Arlie O. Petters

Identifying lenses with small-scale structure. II. Fold lenses

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.

GRAVITATIONAL MICROLENSING NEAR CAUSTICS. I. FOLDS

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.

Morse theory and gravitational microlensing

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− + Πki=1(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).

Light's bending angle due to black holes: from the photon sphere to infinity

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.

Gravitational Microlensing near Caustics. II. Cusps

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 ∝ u-1, where u is the distance of the source from the cusp, whereas perpendicular to this axis, μ ∝ u-2/3. When the source passes through a point u0 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 |u|1/2 for |u| 1, where |u| 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.

Arnold's singularity theory and gravitational lensing

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.

Magnification relations for Kerr lensing and testing Cosmic Censorship

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.

Mathematics of gravitational lensing: multiple imaging and magnification

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.

Multiplane gravitational lensing. II. Global geometry of caustics

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.

Multiplane gravitational lensing. I. Morse theory and image counting

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.