Lognormal Property of Weak‐Lensing Fields

Abstract

The statistical properties of weak-lensing fields are studied quantitatively using ray-tracing simulations. Motivated by an empirical lognormal model that excellently characterizes the probability distribution function of a three-dimensional mass distribution, we critically investigate the validity of the lognormal model in weak-lensing statistics. Assuming that the convergence field κ is approximately described by the lognormal distribution, we present analytic formulae of convergence for the one-point probability distribution function (PDF) and the Minkowski functionals. The validity of the lognormal models is checked in detail by comparing those predictions with ray-tracing simulations in various cold dark matter models. We find that the one-point lognormal PDF can accurately describe the non-Gaussian tails of convergence fields up to ν ~ 10, where ν is the level threshold given by ν ≡ κ/κ21/2, although the systematic deviation from the lognormal prediction becomes manifest at higher source redshift and larger smoothing scales. The lognormal formulae for Minkowski functionals also fit the simulation results when the source redshift is low, zs = 1. Accuracy of the lognormal fit remains good even at small angular scales 2' θ 4', where the perturbation formulae by the Edgeworth expansion break down. On the other hand, the lognormal model enables us to predict higher order moments, i.e., skewness S3,κ and kurtosis S4,κ, and we thus discuss the consistency by comparing the predictions with the simulation results. Since these statistics are very sensitive to the high- and low-convergence tails, the lognormal prediction does not provide a successful quantitative fit. 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 ν 5 for low-zs samples.