%% 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, BenqueBelize},
Year = {2007},
Month = {August},
Key = {fds70670}
}
@book{fds70667,
Author = {A.O. Petters},
Title = {Scientific Reasoning: Student and Teacher
Editions},
Publisher = {BRC, BenqueBelize},
Year = {2007},
Month = {July},
Key = {fds70667}
}
@book{fds70668,
Author = {A.O. Petters},
Title = {PSE Mathematics: Student and Teacher Editions},
Publisher = {BRC, BenqueBelize},
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=11/ref=sr_1_1/0021045375379126},
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
quasiequatorial lensing observables},
Journal = {Journal of Mathematical Physics},
Volume = {52},
Number = {10},
Pages = {102501102501},
Publisher = {AIP Publishing},
Year = {2011},
Month = {October},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.3642616},
Abstract = {In this second paper, we develop an analytical theory of
quasiequatorial lensing by Kerr black holes. In this
setting we solve perturbatively our general lens equation
with displacement given in Paper I, going beyond
weakdeflection 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{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 = {092502092502},
Publisher = {AIP Publishing},
Year = {2011},
Month = {September},
ISSN = {00222488},
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{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},
Pages = {022501022501},
Publisher = {AIP Publishing},
Year = {2011},
Month = {February},
ISSN = {00222488},
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
fourimage region close to elliptic and hyperbolic umbilic
caustics. This is a more global and higher order analog of
the wellknown 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{fds243975,
Author = {Petters, AO},
Title = {Gravity's action on light},
Journal = {Notices of the American Mathematical Society},
Volume = {57},
Number = {11},
Pages = {13921409},
Year = {2010},
Month = {December},
ISSN = {00029920},
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 = {20112046},
Publisher = {Springer Nature},
Year = {2010},
Month = {Fall},
ISSN = {00017701},
url = {http://dx.doi.org/10.1007/s1071401009686},
Abstract = {The mathematical theory of gravitational lensing has
revealed many generic and global properties. Beginning with
multiple imaging, we review Morsetheoretic image counting
formulas and lower bound results, and complexalgebraic
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 microminima. Multiple imaging in
optical geometry and a spacetime setting are treated. We
review global magnification relation results for
modeldependent scenarios and cover recent developments on
universal local magnification relations for higher order
caustics. © 2010 Springer Science+Business Media,
LLC.},
Doi = {10.1007/s1071401009686},
Key = {fds243979}
}
@article{fds243983,
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 = {082501},
Year = {2010},
Month = {Summer},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.3271043},
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 = {fds243983}
}
@article{fds243980,
Author = {Petters, AO and Rider, B and Teguia, AM},
Title = {A mathematical theory of stochastic microlensing. II. Random
images, shear, and the KacRice formula},
Journal = {Journal of Mathematical Physics},
Volume = {50},
Number = {12},
Pages = {122501122501},
Publisher = {AIP Publishing},
Year = {2009},
Month = {December},
ISSN = {00222488},
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 KacRice 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{fds243981,
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 = {023503},
Year = {2009},
Month = {September},
ISSN = {00222488},
url = {http://hdl.handle.net/10161/3308 Duke open
access},
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 higherorder 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 = {fds243981}
}
@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 = {072503072503},
Publisher = {AIP Publishing},
Year = {2009},
Month = {August},
ISSN = {00222488},
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 pointmass 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 pointmass 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{fds243984,
Author = {Aazami, AB and Petters, AO},
Title = {A universal magnification theorem for higherorder caustic
singularities},
Journal = {Journal of Mathematical Physics},
Volume = {50},
Number = {3},
Pages = {032501032501},
Publisher = {AIP Publishing},
Year = {2009},
Month = {Spring},
ISSN = {00222488},
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 fourimage region close to
swallowtail, elliptic umbilic, and hyperbolic umbilic
caustics. This is a more global and higherorder analog of
the wellknown fold and cusp magnification relations, in
which the total signed magnifications in the twoimage
region of the fold and the threeimage 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 oneparameter 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 = {17191721},
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
postNewtonian 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{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 = {15631582},
Publisher = {Springer Nature},
Year = {2007},
Month = {October},
ISSN = {00017701},
url = {http://xxx.lanl.gov/abs/grqc/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
strongdeflection 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
bendingangle 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/s1071400704818},
Key = {fds243997}
}
@article{fds243996,
Author = {Werner, MC and Petters, AO},
Title = {Magnification relations for Kerr lensing and testing cosmic
censorship},
Journal = {Physical Review D},
Volume = {76},
Number = {6},
Pages = {064024},
Publisher = {American Physical Society (APS)},
Year = {2007},
Month = {September},
ISSN = {15507998},
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 postNewtonian
order. We show that there are postNewtonian 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 nearfuture
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
postNewtonian 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{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},
Volume = {73},
Number = {10},
Pages = {104032},
Publisher = {American Physical Society (APS)},
Year = {2006},
Month = {May},
ISSN = {15507998},
url = {http://xxx.lanl.gov/abs/grqc/0603061},
Abstract = {Braneworld gravity is a model that endows physical space
with an extra dimension. In the type II RandallSundrum
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 farfield 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•/1018M)1 of them lie
within our Solar System. These objects, which we call
"attolenses," would produce interference fringes in the
energy spectra of gammaray bursts at energies
E∼100(M•/1018M)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 gammaray
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•1019M 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 postpostNewtonian
metrics},
Journal = {Physical Review D},
Volume = {73},
Number = {4},
Pages = {044024},
Publisher = {American Physical Society (APS)},
Year = {2006},
Month = {March},
ISSN = {15507998},
url = {http://xxx.lanl.gov/abs/grqc/0601053},
Abstract = {We study gravitational lensing by compact objects in gravity
theories that can be written in a postpostNewtonian (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 weakdeflection 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
firstorder 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 nearfuture
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 J07373039. © 2006 The
American Physical Society.},
Doi = {10.1103/PhysRevD.73.044024},
Key = {fds244000}
}
@article{fds305708,
Author = {Keeton, CR and Gaudi, BS and Petters, AO},
Title = {Identifying lenses with smallscale structure. II. Fold
lenses},
Journal = {The Astrophysical Journal},
Volume = {635},
Number = {1 I},
Pages = {3559},
Publisher = {IOP Publishing},
Year = {2005},
Month = {December},
ISSN = {0004637X},
url = {http://dx.doi.org/10.1086/497324},
Abstract = {When the source in a fourimage 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 smallscale 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 fourimage lenses
that admit generic, local analyses exhibit flux ratio
anomalies. The fold and cusp relations do not reveal the
nature of the implied smallscale 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 fourimage 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{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},
Volume = {72},
Number = {10},
Pages = {104006},
Publisher = {American Physical Society (APS)},
Year = {2005},
Month = {November},
ISSN = {15507998},
url = {http://xxx.lanl.gov/abs/grqc/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 weakdeflection 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 weakdeflection limit can be expanded
as a Taylor series in one parameter, namely, the
gravitational radius of the lens object. We take care to
derive coordinateindependent 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 weakdeflection 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 firstorder corrections to
the total magnification and centroid position vanish in all
gravity theories that agree with general relativity in the
weakdeflection limit, but they can remain nonzero in
modified theories that disagree with general relativity in
the weakdeflection limit. For the ReissnerNordström
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{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 = {0004637X},
url = {http://xxx.lanl.gov/abs/astroph/0503452},
Abstract = {When the source in a fourimage 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 smallscale 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 fourimage lenses
that admit generic, local analyses exhibit flux ratio
anomalies. The fold and cusp relations do not reveal the
nature of the implied smallscale 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 fourimage 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{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 = {457464},
Publisher = {Oxford University Press (OUP)},
Year = {2003},
Month = {January},
url = {http://xxx.lanl.gov/ps/astroph/0208500},
Abstract = {The standard treatment of gravitational lensing by a point
