Large-Scale Structure of the Universe and Cosmological ...

p points in cell 1
q points in cell 2
Fig. 36. Structure of the coefficient Cp q in large separation limit: Cp q is given by
the sum of all possible trees joining p points in first cell to q points in the second
with only one crossing line. The sums can be done separately on each side leading
to Cp q = Cp1 Cq 1.
5.12 The Two-Point Density PDF
Perturbation theory can obviously be applied to any combination of the density
taken at different locations. In particular, for sound cosmic error computations
(see Chapter 6) the bivariate density distribution is an important
quantity that has been investigated in some detail.
The object of this sub-section is to present the exact results that have been
obtained at tree-level for the two-point density cumulants [51]. We consider the
joint densities at positions x1 and x2 and we are interested in computing the
cumulants 〈δ p (x1)δ q (x2)〉 c where the field is supposed to be filtered at a given
scale R. In general such cumulants are expected to have quite complicated
expressions, depending on both the smoothing length R and the distance |x1−
x2|. We make here the approximation that the distance between the two points
is large compared to the smoothing scale. In other words, we neglect shortdistance
effects.
Let us define the parameters Cp q with,
Cp q =
〈δ p (x1)δ q (x2)〉 c
〈δ(x1)δ(x2)〉 〈δ 2 〉 p+q−2.
(348)
Because of the tree structure of the correlation hierarchy, we expect the coefficients
Cp q to be finite in both the large distance limit and at leading order in
the variance. This expresses the fact that among all the diagrams that connect
the two cells, the ones that involve only one line between the cells are expected
to be dominant in cases when 〈δ(x1)δ(x2)〉 ≪ 〈δ 2 〉.
The next remarkable property is directly due to the tree structure of the highorder
correlation functions. The coefficients Cp q are dimensionless quantities,
that correspond to some geometrical averages of trees. It is quite easy to realize
(see Fig. 36) that such averages can be factorized into two parts, corresponding
to the end points of the line joining the two cells. In other words one should
have,
Cp q = Cp 1Cq 1. (349)
This factorization property is specific to tree structures. It was encountered
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