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Astron. Astrophys. 354, 767-786 (2000)

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2. Density contrast probability distribution

In order to obtain the probability distribution of the magnification of distant sources by gravitational lensing we need the properties of the density field. Hence we briefly recall here the formalism we use to characterize the density fluctuations. We sh all use the same techniques to derive the statistics of the flux perturbations. It is convenient to express the probability distribution of the density contrast [FORMULA] at scale R and redshift z (here [FORMULA] is the mean universe density) in terms of the many-body correlation functions [FORMULA]. Thus we define the quantities ([FORMULA]):

[EQUATION]

where [FORMULA] is the volume of a spherical sphere of radius R. Next we introduce the generating function

[EQUATION]

with [FORMULA]. Note that the parameters [FORMULA] can also be written in term of the cumulants [FORMULA] of the density contrast [FORMULA] at scale R:

[EQUATION]

Then, one can show (White 1979; Balian & Schaeffer 1989) that the probability distribution of the density contrast [FORMULA] within spheres of size R is:

[EQUATION]

where we note [FORMULA] as [FORMULA]. The relation (4) provides the link of the density probability distribution with the correlation functions, hence with the cumulants of the density field. In the non-linear regime [FORMULA] it is convenient to define the function (inverse Laplace transform):

[EQUATION]

which obeys:

[EQUATION]

From very general considerations (Balian & Schaeffer 1989) one expects the function [FORMULA] to behave as a power-law for large y:

[EQUATION]

and to display a singularity at a small negative value of y:

[EQUATION]

where we neglected less singular terms. From this behaviour of [FORMULA] we have:

[EQUATION]

with [FORMULA]. The interest of the function [FORMULA] is that in the non-linear regime for large density contrasts the density probability distribution can be written as (Balian & Schaeffer 1989):

[EQUATION]

with:

[EQUATION]

Thus, the density probability distribution [FORMULA] shows a power-law behaviour from [FORMULA] up to [FORMULA] with an exponential cutoff above [FORMULA]. The measure of [FORMULA] in numerical simulations allows one to recover [FORMULA] hence [FORMULA] since (5) can be inverted as:

[EQUATION]

The function [FORMULA] has been measured in the non-linear regime for various power-spectra by several authors (Valageas et al. 1999; Bouchet et al. 1991; Colombi et al. 1997; Munshi et al. 1999). In particular, although Colombi et al. (1996) found a small scale-dependence other authors found that the numerical results were consistent with [FORMULA] being scale-independent in the non-linear regime. Thus, in the following we shall use the scaling function [FORMULA] obtained by Valageas et al.(1999) for any redshift. Note that it depends on the power-spectrum and it must be obtained from numerical simulations since there is no known method to derive analytically [FORMULA]. The scale-invariance of [FORMULA], hence of the coefficients [FORMULA], can be interpreted as evidence for the stable-clustering ansatz (Peebles 1980):

[EQUATION]

which was studied in details in Balian & Schaeffer (1989). Here [FORMULA] is the scale-factor and [FORMULA] is the (local) slope of the two-point correlation function. Note that for an initial linear power-spectrum which is a power-law [FORMULA] we ha ve if stable-clustering is valid:

[EQUATION]

The interest of the formulation (4) is that once [FORMULA], or [FORMULA], is known the density probability distribution [FORMULA] can be obtained for any time and scale in the non-linear regime provided that one knows the behaviour of [FORMULA] (and that indeed [FORMULA] is scale-invariant). Note that a similar technique can be used in the quasi-linear regime with a different [FORMULA] obtained by perturbative calculations (Bernardeau 1994).

Then, in order to obtain the properties of the non-linear density field we only need to model the evolution of the two-point correlation function [FORMULA], or of the power-spectrum [FORMULA]. To this order we use the fits given by Peacock & Dodds (1996) which give the non-linear power-spectrum [FORMULA] from its linear counterpart [FORMULA]. Note that this behaviour of [FORMULA] is consistent with the stable-clustering ansatz.

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© European Southern Observatory (ESO) 2000

Online publication: February 25, 2000
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