随机微透镜II的数学理论随机图像、剪切和Kac-Rice公式
A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula
摘要：
继续我们的随机微透镜的数学理论的发展，我们研究了随机剪切和期望数量的随机透镜图像的不同类型。特别地，我们刻画了联合概率密度函数（p.d.f.）的渐近表达式中的前三个前导项。随机剪切张量在一般点在透镜平面由于点质量在无限数量的恒星的限制。根据这个命令，警察局。取决于剪切张量的大小、光学深度以及通过径向位置和恒星质量的组合的平均恒星数量。因此，剪切分量的p.d.f.s被认为是收敛的，在无限数量的恒星的限制，转移柯西分布，这表明，剪切分量有重尾在该限制。渐近概率密度函数在无限多恒星的极限下，给出了剪切星等的一个新的计算公式。扩展到一般的随机分布的透镜，我们采用Kac-Rice公式和Morse理论推导出一般公式的预期的图像总数和预期的鞍形图像的数量。我们进一步推广这些结果，考虑随机源定义在一个可数紧覆盖的光源平面。这样做是为了引入由于一般透镜映射而导致的正奇偶校验图像的{ \it global}预期数量的概念。将结果应用于微透镜，我们计算了在无限多个恒星的极限下，恒星均匀分布的渐近全局期望最小图像数。这个全局期望值是有界的，而全局期望图像数和全局期望鞍图像数随着恒星数的数量级而发散。
英文摘要：
Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (p.d.f.) of the random shear tensor at a general point in the lens plane due to point masses in the limit of an infinite number of stars. Up to this order, the p.d.f. depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the stars' masses. As a consequence, the p.d.f.s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the limit of an infinite number of stars is also presented. Extending to general random distributions of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of {\it global} expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.
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