**4.2. Correlation function**

The two-point correlation function
(*r*) is defined
as the excess
over Poisson of the joint probability of finding objects in two volume
elements separated by *r* and averaged over a very large volume
[42].
We shall use the term
``correlation function'' for
its estimate, determined in a limited volume, and calculate it using
the formula:

(7) |

where < *DD*(*r*) > is the number of pairs of galaxies (or
clusters of galaxies) in the range of distances *r*±*dr*
/ 2, *dr* is
the bin size, < *RR*(*r*) > is the respective number of
pairs in a Poisson sample of points, *n* and *n*_{R}
are the mean number
densities of clusters in respective samples, and brackets
< ... > denote the ensemble average. The summation is over the
whole volume under study, and it is assumed that the galaxy and
Poisson samples have identical shape, volume and selection function.

Both the correlation function and the power spectrum characterise the distribution of galaxies, clusters and superclusters. In an ideal case in the absence of errors, and if both functions are determined in the whole space, they form a mutual pair of Fourier transformations:

(8) | |

(9) |

These formulae are useful when studying theoretical models or results of numerical simulations. For real samples they are of less use since observational errors and selection effects influence these functions in a different way. Also they reflect the spatial distribution of objects differently, thus they complement each other.

There exists a large body of studies of the correlation function of galaxies and clusters. Already early studies have shown that on small scales the correlation function can be expressed as a power law (see [42]):

(10) |

where
1.8 is the power index, and
*r*_{0} 5
*h*^{-1} Mpc is the correlation length. The correlation
function of
clusters of galaxies is similar, but shifted to larger scales, i.e. it
has approximately the same power index but a larger correlation length,
*r*_{0} 25
*h*^{-1} Mpc. On small scales the correlation function
reflects the fractal dimension,
*D* = 3 - ,
of the distribution of galaxies and clusters
[54].
On large scales the
correlation function depends on the distribution of systems of
galaxies.

In order to understand better how different geometries of the distribution of galaxies and clusters are reflected in the properties of the correlation function and power spectrum, we shall construct several mock samples with known geometrical properties, and calculate both functions. Here we use results of the study of geometrical properties of the correlation function [25, 20].

The correlation function is determined by mutual distances of galaxies
(or clusters) in real space, thus this function depends directly
on the structure of galaxy systems themselves (on small separations
which are comparable to sizes of these systems), and on the
distribution of galaxy systems (on separations which exceed the
dimensions of galaxy systems). To see these effects separately we
have constructed two series of mock samples. In the first series we
change the distribution of galaxies on small scales following
[25],
in the second series we change the distribution of systems
themselves [20].
In the first case we consider test particles
as galaxies; they form two populations - clusters and field galaxies.
Clusters are located randomly in a cube of size *L* = 100
*h*^{-1} Mpc; clusters
have a varying number of member galaxies from 12 to 200, and an
abundance and mass distribution in accordance with the observed
cluster mass distribution (see the next subsection). Inside clusters
galaxies are located randomly with an isothermal density distribution.
For the field population we consider three cases. In the first model
there is no field population at all; in Figure 4 this
model is designated gal248 (it contains 248 clusters). The second
model has 200 clusters and 6000 randomly located field galaxies
(designated as g2006t). The third model has also 200 clusters, but
field galaxies are located in filaments - each filament crosses one
cluster in either x, y, or z-axis direction, randomly chosen, and has
30 galaxies; this sample is designated as g20030. The total number of
galaxies in clusters and in the field population in the two last
models is approximately equal.

The distribution of galaxies in a sheet of thickness 15
*h*^{-1} Mpc of the
model g20030 is shown in the upper left panel of
Figure 4. In the other two models the distribution of
clusters is similar, but field galaxies are distributed randomly or
are absent. The correlation functions and power spectra of all three
models are plotted in the middle and lower left panels, respectively.
We see that the slope (power law index) of the correlation function at
small separations is
3 in the pure cluster model, and
1.8 in other two
models. These differences are due to
the difference in the effective fractal dimension:
*D* = 3 -
0 for the pure cluster model
(clusters are spherically
symmetrical); and
*D* 1.2 in a mixture of
spherically
symmetrical clusters and one-dimensional filaments. The power index
depends on the partition of galaxies between the clustered and the
field populations. On scales of
*r* 3
*h*^{-1} Mpc the correlation
function changes sharply. In the pure cluster model it has small
negative values for *r* > 3 *h*^{-1} Mpc, while in
the random field models
(*r*)
0 for *r* > 3
*h*^{-1} Mpc. In the model with filaments the
correlation function is positive but has a smaller power index, of
1.2. The change of the power
index occurs on a scale
equal to the diameter of clusters of galaxies. On larger scale the
behaviour of the correlation function depends on properties of the
field population and on the distribution of clusters. In the absence
of the field there are only a few close neighbours of clusters, hence
the slightly negative value of the correlation function; on larger
scales the level of the correlation function reflects the distribution
of clusters; clusters are distributed randomly and the level is
approximately zero. In the random field model galaxies are located
also in the vicinity of clusters and the zero level of the correlation
function begins immediately beyond the end of cluster galaxies. In
the filamentary model the effective power index of the correlation
function reflects the mean fractal dimension of filaments.

