To draw lessons for cosmology, we need not only the physical properties of individual clusters but also an understanding of how typical the numbers are. The issue here is whether the Abell catalog or any other now available is adequate for the purpose. There are known problems in the catalogs: they contain objects with suspiciously low velocity dispersions, and they miss systems whose X-ray properties might be consistent with massive clusters.

Recently there has been considerable interest in the possible systematic errors this might introduce in estimates of cluster masses and spatial correlations (Sutherland 1988; Kaiser 1989; Dekel et al. 1989; Frenk et al. 1989). The points are well taken but I think the situation is not disastrous: if we take a balanced view, not attempting to push the data too hard, and taking care to look for supporting evidence from tests of reproducibility, we get some believable and useful measures.

The cluster-galaxy cross correlation function, 1 +
_{cg}(*r*),
is the mean number density
of galaxies as a function of distance *r* from a cluster, measured
in units of the large-scale
mean density. The fact that one finds consistent estimates of
_{cg} from
different cluster
distance classes (with reasonable choice of parameters in the luminosity
function) is
evidence that the typical richness of the cluster sample does not vary
substantially with
distance. The number of bright galaxies within the Abell radius
*r*_{a} = 1.5*h*^{-1} Mpc (*H* =
100*h* km sec^{-1} Mpc^{-1}) around a cluster is
larger than expected for a homogeneous distribution by the factor

The original estimate (Seldner and Peebles 1977) is *N(<
r*_{a}) / *nV*_{a} = 360; the reanalysis
by Lilje and
Efstathiou (1988),
which uses better cluster distances and galaxy luminosity
function, is half that. I adopt the mean with twice the weight for the
newer value:

(1) |

The scatter around the mean value of *N(< r*_{a}) surely is
large, even for a given
nominal richness class, because richness estimates are compromised by
groups and clusters
seen in projection. The rms scatter in *N(< r)* from cluster to
cluster is measured
by the cluster-galaxy-galaxy correlation function,
_{cgg}
(Fry and Peebles
1980).
Estimates of
_{cgg} should
be reworked using the better current distance scales and luminosity
function; the old result is

(2) |

This is substantial, but still it indicates that we can trust equation (1) to a factor of two for the overdensity of a typical richness class 1 cluster.

We can compare equation (2) to the scatter in estimates of the velocity
dispersion
*v*^{2} for Abell clusters. The average over the estimates
of *v*^{2} for the 54 *R*
1 clusters in
the Struble and
Rood (1987)
compilation is

(3) |

The last factor is the fractional rms scatter in the estimates of
*v*^{2} among the 54 clusters.
The scatter surely has been inflated by the cosmological redshift
differences of objects
accidentally close in projection (and perhaps suppressed by
over-enthusiastic pruning of
the tails of the velocity distribution). However, the coincidence of the
fractional scatter
in *v*^{2} and the bound for *N* (eq. [2]) is
consistent with the assumption that the scatter
in velocity dispersions is dominated by the scatter in masses of the
clusters rather than measuring errors.

Arguing for the same conclusion is the fact that most clusters are X-ray
sources,
and that where the X-ray gas temperature is known it is about what would
be expected
if the plasma and galaxies had the same temperature. For example, in
Mushotzky's (1984)
sample, clusters with X-ray temperature ~ 10^{0.6} keV
have galaxy line of sight
velocity dispersions in the range 900 ± 300 km sec^{-1},
compared to the expected value
800 km sec^{-1} for equal temperatures. Part of the scatter
surely comes from temperature
differences between plasma and galaxies and from variations of temperature with
position, but the key point for our purpose is that there is not a lot
of room for spurious estimates of the typical cluster velocity dispersion.

As a final check, let us estimate the cosmological mean mass density from these
numbers. Since the velocity dispersion in a cluster tends to drop with
increasing distance
from the center, and equation (3) surely has been somewhat inflated by
random errors in
the measurements of *v*, a reasonable estimate of the rms line of
sight velocity dispersion
at the Abell radius is 750 km sec^{-1}. In the isothermal gas
sphere model, this makes the mass within the Abell radius

(4) |

If galaxies traced the large-scale mass distribution, then the contrast in galaxy counts in equation (1) would be the same as the mass contrast:

With equation (4) this fixes the mean mass density, , which translates to the cosmological density parameter

(5) |

The more direct way to get at this number is to find masses and
luminosities of individual clusters. For example,
Hughes (1989)
finds for the Coma Cluster a mass to light
ratio ~ 300*h* solar units. This multiplied by the mean luminosity
density gives ~ 0.3,
a commonly encountered number from this method. Very similar numbers
follow from masses derived from luminous arcs
(Grossman and
Narayan 1989,
Hammer and Rigault
1989).
The value of this effective
thus is secure, and the
consistency with equation (5) is a positive check of equations (1) and (4).

