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We report ALMA observations of the dust continuum and {\cii} emission of the host galaxy of J0439+1634, a gravitationally lensed quasar at $z=6.5$. Gravitational lensing boosts the source-plane resolution to $\sim0\farcs15$ $(\sim0.8\text{ kpc})$. The lensing model derived from the ALMA data is consistent with the fiducial model in \citet{fan19} based on {\it HST} imaging. The host galaxy of J0439+1634 can be well-fitted by a S\'ersic profile consistent with an exponential disk, both in the far-infrared (FIR) continuum and the {\cii} emission. The overall magnification is $4.53\pm0.05$ for the continuum and $3.44\pm0.05$ for the {\cii} line. The host galaxy of J0439+1634 is a compact ultra-luminous infrared galaxy, with a total star formation rate (SFR) of $1.56\times10^{3}M_\odot/\text{year}$ after correcting for lensing and an effective radius of $0.74$ kpc. The resolved regions in J0439+1634 follow the ``{\cii} deficit," where the {\cii}-to-FIR ratio decreases with FIR surface brightness. The reconstructed velocity field of J0439+1634 appears to be rotation-like. The maximum line-of-sight rotation velocity of 130 km/s at a radius of 2 kpc. However, our data cannot be fit by an axisymmetric thin rotating disk, and the inclination of the rotation axis, $i$, remains unconstrained. We estimate the dynamical mass of the host galaxy to be $7.9\sin^{-2}(i)\times10^{9}M_\odot$. J0439+1634 is likely to have a high gas-mass fraction and an oversized SMBH compared to local relations. The SFR of J0439+1634 reaches the maximum possible values, and the SFR surface density is close to the highest value seen in any star-forming galaxy currently known in the universe., Comment: 14 pages, 7 figures. Accepted by ApJ

We report Northern Extended Millimeter Array (NOEMA) CO($J = 3 - 2$) observations of the dusty star-forming galaxy ACT-S\,J020941+001557 at $z = 2.5528$, which was detected as an unresolved source in the Atacama Cosmology Telescope (ACT) equatorial survey. Our spatially resolved spectral line data support the derivation of a gravitational lens model from 37 independent velocity channel maps using a pixel-based algorithm, from which we infer a velocity-dependent magnification factor $\mu \approx 7-22$ with a luminosity-weighted mean $\left\approx 13$. The resulting source-plane reconstruction is consistent with a rotating disk, although other scenarios cannot be ruled out by our data. After correction for lensing, we derive a line luminosity $L^{\prime}_{\rm CO(3-2)}= (5.53\pm 0.69) \times 10^{10}\,{\rm \,K\,km\,s^{-1}\,pc^{2}}$, a cold gas mass $M_{{\rm gas}}= (3.86 \pm 0.33) \times 10^{10}\,M_{\odot}$, a dynamical mass $M_{\rm dyn}\,{\rm sin}^2\,i = 3.9^{+1.8}_{-1.5} \times 10^{10}\,M_{\odot}$, and a gas mass fraction $f_{\rm gas}\,{\rm csc}^2\,i = 1.0^{+0.8}_{-0.4}$. The line brightness temperature ratio of $r_{3,1}\approx 1.6$ relative to a Green Bank Telescope CO($J=1-0$) detection may be elevated by a combination of external heating of molecular clouds, differential lensing, and/or pointing errors., Comment: 8 pages, 5 figures, accepted to ApJ

Gravitational lensing has emerged as a powerful probe of the matter distribution on subgalactic scales, which itself may contain important clues about the fundamental origins and properties of dark matter. Broadly speaking, two different approaches have been taken in the literature to map the small-scale structure of the Universe using strong lensing, with one focused on measuring the position and mass of a small number of discrete massive subhalos appearing close in projection to lensed images, and the other focused on detecting the collective effect of all the small-scale structure between the lensed source and the observer. In this paper, we follow the latter approach and perform a detailed study of the sensitivity of galaxy-scale gravitational lenses to the ensemble properties of small-scale structure. As in some previous studies, we adopt the language of the substructure power spectrum to characterize the statistical properties of the small-scale density field. We present a comprehensive theory that treats lenses with extended sources as well as those with time-dependent compact sources (such as quasars) in a unified framework for the first time. Our approach uses mode functions to provide both computational advantages and insights about couplings between the lens and source. The goal of this paper is to develop the theory and gain the intuition necessary to understand how the sensitivity to the substructure power spectrum depends on the source and lens properties, with the eventual aim of identifying the most promising targets for such studies., 24 pages + appendices, 12 figures. v2: matches published version

Before turning to lensing by arbitrary mass distributions, we would do well to remember that the Newtonian deflection angle is off from the actual value by a factor of two. Narrowly speaking, the result of this chapter will be that factor. More broadly, though, understanding how gravitational lensing is an inevitable consequence of general relativity puts the following chapters on solid theoretical ground. We begin in Sect. 3.1 with a review of special relativity, including the mathematics of four-dimensional spacetime. The geodesic equation, whose solution describes the trajectory of a particle moving through curved spacetime, is presented in Sect. 3.2. The special case of a spherically symmetric mass distribution, which is the basis of weak-field gravitational lensing, is treated in Sects. 3.3 and 3.4. We conclude in Sect. 3.5 with a discussion of a homogeneous, isotropic universe, which is the setting for the cosmological applications of lensing given in later chapters.

