Stochastic Microlensing: Mathematical Theory and Applications

Abstract

<p>Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory </p><p>of stochastic microlensing. We first construct a natural probability space for stochastic microlensing and characterize the general behaviour of the random time </p><p>delay functions' random critical sets. Next we study stochastic microlensing in two distinct random microlensing scenarios: The uniform stars' distribution with</p><p> constant mass spectrum and the spatial stars' distribution with general mass spectrum. For each scenario, we determine exact and asymptotic (in the large number</p><p> of point masses limit) stochastic properties of the random time delay functions and associated random lensing maps and random shear tensors, including their </p><p>moments and asymptotic density functions. We use these results to study certain random observables, such as random fixed lensed images, random bending angles, </p><p>and random magnifications. These results are relevant to the theory of random </p><p>fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.</p><p>Continuing our development of a mathematical theory of stochastic microlensing, we study the stochastic version of the Image Counting Problem, first considered </p><p>in the non-random setting by Einstein and generalized by Petters. In particular, we employ the Kac-Rice formula and Morse theory to deduce general formulas for </p><p>the expected total number of images and the expected number of saddle images for a general random lensing scenario. We further </p><p>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</p><p> global expected number of positive parity images due to a general lensing map. Applying the result to the uniform stars' distribution random microlensing </p><p>scenario, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars. This global expectation is bounded, </p><p>while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.</p><p>Finally, we outline a framework for the study of stochastic microlensing in the neighbourhood of lensed images. This framework is related to the study of the </p><p>local geometry of a random surface. In our case, the surface is non-Gaussian, and therefore standard literature on the subject does not apply. We explore the case</p><p> of a random gravitational field caused by a random star.</p>