A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula

TL;DR

This paper develops a mathematical framework for stochastic microlensing, analyzing the distribution of shear and the expected number of images using advanced probabilistic tools, revealing heavy-tailed shear distributions and asymptotic image counts.

Contribution

It introduces new asymptotic formulas for shear distributions and expected image counts, extending microlensing theory with rigorous probabilistic and geometric methods.

Findings

Shear components converge to shifted Cauchy distributions with heavy tails.

Asymptotic expected number of minimum images is finite for uniform star distributions.

Expected total and saddle images diverge as the number of stars increases.

Abstract

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.