Publications of Arlie O. Petters    :chronological  alphabetical  combined listing:

%% Books   
@book{fds15387,
   Author = {A. O. Petters and M. C. Werner},
   Title = {Gravitational Lensing and Black Holes},
   Publisher = {Springer, in preparation},
   Year = {2017},
   Month = {Spring},
   Key = {fds15387}
}

@book{fds51036,
   Author = {A. O. Petters and X. Dong},
   Title = {An Introduction to Mathematical Finance: Understanding and
             Building Financial Intuition},
   Series = {SUMAT},
   Publisher = {Springer, in preparation},
   Year = {2016},
   Month = {Winter},
   Key = {fds51036}
}

@book{fds70670,
   Author = {A.O. Petters},
   Title = {Algebra, Geometry, and Trigonometry: Student and Teacher
             Editions},
   Publisher = {BRC, Benque-Belize},
   Year = {2007},
   Month = {August},
   Key = {fds70670}
}

@book{fds70667,
   Author = {A.O. Petters},
   Title = {Scientific Reasoning: Student and Teacher
             Editions},
   Publisher = {BRC, Benque-Belize},
   Year = {2007},
   Month = {July},
   Key = {fds70667}
}

@book{fds70668,
   Author = {A.O. Petters},
   Title = {PSE Mathematics: Student and Teacher Editions},
   Publisher = {BRC, Benque-Belize},
   Year = {2007},
   Key = {fds70668}
}

@book{fds9010,
   Author = {A. O. Petters and H. Levine and J. Wambsganss},
   Title = {Singularity Theory and Gravitational Lensing},
   Series = {Progress in Mathematical Physics, Volume
             21},
   Publisher = {Birkhauser},
   Year = {2001},
   Month = {June},
   url = {http://www.amazon.com/exec/obidos/ASIN/0817636684/qid=1028663985/sr=1-1/ref=sr_1_1/002-1045375-379126},
   Key = {fds9010}
}


%% Papers Published   
@article{fds243976,
   Author = {Aazami, AB and Keeton, CR and Petters, AO},
   Title = {Lensing by Kerr black holes. II: Analytical study of
             quasi-equatorial lensing observables},
   Journal = {Journal of Mathematical Physics},
   Volume = {52},
   Number = {10},
   Pages = {102501},
   Year = {2011},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.3642616},
   Abstract = {In this second paper, we develop an analytical theory of
             quasi-equatorial lensing by Kerr black holes. In this
             setting we solve perturbatively our general lens equation
             with displacement given in Paper I, going beyond
             weak-deflection Kerr lensing to third order in our expansion
             parameter ε, which is the ratio of the angular
             gravitational radius to the angular Einstein radius. We
             obtain new formulas and results for the bending angle, image
             positions, image magnifications, total unsigned
             magnification, and centroid, all to third order in ε and
             including the displacement. New results on the time delay
             between images are also given to second order in ε, again
             including displacement. For all lensing observables we show
             that the displacement begins to appear only at second order
             in ε. When there is no spin, we obtain new results on the
             lensing observables for Schwarzschild lensing with
             displacement. © 2011 American Institute of
             Physics.},
   Doi = {10.1063/1.3642616},
   Key = {fds243976}
}

@article{fds243977,
   Author = {Aazami, AB and Petters, AO and Rabin, JM},
   Title = {Orbifolds, the A, D, E family of caustic singularities, and
             gravitational lensing},
   Journal = {Journal of Mathematical Physics},
   Volume = {52},
   Number = {2},
   Year = {2011},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.3545578},
   Abstract = {We provide a geometric explanation for the existence of
             magnification relations for the An(n = 2), Dn(n = 4), E6,
             E7, E8 family of caustic singularities, which were
             established in recent work. In particular, it was shown that
             for families of general mappings between planes exhibiting
             any of these caustic singularities, and for any noncaustic
             target point, the total signed magnification of the
             corresponding preimages vanishes. As an application to
             gravitational lensing, it was also shown that, independent
             of the choice of a lens model, the total signed
             magnification vanishes for a light source anywhere in the
             four-image region close to elliptic and hyperbolic umbilic
             caustics. This is a more global and higher order analog of
             the well-known fold and cusp magnification relations. We now
             extend each of these mappings to weighted projective space,
             which is a compact orbifold, and show that magnification
             relations translate into a statement about the behavior of
             these extended mappings at infinity. This generalizes
             multidimensional residue techniques developed in previous
             work, and introduces weighted projective space as a new tool
             in the theory of caustic singularities and gravitational
             lensing. © 2011 American Institute of Physics.},
   Doi = {10.1063/1.3545578},
   Key = {fds243977}
}

@article{fds243978,
   Author = {Aazami, AB and Keeton, CR and Petters, AO},
   Title = {Lensing by Kerr black holes. I. General lens equation and
             magnification formula},
   Journal = {Journal of Mathematical Physics},
   Volume = {52},
   Number = {9},
   Pages = {092502},
   Year = {2011},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.3642614},
   Abstract = {We develop a unified, analytic framework for gravitational
             lensing by Kerr black holes. In this first paper, we present
             a new, general lens equation and magnification formula
             governing lensing by a compact object. Our lens equation
             assumes that the source and observer are in the
             asymptotically flat region and does not require a small
             angle approximation. Furthermore, it takes into account the
             displacement that occurs when the light ray's tangent lines
             at the source and observer do not meet on the lens plane. We
             then explore our lens equation in the case when the compact
             object is a Kerr black hole. Specifically, we give an
             explicit expression for the displacement when the observer
             is in the equatorial plane of the Kerr black hole as well as
             for the case of spherical symmetry. © 2011 American
             Institute of Physics.},
   Doi = {10.1063/1.3642614},
   Key = {fds243978}
}

@article{fds243975,
   Author = {Petters, AO},
   Title = {Gravity's action on light},
   Journal = {Notices of the American Mathematical Society},
   Volume = {57},
   Number = {11},
   Pages = {1392-1409},
   Year = {2010},
   ISSN = {0002-9920},
   Key = {fds243975}
}

@article{fds243979,
   Author = {Petters, AO and Werner, MC},
   Title = {Mathematics of gravitational lensing: Multiple imaging and
             magnification},
   Journal = {General Relativity and Gravitation},
   Volume = {42},
   Number = {9},
   Pages = {2011-2046},
   Year = {2010},
   Month = {Fall},
   ISSN = {0001-7701},
   url = {http://dx.doi.org/10.1007/s10714-010-0968-6},
   Abstract = {The mathematical theory of gravitational lensing has
             revealed many generic and global properties. Beginning with
             multiple imaging, we review Morse-theoretic image counting
             formulas and lower bound results, and complex-algebraic
             upper bounds in the case of single and multiple lens planes.
             We discuss recent advances in the mathematics of stochastic
             lensing, discussing a general formula for the global
             expected number of minimum lensed images as well as
             asymptotic formulas for the probability densities of the
             microlensing random time delay functions, random lensing
             maps, and random shear, and an asymptotic expression for the
             global expected number of micro-minima. Multiple imaging in
             optical geometry and a spacetime setting are treated. We
             review global magnification relation results for
             model-dependent scenarios and cover recent developments on
             universal local magnification relations for higher order
             caustics. © 2010 Springer Science+Business Media,
             LLC.},
   Doi = {10.1007/s10714-010-0968-6},
   Key = {fds243979}
}

