Stochastic Microlensing: Mathematical Theory and Applications
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Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory
of stochastic microlensing. We first construct a natural probability space for stochastic microlensing and characterize the general behaviour of the random time
delay functions' random critical sets. Next we study stochastic microlensing in two distinct random microlensing scenarios: The uniform stars' distribution with
constant mass spectrum and the spatial stars' distribution with general mass spectrum. For each scenario, we determine exact and asymptotic (in the large number
of point masses limit) stochastic properties of the random time delay functions and associated random lensing maps and random shear tensors, including their
moments and asymptotic density functions. We use these results to study certain random observables, such as random fixed lensed images, random bending angles,
and random magnifications. These results are relevant to the theory of random
fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.
Continuing our development of a mathematical theory of stochastic microlensing, we study the stochastic version of the Image Counting Problem, first considered
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
the expected total number of images and the expected number of saddle images for a general random lensing scenario. 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
global expected number of positive parity images due to a general lensing map. Applying the result to the uniform stars' distribution random microlensing
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,
while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.
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
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
of a random gravitational field caused by a random star.
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