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

Fig. 43. Example of a computation of the S3, S4 and S5 parameters in the tree
hierarchical model for dark matter halos selected with a varying threshold in x,
defined by Eq. (541). Calculations have been made with the vertex generating function,
ζ(τ) = (1 − τ/κ) −κ with κ = 1.3. For large values of x one explicitly sees the
Sp → p p−2 behavior expected in the high threshold limit.
In this framework a number of important properties and results have been
derived,
(i) the correlation functions of the halo population follow a tree structure
similar to the one of the matter field in the large separation limit (e.g.
when the distances between the halos are much larger than their size);
(ii) the values of the vertices depend only on the internal properties of the
halos, namely on the reduced variable,
x = ρ
ρσ 2;
(541)
(iii) all vertices are growing functions of x and have a specific large x asymptotic
behavior,
ν1(x) ≡b(x) ∼ x (542)
νp(x) ∼b p (x); (543)
The large x limit that has been found for the high-threshold clipping limit
is once again recovered, since we expect in such a model that Sh,p → p p−2
when x → ∞. Property (iii), together with (ii), also holds for halos in the
framework of the Press-Schechter approach, as we shall see in the next section
[see discussion below Eq. (556)].
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