Singularity theory and gravitationallensing / Adie o. Petters, Harold Levine, Ioachim Wambsganss. p. cm.-(Progress in mathematical physics ; v 21) Includes bibliographical references and indexes.

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 = ϑ•/ϑE 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 10 micro-arcseconds, 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.

The inability of standard models to explain the flux ratios in many 4-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 milli-lensing or micro-lensing, we cannot identify the cause of the anomalies using only broad-band 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.

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.