@article{fds243981,
   Author = {Aazami, AB and Petters, AO},
   Title = {A universal magnification theorem. III. Caustics beyond
             codimension 5},
   Journal = {Journal of Mathematical Physics},
   Volume = {51},
   Number = {2},
   Pages = {023503},
   Year = {2010},
   ISSN = {0022-2488},
   url = {http://hdl.handle.net/10161/3308 Duke open
             access},
   Abstract = {In the final paper of this series, we extend our results on
             magnification invariants to the infinite family of A
             n(n≥2), D n(n≥4), E 6, E 7, E 8 caustic singularities.
             We prove that for families of general mappings between
             planes exhibiting any caustic singularity of the A n(n≥2),
             D n(n≥4), E 6, E 7, E 8 family, and for a point in the
             target space lying anywhere in the region giving rise to the
             maximum number of lensed images (real preimages), the total
             signed magnification of the lensed images will always sum to
             zero. The proof is algebraic in nature and relies on the
             Euler trace formula. © 2010 American Institute of
             Physics.},
   Doi = {10.1063/1.3271043},
   Key = {fds243981}
}

@article{fds243980,
   Author = {Petters, AO and Rider, B and Teguia, AM},
   Title = {A mathematical theory of stochastic microlensing. II. Random
             images, shear, and the Kac-Rice formula},
   Journal = {Journal of Mathematical Physics},
   Volume = {50},
   Number = {12},
   Pages = {122501},
   Year = {2009},
   ISSN = {0022-2488},
   url = {http://hdl.handle.net/10161/3370 Duke open
             access},
   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.},
   Doi = {10.1063/1.3267859},
   Key = {fds243980}
}

@article{fds243982,
   Author = {Petters, AO and Rider, B and Teguia, AM},
   Title = {A mathematical theory of stochastic microlensing. I. Random
             time delay functions and lensing maps},
   Journal = {Journal of Mathematical Physics},
   Volume = {50},
   Number = {7},
   Pages = {072503},
   Year = {2009},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.3158854},
   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.},
   Doi = {10.1063/1.3158854},
   Key = {fds243982}
}

@article{fds243983,
   Author = {Aazami, AB and Petters, AO},
   Title = {A universal magnification theorem. II. Generic caustics up
             to codimension five},
   Journal = {Journal of Mathematical Physics},
   Volume = {50},
   Number = {8},
   Pages = {082501},
   Year = {2009},
   Month = {Summer},
   ISSN = {0022-2488},
   url = {http://dx.doi.org/10.1063/1.3179163},
   Abstract = {We prove a theorem about magnification relations for all
             generic general caustic singularities up to codimension
             five: folds, cusps, swallowtail, elliptic umbilic,
             hyperbolic umbilic, butterfly, parabolic umbilic, wigwam,
             symbolic umbilic, second elliptic umbilic, and second
             hyperbolic umbilic. Specifically, we prove that for a
             generic family of general mappings between planes exhibiting
             any of these singularities, and for a point in the target
             lying anywhere in the region giving rise to the maximum
             number of real preimages (lensed images), the total signed
             magnification of the preimages will always sum to zero. The
             proof is algebraic in nature and makes repeated use of the
             Euler trace formula. We also prove a general algebraic
             result about polynomials, which we show yields an
             interesting corollary about Newton sums that in turn readily
             implies the Euler trace formula. The wide field imaging
             surveys slated to be conducted by the Large Synoptic Survey
             Telescope are expected to find observational evidence for
             many of these higher-order caustic singularities. Finally,
             since the results of the paper are for generic general
             mappings, not just generic lensing maps, the findings are
             expected to be applicable not only to gravitational lensing
             but also to any system in which these singularities appear.
             © 2009 American Institute of Physics.},
   Doi = {10.1063/1.3179163},
   Key = {fds243983}
}

@article{fds243984,
   Author = {Aazami, AB and Petters, AO},
   Title = {A universal magnification theorem for higher-order caustic
             singularities},
   Journal = {Journal of Mathematical Physics},
   Volume = {50},
   Number = {3},
   Pages = {032501},
   Year = {2009},
   Month = {Spring},
   ISSN = {0022-2488},
   url = {http://arxiv.org/abs/0811.3447},
   Abstract = {We prove that, independent of the choice of a lens model,
             the total signed magnification always sums to zero for a
             source anywhere in the four-image region close to
             swallowtail, elliptic umbilic, and hyperbolic umbilic
             caustics. This is a more global and higher-order analog of
             the well-known fold and cusp magnification relations, in
             which the total signed magnifications in the two-image
             region of the fold and the three-image region of the cusp
             are both always zero. As an application, we construct a
             lensing observable for the hyperbolic umbilic magnification
             relation and compare it with the corresponding observables
             for the cusp and fold relations using a singular isothermal
             ellipsoid lens. We demonstrate the greater generality of the
             hyperbolic umbilic magnification relation by showing how it
             applies to the fold image doublets and cusp image triplets
             and extends to image configurations that are neither. We
             show that the results are applicable to the study of
             substructure on galactic scales using observed quadruple
             images of lensed quasars. The magnification relations are
             also proven for generic one-parameter families of mappings
             between planes, extending their potential range of
             applicability beyond lensing. © 2009 American Institute of
             Physics.},
   Doi = {10.1063/1.3081055},
   Key = {fds243984}
}

@article{fds243965,
   Author = {Keeton, CR and Petters, AO},
   Title = {Testing theories of gravity with black hole
             lensing},
   Journal = {11th Marcel Grossmann Meeting on Recent Developments in
             Theoretical and Experimental General Relativity, Gravitation
             and Relativistic Field Theories - Proc. of the MG11 Meeting
             on General Relativity},
   Pages = {1719-1721},
   Year = {2008},
   Month = {December},
   Abstract = {The gravitational deflection of light provided one of the
             first observational confirmations of general relativity. Now
             we are considering how gravitational lensing can provide
             novel tests of Einstein's theory, and intriguing
             alternatives. We have developed a comprehensive analytical
             framework for lensing by black holes, and made concrete
             predictions that are testable with existing or planned
             instruments. Two examples: (1) In parametrized
             post-Newtonian models, there are universal relations among
             lensing observables. Observed violations of these relations
             would falsify all PPN models in one fell swoop. (2) In
             braneworld gravity, there could be many primordial black
             holes in our Solar System that would produce interference
             fringes in the energy spectra of gamma ray bursts, which
             could be detected with the GLAST satellite starting in 2007.
             © 2008 World Scientific Publishing Co. Pte.
             Ltd.},
   Key = {fds243965}
}

@article{fds243996,
   Author = {Werner, MC and Petters, AO},
   Title = {Magnification relations for Kerr lensing and testing cosmic
             censorship},
   Journal = {Physical Review D - Particles, Fields, Gravitation, and
             Cosmology},
   Volume = {76},
   Number = {6},
   Pages = {064024},
   Year = {2007},
   ISSN = {1550-7998},
   url = {http://xxx.lanl.gov/abs/0706.0132},
   Abstract = {A Kerr black hole with mass parameter m and angular momentum
             parameter a acting as a gravitational lens gives rise to two
             images in the weak field limit. We study the corresponding
             magnification relations, namely, the signed and absolute
             magnification sums and the centroid up to post-Newtonian
             order. We show that there are post-Newtonian corrections to
             the total absolute magnification and centroid proportional
             to a/m, which is in contrast to the spherically symmetric
             case where such corrections vanish. Hence we also propose a
             new set of lensing observables for the two images involving
             these corrections, which should allow measuring a/m with
             gravitational lensing. In fact, the resolution capabilities
             needed to observe this for the Galactic black hole should in
             principle be accessible to current and near-future
             instrumentation. Since a/m>1 indicates a naked
             singularity, a most interesting application would be a test
             of the cosmic censorship conjecture. The technique used to
             derive the image properties is based on the degeneracy of
             the Kerr lens and a suitably displaced Schwarzschild lens at
             post-Newtonian order. A simple physical explanation for this
             degeneracy is also given. © 2007 The American Physical
             Society.},
   Doi = {10.1103/PhysRevD.76.064024},
   Key = {fds243996}
}

@article{fds243997,
   Author = {Iyer, SV and Petters, AO},
   Title = {Light's bending angle due to black holes: From the photon
             sphere to infinity},
   Journal = {General Relativity and Gravitation},
   Volume = {39},
   Number = {10},
   Pages = {1563-1582},
   Year = {2007},
   ISSN = {0001-7701},
   url = {http://xxx.lanl.gov/abs/gr-qc/0611086},
   Abstract = {The bending angle of light is a central quantity in the
             theory of gravitational lensing. We develop an analytical
             perturbation framework for calculating the bending angle of
             light rays lensed by a Schwarzschild black hole. Using a
             perturbation parameter given in terms of the gravitational
             radius of the black hole and the light ray's impact
             parameter, we determine an invariant series for the
             strong-deflection bending angle that extends beyond the
             standard logarithmic deflection term used in the literature.
             In the process, we discovered an improvement to the standard
             logarithmic deflection term. Our perturbation framework is
             also used to derive as a consistency check, the recently
             found weak deflection bending angle series. We also
             reformulate the latter series in terms of a more natural
             invariant perturbation parameter, one that smoothly
             transitions between the weak and strong deflection series.
             We then compare our invariant strong deflection
             bending-angle series with the numerically integrated exact
             formal bending angle expression, and find less than 1%
             discrepancy for light rays as far out as twice the critical
             impact parameter. The paper concludes by showing that the
             strong and weak deflection bending angle series together
             provide an approximation that is within 1% of the exact
             bending angle value for light rays traversing anywhere
             between the photon sphere and infinity. © 2007 Springer
             Science+Business Media, LLC.},
   Doi = {10.1007/s10714-007-0481-8},
   Key = {fds243997}
}

@article{fds157981,
   Author = {C. Keeton and A.O. Petters},
   Title = {Testing Theories of Gravity with Black Hole
             Lensing},
   Booktitle = {Proceedings of the Ninth Marcel Grossmann Meeting on General
             Relativity, ed. R. Ruffini},
   Year = {2006},
   Month = {Summer},
   Key = {fds157981}
}

@article{fds243985,
   Author = {Keeton, CR and Petters, AO},
   Title = {Formalism for testing theories of gravity using lensing by
             compact objects. III. Braneworld gravity},
   Journal = {Physical Review D - Particles, Fields, Gravitation, and
             Cosmology},
   Volume = {73},
   Number = {10},
   Pages = {104032},
   Year = {2006},
   ISSN = {1550-7998},
   url = {http://xxx.lanl.gov/abs/gr-qc/0603061},
   Abstract = {Braneworld gravity is a model that endows physical space
             with an extra dimension. In the type II Randall-Sundrum
             braneworld gravity model, the extra dimension modifies the
             spacetime geometry around black holes, and changes
             predictions for the formation and survival of primordial
             black holes. We develop a comprehensive analytical formalism
             for far-field black hole lensing in this model, using
             invariant quantities to compute all the geometric optics
             lensing observables: bending angle, image position,
             magnification, centroid, and time delay. We then make the
             first analysis of wave optics in braneworld lensing, working
             in the semiclassical limit. Through quantitative examples we
             show that wave optics offers the only realistic way to
             observe braneworld effects in black hole lensing. We point
             out that if primordial braneworld black holes exist, have
             mass M•, and contribute a fraction fbh of the dark matter,
             then roughly ∼3×105×fbh(M•/10-18M)-1 of them lie
             within our Solar System. These objects, which we call
             "attolenses," would produce interference fringes in the
             energy spectra of gamma-ray bursts at energies
             E∼100(M•/10-18M)-1MeV (which will soon be accessible
             with the GLAST satellite). Primordial braneworld black holes
             spread throughout the Universe could produce similar
             interference effects. If they contribute a fraction Ω• of
             the total energy density, the probability that gamma-ray
             bursts are "attolensed" is at least ∼0.1Ω•. If
             observed, attolensing interference fringes would yield a
             simple upper limit on M•. Detection of a primordial black
             hole with M•10-19M would challenge general relativity and
             favor the braneworld model. Further work on lensing tests of
             braneworld gravity must proceed into the physical optics
             regime, which awaits a description of the full spacetime
             geometry around braneworld black holes. © 2006 The American
             Physical Society.},
   Doi = {10.1103/PhysRevD.73.104032},
   Key = {fds243985}
}

@article{fds244000,
   Author = {Keeton, CR and Petters, AO},
   Title = {Formalism for testing theories of gravity using lensing by
             compact objects. II. Probing post-post-Newtonian
             metrics},
   Journal = {Physical Review D - Particles, Fields, Gravitation, and
             Cosmology},
   Volume = {73},
   Number = {4},
   Pages = {044024},
   Year = {2006},
   ISSN = {1550-7998},
   url = {http://xxx.lanl.gov/abs/gr-qc/0601053},
   Abstract = {We study gravitational lensing by compact objects in gravity
             theories that can be written in a post-post-Newtonian (PPN)
             framework: i.e., the metric is static and spherically
             symmetric, and can be written as a Taylor series in m•/r,
             where m• is the gravitational radius of the compact
             object. Working invariantly, we compute corrections to
             standard weak-deflection lensing observables at first and
             second order in the perturbation parameter ε=•/E, where
             • is the angular gravitational radius and E is the angular
             Einstein ring radius of the lens. We show that the
             first-order corrections to the total magnification and
             centroid position vanish universally for gravity theories
             that can be written in the PPN framework. This arises from
             some surprising, fundamental relations among the lensing
             observables in PPN gravity models. We derive these relations
             for the image positions, magnifications, and time delays. A
             deep consequence is that any violation of the universal
             relations would signal the need for a gravity model outside
             the PPN framework (provided that some basic assumptions
             hold). In practical terms, the relations will guide
             observational programs to test general relativity, modified
             gravity theories, and possibly the cosmic censorship
             conjecture. We use the new relations to identify lensing
             observables that are accessible to current or near-future
             technology, and to find combinations of observables that are
             most useful for probing the spacetime metric. We give
             explicit applications to the galactic black hole,
             microlensing, and the binary pulsar J0737-3039. © 2006 The
             American Physical Society.},
   Doi = {10.1103/PhysRevD.73.044024},
   Key = {fds244000}
}

@article{fds244001,
   Author = {Keeton, C and Gaudi, S and Petters, AO},
   Title = {Identifying Lensing by Substructure II. Fold
             Lenses},
   Journal = {Astrophysical Journal},
   Volume = {635},
   Number = {1 I},
   Pages = {35},
   Year = {2005},
   Month = {November},
   ISSN = {0004-637X},
   url = {http://xxx.lanl.gov/abs/astro-ph/0503452},
   Abstract = {When the source in a four-image gravitational lens system
             lies sufficiently close to a "fold" caustic, two of the
             lensed images lie very close together. If the lens potential
             is smooth on the scale of the separation between the two
             close images, the difference between their fluxes should
             approximately vanish, Rfold = (F+ - F-)/(F+ + F-) ≈ 0.
             (The subscript indicates the image parity.) Violations of
             this "fold relation" in observed lenses are thought to
             indicate the presence of structure on scales smaller than
             the separation between the close images. We present a
             detailed study of the fold relation in realistic smooth
             lenses, finding it to be more subtle and rich than was
             previously realized. The degree to which Rfold can differ
             from zero for smooth lenses depends not only on the distance
             of the source from the caustic, but also on its location
             along the caustic, and then on the angular structure of the
             lens potential (ellipticity, multipole modes, and external
             shear). Since the source position is unobservable, it is
             impossible to say from Rfold alone whether the flux ratios
             in an observed lens are anomalous or not. Instead, we must
             consider the full distribution of Rfold values that can be
             obtained from smooth lens potentials that reproduce the
             separation d1 between the two close images and the distance
             d2 to the next nearest image. (By reducing the image
             configuration to these two numbers, we limit our model
             dependence and obtain a generic analysis.) We show that the
             generic features of this distribution can be understood,
             which means that the fold relation provides a robust probe
             of small-scale structure in lens galaxies. We then compute
             the full distribution using Monte Carlo simulations of
             realistic smooth lenses. Comparing these predictions with
             the data, we find that five of the 12 known lenses with fold
             configurations have flux ratio anomalies: B0712+472, SDSS
             0924+0219, PG 1115+080, B1555+375, and B1933+503. Combining
             this with our previous analysis revealing anomalies in three
             of the four known lenses with cusp configurations, we
             conclude that at least half (8/16) of all four-image lenses
             that admit generic, local analyses exhibit flux ratio
             anomalies. The fold and cusp relations do not reveal the
             nature of the implied small-scale structure, but do provide
             the formal foundation for substructure studies, and also
             indicate which lenses deserve further study. Although our
             focus is on close pairs of images, we show that the fold
             relation can be used - with great care - to analyze all
             image pairs in all 22 known four-image lenses and reveal
             lenses with some sort of interesting structure. © 2005. The
             American Astronomical Society. All rights
             reserved.},
   Doi = {10.1086/497324},
   Key = {fds244001}
}