We have seen the beauty and power of gravitational lensing in both the local and distant universe, but exploiting this rich phenomenon requires a quantitative treatment. Many of the most important concepts in lensing can be gleaned from the axisymmetric case, where the mass distribution of the lens has either spherical or cylindrical symmetry in three dimensions, corresponding to circular symmetry in two dimensions. Strictly speaking, we must work within the confines of general relativity, but a Newtonian derivation of the deflection angle is nevertheless instructive (Sect. 2.1) and requires only a slight mathematical tweak to bring the result in line with relativity. Because the radial extent of a lens is much smaller than either the distance from the observer to the lens or from the lens to the source, we can project along the line of sight to arrive at a two-dimensional problem (Sect. 2.2). Using this “thin lens” approximation, we derive the equation that a deflected light ray must obey and discuss how to determine the positions and magnifications of lensed images. We then use this framework to analyze a few simple yet representative lens models (Sect. 2.3). The final three sections derive conditions for multiple imaging by axisymmetric lenses and discuss how to extend our analysis to non-axisymmetric lenses.

The remainder of this book is concerned with applications of gravitational lensing of cosmological relevance. We proceed by considering ever larger mass and distance scales, starting with galaxy-scale strong lensing in the present chapter. The background sources in this context are point-like quasars or extended galaxies. As we will see in Sect. 6.1, the axisymmetric models of Chap. 2 can be readily adapted to the more general setting of Chap. 4. Only such non-axisymmetric mass distributions can be expected to describe observed lens systems. We noted in Chap. 2 that actual galaxies have finite density everywhere and offered the nonsingular isothermal sphere as an example. We extend that discussion to more general lenses in Sect. 6.2. Even all of these refinements are insufficient for understanding the full variety of observed galaxy lenses. For example, we must account for extended sources (Sect. 6.3) and for lenses that cannot be described by smooth density profiles (Sect. 6.4). Using the properties of observed images to infer the mass distribution of the lens is the subject of Sect. 6.5. Applications of the models and methods described in this chapter are outlined in Sect. 6.6.

Strong gravitational lensing provides a powerful test of Cold Dark Matter (CDM) as it enables the detection and mass measurement of low mass haloes even if they do not contain baryons. Compact lensed sources such as Active Galactic Nuclei (AGN) are particularly sensitive to perturbing subhalos, but their use as a test of CDM has been limited by the small number of systems which have significant radio emission which is extended enough avoid significant lensing by stars in the plane of the lens galaxy, and red enough to be minimally affected by differential dust extinction. Narrow-line emission is a promising alternative as it is also extended and, unlike radio, detectable in virtually all optically selected AGN lenses. We present first results from a WFC3 grism narrow-line survey of lensed quasars, for the quadruply lensed AGN HE0435-1223. Using a forward modelling pipeline which enables us to robustly account for spatial blending, we measure the [OIII] 5007 \AA~ flux ratios of the four images. We find that the [OIII] fluxes and positions are well fit by a simple smooth mass model for the main lens. Our data rule out a $M_{600}>10^{8} (10^{7.2}) M_\odot$ NFW perturber projected within $\sim$1\farcs0 (0\farcs1) arcseconds of each of the lensed images, where $M_{600}$ is the perturber mass within its central 600 pc. The non-detection is broadly consistent with the expectations of $\Lambda$CDM for a single system. The sensitivity achieved demonstrates that powerful limits on the nature of dark matter can be obtained with the analysis of $\sim20$ narrow-line lenses., Comment: Accepted for publication in MNRAS, 15 pages, 8 figures

We present a combined strong and weak lensing analysis of the J085007.6+360428 (J0850) field, which was selected by its high projected concentration of luminous red galaxies and contains the massive cluster Zwicky 1953. Using Subaru/Suprime-Cam $BVR_{c}I_{c}i^{\prime}z^{\prime}$ imaging and MMT/Hectospec spectroscopy, we first perform a weak lensing shear analysis to constrain the mass distribution in this field, including the cluster at $z = 0.3774$ and a smaller foreground halo at $z = 0.2713$. We then add a strong lensing constraint from a multiply-imaged galaxy in the imaging data with a photometric redshift of $z \approx 5.03$. Unlike previous cluster-scale lens analyses, our technique accounts for the full three-dimensional mass structure in the beam, including galaxies along the line of sight. In contrast with past cluster analyses that use only lensed image positions as constraints, we use the full surface brightness distribution of the images. This method predicts that the source galaxy crosses a lensing caustic such that one image is a highly-magnified "fold arc", which could be used to probe the source galaxy's structure at ultra-high spatial resolution ($< 30$ pc). We calculate the mass of the primary cluster to be $\mathrm{M_{vir}} = 2.93_{-0.65}^{+0.71} \times 10^{15}~\mathrm{M_{\odot}}$ with a concentration of $\mathrm{c_{vir}} = 3.46_{-0.59}^{+0.70}$, consistent with the mass-concentration relation of massive clusters at a similar redshift. The large mass of this cluster makes J0850 an excellent field for leveraging lensing magnification to search for high-redshift galaxies, competitive with and complementary to that of well-studied clusters such as the HST Frontier Fields., Accepted for publication in The Astrophysical Journal; 14 pages, 13 figures, 3 tables

We propose a single galaxy gravitational lensing model with a mass density that has a spiral structure. Namely, we extend the arcsine gravitational lens (a truncated singular isothermal elliptical model), adding an additional parameter that controls the amount of spiraling in the structure of the mass density. An important feature of our model is that, even though the mass density is sophisticated, we succeed in integrating the deflection term in closed form using a Gauss hypergeometric function. When the spiraling parameter is set to zero, this reduces to the arcsine lens.

We present ~1" resolution (~2 kpc in the source plane) observations of the CO(1-0), CO(3-2), Halpha, and [N II] lines in the strongly-lensed z=2.26 star-forming galaxy SDSS J0901+1814. We use these observations to constrain the lensing potential of a foreground group of galaxies, and our source-plane reconstructions indicate that SDSS J0901+1814 is a nearly face-on (i~30 degrees) massive disk with r_{1/2}>~4 kpc for its molecular gas. Using our new magnification factors (mu_tot~30), we find that SDSS J0901+1814 has a star formation rate (SFR) of 268^{+63}_{-61} M_sun/yr, M_gas=(1.6^{+0.3}_{-0.2})x10^11x(alpha_CO/4.6) M_sun, and M_star=(9.5^{+3.8}_{-2.8})x10^10 M_sun, which places it on the star-forming galaxy "main sequence." We use our matched high-angular resolution gas and SFR tracers (CO and Halpha, respectively) to perform a spatially resolved (pixel-by-pixel) analysis of SDSS J0901+1814 in terms of the Schmidt-Kennicutt relation. After correcting for the large fraction of obscured star formation (SFR_Halpha/SFR_TIR=0.054^{+0.015}_{-0.014}), we find SDSS J0901+1814 is offset from "normal" star-forming galaxies to higher star formation efficiencies independent of assumptions for the CO-to-H_2 conversion factor. Our mean best-fit index for the Schmidt-Kennicutt relation for SDSS J0901+1814, evaluated with different CO lines and smoothing levels, is n=1.54+/-0.13; however, the index may be affected by gravitational lensing, and we find n=1.24+/-0.02 when analyzing the source-plane reconstructions. While the Schmidt-Kennicutt index largely appears unaffected by which of the two CO transitions we use to trace the molecular gas, the source-plane reconstructions and dynamical modeling suggest that the CO(1-0) emission is more spatially extended than the CO(3-2) emission., ApJ accepted; 40 pages, 24 figures, 8 tables

In gravitational lensing, magnification cross sections characterize the probability that a light source will have magnification greater than some fixed value, which is useful in a variety of applications. The (area) cross section is known to scale as &mu, &minus, 2 for fold caustics and &mu, 2.5 for cusp caustics. We aim to extend the results to higher-order caustic singularities, focusing on the elliptic umbilic, which can be manifested in lensing systems with two or three galaxies. The elliptic umbilic has a caustic surface, and we show that the volume cross section scales as &mu, 2.5 in the two-image region and &mu, 2 in the four-image region, where &mu, is the total unsigned magnification. In both cases our results are supported both numerically and analytically.

In the standard structure formation scenario based on the cold dark matter paradigm, galactic halos are predicted to contain a large population of dark matter subhalos. While the most massive members of the subhalo population can appear as luminous satellites and be detected in optical surveys, establishing the existence of the low mass and mostly dark subhalos has proven to be a daunting task. Galaxy-scale strong gravitational lenses have been successfully used to study mass substructures lying close to lensed images of bright background sources. However, in typical galaxy-scale lenses, the strong lensing region only covers a small projected area of the lens's dark matter halo, implying that the vast majority of subhalos cannot be directly detected in lensing observations. In this paper, we point out that this large population of dark satellites can collectively affect gravitational lensing observables, hence possibly allowing their statistical detection. Focusing on the region of the galactic halo outside the strong lensing area, we compute from first principles the statistical properties of perturbations to the gravitational time delay and position of lensed images in the presence of a mass substructure population. We find that in the standard cosmological scenario, the statistics of these lensing observables are well approximated by Gaussian distributions. The formalism developed as part of this calculation is very general and can be applied to any halo geometry and choice of subhalo mass function. Our results significantly reduce the computational cost of including a large substructure population in lens models and enable the use of Bayesian inference techniques to detect and characterize the distributed satellite population of distant lens galaxies., 21 pages + appendices, 7 figures. v2: Some derivations streamlined, extended appendices. Matches version published in PRD

