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

〈δ 2 〉c = σ 2 = 〈δ 2 〉 − 〈δ〉 2 c
〈δ 3 〉c = 〈δ 3 〉 − 3〈δ 2 〉c〈δ〉c − 〈δ〉 3 c
〈δ 4 〉c = 〈δ 4 〉 − 4〈δ 3 〉c〈δ〉c − 3〈δ 2 〉 2 c − 6〈δ2 〉c〈δ〉 2 c − 〈δ〉4c 〈δ 5 〉c = 〈δ 5 〉 − 5〈δ 4 〉c〈δ〉c − 10〈δ 3 〉c〈δ 2 〉c − 10〈δ 3 〉c〈δ〉 2 c − 15〈δ2 〉 2 c 〈δ〉c
−10〈δ 2 〉c〈δ〉 3 c − 〈δ〉5 c
(130)
In most cases 〈δ〉 = 0 and the above equations simplify considerably. In the
following we usually denote σ 2 the local second order cumulant. The Wick
theorem then implies that in case of a Gaussian field σ 2 is the only nonvanishing
cumulant.
It is important to note that the local PDF is essentially characterized by its
cumulants which constitute a set of independent quantities. This is important
since in most of applications that follow the higher-order cumulants are small
compared to their associated moments. Finally, let’s note that a useful mathematical
property of cumulants is that 〈(bδ) n 〉c = b n 〈δ n 〉c, and 〈(b+δ) n 〉c = 〈δ n 〉c
where b is an ordinary number.
3.3.2 Smoothing
The density distribution is usually smoothed with a filter WR of a given size,
R, commonly a top-hat or a Gaussian window. Indeed, this is required by the
discrete nature of galaxy catalogs and N-body experiments used to simulate
them. Moreover, we shall see later that the scale-free nature of gravitational
clustering implies some remarkable properties about the scaling behavior of
the smoothed density distribution. The quantities of interest are then the
moments 〈δ p
R
δR(x) =
〉 and the cumulants 〈δp
R 〉c of the smoothed density field
WR(x ′ − x)δ(x ′ )d 3 x ′ . (131)
Note that for the top hat window,
〈δ p
R 〉c
=
vR
ξp(x1, . . .,xp) dD x1 . . .d D xp
v p
R
(132)
(where D = 2 or 3 is the dimension of the field) is nothing but the average of
the N-point correlation function over the corresponding cell of volume vR.
For a smooth field, equations in Sect. 3.3.1 are valid for δ as well as δR. Some
corrections are required if δ is a sum of Dirac delta functions as in real galaxy
catalogs. We shall come back to this in Chapter 6.
In the remaining of this chapter, we shall omit the subscript R which stands
for smoothing, but it will be implicitly assumed.
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