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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. In addition, we derive the probability density function (pdf) of a random point-mass potential, which form the constituent of a stochastic microlens potential. We characterize the exact pdf 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 if the normalized time delay function was at a general point of the lens plane. For the large number of point-mass limit, we also prove that the asymptotic pdf of the random lensing map under a specified scaling converges to a bivariate normal distribution. We show analytically that the pdf 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 pdf 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.

© 2009 American Institute of Physics

Received 01 July 2008
Accepted 03 June 2009
Published online 22 July 2009

Acknowledgments:
A.M.T. would like to thank Robert Adler, Jonathan Mattingly, and Andrea Watkins for useful discussions. A.O.P. acknowledges the support of NSF Grant Nos. DMS-0707003 and AST-0434277-02.

Article outline:

I. INTRODUCTION
II. BASIC FRAMEWORK
III. RESULTS ON THE RANDOM TIME DELAY FUNCTION
A. Exact moments and pdfs
B. Asymptotic pdf for the time delay function
IV. RESULTS ON THE RANDOM LENSING MAP
A. Exact expectation and variance of the lensing map components
B. Asymptotic pdf of the lensing map
V. RESULTS OF THE RANDOM BENDING ANGLE AND FIXED POINTS
A. Asymptotic pdf of the bending angle
B. Fixed points
VI. CONCLUSION

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2009-07-22

2016-10-01

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