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Astron. Astrophys. 364, 1-16 (2000)

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3. One-point PDF construction

In this section we succinctly review the theory for the construction of the one-point PDF statistical quantities. In particular we recall the mathematical relationship between the one-point PDF and the moment and cumulant generating functions.

3.1. General formalism

In general one can define [FORMULA] as the generating function of the cumulants of a given local random variable [FORMULA],

[EQUATION]

It is given through a Laplace transform of the one-point PDF of the local density contrast [FORMULA],

[EQUATION]

For hierarchical models, that is when [FORMULA] it is convenient to define [FORMULA] as

[EQUATION]

Then the one-point PDF of [FORMULA] is then given by the inverse Laplace transform (see Balian & Schaeffer 1989),

[EQUATION]

of the moment generating function. Here [FORMULA] is the r.m.s. density fluctuation.

3.2. The projection effects

The relation (1) states that the local convergence can be viewed as the superposition of independent layers of cosmic matter field. The direct calculation of the one-point PDF of such a sum would involve an infinite number of convolution products which makes it intractable. It is more convenient to consider the cumulant generating functions which simply add when different layers are superposed (because Laplace transforms change convolutions into ordinary products).

The projections effects for statistical properties of the local convergence have already been considered in previous papers. It has been shown in particular how the moments of the projected density can be related to the ones of the 3D field in both hierarchical models corresponding to the non-linear regime (Tóth et al. 1989) and in the quasilinear regime (Bernardeau 1995).

More recently it has been shown (Valageas 2000a,b; Munshi & Jain 1999a) that these results could be extended to the full PDF of the projected density. In particular the cumulant generating function of the projected density can be obtained by a simple line-of-sight average of the 3D cumulant generating function.

It is convenient to define the normalized projected density contrast [FORMULA] by:

[EQUATION]

where [FORMULA] is the selection function for the projection effects as a function of the radial distance [FORMULA]:

[EQUATION]

When filtering effects are included we have,

[EQUATION]

where [FORMULA] is a given window function at scale [FORMULA]. A particular case is provided by the normalized aperture mass [FORMULA],

[EQUATION]

The cumulants of the projected density can be related to those of the 3D density fields. Formally they correspond to the ones of the field when it is filtered by a conical shape window. Thus, from (20) we obtain:

[EQUATION]

The computation of such quantities can be made in the small angle approximation. Such approximation is valid when the transverse distances [FORMULA] are much smaller than the radial distances [FORMULA]. In this case the integral (24) is dominated by configurations where [FORMULA]. It permits to make the change of variables [FORMULA] with [FORMULA]. Then, since the correlation length (beyond which the many-body correlation functions are negligible) is much smaller than the Hubble scale [FORMULA] (where [FORMULA] is the Hubble constant at redshift z) the integral over [FORMULA] converges over a small distance of the order of [FORMULA] and the expression (24) can be simplified in,

[EQUATION]

Taking filtering effects into account leads to,

[EQUATION]

Thus the projection effects reduce to

[EQUATION]

where [FORMULA] is the filtered 3D density with a cylindrical filter of transverse size [FORMULA] and depth L (which goes to infinity in (25)).

In particular this result gives the expression of the variance of the filtered projected density contrast,

[EQUATION]

This expression can be re-expressed in terms of the power spectrum [FORMULA] (nonlinear power spectrum), defined in this paper with

[EQUATION]

Then

[EQUATION]

where [FORMULA] is the component of [FORMULA] orthogonal to the radial direction and [FORMULA] is the component along the line-of-sight. In the previous integral, [FORMULA] and [FORMULA], so that when L is large [FORMULA] is negligible compared to [FORMULA] which leads to,

[EQUATION]

where W is the Fourier shape of the 2D window function. This relation holds for the filtered projected density contrast as well as the aperture mass, for which W in (31) is to be replaced by [FORMULA] or [FORMULA].

The formal expression for the higher order moments can be simplified by taking advantage of the so-called scaling laws for the correlation functions. It is in particular natural to assume that,

[EQUATION]

with a coefficient of proportionality, [FORMULA], that depends on both the power spectrum and filter shapes, but not on the power spectrum normalization. For power law spectrum it implies in particular that these coefficients do not depend on the filtering scale. In the coming sections we present in more details the origin of this scaling relation. It allows to define

[EQUATION]

The Eq. (27) then relates the cumulant generating function [FORMULA] for the projected density to the one corresponding to cylindrical filtering effects,

[EQUATION]

with

[EQUATION]

which can be rewritten in terms of the matter fluctuation power spectrum,

[EQUATION]

In this expression we have explicitly written the redshift dependence of the power spectrum. In case of a power law spectrum,

[EQUATION]

it takes a much simpler form given by,

[EQUATION]

The result (34) is the cornerstone of the calculations we present. It allows to relate the cumulant generating function of projected quantities to the ones computed in much simpler geometries.

The difficulty then resides in the computation of [FORMULA] in the regimes we are interested in. Two limit cases are actually accessible to exact calculations. First, one is the quasilinear regime where one can take advantage of the special properties of the perturbative expansion in Lagrangian space to build the cumulant generating function in Eulerian space. These kinds of properties had been used previously for 3D or 2D top-hat filters only. We show here how this method can be extended to the filter (15). Second, in the strongly nonlinear regime one can also do exact numerical calculations when one is assuming the high order correlation functions to follow a tree model. The derivations corresponding to these regimes are presented in the next two sections.

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© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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