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(eta), for the reduced convergence, eta. For a given cosmological model, P( h) the magnification probability distribution P(mu), at redshift z is related to the UPDF by, P(mu) = P(eta)/(2/kappa(min)\), where eta = 1 + (mu - 1)/(2\kappa(min)\), and kappa(min) (the minimum convergence) can be directly computed from the cosmological parameters (Omega(m) and Omega(Lambda)). 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 xi(eta), the variance of eta. 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 Lambda cold dark matter (CDM) model by Wambsganss et al. (1997). Each alternative cosmological model is then described by a single function xi(eta)(z). We find that this method gives P(mu) in excellent agreement with numerical ray-tracing and three-dimensional shear matrix calculations, and we provide numerical fits for three representative models (SCDM, LambdaCDM, 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.