## 2. Density contrast probability distributionIn 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
at scale where is the volume of a
spherical sphere of radius with . Note that the parameters
can also be written in term of the
cumulants of the density contrast
at scale Then, one can show (White 1979; Balian & Schaeffer 1989) that
the probability distribution of the density contrast
within spheres of size where we note as . 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 it is convenient to define the function (inverse Laplace transform): From very general considerations (Balian & Schaeffer 1989) one
expects the function to behave as a
power-law for large and to display a singularity at a small negative value of
where we neglected less singular terms. From this behaviour of we have: with . The interest of the function is that in the non-linear regime for large density contrasts the density probability distribution can be written as (Balian & Schaeffer 1989): with: Thus, the density probability distribution shows a power-law behaviour from up to with an exponential cutoff above . The measure of in numerical simulations allows one to recover hence since (5) can be inverted as: The function 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 being scale-independent in the non-linear regime. Thus, in the following we shall use the scaling function 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 . The scale-invariance of , hence of the coefficients , can be interpreted as evidence for the stable-clustering ansatz (Peebles 1980): which was studied in details in Balian & Schaeffer (1989). Here is the scale-factor and is the (local) slope of the two-point correlation function. Note that for an initial linear power-spectrum which is a power-law we ha ve if stable-clustering is valid: The interest of the formulation (4) is that once , or , is known the density probability distribution can be obtained for any time and scale in the non-linear regime provided that one knows the behaviour of (and that indeed is scale-invariant). Note that a similar technique can be used in the quasi-linear regime with a different 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 , or of the power-spectrum . To this order we use the fits given by Peacock & Dodds (1996) which give the non-linear power-spectrum from its linear counterpart . Note that this behaviour of is consistent with the stable-clustering ansatz. © European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |