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

For the reasons discussed in Sect. 4.1.1, it is convenient to rescale the third
moment and define the skewness parameter S3 (see Sect. 2),
S3 ≡ 〈δ3 〉
〈δ2 34
2 =
〉 7 + O(σ2 ). (229)
The skewness measures the tendency of gravitational clustering to create an
asymmetry between underdense and overdense regions (see Fig. 20). Indeed,
as clustering proceeds there is an increased probability of having large values
of δ (compared to a Gaussian distribution), leading to an enhancement of the
high-density tail of the PDF. In addition, as underdense regions expand and
most of the volume becomes underdense, the maximum of the PDF shifts to
negative values of δ. From Eq. (144) we see that the maximum of the PDF is
in fact reached at
δmax ≈ − S3
2 σ2 , (230)
to first order in σ. We thus see that the skewness factor S3 contains very useful
information on the shape of the PDF.
5.1.2 The Smoothed Case
At this stage however the calculation in Eq. (229) is somewhat academic because
it applies to the statistical properties of the local, unfiltered, density
field. In practice the fields are always observed at a finite spatial resolution
(whether it is in an observational context or in numerical simulations). The
effect of filtering, which amounts to convolving the density field with some
window function, should be taken into account in the computation of S3. The
main difficulty lies in the complexity this brings into the computation of the
angular integral. To obtain the skewness of the local filtered density, δR, one
indeed needs to calculate,
with
〈δ 3
R 〉 = 3 〈
δ (1)
2 R
δ (2)
R 〉 (231)
δ (1)
R =a d 3 kδ(k) exp[ik · x] W3(k1 R), (232)
δ (2)
R =a2
d 3
k1 d 3 k2 δ(k1) δ(k2) exp[i(k1 + k2) · x] ×
F2 (k1,k2) W3(|k1 + k2| R), (233)
where W3(k) is the 3D filtering function in Fourier space. It leads to the
expression for the third moment,
〈δ 3 R〉 = 6 a 4
d 3 k1
d 3 k2 P(k1) P(k2) W3(k1 R) W3(k2 R) ×
F2(k1,k2) W3(|k1 + k2| R), (234)
85