX X­Ray z <
# 0.3 # 450 # 380+
. = Automated Catalog
power law (e.g, #cc(r) # 0) and
on what scales?
How do the amplitude and
shape of #cc(r) vary with cos­
mic time?
Expectations are that, to some de­
gree, ro must depend on the mass
of the cluster: if structure grows by
gravitational amplification of fluctu­
ations in a Gaussian field, then col­
lapsed objects form near peaks in
this field and their clustering will
depend on the height of the peak
(Kaiser 1984; Barnes et al. 1985).
Early data on #gg (r) and #cc(r) led
Szalay & Schramm (1985) to propose
a ``scale­invariant'' form given by
# i (r) =
1
3
(r/d i ) -1.8
and thus ro = 0.54d i where d i
is the mean interobject separation.
Szapudi, Szalay, & Bosch’an (1992)
demonstrated that amplification is a
consequence of enhanced weighting
of dense regions when deriving the
higher moments of the density field
(e.g, #cc(r)). It is important to em­
phasize that the existence of a scale
dependence to the correlation length
does not imply that galaxies and
clusters cannot both be tracers of the
distribution of large--scale structure.
Rather, it suggests that these two
classes of systems trace the underly­
ing matter di#erently.
Bahcall & West (1992; BW92)
proposed that data for a wide range
of catalogs of clusters and galaxies
satisfy a scale­invariant relationship
between ro and d i of the form ro =
0.4d i . Croft & Efstathiou (1994),
however, could not reproduce a re­
lationship between ro and dc (inter­
cluster separation) as strong as the
BW92 result using SCDM N­body
simulations. They found a trend
which showed little dependence of ro
on dc when dc >
# 30h -1 Mpc. A
subsequent analysis by Croft et al.
(1997), using an extended subset of
the richest APM clusters, suggests
that these systems show only a weak
dependence of ro on dc that is consis­
tent with low density CDM models.
In contrast, Walter & Klypin (1996)
were able to reproduce a relationship
between ro and dc which is as strong
as the BW92 result from CHDM
N­body simulations. However, those
same simulations predict a dramatic
decrease in the comoving space den­
sity of clusters as one looks back to
z = 0.5, a prediction which is clearly
not consistent with present observa­
tions (Postman et al. 1996; Carlberg
et al. 1997). See also the results based
on the Virgo Consortium simulations
(Colberg et al. , these proceedings).
Figure 3 summarizes the current
situation. Part of the scatter in the
figure is due to inconsistent compar­
isons between authors. The BW92
dependence of ro = 0.4dc was based
on power law fits to #cc (r) with #
constrained to be 1.8. For the APM
Evolution of Large Scale Structure / Garching August 1998

6 Marc Postman
Figure 3. The spatial correlation length as a function of the intercluster
separation for di#erent cluster samples. At large dc , we show the APM results
when the slope, #, is constrained to be 1.8. The relationships ro = 0.4dc and
ro = 2.5 # dc are shown for comparison. Based on results from Bahcall &
West (1992), Croft et al. (1997), Abadi, Lambas, & Muriel (1998), and this
review.
results, Croft et al. (1997) allow # to
be a free parameter and often obtain
fits with # >
# 2.2 for dc > 60h -1
Mpc. Since there exists a significant
covariance between ro and #, such
comparisons must really be done at
a common slope value. Indeed, when
one fixes # at 1.8, the APM ro at
dc > 60h -1 Mpc do increase as
shown. A more substantial cause for
scatter in Figure 3 is demonstrated
by Eke et al. (1996) who find that ro
can vary by up to 50% depending on
the precise cluster identification pro­
cedure. They could reproduce either
a strong or weak dependence of ro on
dc depending on which cluster iden­
tification method was used and are
able to reconcile the BW92 and the
Croft et al. (1997) findings as con­
sequences of the di#erent selection
procedures used by ACO and by the
APM team.
In sum, a weak dependence of
correlation amplitude on interclus­
ter separation (and richness) is fairly
well­established (ro # # dc or ro #
0.2dc) both theoretically and ob­
servationally. The observational evi­
dence for a stronger dependence (e.g,
ro = 0.4dc) is from the angular
clustering properties of Abell RC #
2 clusters, the clustering of x­ray
bright Abell clusters (Abadi, Lam­
bas, & Muriel 1998) and the super­
cluster correlation function (Bahcall
& Burgett 1986) --- all of which are
derived from the Abell and ACO cat­
alogs. The APM cluster results for
dc > 60h -1 Mpc are based on <
60 clusters and, hence, are subject
to possible systematic e#ects (as is
any small catalog). The results for
dc > 60 thus require confirmation
from larger redshift surveys (e.g., the
extended APM, 2dF, and Sloan Digi­
tal Sky Survey [SDSS]). In any event,
careful attention needs to be paid by
both observers and theorists to the
not so subtle e#ects of the cluster se­
lection process on large--scale struc­
ture statistics before any physical in­
Evolution of Large Scale Structure / Garching August 1998

Clusters as Tracers of Large--Scale Structure 7
ferences are made based on the ob­
served relationship between ro and
dc .