The power spectrum of all three models has a similar shape. On small scales the power index is negative and depends on the clustering law of galaxies in clusters. On larger scales the shape of the power spectrum depends on the distribution of galaxies and clusters on respective scales. Because both clusters and field galaxies are essentially randomly distributed (the location of filaments is also random), the spectrum has a zero power index as expected for a random distribution. The amplitude of the power spectrum depends on the fraction of galaxies in clusters. In the pure cluster model the amplitude is much higher, and the difference in the amplitude depends on the fraction of galaxies in clusters. This effect is similar to the influence of the void population discussed above.

Now we shall discuss models of the cluster distribution. We do not
include the population of galaxies; instead we shall investigate how
different distribution of clusters affects the correlation function
and power spectrum. Here we assume that a fraction of clusters are
located in superclusters, the rest form a field cluster population.
We consider again three models: randomly distributed superclusters,
regularly spaced superclusters, and superclusters formed by the
Voronoi tessellation model. In the Voronoi model centers of voids are
located randomly, and clusters are placed as far from void centres as
possible. These models differ in their degree of regularity of
the distribution of superclusters. The random supercluster model has
no regularity and no built-in scale. The Voronoi model has a
characteristic scale - the mean diameter of voids (determined by the
number of voids in the sample volume), but no regularity in the
distribution of voids. In the regular model superclusters are located
randomly along rods which form a regular rectangular grid of step size
120 *h*^{-1} Mpc; this scale defines the mean size of voids
between
superclusters, and also puts voids to a semiregular honeycomb-like
lattice. In addition a field population of isolated randomly located
clusters is present in this model.

The distribution of clusters of the random, Voronoi, and regular model
are shown in the upper panels of Figure 4 and
5; correlation functions are given in the middle
panels, and
power spectra in the lower panels. We see that on small scales all
correlation functions are identical; power spectra on small
wavenumbers are also identical. This is due to the fact that on these
scales both functions are determined by the distribution of clusters
*within* superclusters; and superclusters in all models were
generated using the same algorithm as in generating galaxies in
clusters. On larger scales there are important differences between
models. In the random supercluster model the correlation function
approaches zero at *r* > 80 *h*^{-1} Mpc. In the
Voronoi model it has a minimum around
*r* 80
*h*^{-1} Mpc, followed by a secondary maximum at
*r* 150
*h*^{-1} Mpc; thereafter it approaches zero. The
correlation
function of the regular rod model is oscillating: it has a series of
regularly spaced maxima and minima with a period of ~ 120
*h*^{-1} Mpc;
the amplitude of oscillations decreases with increasing separation.
The power spectrum of the random supercluster model is flat and
featureless on large scales, while in the Voronoi and regular models it has
a sharp maximum at wavenumber, which corresponds to the mean diameter
of voids in models and to the period of oscillations of the
correlation function. The shape of the power spectrum on large scales
of these two models is, however, different.

These mock samples illustrate properties of the correlation function
and power spectrum on small and large scales and their dependence on
the distribution of galaxies and clusters within systems and on the
distribution of systems themselves. The correlation function of
clusters of galaxies in rich superclusters is shown in the right panel
of Figure 3. This function is
oscillating with a
period which corresponds to the maximum of the power spectrum at
*k* = 0.05 *h* Mpc^{-1}, seen in the left panel of the
same Figure. We note
that a periodicity of the distribution of high-density regions with
the same period has been observed in the direction of the galactic poles
by Broadhurst [8].
All these facts suggests that there
exists a preferred scale of ~ 130 *h*^{-1} Mpc in the
Universe, and
possibly also some regularity in the distribution of the
supercluster-void network.