The arrival times, positions, and fluxes of multiple images in strong lens systems can be used to infer the presence of dark subhalos in the deflector, and thus test predictions of cold dark matter models. However, gravitational lensing does not distinguish between perturbations to a smooth gravitational potential arising from baryonic and non-baryonic mass. In this work, we quantify the extent to which the stellar mass distribution of a deflector can reproduce flux ratio and astrometric anomalies typically associated with the presence of a dark matter subhalo. Using Hubble Space Telescope images of nearby galaxies, we simulate strong lens systems with real distributions of stellar mass as they would be observed at redshift $z_d=0.5$. We add a dark matter halo and external shear to account for the smooth dark matter field, omitting dark substructure, and use a Monte Carlo procedure to characterize the distributions of image positions, time delays, and flux ratios for a compact background source of diameter 5 pc. By convolving high-resolution images of real galaxies with a Gaussian PSF, we simulate the most detailed smooth potential one could construct given high quality data, and find scatter in flux ratios of $\approx 10\%$, which we interpret as a typical deviation from a smooth potential caused by large and small scale structure in the lensing galaxy. We demonstrate that the flux ratio anomalies arising from galaxy-scale baryonic structure can be minimized by selecting the most massive and round deflectors, and by simultaneously modeling flux ratio and astrometric data., Comment: 26 pages, 22 figures, submitted to MNRAS

I review methods for modeling gravitational lens systems comprising multiple images of a background source surrounding a foreground galaxy. In a Bayesian framework, the likelihood is driven by the nature of the data, which in turn depends on whether the source is point-like or extended. The prior encodes astrophysical expectations about lens galaxy mass distributions, either through a careful choice of model families, or through an explicit Bayesian prior applied to under-constrained free-form models. We can think about different lens modeling methods in terms of their choices of likelihoods and priors.

[Abridged] Many current and future astronomical surveys will rely on samples of strong gravitational lens systems to draw conclusions about galaxy mass distributions. We use a new strong lensing pipeline (presented in Paper I of this series) to explore selection biases that may cause the population of strong lensing systems to differ from the general galaxy population. Our focus is on point-source lensing by early-type galaxies with two mass components (stellar and dark matter) that have a variety of density profiles and shapes motivated by observational and theoretical studies of galaxy properties. We seek not only to quantify but also to understand the physics behind selection biases related to: galaxy mass, orientation and shape; dark matter profile parameters such as inner slope and concentration; and adiabatic contraction. We study how all of these properties affect the lensing Einstein radius, total cross-section, quad/double ratio, and image separation distribution. We find significant (factors of several) selection biases with mass; orientation, for a given galaxy shape at fixed mass; cusped dark matter profile inner slope and concentration; concentration of the stellar and dark matter deprojected Sersic models. Interestingly, the intrinsic shape of a galaxy does not strongly influence its lensing cross-section when we average over viewing angles. Our results are an important first step towards understanding how strong lens systems relate to the general galaxy population., 26 pages, 15 figures; paper I at arXiv:0808.2493; accepted for publication in MNRAS (minor revisions); PDF file with full resolution figures at http://www.sns.ias.edu/~rmandelb/paper2.pdf

In strong gravitational lensing, the multiple images we see correspond to light rays that leave the source in slightly different directions. If the source emission is anisotropic, the images may differ from conventional lensing predictions (which assume isotropy). To identify scales on which source anisotropy may be important, we study the angle delta between the light rays emerging from the source, for different lensing configurations. If the lens has a power law profile M propto R^gamma, the angle delta initially increases with lens redshift and then either diverges (for a steep profile gamma1) as zl approaches zs. The scaling with lens mass is roughly delta propto M^(1/(2-gamma)). The results for an NFW profile are qualitatively similar to those for a shallow power law, with delta peaking at about half the redshift of the source (not half the distance). In practice, beaming could modify the statistics of beamed sources lensed by massive clusters: for an opening angle theta_jet, there is a probability as high as P ~ 0.02-0.07 (theta_jet/{0.5} deg)^-1 that one of the lensed images may be missed (for 2 < zs < 6). Differential absorption within Active Galactic Nuclei could modify the flux ratios of AGNs lensed by clusters; a sample of AGNs lensed by clusters could provide further constraints on the sizes of absorbing regions. Source anisotropy is not likely to be a significant effect in galaxy-scale strong lensing., minor changes, accepted to MNRAS

We show that dark matter substructure in galaxy-scale halos perturbs the time delays between images in strong gravitational lens systems. The variance of the effect depends on the subhalo mass function, scaling as the product of the substructure mass fraction and a characteristic mass of subhalos (namely /). Time delay perturbations therefore complement gravitational lens flux ratio anomalies and astrometric perturbations by measuring a different moment of the subhalo mass function. Unlike flux ratio anomalies, "time delay millilensing" is unaffected by dust extinction or stellar microlensing in the lens galaxy. Furthermore, we show that time delay ratios are immune to the radial profile degeneracy that usually plagues lens modeling. We lay out a mathematical theory of time delay perturbations and find it to be tractable and attractive. We predict that in "cusp" lenses with close triplets of images, substructure may change the arrival-time order of the images (compared with smooth models). We discuss the possibility that this effect has already been observed in RX J1131-1231., Comment: accepted in ApJ; expanded discussion of macromodel uncertainties

