Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

The testing of theories of **structure formation**
using observations of the **large-scale structure** of the distribution of
galaxies requires a statistical approach. Theoretical studies of the
problem of structure formation generally consist of performing
numerical ** N-body simulations** on powerful computers. Such
simulations
show how

One of the simplest (and most commonly used) statistical methods
appropriate for the analysis of galaxy clustering observations is the
*correlation function* or, more accurately, the *two-point correlation
function*. This measures the statistical tendency for galaxies to occur
in pairs rather than individually. The correlation function, usually
denoted by *(r)*,
measures the number of pairs of galaxies found at a
separation r compared with how many such pairs would be found if
galaxies were distributed at random throughout space. More formally,
the probability of finding two galaxies in small volumes
d*V*_{1} and d*V*_{2}
separated by a distance *r* is defined to be be

d*P* = *n*^{2} (1 + *(r)*) d*V*_{1} d*V*_{2}

where *n* is the average density of galaxies per unit volume. A positive
value of *(r)*
thus indicates that there are more pairs of galaxies
with a separation r than would occur at random; galaxies are then said
to be clustered on the scale *r*. A negative value indicates that
galaxies tend to avoid each other; they are then said to be
*anticlustered*. A completely random distribution, usually called a
*Poisson distribution*, has *(r)* = 0 for all values of *r*.

Estimates of the correlation function of galaxies indicate that
*(r)*
is a power-law function of *r*:

*(r)* (*r/r*_{0})^{-1.8}

where the constant *r*_{0} is usually called the
*correlation length*. The
value of *r*_{0} depends slightly on the type of galaxy
chosen, but is
around 5 Mpc for bright galaxies. This behaviour indicates that these
galaxies are highly clustered on scales of up to several tens of
millions of light years in a roughly **fractal** pattern. On larger
scales, however,
*(r)* becomes negative, indicating the presence of
large voids (see large-scale structure). The correlation function
*(r)*
is mathematically related to the **power spectrum** *P(k)* by a Fourier
transformation; the function *P(k)* is also used as a descriptor of
clustering on large scales.

FURTHER READING: Peebles, P.J.E., The Large-Scale Structure of the Universe (Princeton University Press, Princeton, 1980).