@article{fds244002,
   Author = {Keeton, CR and Petters, AO},
   Title = {Formalism for testing theories of gravity using lensing by
             compact objects: Static, spherically symmetric
             case},
   Journal = {Physical Review D - Particles, Fields, Gravitation, and
             Cosmology},
   Volume = {72},
   Number = {10},
   Pages = {104006},
   Year = {2005},
   ISSN = {1550-7998},
   url = {http://xxx.lanl.gov/abs/gr-qc/0511019},
   Abstract = {We are developing a general, unified, and rigorous
             analytical framework for using gravitational lensing by
             compact objects to test different theories of gravity beyond
             the weak-deflection limit. In this paper we present the
             formalism for computing corrections to lensing observables
             for static, spherically symmetric gravity theories in which
             the corrections to the weak-deflection limit can be expanded
             as a Taylor series in one parameter, namely, the
             gravitational radius of the lens object. We take care to
             derive coordinate-independent expressions and compute
             quantities that are directly observable. We compute series
             expansions for the observables that are accurate to second
             order in the ratio ε= •/ E of the angle subtended by the
             lens's gravitational radius to the weak-deflection Einstein
             radius, which scales with mass as ε M•1/2. The positions,
             magnifications, and time delays of the individual images
             have corrections at both first and second order in ε, as
             does the differential time delay between the two images.
             Interestingly, we find that the first-order corrections to
             the total magnification and centroid position vanish in all
             gravity theories that agree with general relativity in the
             weak-deflection limit, but they can remain nonzero in
             modified theories that disagree with general relativity in
             the weak-deflection limit. For the Reissner-Nordström
             metric and a related metric from heterotic string theory,
             our formalism reveals an intriguing connection between
             lensing observables and the condition for having a naked
             singularity, which could provide an observational method for
             testing the existence of such objects. We apply our
             formalism to the galactic black hole and predict that the
             corrections to the image positions are at the level of
             10μarcs (microarcseconds), while the correction to the time
             delay is a few hundredths of a second. These corrections
             would be measurable today if a pulsar were found to be
             lensed by the galactic black hole, and they should be
             readily detectable with planned missions like MAXIM. © 2005
             The American Physical Society.},
   Doi = {10.1103/PhysRevD.72.104006},
   Key = {fds244002}
}

@article{fds305708,
   Author = {Keeton, CR and Gaudi, BS and Petters, AO},
   Title = {Identifying lenses with small-scale structure. II. Fold
             lenses},
   Journal = {The Astrophysical Journal},
   Volume = {635},
   Number = {1 I},
   Pages = {35-59},
   Year = {2005},
   ISSN = {0004-637X},
   url = {http://dx.doi.org/10.1086/497324},
   Abstract = {When the source in a four-image gravitational lens system
             lies sufficiently close to a "fold" caustic, two of the
             lensed images lie very close together. If the lens potential
             is smooth on the scale of the separation between the two
             close images, the difference between their fluxes should
             approximately vanish, Rfold = (F+ - F-)/(F+ + F-) ≈ 0.
             (The subscript indicates the image parity.) Violations of
             this "fold relation" in observed lenses are thought to
             indicate the presence of structure on scales smaller than
             the separation between the close images. We present a
             detailed study of the fold relation in realistic smooth
             lenses, finding it to be more subtle and rich than was
             previously realized. The degree to which Rfold can differ
             from zero for smooth lenses depends not only on the distance
             of the source from the caustic, but also on its location
             along the caustic, and then on the angular structure of the
             lens potential (ellipticity, multipole modes, and external
             shear). Since the source position is unobservable, it is
             impossible to say from Rfold alone whether the flux ratios
             in an observed lens are anomalous or not. Instead, we must
             consider the full distribution of Rfold values that can be
             obtained from smooth lens potentials that reproduce the
             separation d1 between the two close images and the distance
             d2 to the next nearest image. (By reducing the image
             configuration to these two numbers, we limit our model
             dependence and obtain a generic analysis.) We show that the
             generic features of this distribution can be understood,
             which means that the fold relation provides a robust probe
             of small-scale structure in lens galaxies. We then compute
             the full distribution using Monte Carlo simulations of
             realistic smooth lenses. Comparing these predictions with
             the data, we find that five of the 12 known lenses with fold
             configurations have flux ratio anomalies: B0712+472, SDSS
             0924+0219, PG 1115+080, B1555+375, and B1933+503. Combining
             this with our previous analysis revealing anomalies in three
             of the four known lenses with cusp configurations, we
             conclude that at least half (8/16) of all four-image lenses
             that admit generic, local analyses exhibit flux ratio
             anomalies. The fold and cusp relations do not reveal the
             nature of the implied small-scale structure, but do provide
             the formal foundation for substructure studies, and also
             indicate which lenses deserve further study. Although our
             focus is on close pairs of images, we show that the fold
             relation can be used - with great care - to analyze all
             image pairs in all 22 known four-image lenses and reveal
             lenses with some sort of interesting structure. © 2005. The
             American Astronomical Society. All rights
             reserved.},
   Doi = {10.1086/497324},
   Key = {fds305708}
}

@article{fds243998,
   Author = {Keeton, C and Gaudi, S and Petters, AO},
   Title = {Identifying Lensing by Substructure I. Cusp
             Lenses},
   Journal = {Astrophys. J.},
   Volume = {598},
   Number = {1 I},
   Pages = {138},
   Year = {2003},
   ISSN = {0004-637X},
   url = {http://xxx.lanl.gov/abs/astro-ph/0210318},
   Abstract = {The inability of standard models to explain the flux ratios
             in many four-image gravitational lens systems has been
             presented as evidence for significant small-scale structure
             in lens galaxies. That claim has generally relied on
             detailed lens modeling, so it is both model dependent and
             somewhat difficult to interpret. We present a more robust
             and generic method for identifying lenses with small-scale
             structure. For a close triplet of images created when the
             source lies near an ideal cusp catastrophe, the sum of the
             signed magnifications should exactly vanish, independent of
             any global properties of the lens potential. For realistic
             cusps, the magnification sum vanishes only approximately,
             but we show that it is possible to place strong upper bounds
             on the degree to which the magnification sum can deviate
             from zero. Lenses with flux ratio "anomalies," or fluxes
             that significantly violate the upper bounds, can be said
             with high confidence to have structure in the lens potential
             on scales of the image separation or smaller. Five observed
             lenses have such flux ratio anomalies: B2045+265 has a
             strong anomaly at both radio and optical/near-IR
             wavelengths; B0712+472 has a strong anomaly at
             optical/near-IR wavelengths and a marginal anomaly at radio
             wavelengths; 1RXS J1131-1231 has a strong anomaly at optical
             wavelengths; RX J0911+0551 appears to have an anomaly at
             optical/near-IR wavelengths, although the conclusion in this
             particular lens is subject to uncertainties in the typical
             strength of octopole density perturbations in early-type
             galaxies; and finally, SDSS J0924+0219 has a strong anomaly
             at optical wavelengths. Interestingly, analysis of the cusp
             relation does not reveal a significant anomaly in B1422+231,
             even though this lens is known to be anomalous from detailed
             modeling. Methods that are more sophisticated (and less
             generic) than the cusp relation may therefore be necessary
             to uncover flux ratio anomalies in some systems. Although
             these flux ratio anomalies might represent either
             millilensing or microlensing, we cannot identify the cause
             of the anomalies using only broadband flux ratios in
             individual lenses. Rather, the conclusion we can draw is
             that the lenses have significant structure in the lens
             potential on scales comparable to or smaller than the
             separation between the images. Additional arguments must be
             invoked to specify the nature of this small-scale
             structure.},
   Doi = {10.1086/378934},
   Key = {fds243998}
}

