© CAMBRIDGE UNIVERSITY PRESS 1999 |

**7. The Expanding Search for Homogeneity**

A paradox!

A most ingenious paradox!

We've quips and quibbles heard in flocks,

But none to beat this paradox!

Gilbert and Sullivan

Over all scales on which astronomers had looked, from planets to clusters of galaxies, the distribution was nonuniform. Yet, the intuition of great observers like Hubble suggested there would be a scale without structure. Where was it?

Perhaps it was nowhere; perhaps it was everywhere. Every scale
might be clustered, but in the same manner, so there was no
distinction between scales. This self-similar picture is more subtle
than the idea that the Universe suddenly becomes uniform at a
particular very large scale. Carpenter (1938) suggested that it
followed from his earlier (Carpenter, 1931) discovery of a single
relation between the number of galaxies in a cluster and the cluster
radius. He refined this relation in 1938 to the form *N*
*R*_{c}^{1.5}. Although he considered this to be an
upper bound of the *N - R*_{c}
distribution, it also provides a good fit through his data. Since it
applied over three orders of magnitude in *N*, he concluded in the
later paper "that there is no essential distinction between the large, rich
clusters and the smaller groups, the total populations and other
characteristics varying gradually from one extreme to the other."

Without realizing it, Carpenter had found the first evidence
for the form of the two-point correlation function
(*r*) and,
indirectly, for the fractal nature of the galaxy distribution. From
Equation (6.8), the excess number of galaxies (above the average
number in a Poisson distribution) around a galaxy is
*N*_{c} = 4
*r*^{2}
(*r*) *dr*.
Over small scales where
>> 1 we
expect *N*_{c}
*N*, the
number of galaxies in the cluster. Therefore if
(*r*) is a power
law, as Carpenter might have expected in his self-similar picture,
then *N*_{c}
*r*^{1.5} implies
*r*^{-1.5}.

This is not far from the typical modem value
*r*^{-1.8}.
Furthermore, if the number density of galaxies were uniform, we would
expect *N*
*r*^{3}. In a more general fractal distribution *N*
~ *r*^{3-d}.
(see Chapter 12): 50 Carpenter essentially found the fractal dimension
*d* = 1.5. Actually the situation is not quite so simple because the
form of
(*r*) and
the fractal dimension both turn out to depend upon
scale (see Chapters 12, 14, and 20).

Zwicky (1937, 1938) came to a similar conclusion that nearly all galaxies were clustered on all scales, rather than most galaxies being part of a fairly uniform field distribution with a sprinkling of dense clusters. He also described the concentration of elliptical galaxies in clusters, suggesting this was caused by mass segregation. And he attempted to calculate the spatial distribution function of galaxies, assuming a Boltzmann energy distribution and no statistical evolution of the system of galaxies. Although this missed the essential gravitational physics of the problem, like the attempts of Lemaitre and Milne, and although Zwicky's program could not be calculated, it helped open a significant question: Could statistical mechanics describe the galaxy distribution?

Thus by the late 1950s there was a consensus that galaxies did not have a Poisson distribution, even though the description of their clustering was unclear. Disputes centered over whether the essential feature was self-similar hierarchical clustering, or a few large clusters in a uniform field, or the specific distributions of cluster forms and separations that Neyman and Scott proposed, or fits to ad hoc forms of the two-point correlation function. Relations among some of these descriptions were beginning to emerge along with a rudimentary understanding of their underlying physics.

At this stage, the scale expanded. Why not repeat the galaxy clustering analyses with the clusters themselves, considering each cluster as a point mass? Are the clusters clustered? And so the great superclustering controversy began.

It was abetted by the construction of two different cluster catalogs, each defining clusters differently and subject to different selection effects. The discussion almost mimicked the earlier controversy between Hubble and Shapley, with their two different galaxy catalogs. Previously the discovery of clusters, like the earliest discoveries of galaxies, had been a rather haphazard affair. As usual, the most prominent and often least representative ones were noticed firstl To systematize this process, Abell (1958) examined 879 pairs of photographs in the National Geographic Society-Palomar Observatory Sky Survey made with the 48-inch Palomar Schmidt Telescope. It covered the sky from -27° to the north celestial pole, dividing it into 6.°6 square fields with 0.°6 overlaps along common edges. Abell was one of the main observers. One photograph of each field was most sensitive to red light, the other to blue.

