A Universal Probability Distribution Function for Weak-lensing Amplification

Abstract

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, 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 ${\Lambda}$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 provide numerical fits for three representative models (SCDM, $\Lambda$CDM, 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.