@article{fds244003,
   Author = {Petters, AO},
   Title = {On relativistic corrections to microlensing effects:
             Applications to the Galactic black hole},
   Journal = {Monthly Notices of the Royal Astronomical
             Society},
   Volume = {338},
   Number = {2},
   Pages = {457-464},
   Year = {2003},
   url = {http://xxx.lanl.gov/ps/astro-ph/0208500},
   Abstract = {The standard treatment of gravitational lensing by a point
             mass lens M is based on a weak-field deflection angle α̂ =
             2/xo, where X0 = r0c/2GM with r0 being the distance of
             closest approach to the mass of a lensed light ray. It was
             shown that for a point mass lens, the total magnification
             and image centroid shift of a point source remain unchanged
             by relativistic corrections of second order in 1/X 0. This
             paper considers these issues analytically, taking into
             account the relativistic images, under three assumptions
             A1-A3, for a Schwarzschild black hole lens with a background
             point and extended sources having arbitrary surface
             brightness profiles. The assumptions are A1, the source is
             close to the line of sight and lies in the asymptotically
             flat region outside the black hole lens; A2, the
             observer-lens and lens-source distances are significantly
             greater than the impact parameters of the lensed light rays;
             and A3, the distance of closest approach of any light ray
             that does not wind around the black hole on its travel from
             the source to the observer, lies in the weak-field regime
             outside the black hole. We apply our results to the Galactic
             black hole for lensing scenarios where A1-A3 hold. We show
             that a single factor characterizes the full relativistic
             correction to the weak-field image centroid and
             magnification. As the lens-source distance increases, the
             relativistic correction factor strictly decreases. In
             particular, we find that for point and extended sources
             approximately 10 pc behind the black hole, which is a
             distance significantly outside the tidal disruption radius
             of a Sun-like source, the relativistic correction factor is
             minuscule, of the order of 10 -14. Therefore, for standard
             lensing configurations, any detectable relativistic
             corrections to microlensing by the Galactic black hole will
             most likely have to come from sources significantly closer
             to the black hole.},
   Doi = {10.1046/j.1365-8711.2003.06065.x},
   Key = {fds244003}
}

@article{fds305707,
   Author = {Keeton, CR and Gaudi, BS and Petters, AO},
   Title = {Identifying lenses with small-scale structure. I. Cusp
             lenses},
   Journal = {The Astrophysical Journal},
   Volume = {598},
   Number = {1 I},
   Pages = {138-161},
   Year = {2003},
   ISSN = {0004-637X},
   url = {http://dx.doi.org/10.1086/378934},
   Abstract = {The inability of standard models to explain the flux ratios
             in many four-image gravitational lens systems has been
             presented as evidence for significant small-scale structure
             in lens galaxies. That claim has generally relied on
             detailed lens modeling, so it is both model dependent and
             somewhat difficult to interpret. We present a more robust
             and generic method for identifying lenses with small-scale
             structure. For a close triplet of images created when the
             source lies near an ideal cusp catastrophe, the sum of the
             signed magnifications should exactly vanish, independent of
             any global properties of the lens potential. For realistic
             cusps, the magnification sum vanishes only approximately,
             but we show that it is possible to place strong upper bounds
             on the degree to which the magnification sum can deviate
             from zero. Lenses with flux ratio "anomalies," or fluxes
             that significantly violate the upper bounds, can be said
             with high confidence to have structure in the lens potential
             on scales of the image separation or smaller. Five observed
             lenses have such flux ratio anomalies: B2045+265 has a
             strong anomaly at both radio and optical/near-IR
             wavelengths; B0712+472 has a strong anomaly at
             optical/near-IR wavelengths and a marginal anomaly at radio
             wavelengths; 1RXS J1131-1231 has a strong anomaly at optical
             wavelengths; RX J0911+0551 appears to have an anomaly at
             optical/near-IR wavelengths, although the conclusion in this
             particular lens is subject to uncertainties in the typical
             strength of octopole density perturbations in early-type
             galaxies; and finally, SDSS J0924+0219 has a strong anomaly
             at optical wavelengths. Interestingly, analysis of the cusp
             relation does not reveal a significant anomaly in B1422+231,
             even though this lens is known to be anomalous from detailed
             modeling. Methods that are more sophisticated (and less
             generic) than the cusp relation may therefore be necessary
             to uncover flux ratio anomalies in some systems. Although
             these flux ratio anomalies might represent either
             millilensing or microlensing, we cannot identify the cause
             of the anomalies using only broadband flux ratios in
             individual lenses. Rather, the conclusion we can draw is
             that the lenses have significant structure in the lens
             potential on scales comparable to or smaller than the
             separation between the images. Additional arguments must be
             invoked to specify the nature of this small-scale
             structure.},
   Doi = {10.1086/378934},
   Key = {fds305707}
}

@article{fds243971,
   Author = {Gaudi, BS and Petters, AO},
   Title = {Gravitational microlensing near caustics. I.
             Folds},
   Journal = {The Astrophysical Journal},
   Volume = {574},
   Number = {2 I},
   Pages = {970-984},
   Year = {2002},
   ISSN = {0004-637X},
   url = {http://xxx.lanl.gov/abs/astro-ph/0112531},
   Abstract = {We study the local behavior of gravitational lensing near
             fold catastrophes. Using a generic form for the lensing map
             near a fold, we determine the observable properties of the
             lensed images, focusing on the case in which the individual
             images are unresolved, i.e., microlensing. Allowing for
             images not associated with the fold, we derive analytic
             expressions for the photometric and astrometric behavior
             near a generic fold caustic. We show how this form reduces
             to the more familiar linear caustic, which lenses a nearby
             source into two images that have equal magnification,
             opposite parity, and are equidistant from the critical
             curve. In this case, the simplicity and high degree of
             symmetry allow for the derivation of semianalytic
             expressions for the photometric and astrometric deviations
             in the presence of finite sources with arbitrary surface
             brightness profiles. We use our results to derive some basic
             properties of astrometric microlensing near folds; in
             particular, we predict, for finite sources with uniform and
             limb-darkening profiles, the detailed shape of the
             astrometric curve as the source crosses a fold. We find that
             the astrometric effects of limb darkening will be difficult
             to detect with the currently planned accuracy of the Space
             Interferometry Mission for Galactic bulge sources; however,
             this also implies that astrometric measurements of other
             parameters, such as the size of the source, should not be
             compromised by an unknown amount of limb darkening. We
             verify our results by numerically calculating the expected
             astrometric shift for the photometrically well-covered
             Galactic binary lensing event OGLE-1999-BUL-23, finding
             excellent agreement with our analytic expressions. Our
             results can be applied to any lensing system with fold
             caustics, including Galactic binary lenses and quasar
             microlensing.},
   Doi = {10.1086/341063},
   Key = {fds243971}
}