Gravitational lensing assists in the detection of quasar hosts by amplifying and distorting the host light away from the unresolved quasar core images. We present the results of HST observations of 30 quasar hosts at redshifts 1 < z < 4.5. The hosts are small in size (r_e 1.7 is a factor of 3--6 higher than the local value. But, depending on the stellar content the ratio may decline at z>4 (if E/S0-like), flatten off to 6--10 times the local value (if Sbc-like), or continue to rise (if Im-like). We infer that galaxy bulge masses must have grown by a factor of 3--6 over the redshift range 3>z>1, and then changed little since z~1. This suggests that the peak epoch of galaxy formation for massive galaxies is above z~1. We also estimate the duty cycle of luminous AGNs at z>1 to be ~1%, or 10^7 yrs, with sizable scatter., Comment: 8 pages, 6 figures, review article with C. Impey at the conference on "QSO Host Galaxies: Evolution and Environment", Aug. 29-Sep. 2, 2005, Lorentz Center, Leiden, The Netherlands

著者人数: 18名, Accepted: 2006-04-25, 資料番号: SA1000374000

When the source in a 4-image gravitational lens lies close to a "fold" caustic, two of the lensed images lie close together. If the lens potential is smooth on the scale of the separation between the two close images, then the difference between their fluxes should approximately vanish, Rfold = (F1-F2)/(F1+F2) \approx 0. 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 study the fold relation and find it to be more subtle and rich than was previously realized. The degree to which Rfold can differ from zero for realistic smooth lenses depends not only on the distance of the source from the caustic, but also on its location along the caustic, and on the angular structure of the lens potential. Thus, it is impossible to say from Rfold alone whether observed flux ratios are anomalous or not. Instead, we must consider the full distribution of Rfold values that can be obtained from smooth potentials that reproduce the separation between the two close images and the distance to the next nearest image. We analyze the generic and specific features of this distribution, and then show that 5 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 3 of the 4 known lenses with cusp configurations, we conclude that at least half (8/16) of all 4-image lenses that admit generic, local analyses exhibit flux ratio anomalies indicative of small-scale structure. 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. [Abridged], Comment: accepted in ApJ; 24 pages, color figures; new data added for SDSS0924, PG1115, and B1422; discussion in Section 6 trimmed for clarity; other minor clarifications added

We study the effects of ellipticity in lens galaxies and external tidal shear from neighboring objects on the statistics of strong gravitational lenses. For isothermal lens galaxies normalized so that the Einstein radius is independent of ellipticity and shear, ellipticity {\it reduces} the lensing cross section slightly, and shear leaves it unchanged. Ellipticity and shear can significantly enhance the magnification bias, but only if the luminosity function of background sources is steep. Realistic distributions of ellipticity and shear {\it lower} the total optical depth by a few percent for most source luminosity functions, and increase the optical depth only for steep luminosity functions. The boost in the optical depth is noticeable (>5%) only for surveys limited to the brightest quasars (L/L_* > 10). Ellipticity and shear broaden the distribution of lens image separations but do not affect the mean. Ellipticity and shear naturally increase the abundance of quadruple lenses relative to double lenses, especially for steep source luminosity functions, but the effect is not enough (by itself) to explain the observed quadruple-to-double ratio. With such small changes to the optical depth and image separation distribution, ellipticity and shear have a small effect on cosmological constraints from lens statistics: neglecting the two leads to biases of just Delta Omega_M = 0.00 \pm 0.01 and Delta Omega_Lambda = -0.02 \pm 0.01 (where the errorbars represent statistical uncertainties in our calculations)., Optical depth normalization discussed. Matches the published version

We combine the Cosmic Lens All-Sky Survey (CLASS) with new Sloan Digital Sky Survey (SDSS) data on the local velocity dispersion distribution function of E/S0 galaxies, $\phi(\sigma)$, to derive lens statistics constraints on $\Omega_\Lambda$ and $\Omega_m$. Previous studies of this kind relied on a combination of the E/S0 galaxy luminosity function and the Faber-Jackson relation to characterize the lens galaxy population. However, ignoring dispersion in the Faber-Jackson relation leads to a biased estimate of $\phi(\sigma)$ and therefore biased and overconfident constraints on the cosmological parameters. The measured velocity dispersion function from a large sample of E/S0 galaxies provides a more reliable method for probing cosmology with strong lens statistics. Our new constraints are in good agreement with recent results from the redshift-magnitude relation of Type Ia supernovae. Adopting the traditional assumption that the E/S0 velocity function is constant in comoving units, we find a maximum likelihood estimate of $\Omega_\Lambda = 0.74$--0.78 for a spatially flat unvierse (where the range reflects uncertainty in the number of E/S0 lenses in the CLASS sample), and a 95% confidence upper bound of $\Omega_\Lambda, Comment: 35 pages, 15 figures, to be published in ApJ

