%% 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}, Doi = {10.1093/mnras/277.4.1399}, 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} }