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

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4. The quasi-linear regime

The reason rigorous calculations can be carried on in this regime is that the cumulant generating function for a cylindrical shape is exactly the one corresponding to the 2D dynamics (Bernardeau 1995). In other words, in this case

[EQUATION]

at leading order. In this regime the problem then reduces to the computation of cumulant generating functions for compensated filters for the 2D dynamics. The latter calculation is presented in the following paragraphs. This is a long and technical calculation that leads to the formulae (85-87).

The calculations we present follow what has been done for the top-hat window filters, with the complication introduced by the use of two such filters instead of one to build the compensated filter.

4.1. 2D statistics in Lagrangian space

The generating function for the compensated filter (13) can be built from the generating function of the joint density PDF for two concentric cells of different radius. This quantity will be obtained in Eulerian space from the one in Lagrangian space through a Lagrangian-Eulerian mapping. More precisely we consider the joint PDF of two reduced volumes defined as the comoving volume [FORMULA] occupied by some matter expressed in units of the volume V it occupied initially,

[EQUATION]

The Eulerian overdensity of this matter region will then be given by the inverse of v,

[EQUATION]

The calculation will be made in two steps. First we present the derivation of the cumulant generating function in Lagrangian space, then the mapping from Lagrangian to Eulerian space.

In a Lagrangian description v corresponds to the Jacobian of the transform from the initial coordinates [FORMULA] in Lagrangian space to the ones in real space [FORMULA],

[EQUATION]

The construction of the volume PDF is then based on the geometrical properties of the Jacobian perturbative expansion. Its expansion with respect to the initial density fluctuations (in the rest of this subsection we consider 2D dynamics) reads

[EQUATION]

Each term of this expansion can be written in terms of the initial Fourier modes of the linear density field,

[EQUATION]

where [FORMULA] describes the time dependence of the linear growing mode. The central issue is the way the geometrical kernel [FORMULA] behaves when geometrical effects are taken into account. At leading order in perturbation theory (that is when only "tree order" terms are taken into account) that amounts to compute terms of the form

[EQUATION]

where [FORMULA] is the angle of the [FORMULA] wave vector. There exists a central property, valid for top-hat filters only, which states that (Bernardeau 1995),

[EQUATION]

where [FORMULA] is defined in (14). This result extends the one obtained in Bernardeau (1994) for the 3D dynamics.

At leading order in Perturbation Theory any cumulant of the form [FORMULA] involves only products of such quantities. The sort of commutation rule given in the previous equation implies that these cumulants can be computed without explicitly taking into account the filtering effects. It means that any cumulant of the form [FORMULA] can be built with a tree shape construction with two different kinds of end points. Formally the generating function of such cumulants

[EQUATION]

reads,

[EQUATION]

where [FORMULA] are the second moment of the linear density contrasts between two cells of fixed Lagrangian radii [FORMULA] and [FORMULA],

[EQUATION]

They are, for the Lagrangian variables [FORMULA], fixed parameters that depend only on the power spectrum shape (and on the set of cell radii chosen at the beginning). The function [FORMULA] is the generating function of the angular averages of [FORMULA],

[EQUATION]

Note that [FORMULA] describes the density contrast of a spherical density fluctuation of linear over-density [FORMULA] for the 2D dynamics. The exact form of [FORMULA] is therefore known for any cosmological model.

The relation (48) can actually be generalized to an arbitrary number of cells in a straightforward way,

[EQUATION]

The latter relation can be rewritten in an equivalent way as

[EQUATION]

where [FORMULA] is the inverse matrix to [FORMULA]. The generating function [FORMULA] can then be written,

[EQUATION]

It gives the generating function of the reduced volume generating function of an arbitrary number of concentric cells. Note that however the known geometrical properties of the Lagrangian expansion terms do not allow to extend these results to non-concentric cells. Thus, our method actually apply to any filter which is axisymmetric (in general one would need an infinite number of cells but in practice numerical discretization always leads to a finite number of concentric shells). It is worth noting that the relation (55) gives,

[EQUATION]

which in turns leads to,

[EQUATION]

4.2. Saddle point approximation and leading order cumulant generating function

In this subsection we explicit the formal relationship between the generating function computed at leading order and the shape of the multivalued density probability distribution function.

The joint PDF is formally given by (this is an extension of Eq. 19)

[EQUATION]

In case of a small variance, the expression of the joint density is obtained through the saddle point approximation. The saddle point conditions read,

[EQUATION]

which gives implicitly the values of [FORMULA] at the saddle point position in terms of [FORMULA]. Taking advantage of the property (59), one gets,

[EQUATION]

It implies that with the saddle point position the expression of the joint PDF is (not taking into account prefactors),

[EQUATION]

with the mapping (62). This is exactly what one would expect for Gaussian initial conditions, [FORMULA] being the linearly extrapolated density contrasts at the chosen scales.

