## 1. IntroductionUsing Type Ia supernovae as standard candles, it is possible to derive the cosmological parameters and from the observed magnitude-redshift relation (e.g. Perlmutter et al.1999). However, several sources of uncertainty (e.g. redshift evolution of the luminosity of SNeIa) can affect this method. In particular, the apparent magnitude of distant supernovae can be distorted by gravitational lensing by the density fluctuations along the line of sight. In view of the importance of the measure of and from SNeIa, it is of interest to get a good estimate of the effect of weak gravitational lensing. Moreover, since the latter is directly linked to the matter distribution in the universe, one might use this effect to get some information on the large-scale structure of the universe itself. Note that contrary to the usual weak lensing statistics obtained when one considers the filtered distortions realized on large angular scales to probe the quasi-linear regime (e.g. Bernardeau et al.1997), the effects studied in this article which are relevant to SNeIa come from strongly non-linear scales ( kpc). Several authors have already studied some aspects of weak gravitational lensing in this non-linear regime, by analytical means (e.g. Frieman 1997; Kantowski 1998; Metcalf 1999; Hui 1999) or numerical simulations (e.g. Wambsganss 1997; Jain et al.1999). Here we present an analytical calculation of the probability distribution of the magnification due to weak lensing, from a model of the non-linear density field which has already been compared with numerical simulations of structure formation. Thus, we can directly express the properties of the weak lensing magnification in terms of the characteristics of the underlying density field. In particular, we recover the non-gaussian behaviour of the magnification as seen in previous numerical studies. Then we show that the the fluctuations of the magnification ( at ) lead to a significant uncertainty for ( for two observations at and , but this error decreases as the number of observations increases). This article is organized as follows. In Sect. 2 we recall the description we use for the density field. Next, in Sect. 3 we derive the probability distribution of the magnifition of distant sources by weak gravitational lensing. We present in Sect. 4 the numerical results we obtain for three cosmologies (critical, open and low-density flat universes) as well as the dependence on the cosmological parameters of the amplitude of the fluctuations of the magnification. Then, in Sect. 5 we compare our results with another approach, used by several authors (e.g. Porciani & Madau 1999), where the density field is described as a collection of smooth virialized halos. In particular, we point out the limitations of this method (it is restricted to large magnifications and it leads to some inconsistencies with some results from numerical simulations). Next, in Sect. 6 we show how weak lensing can affect the measure of and from SNeIa. Finally, in Sect. 7 we present the bias due to weak lensing, that is the distortion of the luminosity function of SNeIa © European Southern Observatory (ESO) 2000 Online publication: February 25, 2000 |