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*Astron. Astrophys. 354, 767-786 (2000)*
## 8. Conclusion
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
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