mass lens M is based on a weakfield 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
A1A3, 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
observerlens and lenssource 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 weakfield regime
outside the black hole. We apply our results to the Galactic
black hole for lensing scenarios where A1A3 hold. We show
that a single factor characterizes the full relativistic
correction to the weakfield image centroid and
magnification. As the lenssource 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 Sunlike 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.13658711.2003.06065.x},
Key = {fds244003}
}
@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 = {0004637X},
url = {http://xxx.lanl.gov/abs/astroph/0210318},
Abstract = {The inability of standard models to explain the flux ratios
in many fourimage gravitational lens systems has been
presented as evidence for significant smallscale 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 smallscale
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/nearIR
wavelengths; B0712+472 has a strong anomaly at
optical/nearIR wavelengths and a marginal anomaly at radio
wavelengths; 1RXS J11311231 has a strong anomaly at optical
wavelengths; RX J0911+0551 appears to have an anomaly at
optical/nearIR wavelengths, although the conclusion in this
particular lens is subject to uncertainties in the typical
strength of octopole density perturbations in earlytype
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 smallscale
structure.},
Doi = {10.1086/378934},
Key = {fds243998}
}
@article{fds305707,
Author = {Keeton, CR and Gaudi, BS and Petters, AO},
Title = {Identifying lenses with smallscale structure. I. Cusp
lenses},
Journal = {The Astrophysical Journal},
Volume = {598},
Number = {1 I},
Pages = {138161},
Publisher = {IOP Publishing},
Year = {2003},
ISSN = {0004637X},
url = {http://dx.doi.org/10.1086/378934},
Abstract = {The inability of standard models to explain the flux ratios
in many fourimage gravitational lens systems has been
presented as evidence for significant smallscale 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 smallscale
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/nearIR
wavelengths; B0712+472 has a strong anomaly at
optical/nearIR wavelengths and a marginal anomaly at radio
wavelengths; 1RXS J11311231 has a strong anomaly at optical
wavelengths; RX J0911+0551 appears to have an anomaly at
optical/nearIR wavelengths, although the conclusion in this
particular lens is subject to uncertainties in the typical
strength of octopole density perturbations in earlytype
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 smallscale
structure.},
Doi = {10.1086/378934},
Key = {fds305707}
}
@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 = {55785611},
Publisher = {AIP Publishing},
Year = {2002},
Month = {November},
ISSN = {00222488},
url = {http://xxx.lanl.gov/abs/astroph/0208135},
Abstract = {Very little attention has been paid to the properties of
optical wavefronts and caustic surfaces due to gravitational
lensing. Yet the wavefrontbased 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 weakfield, thinscreen
gravitational lensing. These equations are all related to
the potential of the lens. In the process, we also show that
the standard singleplane 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 spaceborne 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{fds243971,
Author = {Gaudi, BS and Petters, AO},
Title = {Gravitational microlensing near caustics. I.
Folds},
Journal = {The Astrophysical Journal},
Volume = {574},
Number = {2 I},
Pages = {970984},
Publisher = {IOP Publishing},
Year = {2002},
Month = {August},
ISSN = {0004637X},
url = {http://xxx.lanl.gov/abs/astroph/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
limbdarkening 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 wellcovered
Galactic binary lensing event OGLE1999BUL23, 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 = {468489},
Publisher = {IOP Publishing},
Year = {2002},
ISSN = {0004637X},
url = {http://arxiv.org/abs/astroph/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 u1, where u is the distance of the source
from the cusp, whereas perpendicular to this axis, μ α
u2/3. 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 u101/2 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
limbdarkened 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 cuspcrossing Galactic binary
lens event MACHO1997BUL28. 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{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 = {10111023},
Publisher = {AIP Publishing},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1063/1.532367},
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 pointmass lenses with continuous matter and
shear. Counting formulas and bounds on the number of fixed
points are determined. The results include an oddnumber
fixedpoint 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.},
Doi = {10.1063/1.532367},
Key = {fds243970}
}
@article{fds9006,
Author = {A. O. Petters and F.J. Wicklin},
Title = {Counting Formulas and Bounds on Number of Fixed Points Due
to PointMass 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 = {627634},