4.2 The Zero Crossing of #(r)
Deviations in #(r) from a single
power law behavior are, in principle,
sensitive tests for the shape of the
primordial fluctuation power spec­
trum. One such deviation is the scale
at which the correlation amplitude
goes to zero (Klypin & Rhee 1994;
KR94). The amplitude of #cc(r) ap­
pears to be positive at least out
to 40h -1 Mpc and possibly out to
scales of 60h -1 Mpc (Postman et al.
1992; Olivier et al. 1993; KR94; Dal­
ton et al. 1994b; Boehringer et al.
1998). However, on scales from 60 -
100h -1 Mpc, it is very unlikely (2 -
3# level) that #cc(r) > 0. This re­
sult is seen in the Abell/ACO, APM,
and REFLEX (x­ray selected) clus­
ter catalogs. Systematic e#ects, such
as the integral constraint for a finite
sample (which forces #(r) to even­
tually become negative) or small er­
rors in the mean cluster number den­
sity, appear not to be large enough
to fully explain this zero crossing.
One important implication of this re­
sult may be that #gg (r) is also posi­
tive out to at least 40h -1 Mpc (Sza­
pudi et al. 1992). Indeed, prelimi­
nary results from the 2dF redshift
survey (Maddox, these proceedings)
and the ESO Slice Project (Guzzo,
these proceedings) both find posi­
tive #gg (r) out to at least 35h -1
Mpc. Such observations put severe
constraints on CDM models. As
noted by KR94, #CDM models (e.g,
Kofman, Gnedin, & Bahcall 1993)
predict a zero crossing at rz =
16.5(# h 2 ) -1 Mpc. If the above ob­
servations hold up, then this suggests
that# h lies in the range 0.28 - 0.41.
4.3 The Cluster Power Spec­
trum
The power spectrum of clusters,
P (k), provides a complementary con­
straint on their clustering proper­
ties: broad features in the correla­
tion function are narrow in Fourier
space and vice versa. Furthermore,
errors in P (k) are easier to esti­
mate correctly and the results are
somewhat less sensitive to uncertain­
ties in the mean space density of
clusters than those for #(r). Cur­
rent constraints on the cluster power
spectrum are shown in Figure 4.
The shape of cluster power spec­
trum, like it's inverse Fourier trans­
form #(r), is consistent with the
shape of the power spectrum of op­
tical, IRAS, and radio galaxies at
k > 0.04h Mpc -1 (Peacock & Dodds
1994; Einasto et al. 1997; Retzla#
et al. 1997; Tadros et al. 1998). This
suggests, again, that clusters and
galaxies are tracing similar perturba­
tions in the matter distribution. The
turnover in P (k) is detected (but not
with high significance) for k < 0.03h
Mpc -1 . The SDSS, 2dF, and other
large redshift surveys should even­
tually yield dramatically improved
constraints on the turnover, a fea­
ture which depends upon the hori­
zon scale at the epoch of matter­
radiation equilibrium.