Gravitational lensing can amplify the apparent brightness of distant sources. Images that are highly magnified are often part of multiply-imaged systems, but we consider the possibility of having large magnifications without additional detectable images. In rare but non-negligible situations, lensing can produce a singly highly magnified image; this phenomenon is mainly associated with massive cluster-scale halos (>~1e13.5 Msun). Alternatively, lensing can produce multiply-imaged systems in which the extra images are either unresolved or too faint to be detectable. This phenomenon is dominated by galaxies and lower-mass halos (10 would lack detectable extra images, with 5-10% being true singly-imaged systems. The maximum of 29% is reached only in the unlikely event that all low-mass (10) multiply-imaged z~6 quasars in the SDSS database that have not yet been identified, which seems unlikely. In other words, lensing cannot explain the brightnesses of the z~6 quasars, and models that invoke lensing to avoid having billion-Msun black holes in the young universe are not viable., Comment: accepted in ApJ; 15 emulateapj pages; small revisions to clarify the text

We report the discovery of a new gravitationally lensed quasar from the Sloan Digital Sky Survey, SDSS J090334.92+502819.2. This object was targeted for SDSS spectroscopy as a Luminous Red Galaxy (LRG), but manual examination of the spectrum showed the presence of a quasar at z= 3.6 in addition to a red galaxy at z=0.388, and the SDSS image showed a second possible quasar image nearby. Follow-up imaging and spectroscopy confirmed the lensing hypothesis. In images taken at the ARC 3.5-meter telescope, two quasars are separated by 2.8 arc-seconds; the lensing galaxy is clearly seen and is blended with one of the quasar images. Spectroscopy taken at the Keck II telescope shows that the quasars have identical redshifts of z=3.6 and both show the presence of the same broad absorption line-like troughs. We present simple lens models which account for the geometry and magnifications. The lens galaxy lies near two groups of galaxies and may be a part of them. The models suggest that the groups may contribute considerable shear and may have a strong effect on the lens configuration.

The magnifications of the images in a strong gravitational lens system are sensitive to small mass clumps in the lens potential; this effect has been used to infer the amount of substructure in galaxy dark matter halos. I study the theory of substructure lensing to identify important general features, and to compute analytic cross sections that will facilitate further theoretical studies. I show that the problem of a clump anywhere along the line of sight to a lens can be mapped onto an equivalent problem of a clump in a simple convergence and shear field; clumps at arbitrary redshifts are therefore not hard to handle in calculations. For clumps modeled as singular isothermal spheres (SIS), I derive simple analytic estimates of the cross section for magnification perturbations of a given strength. The results yield two interesting conceptual points. First, lensed images with positive parity are always made brighter by SIS clumps; images with negative parity can be brightened but are much more likely to be dimmed. Second, the clumps need not lie within the lens galaxy; they can be moved in redshift by several tenths and still have a significant lensing effect. Isolated small halos are expected to be common in hierarchical structure formation models, but it is not yet known whether they are abundant enough compared with clumps inside lens galaxies to affect the interpretation of substructure lensing., accepted in ApJ; minor text changes from original version

We report the serendipitous discovery of the first gravitationally lensed quasar candidate from Pan-STARRS. The grizy images reveal four point-like images with magnitudes between 14.9 mag and 18.1 mag. The colors of the point sources are similar, and they are more consistent with quasars than with stars or galaxies. The lensing galaxy is detected in the izy bands, with an inferred photometric redshift of ~0.6, lower than that of the point sources. We successfully model the system with a singular isothermal ellipsoid with shear, using the relative positions of the five objects as constraints. While the brightness ranking of the point sources is consistent with that of the model, we find discrepancies between the model-predicted and observed fluxes, likely due to microlensing by stars and millilensing due to the dark matter substructure. In order to fully confirm the gravitational lens nature of this system and add it to the small but growing number of the powerful probes of cosmology and astrophysics represented by quadruply lensed quasars, we require further spectroscopy and high-resolution imaging., Comment: 29 pages, 7 figures, accepted in ApJ

Matter near a gravitational lens galaxy or projected along the line of sight (LOS) can affect strong lensing observables by more than contemporary measurement errors. We simulate lens fields with realistic three-dimensional mass configurations (self-consistently including voids), and then fit mock lensing observables with increasingly complex lens models to quantify biases and uncertainties associated with different ways of treating the lens environment (ENV) and LOS. We identify the combination of mass, projected offset, and redshift that determines the importance of a perturbing galaxy for lensing. Foreground structures have a stronger effect on the lens potential than background structures, due to nonlinear effects in the foreground and downweighting in the background. There is dramatic variation in the net strength of ENV/LOS effects across different lens fields; modeling fields individually yields stronger priors for $H_0$ than ray tracing through $N$-body simulations. Models that ignore mass outside the lens yield poor fits and biased results. Adding external shear can account for tidal stretching from galaxies at redshifts $z \ge z_{\rm lens}$, but it requires corrections for external convergence and cannot reproduce nonlinear effects from foreground galaxies. Using the tidal approximation is reasonable for most perturbers as long as nonlinear redshift effects are included. Even then, the scatter in $H_0$ is limited by the lens profile degeneracy. Asymmetric image configurations produced by highly elliptical lens galaxies are less sensitive to the lens profile degeneracy, so they offer appealing targets for precision lensing analyses in future surveys like LSST and Euclid., Comment: 26 pages, 19 figures

