3.3 Higher-order correlation
functions
One of the most direct methods to evaluate the deviation from
Gaussianity is to compute the higher-order correlation functions.
Suppose that
now labels the position of the
-th object (galaxy). Then the two-point correlation
function 
is defined
also in terms of the joint probability of the pair of objects
located in the volume elements of
and
,
where
is the mean number density of the objects. This
definition is generalized to three- and four-point correlation
functions,
and 
, in a straightforward
manner:
Apparently
,
, and
are symmetric with respect to the change of the
indices. Define the following quantities with the same symmetry
properties:
Then it is not unreasonable to suspect that the following relations
hold:
where
,
,
, and
are constants. In fact, the analysis of the
two-dimensional galaxy catalogues [68] revealed
The generlization of those relations for
-point correlation
functions is suspected to hold generally,
and is called the hierarchical clustering
ansatz. Cosmological
-body simulations
approximately support the validity of the above ansatz, but also
detect the finite deviation from it [82
].