The amplitude of
P (k) for Abell/ACO clusters is, on
average, a factor of # 2 - 3 higher
than that for APM clusters, consis­
tent with di#erences seen in their re­
spective #(r). In turn, the APM clus­
ter P (k) amplitude is about 6 - 8â
higher than that derived for galax­
ies in the Las Campanas Redshift
Survey (Lin et al. 1996; LCRS). The
observed shape of P (k) is reason­
ably well represented by MDM mod­
els (0.2 <
## # <
# 0.3), low--density
CDM
models(# h # 0.3 ± 0.1),
and/or #CDM models (# # 0.3)
(Borgani et al. 1996). The appar­
ently strong feature seen at 120 ±
15h -1 Mpc in the Abell/ACO P (k)
(Einasto et al. 1997) is not seen in the
P (k) derived from either APM clus­
ters (Tadros et al. 1998) or REFLEX
x­ray selected clusters (Boehringer
et al. 1998). A subsequent analysis
of the Abell/ACO P (k) by Ret­
zla# et al. (1998) find that the fea­
ture is not statistically significant
when sample variance is properly ac­
counted for. None the less, a statis­
tically significant feature is detected
in the 2D LCRS P (k) at around
100h -1 Mpc (Landy et al. 1996) and
spikes continue to be found in the
galaxy redshift distribution on sim­
ilar scales in new pencil beam sur­
veys (Broadhurst et al. 1995). These
features, which are not reproduced
by most non­baryonic matter dom­
inated models, are presumably due
to characteristic scales of voids and
Evolution of Large Scale Structure / Garching August 1998

8 Marc Postman
Figure 4. Power spectra for APM galaxies (Maddox et al. (1996)), APM
clusters (Tadros et al. (1998)), and Abell/ACO clusters (large squares are
Retzla# et al. (1997); small squares are Einasto et al. (1997)). The galaxy
power spectrum has been normalized to match the amplitude of the APM
cluster power spectrum. P (k) for two models also shown.
sheets in the galaxy distribution. The
lack of a significant detection of this
feature in cluster power spectra may
be a consequence their sparser sam­
pling of the density field. For in­
stance, a direct comparison between
the galaxy distribution from the ex­
tended (R # 15.4, N gal # 6000)
CFA redshift survey (Geller 1998)
and the Abell cluster distribution in
the same volume, reveals that sev­
eral prominent features in the galaxy
distribution which contribute to the
peaks originally found by Broadhurst
et al. (1990) are not traced by the
clusters.
Narrow, large amplitude features
in P (k) are surprising yet intrigu­
ing. There is presently no theoreti­
cal consensus on the origin of pre­
ferred scale lengths. Eisenstein et al.
(1998) hypothesize that such excess
power could be a consequence of
baryonic acoustic oscillations in adi­
abatic models. However, they note
that this would require a substan­
tial error in currently favored values
of cosmological parameters. Einasto
et al. (1997) simply conclude that our
present understanding of the forma­
tion of large--scale structure requires
substantial revision.
5 High­Order (N # 3) Statis­
tics
Higher order statistics potentially
provide some of the best constraints
on the degree of biasing (e.g, Jing
1997) and, thus, on the reliability of
clusters as tracers of the mass. The
high­order moments of cluster dis­
tributions have already been shown
to be non­zero. For example, ACO
clusters exhibit hierarchical cluster­
ing behavior given by
# N (r 1 , ..., r N ) =
# #
Q (#)
N # ij
# N-1 # 2 (r ij )
Evolution of Large Scale Structure / Garching August 1998

Clusters as Tracers of Large--Scale Structure 9
up to 6th order (Cappi & Mauro­
gordato 1995) with Q 3 # 1.0. APM
clusters display a similar hierarchi­
cal behavior, at least up to 4th order
(Gaztananga, Croft, & Dalton 1995).
The deprojected SN 's (# N =
SN # N-1
2 ) for APM galaxies (Gaz­
tanaga 1994) and ACO clusters are
quite similar in amplitude which sug­
gests that ACO clusters and APM
galaxies are sampling the same un­
derlying matter distribution but that
the biasing between galaxies and
clusters is non­linear.The S 3,4 values
for ACO and APM clusters are (3.1,
22) and (# 2, # 8), respectively.
6 Very Large Scale (> 200h -1
Mpc) Structure
The large intercluster spacing and
the enhanced amplitude of their clus­
tering makes the study of structure
on very large scales possible, in prin­
ciple. In practice, the signals on these
scales are small and errors in mod­
eling systematic e#ects such as pho­
tometric zeropoint variations, sam­
ple variance, or Galactic reddening
can yield artificial signals of compa­
rable amplitudes. Tully (1987) first
proposed the detection of very large­
scale alignment of the local Abell
cluster distribution with the Super­
galactic plane. Postman et al. (1989)
countered that this e#ect (based on
available data at the time) was not
statistically significant (< 2#) and
could be expected from sample with
the observed #(r). Tully et al. (1992)
extended their analysis to a full­sky
sample of Abell and ACO clusters
and still found an alignment with
the Supergalactic plane, claimed to
be significant at the 6­sigma level,
extending to scales of 450h -1 Mpc.
The structure is only a small ampli­
tude fluctuation (##/# <
# 0.015), if
indeed real. Scaramella (1992) also
used Abell/ACO clusters to study
power on 600h -1 Mpc scales and
found relatively low values for for
density fluctuations, consistent with
limits on the fluctuations in the cos­
mic microwave background (CMB).