Abell took the view that very rich clusters are superimposed on a background of field galaxies whose surface density varies. To qualify for his catalog, each cluster had to meet four criteria:

- Richness. The number of galaxies with magnitude
*m**m*_{3}+ 2 is at least fifty, where*m*_{3}is the magnitude of the third brightest member of the cluster. Choosing the third brightest reduces the chance of confusing a nearby field galaxy with the cluster but introduces some uncertainty until the redshifts are measured. - Compactness. The cluster members must be within a given linear
radial distance of its center. Abell chose 1.5 Mpc (adjusted to a
Hubble constant of
*H*_{0}= 100 km s^{-1}Mpc^{-1}where one megaparsec = 3.1 × 10^{24}cm), claiming the results were insensitive to the exact distance since this scale is generally larger than the main concentration of galaxies. - Distance. So that clusters do not extend over much more than one
6.°6
plate, they must have a redshift greater than 6,000 km s
^{-1}. Thus Virgo is eliminated but Coma squeaks in. The upper limit is 60,000 km s^{-1}, set by requiring galaxies brighter than*m*_{3}+ 2 to be easily visible above the*m*= 20 plate limit. For*H*_{0}= 100 km s^{-1}Mpc^{-1}this range of distances is 60-600 Mpc. - Galactic latitude. Regions of low galactic latitude where large numbers of stars and obscuration from interstellar dust in our own galaxy interfered with the identification of clusters were excluded.

With these criteria, Abell found 2,712 clusters of which 1,682 met more stringent conditions for a more statistically homogeneous sample. Until galaxy redshifts are measured, however, it is uncertain whether any individual cluster is an impostor produced by the chance superposition of two less rich clusters along the line of sight. Moreover, despite choosing m3 for the richness criterion, the catalog may still be incomplete because bright field galaxies in the line of sight increase the threshold definition of a rich cluster. Subsequent analyses (e.g., Lucey, 1983) have shown that these effects are not negligible.

By counting the number of clusters in cells of equal area,
Abell (1958) found they were clumped much more strongly than a Poisson
distribution. The clumping could not be dominated by galactic
obscuration because it persisted around the polar caps (|*b*|
60°)
and because the visibility of the more distant clusters (distance
groups 5 and 6) did not depend on their distance. More distant
clusters are often visible in regions with few closer clusters, and
vice versa. Moreover, counting with cells of different sizes gives
the scale on which the cluster distribution departs most from Poisson.
Abell found this maximum clumping scale varies roughly inversely as
the distance, suggesting a linear scale of about 45 Mpc
(*H*_{0} = 100) for superclustering.

Zwicky (1957) now reached the opposite conclusion: Clustering stops with clusters; larger scales are uniform. His catalog (Zwicky, Herzog, and Wild, 1961) selects clusters by using the eye to estimate regions of the Palomar Sky Survey where the surface number density of galaxies is twice the nearby density. Subtracting the average density of nearby field galaxies from the density within the contour around twice that average gives the number of galaxies in that cluster. Distant faint clusters are less likely to be included in this catalog. Zwicky's (1967) clusters showed much less statistical tendency to cluster on larger scales, leading him to deny the existence of superclustering. But he did realize that many clusters were loosely grouped into larger aggregates. Whether these were superclusters was a largely semantic distinction. Indeed the whole controversy was obscured by the uncertain physical nature and dynamics of these clusters and superclusters. Their rather arbitrary observational definitions had no known relation to any underlying physical properties.

To try to clarify the situation from a more rigorous point of
view, Kiang (1967) generalized the Neyman and Scott galaxy clustering
models and applied them to the clustering of Abell's clusters. The
index of dumpiness (Equation 8.5) and the two-dimensional correlation
function, *w*(),
for clusters (see Equation 14.36) did not fit these
models any better than the galaxies had. Models with superclusters of
a given size did not fit the observations on more than one
scale. Kiang thus suggested the hypothesis of indefinite clustering:
"The indications are that galaxies are clustered on *all* scales, that
there is no preferred size of clusters, and that the clusters have
little physical individuality . ... The hypothesis of indefinite
clustering reinstates, in a sense, the *galaxy* as the ultimate
building
brick of the Universe. But there is no going back to the view of no
clustering. Galaxies are certainly clustered, but we have not
succeeded in finding any preferred sizes of clusters." Evidence began
to accumulate for Carpenter's earlier view.