@article{fds243972,
   Author = {Gaudi, BS and Petters, AO},
   Title = {Gravitational microlensing near caustics. II.
             Cusps},
   Journal = {The Astrophysical Journal},
   Volume = {580},
   Number = {1 I},
   Pages = {468-489},
   Year = {2002},
   ISSN = {0004-637X},
   url = {http://arxiv.org/abs/astro-ph/0206162v2},
   Abstract = {We present a rigorous, detailed study of the generic,
             quantitative properties of gravitational lensing near cusp
             catastrophes. Concentrating on the case in which the
             individual images are unresolved, we derive explicit
             formulas for the total magnification and centroid of the
             images created for sources outside, on, and inside the
             cusped caustic. We obtain new results on how the image
             magnifications scale with respect to separation from the
             cusped caustic for arbitrary source positions. Along the
             axis of symmetry of the cusp, the total magnification μ
             scales as μ α u-1, where u is the distance of the source
             from the cusp, whereas perpendicular to this axis, μ α
             u-2/3. When the source passes through a point u 0 on a fold
             arc abutting the cusp, the image centroid has a jump
             discontinuity; we present a formula for the size of the jump
             in terms of the local derivatives of the lens potential and
             show that the magnitude of the jump scales as |u10|1/2 for
             |u10| ≪ 1, where |u10| is the horizontal distance between
             u0 and the cusp. The total magnifications for a small
             extended source located both on and perpendicular to the
             axis of symmetry are also derived, for both uniform and
             limb-darkened surface brightness profiles. We find that the
             difference in magnification between a finite and point
             source is ≲5% for separations of ≲2.5 source radii from
             the cusp point, while the effect of limb darkening is ≲1%
             in the same range. Our predictions for the astrometric and
             photometric behavior of both pointlike and finite sources
             passing near a cusp are illustrated and verified using
             numerical simulations of the cusp-crossing Galactic binary
             lens event MACHO-1997-BUL-28. Our results can be applied to
             any microlensing system with cusp caustics, including
             Galactic binary lenses and quasar microlensing; we discuss
             several possible applications of our results to these
             topics.},
   Doi = {10.1086/343114},
   Key = {fds243972}
}

@article{fds243999,
   Author = {Frittelli, S and Petters, AO},
   Title = {Wavefronts, caustic sheets, and caustic surfing in
             gravitational lensing},
   Journal = {Journal of Mathematical Physics},
   Volume = {43},
   Number = {11},
   Pages = {5578-5611},
   Year = {2002},
   ISSN = {0022-2488},
   url = {http://xxx.lanl.gov/abs/astro-ph/0208135},
   Abstract = {Very little attention has been paid to the properties of
             optical wavefronts and caustic surfaces due to gravitational
             lensing. Yet the wavefront-based point of view is natural
             and provides insights into the nature of the caustic
             surfaces on a gravitationally lensed lightcone. We derive
             analytically the basic equations governing the wavefronts,
             lightcones, caustics on wavefronts, and caustic surfaces on
             lightcones in the context of weak-field, thin-screen
             gravitational lensing. These equations are all related to
             the potential of the lens. In the process, we also show that
             the standard single-plane gravitational lensing map extends
             to a new mapping, which we call a wavefront lensing map.
             Unlike the standard lensing map, the Jacobian matrix of a
             wavefront lensing map is not symmetric. Our formulas are
             then applied to caustic "surfing." By surfing a caustic
             surface, a space-borne telescope can be fixed on a
             gravitationally lensed source to obtain an observation of
             the source at very high magnification over an extended time
             period, revealing structure about the source that could not
             otherwise be resolved. Using our analytical expressions for
             caustic sheets, we present a scheme for surfing a caustic
             sheet of a lensed source in rectilinear motion. Detailed
             illustrations are also presented of the possible types of
             wavefronts and caustic sheets due to nonsingular and
             singular elliptical potentials, and singular isothermal
             spheres, including an example of caustic surfing for a
             singular elliptical potential lens. © 2002 American
             Institute of Physics.},
   Doi = {10.1063/1.1511790},
   Key = {fds243999}
}

@article{fds9789,
   Author = {A. O. Petters},
   Title = {Stable Lens Systems, Lensed Image Magnification,and
             Magnification Cross Sections},
   Journal = {Proceedings of the Ninth Marcel Grossmann Meeting on General
             Relativity, eds. V. Gurzadyan, R. T. Jantzen, and R.
             Ruffini},
   Publisher = {World Scientific},
   Address = {Singapore},
   Year = {2001},
   Key = {fds9789}
}

@article{fds9790,
   Author = {S. Frittelli and A. O. Petters},
   Title = {Wavefront Singularities due to an Elliptical
             Potential},
   Journal = {Proceedings of the Ninth Marcel Grossmann Meeting on General
             Relativity, eds. V. Gurzadyan, R. T. Jantzen, and R.
             Ruffini},
   Publisher = {World Scientific},
   Address = {Singapore},
   Year = {2001},
   Key = {fds9790}
}

@article{fds9791,
   Author = {B. S. Gaudi and A. O. Petters},
   Title = {Center of Light Curves for Whitney Fold and
             Cusp},
   Journal = {Proceedings of the Ninth Marcel Grossmann Meeting on General
             Relativity, eds. V. Gurzadyan, R. T. Jantzen, and R. Ruffini
             (World Scientific,Singapore)},
   Year = {2001},
   Key = {fds9791}
}

@article{fds243970,
   Author = {Petters, AO and Wicklin, FJ},
   Title = {Fixed points due to gravitational lenses},
   Journal = {Journal of Mathematical Physics},
   Volume = {39},
   Number = {2},
   Pages = {1011-1023},
   Year = {1998},
   Abstract = {A fixed point of a gravitational lensing map represents
             those positions from which a pointlike light source has a
             lensed image that, despite gravitational lensing,
             corresponds to the original position of the source. In this
             paper we study fixed points of lensing maps due to a generic
             gravitational lens with applications to nonsingular isolated
             lenses, and to point-mass lenses with continuous matter and
             shear. Counting formulas and bounds on the number of fixed
             points are determined. The results include an odd-number
             fixed-point theorem for nonsingular isolated deflectors.
             Information on the positions of fixed points are found for
             the case of point masses on a lens plane with and without
             shear. The methods of the paper are based on Morse theory,
             complex variables, and resultants. © 1998 American
             Institute of Physics.},
   Key = {fds243970}
}

@article{fds9006,
   Author = {A. O. Petters and F.J. Wicklin},
   Title = {Counting Formulas and Bounds on Number of Fixed Points Due
             to Point-Mass Lenses},
   Journal = {Proceedings of the Eighth Marcel Grossmann Meeting on
             General Relativity 1997, ed. R. Ruffini (World Scientific,
             Singapore)},
   Year = {1997},
   Month = {Summer},
   Key = {fds9006}
}

@article{fds9007,
   Author = {S. Mao and A. O. Petters and H. Witt},
   Title = {Properties of Point Mass Lenses on a Regular Polygon and the
             Problem of Maximum Number of Lensed Images},
   Journal = {in Proceedings of the Eighth Marcel Grossman Meeting on
             General Relativity, ed. R. Ruffini (World Scientific,
             Singapore)},
   Year = {1997},
   Month = {Summer},
   Key = {fds9007}
}

@article{fds9008,
   Author = {A. O. Petters},
   Title = {Some Global Results on Gravitational Lensing},
   Journal = {Proceedings of the Eight Marcel Grossman Meeting on General
             Relativity, ed. R. Ruffini (World Scientific,
             Singapore)},
   Year = {1997},
   Month = {Summer},
   Key = {fds9008}
}

@article{fds243969,
   Author = {Petters, AO},
   Title = {Curvature of caustics and singularities of gravitational
             lenses},
   Journal = {Nonlinear Analysis: Theory, Methods & Applications},
   Volume = {30},
   Number = {1},
   Pages = {627-634},
   Year = {1997},
   ISSN = {0362-546X},
   Key = {fds243969}
}

