Lognormal Property of Weak‐Lensing Fields

Abstract

The statistical property of the weak lensing fields is studied quantitatively using the ray-tracing simulations. Motivated by the empirical lognormal model that characterizes the probability distribution function(PDF) of the three-dimensional mass distribution excellently, we critically investigate the validity of lognormal model in the weak lensing statistics. Assuming that the convergence field, $\kappa$, is approximately described by the lognormal distribution, we present analytic formulae of convergence for the one-point PDF and the Minkowski functionals. Comparing those predictions with ray-tracing simulations in various cold dark matter models, we find that the one-point lognormal PDF can describe the non-Gaussian tails of convergence fields accurately up to $u\sim10$, where $u$ is the level threshold given by $u\equiv\kappa/\var^{1/2}$, although the systematic deviation from lognormal prediction becomes manifest at higher source redshift and larger smoothing scales. The lognormal formulae for Minkowski functionals also fit to the simulation results when the source redshift is low. Accuracy of the lognormal-fit remains good even at the small angular scales, where the perturbation formulae by Edgeworth expansion break down. On the other hand, lognormal models does not provide an accurate prediction for the statistics sensitive to the rare events such as the skewness and the kurtosis of convergence. We therefore conclude that the empirical lognormal model of the convergence field is safely applicable as a useful cosmological tool, as long as we are concerned with the non-Gaussianity of $u\simlt5$ for low source redshift samples.