Publisher = {Elsevier BV},
Year = {1997},
Month = {January},
ISSN = {0362546X},
url = {http://dx.doi.org/10.1016/S0362546X(97)000680},
Doi = {10.1016/S0362546X(97)000680},
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 = {16051613},
Publisher = {AIP Publishing},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.1063/1.531818},
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
threedimensional distribution of g point masses on g lens
planes creates at most 2(22(g1)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 pointmass 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 pointmass gravitational lens
systems. © 1997 American Institute of Physics.},
Doi = {10.1063/1.531818},
Key = {fds243988}
}
@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 = {29202933},
Publisher = {AIP Publishing},
Year = {1996},
Month = {June},
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 nonnormalized 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(g1). Consequences of
these bounds for the global geometry of caustics are
discussed. It is also shown that if γ≥0 and σcis
sufficiently large, then all cusps can be eliminated, that
is, Ncusps=0. The paper also includes equations for
calculating all the bicaustics (i.e., curves yielding the
positions of cusps during a oneparameter evolution) of a
single pointmass lens with continuous matter and shear. The
methods of the paper are based on a new approach to
pointmass gravitational lensing using complex quantities
and the theory of resultants. © 1996 American Institute of
Physics.},
Doi = {10.1063/1.531630},
Key = {fds305706}
}
@article{fds243992,
Author = {Fetters, AO},
Title = {Lower bounds on image magnification in gravitational
lensing},
Journal = {Proceedings. Mathematical, Physical, and Engineering
Sciences},
Volume = {452},
Number = {1949},
Pages = {14751490},
Publisher = {The Royal Society},
Year = {1996},
Month = {January},
ISSN = {13645021},
url = {http://dx.doi.org/10.1098/rspa.1996.0075},
Abstract = {A rigorous study of lower bounds on image magnification in
singleplane gravitational lensing is presented. These
bounds are determined for the total magnification of point
sources undergoing lensing by a general singleplane
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 nonsingular and singular
deflectors. The latter lenses are considered in detail for
the cases of pointmass 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.},
Doi = {10.1098/rspa.1996.0075},
Key = {fds243992}
}
@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 = {29202933},
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 nonnormalized 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(g1). 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 bicaustics (i.e., curves yielding the
positions of cusps during a oneparameter evolution) of a
single pointmass lens with continuous matter and shear. The
methods of the paper are based on a new approach to
pointmass 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 In DoublePlane Lensing},
Journal = {Symposium International Astronomical Union},
Volume = {173},
Pages = {283284},
Publisher = {Cambridge University Press (CUP)},
Year = {1996},
url = {http://dx.doi.org/10.1017/s0074180900231562},
Abstract = {<jats:p>Consider two point masses<jats:italic>m</jats:italic><jats:sub>1</jats:sub>and<jats:italic>m</jats:italic><jats:sub>2</jats:sub>on
distinct planes with respective shears γ<jats:sub>1</jats:sub>,
γ<jats:sub>2</jats:sub>and continuous matter having
densities κ<jats:sub>1</jats:sub>and κ<jats:sub>2</jats:sub>.
It is assumed that the lens equation is as
follows:<jats:dispformula id="S0074180900231562_eqnU1">??</jats:dispformula>where<jats:dispformula
id="S0074180900231562_eqnU2">??</jats:dispformula></jats:p>},
Doi = {10.1017/s0074180900231562},
Key = {fds243990}
}
@article{fds243991,
Author = {Petters, AO},
Title = {A Cusp—Counting Formula For Caustics Due To Multiplane
Gravitational Lensing},
Journal = {Symposium International Astronomical Union},
Volume = {173},
Number = {173},
Pages = {281282},
Publisher = {Cambridge University Press (CUP)},
Year = {1996},
ISSN = {00741809},
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},
Abstract = {<jats:p>Consider a gravitational lens system
with<jats:italic>K</jats:italic>planes. If light rays are
traced back from the observer to the light source plane,
then the points on the first lens plane where a light ray
either terminates, or, passes through and terminates before
reaching the light source plane, are “obstruction
points.” More precisely, tracing rays back to the source
plane induces a<jats:italic>K</jats:italic><jats:italic>plane
lensing map η</jats:italic>:<jats:italic>U</jats:italic>⊆<jats:bold>R</jats:bold><jats:sup>2</jats:sup>→<jats:bold>R</jats:bold><jats:sup>2</jats:sup>of
the form η(<jats:bold>x</jats:bold><jats:sub>1</jats:sub>)
=<jats:bold>x</jats:bold><jats:sub>1</jats:sub>−∑<jats:sub>i=1</jats:sub><jats:sup><jats:italic>k</jats:italic></jats:sup>α<jats:sub><jats:italic>i</jats:italic></jats:sub>(<jats:bold>x</jats:bold><jats:sub><jats:italic>i</jats:italic></jats:sub>(<jats:bold>x</jats:bold><jats:sub><jats:italic>i</jats:italic></jats:sub>)).