There are three notable results from these observations. First, there is
a hard upper
cutoff in cluster masses, as evidenced by the fact that the
Struble-Rood
(1987) catalog
lists just one cluster with estimated *v*^{2} ~ three times
the mean in equation (3). That
is, Nature is adept at placing the mass in equation (4) into a radius of
1.5*h*^{-1} Mpc, but
quite reluctant to place three times that amount in the same
volume. (The analogous
effect for galaxies is the upper cutoff in circular velocity at *r*
~ 10 kpc at about twice
that of our galaxy.) This cluster mass cutoff might be expected in
theories for cluster
formation out of Gaussian primeval density fluctuations, because the
upper envelope of
Gaussian fluctuations is tightly bounded. In cosmic string theories the
effect is more
problematic, because clusters are supposed to be seeded by loops that
have a broad range of masses. However,
Zurek (1988)
argues that with suitable parameter choices
the predicted spread in cluster masses may agree with the observations.

Second, the mass *M*_{d} in equation (4) is at the upper
end of the range of masses *M*_{x} of the X-ray producing gas
(Jones and Foreman
1984).
The fact that *M*_{x} typically is
less than *M*_{d} illustrates the familiar point that
clusters generally are dominated by dark
mass. It will be interesting to see whether there are clusters in which
the dynamical
mass within the Abell radius may be accounted for by the mass needed to
produce the
X-rays. If so, then theories that assume the universe is dominated by
weakly interacting
matter initially well mixed with the baryons would a require
considerable settling of the
plasma, an effect that might be testable by the methods described at
these Proceedings by Evrard.

Third, as we have noted in equation (5), if the mean mass per galaxy in clusters applied to all galaxies, the mean mass density would be less than the simple (and currently fashionable) Einstein-de Sitter cosmology with negligibly small space curvature and cosmological constant. The possible relation to galaxy formation theories is discussed in Section II below.

Galaxy formation theories also are tested by the spatial clustering of
clusters, as
measured by the usual N-point correlation functions. As for any other
statistic, the way
to decide whether the cluster-cluster two-point function,
_{cc}, is
reliably detected is to
test for reproducibility of results from independent samples and from
different ways of
analyzing the same sample. The hope is that these different approaches
are differently
affected by systematic errors, so errors would be revealed as
significant discrepancies
in the statistic. Thus Mike Hauser and I decided that we had a
believable detection of
clustering because we found that the two-point angular function,
*w*_{cc}(), in the
Abell (1958)
catalog scales with distance about as expected for a
spatially homogeneous random process.

Bahcall and Soneira (1983) introduced the use of the standard fitting form for the two-point function:

(6) |

The angular two-point function gives *hr*_{cc} ~ 28 ±
7 Mpc (Hauser and
Peebles 1973).
Bahcall's (1988)
estimate, based mainly on redshift samples, is
23 ± 3 Mpc. The angular
and redshift correlation functions use the same catalog but in different
ways, that one
might have expected would be differently affected by spurious clustering
introduced
in the discovery of the clusters. The rough consistency of these
estimates of *r*_{cc} thus
argues for the reality of the clustering. The same comments apply to the
deeper redshift sample of
Postman (1989),
which gives 19^{+7}_{-4}, and
the X-ray selected sample of
Lahav et al. (1989),
which gives 19 ± 4 Mpc. The mean of these four estimates is

(7) |

where I have taken the precaution of doubling the formal standard deviation of the mean.

Dekel et al. (1989)
argue that the clustering of Abell clusters may be an artifact of
projection contamination. The case for contamination at some level is
persuasive, but of
course their model for the effect need not usefully approximate the way
people actually
discover clusters. My conclusion from the reproducibility of
*r*_{cc} is that projection effects
are a second order correction, that there is a strong case for the
reality of clustering at
the level of equation (7). With the Hubble constant *H* = 50 km
sec^{-1} Mpc^{-1} favored by the biased cold dark matter theory
(Frenk et al. 1989),
the mass autocorrelation
function vanishes at *r* ~ 70 Mpc, about 50 percent larger than
*r*_{cc}. Thus the cold
dark matter theory does not directly contradict equation (7), but in
some estimates the margin is uncomfortably close
(Blumenthal, Dekel
and Primack 1988).