**3.3. Counts in cells**

The counts of galaxies in cells are related to the correlation
functions of all orders and potentially provide an important means of
testing for the presence of voids in a sample of galaxies. The
relationship between the probability *P*_{N}(*V*) of
finding *N* galaxies in a
sample volume *V* and the correlation functions of all orders was given
by White (1979).
The particular case *P*_{0}(*V*) is called the *Void
Probability Function*, `VPF' for short, and is thought to be a
sensitive discriminator of clustering models.

The probability that a volume *V*, randomly selected in a sample of
points having mean number density *n*_{0}, will contain no
galaxies was first given by
White (1979)

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*P*_{0} depends on the mean density of the sample, and in
fact it can only
depend on the product *n*_{0}*V*. The scale *a*
is given in terms of the correlation functions of the distribution:

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Here _{i} is the
*i*-point correlation function of (*i* - 1) coordinates and
is determined on linear scales by (among other things) the power
spectrum of the primordial density fluctuations. For purely Gaussian
fluctuations the sum in a is cut off beyond the second term, but as we
discussed in the section on correlation function, gravitational
evolution destroys the Gaussian character of fluctuations. If we wish
to compute *P*_{0}(*V*) in a general case we are
forced to make an *ansatz*
about the relationship between second and higher order correlation
functions either through BBGKY hierarchies or by intelligent guesswork
(Schaeffer, 1985;
Fry 1986).
The data can then be used to test this
hypothesis. The VPF was first studied observationally by
Maurogordato and
Lachieze-Rey (1987)
who were able to confirm the Schaeffer scaling
relations. The recent article by
Einasto et al. (1991)
provides a
clear exposition of what the Void probability function actually
measures.
Cappi, Maurogodato and
Lachieze-Rey (1991)
have confirmed
that the VPF of the distribution of rich galaxy clusters shows scaling
behaviour up to a scale of 50 *h*^{-1} Mpc.

*P*_{0}(*V*; *n*_{0}) should be
distinguished carefully from the probability of
finding a void of the kind that has been identified as a feature of
the large scale galaxy surveys. The VPF describes the probability that
a randomly placed sphere of a given volume *V* contains a given number
of galaxies - not the probability of finding a region of volume *V*
which is devoid of galaxies.

The probability of finding *N* galaxies in a randomly selected volume
*V*, *P*_{N}(*V*) has been discussed in terms of
quite general scaling hypotheses by Balian and Schaeffer
(1989a,
b).
Balian and Schaeffer
were able to compute the properties of the counts-in-cells
distribution *P*_{N}(*V*) on the hypothesis that the
higher order correlation
functions are related to the two-point correlation function through
rather general scaling hierarchies. The CfA survey data appears to
support both the form of *P*_{N}(*V*) and the
Balian-Schaeffer scaling hypothesis
(Maurogordato and
Lachieze-Rey, 1987;
Alimi, Blanchard and
Schaeffer, 1990).
There is an extensive analysis of galaxy counts in cells by
Fry et al. (1989).

An alternative approach to the counts in cells distribution was taken by Saslaw and Hamilton (1984) who argued that

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The value of the constant
(called
*b* by Saslaw and Hamilton, but we
wish to avoid confusion with the bias parameter) is from
Crane and Saslaw's (1986)
analysis of the Zwicky catalogue of galaxies. The
parameter is
interpreted physically by Saslaw and Hamilton as being the ratio
=
- *W* / 2*K* of the gravitational correlation energy, *W*,
to the kinetic energy in peculiar motions, *K*. In fact,
could
depend on scale and will certainly depend on time. This distribution
function is discussed at length in
Itoh et al. (1990a,
b).

What is interesting is that this distribution function fits N-body
models rather well
(Suto, Itoh and Inagaki 1988,
1990),
provided that
depends on
scale as
1 - (*r*)
*r*^{- (3 -
)/2},
= 1.8
being the slope of the two-point correlation function.
Itoh (1990)
has an interesting
discussion of the relationship between the Saslaw-Hamilton
distribution function and the fractal dimensions *D*_{q} of
a set having
this distribution function, though he somehow ends up with a set whose
Hausdorff dimension *D*_{0} is smaller than the correlation
dimension *D*_{2}.