In strong gravitational lens systems, the light bending is usually dominated by one main galaxy, but may be affected by other mass along the line of sight (LOS). Shear and convergence can be used to approximate the contributions from less significant perturbers (e.g. those that are projected far from the lens or have a small mass), but higher order effects need to be included for objects that are closer or more massive. We develop a framework for multiplane lensing that can handle an arbitrary combination of tidal planes treated with shear and convergence and planes treated exactly (i.e., including higher order terms). This framework addresses all of the traditional lensing observables including image positions, fluxes, and time delays to facilitate lens modelling that includes the non-linear effects due to mass along the LOS. It balances accuracy (accounting for higher-order terms when necessary) with efficiency (compressing all other LOS effects into a set of matrices that can be calculated up front and cached for lens modelling). We identify a generalized multiplane mass sheet degeneracy, in which the effective shear and convergence are sums over the lensing planes with specific, redshift-dependent weighting factors., Comment: 13 pages, 2 figures

Gravitational lens modeling of spatially resolved sources is a challenging inverse problem with many observational constraints and model parameters. We examine established pixel-based source reconstruction algorithms for de-lensing the source and constraining lens model parameters. Using test data for four canonical lens configurations, we explore statistical and systematic uncertainties associated with gridding, source regularisation, interpolation errors, noise, and telescope pointing. Specifically, we compare two gridding schemes in the source plane: a fully adaptive grid that follows the lens mapping but is irregular, and an adaptive Cartesian grid. We also consider regularisation schemes that minimise derivatives of the source (using two finite difference methods) and introduce a scheme that minimises deviations from an analytic source profile. Careful choice of gridding and regularisation can reduce "discreteness noise" in the $\chi^2$ surface that is inherent in the pixel-based methodology. With a gridded source, some degree of interpolation is unavoidable, and errors due to interpolation need to be taken into account (especially for high signal-to-noise data). Different realisations of the noise and telescope pointing lead to slightly different values for lens model parameters, and the scatter between different "observations" can be comparable to or larger than the model uncertainties themselves. The same effects create scatter in the lensing magnification at the level of a few percent for a peak signal-to-noise ratio of 10, which decreases as the data quality improves., Comment: 20 pages, 18 figures, accepted to MNRAS, see http://physics.rutgers.edu/~tagoreas/papers/ for high resolution images

HST V and I-band observations show that the gravitational lens B1359+154 consists of six images of a single z_s=3.235 radio source and its star-forming host galaxy, produced by a compact group of galaxies at z_l = 1. VLBA observations at 1.7 GHz strongly support this conclusion, showing six compact cores with similar low-frequency radio spectra. B1359+154 is the first example of galaxy-scale gravitational lensing in which more than four images are observed of the same background source. The configuration is due to the unique lensing mass distribution: three primary lens galaxies lying on the vertices of a triangle separated by 0.7 arcsec (4/h kpc), inside the 1.7 arcsec diameter Einstein ring defined by the radio images. The gravitational potential has additional extrema within this triangle, creating a pair of central images that supplement the ``standard'' four-image geometry of the outer components. Simple mass models consisting of three lens galaxies constrained by HST and VLBA astrometry naturally reproduce the observed image positions but must be finely-tuned to fit the flux densities., 28 pages including 6 figures, ApJ submitted

We have obtained and modeled NICMOS images of the Einstein ring lens system MG J1131+0456, which show that its lens galaxy is an H = 18.6 mag, transparent, early-type galaxy at a redshift of zl 0.84; it has a major axis effective radius Re = 07 ? 01, projected axis ratio b/a = 0.6 ? 0.1, and major axis P.A. = 55? ? 9?. The lens is the brightest member of a group of at least seven galaxies with similar R-I and I-H colors, and the two closest group members produce sufficient tidal perturbations to explain the shape of the ring. The host galaxy of the MG J1131+0456 source is a zs 2 extremely red object (ERO) that is lensed into optical and infrared rings of dramatically different morphologies. These differences imply a strongly wavelength-dependent source morphology that could be explained by embedding the host in a larger, dusty disk. At 1.6 ?m (H), the ring is spectacularly luminous, with a total observed flux of H = 17.4 mag and a demagnified flux of 19.3 mag, corresponding to a 1-2 L* galaxy at the probable source redshift of zs 2. Thus, it is primarily the stellar emission of the radio source host galaxy that produces the overall colors of two of the reddest radio lenses, MG J1131+0456 and JVAS B1938+666, aided by the suppression of optical active galactic nucleus emission by dust in the source galaxy. The dusty lens hypothesis ? that many massive early-type galaxies with 0 zl 1 have large, uniform dust opacities ? is ruled out.