4.3. Lagrangian-Eulerian mapping

To relate Lagrangian and Eulerian space, one uses the same trick as in Bernardeau (1994), that is,

[EQUATION]

The leading order cumulant generating function can then be obtained by an identification of the exponential term when one uses the saddle point approximation. The variables are however now changed in [FORMULA] which are related to [FORMULA] with

[EQUATION]

Moreover the variables [FORMULA] enter also the expression of the cell correlation coefficients [FORMULA] since they are in Eq. (51) computed for a fixed mass scale and not for a fixed Eulerian space radius. As a result, the coefficient [FORMULA] expressed in terms of [FORMULA] should be understood as function of the variable [FORMULA] through

[EQUATION]

where [FORMULA] are all kept fixed.

The cumulant generating function in Eulerian space is obtained also with a saddle point approximation in the computation of

[EQUATION]

which leads to,

[EQUATION]

with the stationary conditions,

[EQUATION]

where the partial derivatives should then be understood for fixed radius [FORMULA] and [FORMULA]. The relation (69), together with the conditions (70) gives the formal expression of the cumulant generating function in Eulerian space.

The case we are interested in,

[EQUATION]

if

[EQUATION]

corresponds to 2 cells, if [FORMULA] is built with the filter (15). Then the generating function for [FORMULA] is obtained with a peculiar choice for [FORMULA],

[EQUATION]

It is actually convenient to define,

[EQUATION]

and

[EQUATION]

where [FORMULA] is defined by

[EQUATION]

With this choice of variable, [FORMULA] does not depend formally on the variance but only on y. To be more specific one has finally,

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are considered as function of [FORMULA] and [FORMULA] through the variables [FORMULA] and [FORMULA]. the saddle point conditions then read,

[EQUATION]

In this case the function [FORMULA] can be numerically calculated. We have restricted our calculations to the case where the power spectrum follows a power law behavior with index [FORMULA]. To do the numerical computations we also use a simplified expression for the 2D spherical collapse dynamics,

[EQUATION]

with

[EQUATION]

The resulting function [FORMULA] is shown in Fig. 2, together with the functions [FORMULA] and [FORMULA].

[FIGURE] Fig. 2. Shape of the functions [FORMULA] and [FORMULA] for the two-cell compensated window function ([FORMULA]). The computations have been done for a power law spectrum for [FORMULA]. The solution for [FORMULA] has been extended beyond the singularity to show explicitly that the singularities are due to double solution in y.

4.4. Properties of the cumulant generating function

These figures clearly show that the function [FORMULA] has two singularities on the real axis. This is to be compared to what is encountered for counts-in-cells statistics where only one singular point is expected. The numerical resolutions have been extended slightly beyond the singularities to show that they are due to the resolution of the implicit equations in [FORMULA] that have multiple solution in y. As a result the generic behavior near any of such singularity is

[EQUATION]

This behavior directly induces exponential cutoffs for the shape of the density PDF (see Balian & Schaeffer 1989; Bernardeau & Schaeffer 1992). In case of compensated filter, the fact that we obtain 2 singularities, induces two exponential cut-offs on both side of the PDF as it appears clearly on the results presented in Sect. 6 (see also Valageas 2000c).

In a phenomenological way the function [FORMULA] can be described by an effective vertex generating function [FORMULA] so that,

[EQUATION]

Numerically the effective vertex generating function is well described by a fifth order polynomial,

[EQUATION]

which is regular, the expected singular behavior for [FORMULA] being induced by Eq. (86) (see Fig. 3). Note that [FORMULA] can be viewed as a generating function of effective vertices. It implies that for instance the skewness of the 2D (or equivalently cylindrical) compensated filtered density is

[EQUATION]

for such power law spectrum shape. It is however important to have in mind that the shape of [FORMULA] as well as the skewness depend on the window function normalization . The calculations have been given here for the [FORMULA] filter defined in Eq. (15). For the [FORMULA] filter, Eq. (12), [FORMULA] for instance would have been about 1.459 times larger, because of the normalization discrepancy between the two filters.

[FIGURE] Fig. 3. Geometrical representation of Eq. (86). The function [FORMULA] is given by the intersection of the line [FORMULA] with the curve [FORMULA]. Singularities appear when both curves are tangential, as shown in the figure.

The skewness of [FORMULA] is then related to this one through a simple projection factor,

[EQUATION]

This latter relation is actually valid in both the quasilinear and the nonlinear regime.

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© European Southern Observatory (ESO) 2000

Online publication: December 15, 2000
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