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 $\nu\sim10$, where $\nu$ is the level threshold given by $\nu\equiv\kappa/\var^{1/2}$, 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 $\nu\simlt5$ for low source redshift samples.
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机译：使用光线跟踪模拟定量研究了弱透镜场的统计特性。基于极好的描述三维质量分布的概率分布函数（PDF）的经验对数正态模型，我们批判性地研究了对数正态模型在弱透镜统计中的有效性。假设收敛域$ \ kappa $由对数正态分布近似描述，我们给出了单点PDF和Minkowski函数的收敛解析公式。将这些预测与各种冷暗物质模型中的光线跟踪模拟进行比较，我们发现单点对数正态PDF可以准确描述高达$ \ nu \ sim10 $的会聚场的非高斯尾部，其中$ \ nu $为尽管在较高的源红移和较大的平滑比例下，与对数正态预测的系统偏差变得明显，但由$ \ nu \ equiv \ kappa / \ var ^ {1/2} $给出的级别阈值。当源红移较低时，Minkowski泛函的对数正态公式也适合模拟结果。即使在较小的角度范围内，对数正态拟合的准确性仍然很好，在这种情况下，Edgeworth扩展的扰动公式会失效。另一方面，对数正态模型不能为对诸如偏度和收敛峰度之类的稀有事件敏感的统计数据提供准确的预测。因此，我们得出的结论是，只要我们关注低源红移样本的\\ nu \ simlt5 $的非高斯性，收敛场的经验对数正态模型就可以安全地用作有用的宇宙学工具。
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