@article{fds243988,
   Author = {Petters, AO},
   Title = {Multiplane gravitational lensing. III. Upper bound on number
             of images},
   Journal = {Journal of Mathematical Physics},
   Volume = {38},
   Number = {3},
   Pages = {1605-1613},
   Year = {1997},
   Abstract = {The total number of lensed images of a light source
             undergoing gravitational lensing varies as the source
             traverses a caustic network. It is rigorously shown that for
             a pointlike light source not on any caustic, a
             three-dimensional distribution of g point masses on g lens
             planes creates at most 2(22(g-1)-1) lensed images of the
             source (g≥2). This complements previous work [Paper I, J.
             Math. Phys. 36, 4263 (1995)] that showed at least 2g lensed
             images occur. Application of the upper bound to the global
             geometry of caustics is also presented. Our methods are
             based on a complex formulation of point-mass gravitational
             lensing and techniques from the theory of resultants. The
             latter yields a new approach to studying upper bounds on
             number of lensed images due to point-mass gravitational lens
             systems. © 1997 American Institute of Physics.},
   Key = {fds243988}
}

@article{fds9001,
   Author = {A. O. Petters},
   Title = {Mathematical Aspects of Gravitational Lensing},
   Journal = {Proceedings of the Seventh Marcel Grossman Meeting on
             General Relativity, Vol. B, eds. R.T. Jantzen and G.M.
             Keiser (World Scientific, Singapore)},
   Year = {1996},
   Key = {fds9001}
}

@article{fds243989,
   Author = {Petters, AO and Witt, H},
   Title = {Bounds on Number of Cusps Due to Point Mass Gravitional
             Lenses with Continuous Matter and Shear},
   Journal = {J. Math. Phys.},
   Volume = {37},
   Number = {2920},
   Pages = {2920-2933},
   Year = {1996},
   url = {http://dx.doi.org/10.1063/1.531630},
   Abstract = {Generic caustics in gravitational lensing occur locally
             either as folds or cusps. This paper rigorously proves that
             the total number of cusps, Ncusps, due to g point masses on
             a single plane having non-normalized external shear γ>0
             and continuous matter with constant density σc, is bounded
             as follows: 0≤Ncusps≤12g2. For vanishing shear γ=0 we
             obtain the result 0≤Ncusps≤12g(g-1). Consequences of
             these bounds for the global geometry of caustics are
             discussed. It is also shown that if γ≥0 and σc is
             sufficiently large, then all cusps can be eliminated, that
             is, Ncusps=0. The paper also includes equations for
             calculating all the bi-caustics (i.e., curves yielding the
             positions of cusps during a one-parameter evolution) of a
             single point-mass lens with continuous matter and shear. The
             methods of the paper are based on a new approach to
             point-mass gravitational lensing using complex quantities
             and the theory of resultants. © 1996 American Institute of
             Physics.},
   Doi = {10.1063/1.531630},
   Key = {fds243989}
}

@article{fds243990,
   Author = {Petters, AO and Wicklin, FJ},
   Title = {New Caustic Phenomena for Caustics in Double-Plane
             Lensing},
   Journal = {Astrophysical Applications of Gravitational Lensing, eds.
             C.S. Kochanek and J.N. Hewitt (Klumer Academic,
             Dordrecht)},
   Year = {1996},
   Key = {fds243990}
}

@article{fds243991,
   Author = {Petters, AO},
   Title = {A cusp - Counting formula for caustics due to multiplane
             gravitational lensing},
   Journal = {Iau Symposia},
   Number = {173},
   Pages = {281-282},
   Year = {1996},
   ISSN = {0074-1809},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1996BF25F00078&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Key = {fds243991}
}

@article{fds243992,
   Author = {Fetters, AO},
   Title = {Lower bounds on image magnification in gravitational
             lensing},
   Journal = {Proceedings of the Royal Society of London: Mathematical,
             Physical and Engineering Sciences},
   Volume = {452},
   Number = {1949},
   Pages = {1475-1490},
   Year = {1996},
   ISSN = {1364-5021},
   Abstract = {A rigorous study of lower bounds on image magnification in
             single-plane gravitational lensing is presented. These
             bounds are determined for the total magnification of point
             sources undergoing lensing by a general single-plane
             gravitational lens. The lower bounds are expressed as a
             function of the number of images of the source, the number
             of obstruction points of the deflector potential, and mass
             density of the lens. In particular, our lower bounds adjust
             according to the multiplicity of the region of the caustic
             network where the lensed source is located. The results for
             the general lens are then used to find lower bounds on the
             total magnification due to non-singular and singular
             deflectors. The latter lenses are considered in detail for
             the cases of point-mass deflectors with shear and continuous
             matter (subcritical, strongly sheared, supercritical, and
             critical cases). Automatic with this study are general
             results on image counting and a discussion of the
             magnifications and trajectories of images of a lensed source
             as the source moves to 'infinity'. The paper uses
             Morsetheoretic tools, yielding a new approach to the study
             of lower bounds on image magnification in gravitational
             lensing. © 1996 The Royal Society.},
   Key = {fds243992}
}

@article{fds305706,
   Author = {Petters, AO and Witt, HJ},
   Title = {Bounds on number of cusps due to point mass gravitational
             lenses},
   Journal = {Journal of Mathematical Physics},
   Volume = {37},
   Number = {6},
   Pages = {2920-2933},
   Year = {1996},
   url = {http://dx.doi.org/10.1063/1.531630},
   Abstract = {Generic caustics in gravitational lensing occur locally
             either as folds or cusps. This paper rigorously proves that
             the total number of cusps, Ncusps, due to g point masses on
             a single plane having non-normalized external shear γ>0
             and continuous matter with constant density σc, is bounded
             as follows: 0≤Ncusps≤12g2. For vanishing shear γ=0 we
             obtain the result 0≤Ncusps≤12g(g-1). Consequences of
             these bounds for the global geometry of caustics are
             discussed. It is also shown that if γ≥0 and σc is
             sufficiently large, then all cusps can be eliminated, that
             is, Ncusps=0. The paper also includes equations for
             calculating all the bi-caustics (i.e., curves yielding the
             positions of cusps during a one-parameter evolution) of a
             single point-mass lens with continuous matter and shear. The
             methods of the paper are based on a new approach to
             point-mass gravitational lensing using complex quantities
             and the theory of resultants. © 1996 American Institute of
             Physics.},
   Doi = {10.1063/1.531630},
   Key = {fds305706}
}

@article{fds243993,
   Author = {Petters, AO and Wicklin, FJ},
   Title = {Caustics of the double-plane two-point-mass gravitational
             lens with continuous matter and shear},
   Journal = {Monthly Notices of the Royal Astronomical
             Society},
   Volume = {277},
   Number = {4},
   Pages = {1399-1403},
   Year = {1995},
   Month = {Summer},
   ISSN = {0035-8711},
   url = {http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995TL36100019&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=47d3190e77e5a3a53558812f597b0b92},
   Key = {fds243993}
}

@article{fds243994,
   Author = {Petters, AO},
   Title = {Multiplane gravitational lensing. II. Global geometry of
             caustics},
   Journal = {Journal of Mathematical Physics},
   Volume = {36},
   Number = {8},
   Pages = {4276-4295},
   Year = {1995},
   ISSN = {0022-2488},
   Abstract = {The global geometry of caustics due to a general multiplane
             gravitational lens system is investigated. Cusp-counting
             formulas and total curvatures are determined for individual
             caustics as well as whole caustic networks. The notion of
             light path obstruction points is fundamental in these
             studies. Lower bounds are found for such points and are used
             to get upper bounds for the total curvature. Curvature
             functions of caustics are also treated. All theorems
             obtained do not rely on the detailed nature of any specific
             potential assumed as a gravitational lens model, but on the
             overall differential-topological properties of general
             potentials. The methods employed are based on the following:
             Morse theory, projectivized rotation numbers, the
             Fabricius-Bjerre-Halpern formula, Whitney's rotation number
             formula, Seifert decompositions, and the Gauss-Bonnet
             theorem. © 1995 American Institute of Physics.},
   Key = {fds243994}
}