If Abell's clusters are themselves clustered, this should be
detectable unambiguously in their two-point correlation function
(*r*),
independent of any a priori model. And if they are clustered on
all scales, then
(*r*) should
be a scale-free power law. With this
motivation, along with some earlier ideas from a phase transition
approach (Saslaw, 1968) to gravitational clustering, Kiang and Saslaw
(1969) analyzed the three-dimensional
(*r*) for
Abell's clusters. Using rather crude magnitude estimates of distance, they
found clear evidence for correlations over scales of 50-100 Mpc (with
*H*_{0} = 100). However, their spatial smoothing over 50
Mpc cubes,
necessitated by the inhomogeneity of the cluster distribution, made it
impossible to find the functional form of
(*r*)
accurately.

Even as the controversy over superclustering and homogeneity
continued during the next several years (de Vaucouleurs, 1970, 1971),
the seeds of its resolution had already been planted. Discovery of the
2.7 degree cosmic microwave background (Penzias and Wilson, 1965) was
quickly followed by tight limits on its angular variation
(cf. Partridge, 1995). Although the connection of this radiation with
the early universe remained controversial for many years, nearly all
pockets of resistance vanished after the definitive *COBE (Cosmic
Background Explorer)* satellite observations (Mather et al., 1990)
showed it had a very accurate blackbody spectrum in the measured
range. This relic of the thermal equilibrium between matter and
radiation in the early Universe is now known to be homogeneous at the
level of
*T / T*
10^{-5} over
angular scales of 7' (Bowyer & Leinert,
1990). Similar upper limits to inhomogeneity apply over scales of
about 3' to 10°. These largest homogeneous scales are greater than the
distances over which parts of the Universe can interact before matter
and radiation decouple at *z*
10^{3}. Smaller amplitude inhomogeneities are,
however, found over a range of spatial scales. For detailed
discussions of the cosmic microwave background and its implications,
see Partridge (1995) and more recent reviews of this rapidly
developing subject.

Discovery of the blackbody spectrum and essentially uniform surface brightness of the cosmic microwave background with very small amplitude fluctuations strongly constrains the nature, size, and development of any inhomogeneities. Regions of varying density distort the microwave background both by gravitational interaction and by their different levels of ionization and radiation. The almost complete uniformity on the largest scale raises two questions: How could regions of the Universe that were outside each other's horizons (i.e., not causally connected) reach almost the same state? and what do the small radiation inhomogeneities imply for galaxy formation and clustering? Current answers to these questions involve inflationary models of the early universe and their initial spectrum of perturbations. Their results are still controversial and their exploration continues. They seem more relevant to galaxy formation than to subsequent clustering.

Direct evidence for statistical homogeneity in the
distribution of matter came from the first accurate measurement of the
galaxy two-point correlation function. Totsuji and Kihara (1969)
solved this long-standing problem. The abstract of their paper reads:
"The correlation function for the spatial distribution of galaxies in
the universe is determined to be (*r*_{0} /
*r*)^{1.8}, *r* being the distance
between galaxies. The characteristic length *r*_{0} is 4.7
Mpc. This
determination is based on the distribution of galaxies brighter than
the apparent magnitude 19 counted by Shane and Wirtanen (1967). The
reason why the correlation function has the form of the inverse power
of r is that the universe is in a state of `neutral' stability."

