In this article, we have shown how one can obtain the probability distribution of the magnification of distant sources by weak gravitational lensing, using a realistic description of the density field which has already been checked against numerical simulations of structure formation within hierarchical scenarios. Thus, this work improves the results obtained by previous analytical studies which used for instance a "Swiss cheese" model to describe the universe (Kantowski 1998) or considered a collection of virialized halos amid an empty space. We recover the behaviour observed in numerical simulations, which is not surprising since we consider similar density fields. Thus, the probability distribution of the magnification shows a maximum at a value slightly smaller than the mean and it shows an extended large µ tail. Moreover, the variance increases at larger redshifts while the deviation from a gaussian gets higher at lower redshifts. The advantage of our approach is that we obtain a direct connection of the weak lensing properties with the characteristics of the underlying non-linear density field. In particular, the non-gaussian behaviour of the magnification is expressed in terms of the non-gaussian properties of the density field , through its many-body correlation functions.
Then, we have applied our results to the magnification of distant Type Ia supernovae. We have shown that the inaccuracy introduced by weak lensing in the derivation of the cosmological parameters is not negligible : for two observations at and . However, observations can unambiguously discriminate between and . Moreover, in the case of a low-density universe one can clearly discriminate between and . Besides, the accuracy increases as the number of SNeIa gets larger (there are already 42 available SNeIa, see Perlmutter et al.1999). On the other hand, if it were possible to measure the distortions due to weak lensing one would obtain some valuable information on the properties of the underlying non-linear density field, since we have shown that the probability distribution of the magnification can be directly expressed in terms of the probability distribution of the density contrast at the non-linear scale (typical of present galaxies) where the local slope of the initial linear power-spectrum is . However, this would require a rather high accuracy of the observations, as we have shown that the probability distribution of the magnification is not very extended (the typical deviation is of order at , hence mag ). A more detailed discussion of the properties of the p.d.f. and of the convergence and the aperture mass is presented in other articles (Valageas 1999b; Valageas 2000; Bernardeau & Valageas 2000).
© European Southern Observatory (ESO) 2000
Online publication: February 25, 2000