The study of very large scale struc­
ture performed directly from cluster
redshift surveys will not likely ad­
vance much further until automated
wide--area, homogeneous cluster cat­
alogs, like those expected from the
SDSS or the extended APM sur­
vey (#900 clusters; Tadros 1998), be­
come available.
7 The Large­scale Velocity
Field
A complementary approach (and
perhaps a more promising one given
current cluster catalogs) to studying
very large--scale structure using clus­
ters is through the mapping of the
large--scale velocity field. Currently,
at least 7 independent cluster­based
peculiar velocity surveys, all reaching
scales of 100h -1 Mpc or larger, are
either complete or in progress (see
Table 2). Inferences about the under­
lying mass distribution from pecu­
liar velocity surveys are less suscep­
tible to incompleteness e#ects and
radial density gradients than those
from redshift surveys. However, pe­
culiar velocity surveys require highly
accurate photometric and spectro­
scopic calibrations and extremely ho­
mogeneous data (see Strauss, these
proceedings). Careful characteriza­
tion of the systematic errors and the
e#ects of sparse sampling are also re­
quired (e.g, Lauer & Postman 1994;
Feldman & Watkins 1994). Figure 5
summarizes the current constraints
on bulk flow amplitudes from both
galaxy and cluster based surveys.
The constraints on the largest scales
are nearly all from cluster­based sur­
veys. Included in the plot are two
new results. The exciting results of
a 700 km/s flow at 8000 km/s depth
from Hudson et al. are discussed else­
where in these proceedings. Willick
(1998) reports the measurement of
v Bulk = 900 ± 375 km/s (1# er­
ror) in the redshift range 9000 #
cz # 12, 000 km/s based on a Tully­
Fisher survey of 15 rich Abell clus­
ters. Neither of these two results
are consistent with the direction of
Lauer & Postman (1994; LP) re­
sult. They may, however, be con­
sistent with each other. Indeed, no
other work to date has corroborated
the LP bulk flow (see also Wegner
et al. (1998) and Saglia, these pro­
ceedings). This may suggest that ei­
ther the original LP BCG sample is
a statistical fluke or an additional
parameter is required for accurate
BCG distance estimation (e.g, Hud­
son & Ebeling 1997). The extended
Evolution of Large Scale Structure / Garching August 1998

10 Marc Postman
Figure 5. The amplitude of derived bulk flows from recent galaxy and cluster
based peculiar velocity surveys. The results from cluster­based surveys are
indicated by the filled data points.
BCG survey by Lauer, Postman, &
Strauss (1999) will provide a good
test. In contrast, Dale et al. (1997)
find no evidence for a large bulk flow
at 8000 km/s. The inconsistent re­
sults for the amplitude and direc­
tion of large--scale bulk flows argue
that the convergence scale is not yet
well--established. However, the qual­
ity and quantity of data from the on­
going surveys, including promising
results from space­based SBF stud­
ies, should be su#ciently good that
much better constraints will be avail­
able within the next two years or so.
8 The Cluster Mass Function
at z > 0.7
The advent of the Keck tele­
scope and the Low­Resolution Imag­
ing Spectrographhave revolutionized
spectroscopic surveys of distant (z >
0.7) clusters. This capability has now
enabled us (Oke, Postman, & Lu­
bin 1998) to provide a preliminary
constraint on the normalization of
the cluster mass function (CMF) in
range 0.76 < z < 0.92 based on
data for 3 clusters and between 22
and 36 cluster members for each sys­
tem. The specifics of the mass esti­
mation techniques are described in
Postman, Lubin, Oke (1998). The
new constraints on the CMF are
based on the clusters CL1324+3011
(z = 0.76), CL1604+4304 (z = 0.90),
and CL1604+4321 (z = 0.92). The
latter two are part of a superclus­
ter. All 3 clusters are from the Gunn,
Hoessel, Oke (1986; GHO) catalog
and their kinematic masses are all
in excess of 4.5 â10 14 h -1 M# within
their central 1h -1 Mpc regions. If we
make the quite conservative assump­
tion that these are the only 3 clusters
this massive within the entire GHO
catalog (# 72 deg 2 ), then we find
that,
for# # = 0.2, a lower limit on
the CMF in the range 0.7 < z < 1
N(# M = 4.5 â 10 14 h -1 M# ) >
1.1 â 10 -7 h 3 Mpc -3
This constraint is consistent with es­
timates made by Bahcall, Fan, & Cen
Evolution of Large Scale Structure / Garching August 1998

Clusters as Tracers of Large--Scale Structure 11
Table 2. Current Cluster­based Peculiar Velocity Surveys
Investigators D.I. N clus Depth (km/s) # D /cluster
Dale et al. (1997) TF # 50 18,000 # 7%
Gibbons, Fruchter, Bothun (1998) FP 20 11,000 # 6%
Hudson et al. (1998; SMAC) FP # 60 12,000 7%
Lauer & Postman (1994) BCG 119 15,000 16%
Lauer, Postman, Strauss (1999) BCG # 500 24,000 16%
Tonry et al. (1998) SBF 11 10,000 <
# 5%
Wegner et al. (1998; EFAR) FP 84 15,000 8%
Willick (1998) TF 15 12,000 # 5%
(1997) and provides additional ob­
servational support for a low­density
universe. While this constraint is
relatively crude (± factors of 2 -
20), the discovery of similarly mas­
sive systems (e.g, MS1054­03 and
MS1137+66) at similarly high red­
shifts will likely continue to grow
as observations of distant clusters
progress.
9 Cosmological Implications
and Future Developments
The general conclusions one can
draw from the ensemble of cluster
data and simulations discussed above
are that
1. Clusters are reliable tracers of the
underlying mass but trace it dif­
ferently from galaxies. In particu­
lar, clusters trace the large--scale
structure sparsely since they are
relatively rare objects. The bias­
ing between galaxies and clusters
is non­linear and is dependent on
their intrinsic properties (e.g, the
central mass of the cluster).
2. The statistically significant power
seen in the cluster distribution
on scales between 30 - 60h -1
Mpc implies that galaxies are also
likely to exhibit correlations on
the same scales. Indeed, the larger
galaxy redshift surveys (LCRS,
2dF, ESO Slice survey) now con­
firm that #gg (r) > 0 at least to
35h -1 Mpc.
3. The cluster observations seem to
favor MDM models (0.2 <
## #
<
#
0.3), low--density CDM models
(# h # 0.3 ± 0.1), and/or #CDM
models (# # 0.3). The exception
would be if the large--amplitude,
large--scale bulk flows persist, in
which case somewhat higher val­
ues
for# are required.
4. Massive (few â10 14 h -1 M# ) clus­
ters exist at z > 0.7 in an abun­
dance that is hard to reconcile
with# # = 1 models.
There is still much we need to learn
about large--scale structure forma­
tion and evolution and a number of
exciting developments over the next
3 to 5 years will help. New, larger
objective cluster catalogs will soon
be available from surveys such as
the SDSS, 2dF, and extended APM.
Using them, we should be able to
constrain the cluster power spec­
trum with unprecedented accuracy
to scales approaching 1 Gpc. These
catalogs will also enable more ex­
tensive, direct comparisons between
the galaxy and cluster distributions
in identical volumes and will allow
us to establish, with significantly
better accuracy, the dependence of
the clustering properties of clusters
on their intrinsic parameters. Joint
x­ray/optical cluster searches (e.g,
Donahue et al. 1999) should eluci­
date the nature of cluster evolution
at intermediate redshifts (z <
# 1).
Deep, wide­area galaxy surveys (e.g,
Postman et al. 1998b; Jannuzzi, Dey,
et al. 1998) will provide important
and new measurements of the evo­
lution of large--scale structure out
to z = 1 and beyond. These same
surveys, coupled with deeper x­ray
surveys, should prove profitable for
the continued identification of mas­
sive clusters with z >
# 0.8, with the
corresponding implications for struc­
ture formation models. The comple­
tion of several independent cluster­
based peculiar velocity surveys which
Evolution of Large Scale Structure / Garching August 1998

12 Marc Postman
all probe # 100-200h -1 Mpc scales
but with di#erent techniques should,
hopefully, provide a better consen­
sus on the convergence scale and the
origin of the CMB dipole motion.
Lastly, but as important as any of the
above observational e#orts, the new
billion particle simulations, like those
being pioneered by the Virgo Con­
sortium, with high spatial resolution
and spanning a large dynamic range
in cosmic time will provide much
more accurate and refined model pre­
dictions.
Acknowledgments
I thank Tod Lauer, Michael
Strauss, Istv’an Szapudi, Michael Vo­
geley, Neta Bahcall, and Harald
Ebeling for the lively discussions on
various aspects of this review. A spe­
cial thanks to Je# Willick for allow­
ing me to be the first to publicly
present his preliminary bulk flow re­
sult and to Helen Tadros for provid­
ing an electronic version of the APM
cluster P (k).
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