Deep physical insight into the gravitational many-body problem
- usually a good way to short-circuit complicated mathematical
formalism - led Totsuji and Kihara to their conclusion. Previous
guesses at an exponential or Gaussian form for the correlation
function had been based on analogies with a stable inhomogeneous fluid
where (*r*)
decreases exponentially (Rubin, 1954) or on the cluster
models where a Gaussian form (Limber, 1953, 1954, 1957; Neyman, 1962)
was believed to have an a priori plausibility and simplicity. However,
a statistically homogeneous system of gravitating objects in an
unbounded expanding universe is not in stable equilibrium but is in a
state more analogous to a phase transition (Saslaw, 1968, 1969).
During this type of phase transition, at a critical point where forces
acting on the objects are almost evenly balanced, any small
perturbations grow relatively slowly with time, as Lifshitz (1946) and
Bonnor (1957) had found. Under these conditions
(*r*) -
*r*^{-} over
a substantial range of distances where it is scale free and
self-similar. This provided a theoretical structure for the earlier
discoveries of Carpenter (1938) and Kiang (1967). A number of views
were beginning to gell. Galaxy clustering could be considered to be a
phase transition from a Poisson distribution to a correlated
distribution, slowly developing on larger and larger scales as the
universe expands.

With this insight, to which Kihara told me he was led by his
earlier work in molecular and plasma physics, he and Totsuji reworked
the Neyman-Scott correlation analysis. They obtained a relation
between the numbers of galaxies, *N*_{1} and
*N*_{2}, in any two regions of the
sky separated by angular distance
_{12}, the
overall average <*N*>, and a function
*J*()
of geometric integrals (see also Chapters 14 and 20):

(7.1) |

The numerator is just the average of correlated fluctuations around
the mean taken over all pairs of areas separated by
_{12}; the
denominator is for normalization. (The <*N*> term was the one
Limber had initially left out; it comes from
(**r**) in 6.10.) The
right-hand side is a function just of
, arising
from angular integrals over the cones in
which projected numbers are counted. Totsuji and Kihara also showed
that if the three-dimensional two-point correlation function is a
power law with exponent
, the
two-dimensional correlations observed
on the sky will also have this form with
' =
- 1,
that is, *w*() =
() ~
^{-(-1)}.

The left-hand side of (7. 1) was measured directly from the Lick
catalog and compared with the theoretical calculation of
*J*(),
both being functions of the separation (*x*^{2} +
*y*^{2})^{1/2} on the sky.
Figure 7.1
from Totsuji and Kihara's paper (they used *S* instead of
) shows the
results with the left-hand side of (7.1) as the ordinate and the solid
lines representing
*J*()
for different values of
. The
*x*-axis is
angular separation. The solid circles are the observed values that
Totsuji and Kihara found using l° × 1° squares; the open
circles and
crosses are values that Neyman, Scott, and Shane (1953, 1956) had
previously determined for l° × 1° and 10' × 10' squares
respectively. Evidently there is good agreement for
= 1.8
± 0.1,
which also agrees with more modem values. To determine the scale
*r*_{0},
Totsuji and Kihara had to assume a luminosity function for galaxies as
well as an average galactic extinction. Employing the Shane-Wirtanen
(1967) Gaussian luminosity function, their measurement of average
surface density fluctuations <(*N* -
<*N*>)^{2}> then
gave *r*_{0} = 4.7 Mpc. also in agreement with modern values.

The two fundamental hypotheses behind these results are that
gravitational clustering resembles a neutral critical point of a phase
transition, so that
(*r*) is a
power law, and that the galaxy
distribution is statistically uniform and thus <*N*> does not
itself depend on *r*. An unconditional average density
<*N*> can therefore be
found either by averaging around many random points in space (not just
around galaxies) or by simply dividing the total number of galaxies in
the sample by its total volume. Agreement with observations suggests
that over regions of order 10 Mpc, large enough to be fair samples and
including obvious clusters as well as sparser field populations, the
distribution is statistically uniform. (Causal structures that may
exist on larger scales are small perturbations of the total density.)
Since this result is independent of detailed a priori models for
clusters, it is a clear measure of homogeneity. Of course it may not
be unique; particular models of clustering might reproduce the
observations just as well. To be viable, though, they would need to
have a similar economy of hypotheses or some other redeeming features
such as better agreement with more discriminating observations.

It took several years before the importance of these results became widely recognized, partly because they were not well-known, partly because astronomers still thought mainly in terms of clusters rather than correlation functions, and partly because the applicability of ideas from thermodynamics and phase transitions was unfamiliar. Nonetheless, these results ushered in our modem understanding of galaxy clustering, and they provide a convenient conclusion to my historical sketch. Galaxy clustering was soon to change from a minor byway on the periphery of astronomy into a major industry.