A Mathematical Theory of Stochastic Microlensing I. Random Time-Delay Functions and Lensing Maps

Abstract

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. Beginning with the random time delay function and associated lensing map, we determine exact expressions for the mean and variance of these transformations. We characterize the exact p.d.f. of a normalized random time delay function at the origin, showing that it is a shifted gamma distribution, which also holds at leading order in the limit of a large number of point masses at a general point of the lens plane. For the large number of point masses limit, we also prove that the asymptotic p.d.f. of the random lensing map under a specified scaling converges to a bivariate normal distribution. We show analytically that the p.d.f. of the random scaled lensing map at leading order depends on the magnitude of the scaled bending angle due purely to point masses as well as demonstrate explicitly how this radial symmetry is broken at the next order. Interestingly, we found at leading order a formula linking the expectation and variance of the normalized random time delay function to the first Betti number of its domain. We also determine an asymptotic p.d.f. for the random bending angle vector and find an integral expression for the probability of a lens plane point being near a fixed point. Lastly, we show explicitly how the results are affected by location in the lens plane. The results of this paper 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.