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A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac–Rice formula

### 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 (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf 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 star’s mass. As a consequence, the pdf’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 pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) 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 *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.

© 2009 American Institute of Physics

Received 31 July 2008
Accepted 03 November 2009
Published online 15 December 2009

Acknowledgments:
We thank the referees for invaluable feedback that strengthened the paper. A.M.T. would like to thank A. Aazami, R. Adler, J. Mattingly, and A. Watkins for helpful discussions. A.O.P. acknowledges the support of NSF Grant Nos. DMS-0707003 and AST-0434277-02. Part of this work was done at the Petters Research Institute in Belize.

Article outline:

I. INTRODUCTION
II. BASICS
III. RANDOM SHEAR DUE TO POINT MASSES
IV. EXPECTED NUMBER OF IMAGES: GENERAL CASE
A. Expected number of positive parity images: General case and the Kac–Rice formula
B. Expected number of images and saddle images: General case
C. Global expectation: General case
V. GLOBAL EXPECTED NUMBER OF IMAGES: MICROLENSING CASE
VI. CONCLUSION

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2009-12-15

2016-10-22

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