We then define an<jats:italic>obstruction
point</jats:italic>of η to be a point<jats:bold>a</jats:bold>of<jats:italic>U</jats:italic>where
lim<jats:sub><jats:bold>x</jats:bold>1→<jats:bold>a</jats:bold></jats:sub>α<jats:sub><jats:italic>i</jats:italic></jats:sub>(<jats:bold>x</jats:bold><jats:sub><jats:italic>i</jats:italic></jats:sub>(<jats:bold>x</jats:bold><jats:sub>1</jats:sub>))
= ∞ for some “deflection angle” α<jats:sub><jats:italic>i</jats:italic></jats:sub>.</jats:p>},
Doi = {10.1017/s0074180900231550},
Key = {fds243991}
}
@article{fds243993,
Author = {Petters, AO and Wicklin, FJ},
Title = {Caustics of the doubleplane twopointmass gravitational
lens with continuous matter and shear},
Journal = {Monthly Notices of the Royal Astronomical
Society},
Volume = {277},
Number = {4},
Pages = {13991403},
Publisher = {Oxford University Press (OUP)},
Year = {1995},
Month = {Summer},
ISSN = {00358711},
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 = {42764295},
Publisher = {AIP Publishing},
Year = {1995},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.530962},
Abstract = {The global geometry of caustics due to a general multiplane
gravitational lens system is investigated. Cuspcounting
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 differentialtopological properties of general
potentials. The methods employed are based on the following:
Morse theory, projectivized rotation numbers, the
FabriciusBjerreHalpern formula, Whitney's rotation number
formula, Seifert decompositions, and the GaussBonnet
theorem. © 1995 American Institute of Physics.},
Doi = {10.1063/1.530962},
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 = {42634275},
Publisher = {AIP Publishing},
Year = {1995},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.530961},
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
timedelay 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.},
Doi = {10.1063/1.530961},
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, FontenanySt Cloud},
Year = {1994},
Key = {fds8995}
}
@article{fds243966,
Author = {Witt, HJ and Petters, AO},
Title = {Singularities of the one and twopoint mass gravitational
lens},
Journal = {Journal of Mathematical Physics},
Volume = {34},
Number = {9},
Pages = {40934111},
Publisher = {AIP Publishing},
Year = {1993},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.530029},
Abstract = {A detailed study of when a change in the number of caustics
and cusps occurs for one and twopoint 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.},
Doi = {10.1063/1.530029},
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 = {35553581},
Publisher = {AIP Publishing},
Year = {1993},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.530045},
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, timedelay image surfaces, big caustics, and
bicaustics. The results of each classification are discussed
and graphically illustrated. © 1993 American Institute of
Physics.},
Doi = {10.1063/1.530045},
Key = {fds243987}
}
@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 = {47814808},
Publisher = {AIP Publishing},
Year = {1993},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.530321},
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 timedelay 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.},
Doi = {10.1063/1.530321},
Key = {fds305705}
}
@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 = {00222488},
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 timedelay 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{fds243986,
Author = {Petters, AO},
Title = {Morse theory and gravitational microlensing},
Journal = {Journal of Mathematical Physics},
Volume = {33},
Number = {5},
Pages = {19151931},
Publisher = {AIP Publishing},
Year = {1992},
Month = {January},
ISSN = {00222488},
url = {http://dx.doi.org/10.1063/1.529667},
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 singleplane microlens system having compact
bodies plus either subcritical or supercritical continuously
distributed matter. An imagecounting 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 Mis the
total number of critical points of odd index of the
timedelay map and giis 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.},
Doi = {10.1063/1.529667},
Key = {fds243986}
}
@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}
}
%% Other
@misc{fds47735,
Author = {A.O. Petters},
Title = {Singularities in Gravitational Microlensing, Ph.D.
Thesis},
Journal = {MIT, Department of Mathematics},
Year = {1991},
Key = {fds47735}
}