@article{fds243995,
   Author = {Petters, AO},
   Title = {Multiplane gravitational lensing. I. Morse theory and image
             counting},
   Journal = {Journal of Mathematical Physics},
   Volume = {36},
   Number = {8},
   Pages = {4263-4275},
   Year = {1995},
   ISSN = {0022-2488},
   Abstract = {The image counting problem for gravitational lensing by
             general matter deflectors distributed over finitely many
             lens planes is considered. Counting formulas and lower
             bounds are found via Morse theory for the number of images
             of a point source not on a caustic. Images are counted
             within a compact region D not necessarily assumed to
             properly contain the deflector space. In addition, it is
             shown that Morse theory is applicable because multiplane
             time-delay maps Ty generically satisfy the Morse boundary
             conditions relative to D. All results obtained depend only
             on the topological properties induced in the lens planes by
             the deflector potentials and the behavior of grad Ty at
             boundary points of D. © 1995 American Institute of
             Physics.},
   Key = {fds243995}
}

@article{fds8995,
   Author = {H. Levine and A. O. Petters},
   Title = {Singularities and Gravitational Lensing},
   Journal = {Passion des Formes: Hommage a Rene Thom,Vol. 1 (M. Porte,
             ed.) E.N.S. Edition, Fontenany-St Cloud},
   Year = {1994},
   Key = {fds8995}
}

@article{fds243966,
   Author = {Witt, HJ and Petters, AO},
   Title = {Singularities of the one- and two-point mass gravitational
             lens},
   Journal = {Journal of Mathematical Physics},
   Volume = {34},
   Number = {9},
   Pages = {4093-4111},
   Year = {1993},
   ISSN = {0022-2488},
   Abstract = {A detailed study of when a change in the number of caustics
             and cusps occurs for one- and two-point mass gravitational
             lens lying on a single plane with continuously distributed
             matter and an external shear are presented herein. The
             equations for the positions of the cusps generated by such
             lens systems are investigated in detail. This method is
             based on a new approach using complex quantities and applies
             recent results on cusp counting. © 1993 American Institute
             of Physics.},
   Key = {fds243966}
}

@article{fds243987,
   Author = {Petters, AO},
   Title = {Arnold's singularity theory and gravitational
             lensing},
   Journal = {Journal of Mathematical Physics},
   Volume = {34},
   Number = {8},
   Pages = {3555-3581},
   Year = {1993},
   ISSN = {0022-2488},
   Abstract = {Caustics in gravitational lensing are formulated from a
             symplectic geometric viewpoint. Arnold's singularity theory
             is then used to give a rigorous local classification of
             generic gravitational lensing caustics and their evolutions.
             A local classification is also presented of generic image
             surfaces, time-delay image surfaces, big caustics, and
             bicaustics. The results of each classification are discussed
             and graphically illustrated. © 1993 American Institute of
             Physics.},
   Key = {fds243987}
}

@article{fds244004,
   Author = {Levine, H and Petters, AO},
   Title = {New Caustic Singularities in Multiple Lens Plane
             Gravitational Lensing},
   Journal = {Astron. Astrophys.},
   Volume = {272},
   Number = {L17},
   Year = {1993},
   Key = {fds244004}
}

@article{fds244005,
   Author = {Levine, H and Petters, AO and Wambsganss, J},
   Title = {Applications of Singularity Theory to Gravitational
             Lensing},
   Journal = {J. Math. Phys.},
   Volume = {34},
   Number = {10},
   Pages = {4781},
   Year = {1993},
   ISSN = {0022-2488},
   Abstract = {The basic local and global features of stable multiple plane
             gravitational lens systems are investigated using tools from
             singularity theory. All stable multiple plane time-delay and
             lensing maps are classified, and the following global facts
             are proven under the weaker assumption of local stability.
             First, every locally stable multiple plane lensing map has
             an even number of cusps whether the associated deflector is
             singular or not. Second, for nonsingular deflectors the sum
             of the projectivized rotation numbers of its caustics is
             zero, while for singular ones it is negative and even.
             Third, if the deflector has g point masses on a single
             plane, then g is given by the formula g=-1/2Σcr(c), where
             r(c) is the projectivized rotation number of the critical
             curve c and the sum runs through all critical curves.
             Fourth, explicit counting formulas and bounds are found for
             the number of cusps for certain caustic networks. Finally,
             the latter yields that two point masses on a single lens
             plane will generate at least six cusps. However, if the
             masses are put genetically on separate lens planes, then
             there are at least eight cusps. © 1993 American Institute
             of Physics.},
   Key = {fds244005}
}

@article{fds305705,
   Author = {Levine, HI and Petters, AO and Wambsganss, J},
   Title = {Applications of singularity theory to gravitational lensing.
             I. Multiple lens planes},
   Journal = {Journal of Mathematical Physics},
   Volume = {34},
   Number = {10},
   Pages = {4781-4808},
   Year = {1993},
   ISSN = {0022-2488},
   Abstract = {The basic local and global features of stable multiple plane
             gravitational lens systems are investigated using tools from
             singularity theory. All stable multiple plane time-delay and
             lensing maps are classified, and the following global facts
             are proven under the weaker assumption of local stability.
             First, every locally stable multiple plane lensing map has
             an even number of cusps whether the associated deflector is
             singular or not. Second, for nonsingular deflectors the sum
             of the projectivized rotation numbers of its caustics is
             zero, while for singular ones it is negative and even.
             Third, if the deflector has g point masses on a single
             plane, then g is given by the formula g=-1/2Σcr(c), where
             r(c) is the projectivized rotation number of the critical
             curve c and the sum runs through all critical curves.
             Fourth, explicit counting formulas and bounds are found for
             the number of cusps for certain caustic networks. Finally,
             the latter yields that two point masses on a single lens
             plane will generate at least six cusps. However, if the
             masses are put genetically on separate lens planes, then
             there are at least eight cusps. © 1993 American Institute
             of Physics.},
   Key = {fds305705}
}

@article{fds9374,
   Author = {A. O. Petters and D. Spergel},
   Title = {An Analytical Approach to Quasar Variability due to
             Microlensing},
   Journal = {Gravitational Lenses, eds. R. Kayser, T. Schramm, and L.
             Nieser (Lecture Notes in Physics, 406, Springer,
             Ber1in)},
   Year = {1992},
   Key = {fds9374}
}

@article{fds9375,
   Author = {A. O. Petters},
   Title = {Morse Theory and Gravitational Microlensing},
   Journal = {Gravitational Lenses, eds. R. Kayser, T. Schramm, and L.
             Nieser (Lecture Notes in Physics 406, Springer,
             Ber1in)},
   Year = {1992},
   Key = {fds9375}
}

@article{fds243986,
   Author = {Petters, AO},
   Title = {Morse theory and gravitational microlensing},
   Journal = {Journal of Mathematical Physics},
   Volume = {33},
   Number = {5},
   Pages = {1915-1931},
   Year = {1992},
   ISSN = {0022-2488},
   Abstract = {Morse theory is used to rigorously obtain counting formulas
             and lower bounds for the total number of images of a
             background point source, not on a caustic, undergoing
             lensing by a single-plane microlens system having compact
             bodies plus either subcritical or supercritical continuously
             distributed matter. An image-counting formula is also found
             for the case when external shear is added. In addition, it
             is proven that a microlens system consisting of k lens
             planes will generate N = 2M- + Πi=1k(1 - gi) images of a
             background point source not on a caustic, where M- is the
             total number of critical points of odd index of the
             time-delay map and gi is the number of stars on the ith lens
             plane. Morse theoretic tools also yield that the smallest
             value N can have is Πi=1k(1 + gi). © 1992 American
             Institute of Physics.},
   Key = {fds243986}
}


%% Other   
@misc{fds47735,
   Author = {A.O. Petters},
   Title = {Singularities in Gravitational Microlensing, Ph.D.
             Thesis},
   Journal = {MIT, Department of Mathematics},
   Year = {1991},
   Key = {fds47735}
}