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A mathematical theory of stochastic microlensing. I. Random time delay functions and lensing maps

J. Math. Phys. 50, 072503 (2009); doi:10.1063/1.3158854

Published 22 July 2009

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A. O. Petters,1 B. Rider,2 and A. M. Teguia3
1Department of Mathematics and Department of Physics, Duke University, Science Drive, Durham, North Carolina 27708, USA
2Department of Mathematics, University of Colorado at Boulder, Campus Box 395 Boulder, Colorado 80309, USA
3Department of Mathematics, Duke University, Science Drive, Durham, North Carolina 27708, USA
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
History: Received 1 July 2008; accepted 3 June 2009; published 22 July 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/072503/1
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KEYWORDS and PACS

Keywords
PACS
  • 95.30.Sf
    Relativity and gravitation in astrophysics
  • 95.35.+d
    Dark matter (stellar, interstellar, galactic, and cosmological)
  • 98.62.Sb
    Gravitational lenses and luminous arcs
  • 05.40.-a
    Fluctuation phenomena, random processes, noise, and Brownian motion
  • 02.50.-r
    Probability theory, stochastic processes, and statistics
  • YEAR: 2009

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
Publisher:
AIP is a member of CrossRef AIP

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