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Weak gravitational lensing effects on the determination of $Ω_m$ and $Ω_Λ$ from SNeIa

In this article we present an analytic calculation of the probability distribution of the magnification of distant sources due to weak gravitational lensing from non-linear scales. We use a realistic description of the non-linear density field, which has already been compared with numerical simulations. Then, we can directly express the p.d.f. $P(μ)$ of the magnification in terms of the p.d.f. of the density contrast realized on non-linear scales (typical of galaxies) where the local slope of the initial linear power-spectrum is $n=-2$. We recover the behaviour seen by numerical simulations: $P(μ)$ peaks at a value slightly smaller than the mean $<μ>=1$ and it shows an extended large $μ$ tail (as described in another article our predictions also show a good quantitative agreement with results from N-body simulations for a finite smoothing angle). Then, we study the effects of weak lensing on the derivation of the cosmological parameters from SNeIa. We show that the inaccuracy introduced by weak lensing is not negligible: $ΔΩ_m \ga 0.3$ for two observations at $z_s=0.5$ and $z_s=1$. However, observations can unambiguously discriminate between $Ω_m=0.3$ and $Ω_m=1$. Moreover, in the case of a low-density universe one can clearly distinguish an open model from a flat cosmology (besides, the error decreases as the number of observed SNeIa increases). Since distant sources are more likely to be ``demagnified'' the most probable value of the observed density parameter $Ω_m$ is slightly smaller than its actual value. On the other hand, one may obtain some valuable information on the properties of the underlying non-linear density field from the measure of weak lensing distortions.