A Universal Probability Distribution Function for Weak-lensing Amplification

We present an approximate form for the weak lensing magnification distribution of standard candles, valid for all cosmological models, with arbitrary matter distributions, over all redshifts. Our results are based on a universal probability distribution function (UPDF), P (η), for the reduced convergence, η. For a given cosmological model, the magnification probability distribution, P (μ), at redshift z is related to the UPDF by P (μ) = P (η)/2 |κmin|, where η = 1 + (μ − 1)/(2|κmin|), and κmin (the minimum convergence) can be directly computed from the cosmological parameters (Ωm and ΩΛ). We show that the UPDF can be well approximated by a three-parameter stretched Gaussian distribution, where the values of the three parameters depend only on ξη, the variance of η. In short, all possible weak lensing probability distributions can be well approximated by a one-parameter family. We establish this family, normalizing to the numerical ray-shooting results for a ΛCDM model by Wambsganss et al. (1997). Each alternative cosmological model is then described by a single function ξη(z). We find that this method gives P (μ) in excellent agreement with numerical ray-shooting calculations, and provide numerical fits for three representative models (SCDM, ΛCDM, and OCDM). Our results provide an easy, accurate, and efficient method to calculate the weak lensing magnification distribution of standard candles, and should be useful in the analysis of future high-redshift supernova data. Subject headings: cosmology: observations—cosmology: theory—gravitational lensing