We determine 37 differential extinctions in 23 gravitational lens galaxies over the range 0 < z_l < 1. Only 7 of the 23 systems have spectral differences consistent with no differential extinction. The median differential extinction for the optically-selected (radio-selected) subsample is E(B-V)=0.04 (0.06) mag. The extinction is patchy and shows no correlation with impact parameter. The median total extinction of the bluest images is E(B-V)=0.08 mag, although the total extinction distribution is dominated by the uncertainties in the intrinsic colors of quasars. The directly measured extinction distributions are consistent with the mean extinction estimated by comparing the statistics of quasar and radio lens surveys, thereby confirming the need for extinction corrections when using the statistics of lensed quasars to estimate the cosmological model. A disjoint subsample of two face-on, radio-selected spiral lenses shows both high differential and total extinctions, but standard dust-to-gas ratios combined with the observed molecular gas column densities overpredict the amount of extinction by factors of 2-5. For several systems we can estimate the extinction law, ranging from R_V=1.5+/-0.2 for a z_l=0.96 elliptical, to R_V=7.2+/-0.1 for a z_l=0.68 spiral. For the four radio lenses where we can construct non-parametric extinction curves we find no evidence for gray dust over the IR-UV wavelength range. The dust can be used to estimate lens redshifts with reasonable accuracy, although we sometimes find two degenerate redshift solutions.

We examine models of the mass distribution for the first known case of gravitational lensing. Several new sets of constraints are used, based on recent observations. We remodel the VLBI observations of the radio jets in the two images of Q0957+561, showing that the previously derived positions and uncertainties were incorrect. We use as additional constraints the candidate lensed pairs of galaxies identified recently with the Hubble Space Telescope. We explore a wider range of lens models than before, and find that the Hubble constant is not tightly constrained once elliptical lens models are considered. We also discuss the systematic uncertainties caused by the cluster containing the lens galaxy. We conclude that additional observations of the candidate lensed galaxies as well as direct measurements of the cluster mass profile are needed before an accurate value of the Hubble constant can be derived from this lens., 28 pages, Latex + 4 figures, accepted for ApJ. Final version, with minor changes. Model parameters in Table 4 corrected

We combine photometry and lens modeling to study the properties of 17 gravitational lens galaxies between z=0.1 and 1. Most of the lens galaxies are passively evolving early-type galaxies, with a few spirals. The colors, scale lengths, and ellipticities of lens galaxies are similar to those of the general population of early-type galaxies, although there may be a deficit of apparently round lens galaxies produced by the inclination dependence of lensing cross sections. The projected mass distributions are aligned with the projected light distributions to, Comment: 40 pages, 6 tables, 8 figures

We present a high-resolution spectrum of the high redshift, lensed quasar Q1208+1101, obtained with the echellette spectrograph on the Multiple Mirror Telescope. We examine the new and published spectra and provide an updated list of high-confidence metal-line absorption systems at z=1.1349, 2.8626, 2.9118, 2.9136, 2.9149. Combining this with a simple model of the gravitational lens system allows us to constrain the possible lens redshifts. The high-redshift (z > 2.5) and low-redshift (z < 0.4) candidates can be ruled out with high confidence. The current spectra effectively probe about 40% of the redshift range in which the lens is expected. In that range, there is only one known metal-line absorption system, an MgII absorber at z=1.1349. We consider the possibility that this system is the lensing galaxy and discuss the implied parameters of the galaxy., Latex 24 pages, 6 figures. accepted for publication in Astrophysical Journal

Gravitational lensing is a powerful tool for the study of the distribution of dark matter in the Universe. The cold-dark-matter model of the formation of large-scale structures predicts the existence of quasars gravitationally lensed by concentrations of dark matter so massive that the quasar images would be split by over 7 arcsec. Numerous searches for large-separation lensed quasars have, however, been unsuccessful. All of the roughly 70 lensed quasars known, including the first lensed quasar discovered, have smaller separations that can be explained in terms of galaxy-scale concentrations of baryonic matter. Although gravitationally lensed galaxies with large separations are known, quasars are more useful cosmological probes because of the simplicity of the resulting lens systems. Here we report the discovery of a lensed quasar, SDSS J1004+4112, which has a maximum separation between the components of 14.62 arcsec. Such a large separation means that the lensing object must be dominated by dark matter. Our results are fully consistent with theoretical expectations based on the cold-dark-matter model., 10 pages, 3 figures, to appear in the 18th&25th Dec issue of Nature (Letters to Nature)

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., 11 pages; final published version

We present the second report of our systematic search for strongly lensed quasars from the data of the Sloan Digital Sky Survey (SDSS). From extensive follow-up observations of 136 candidate objects, we find 36 lenses in the full sample of 77,429 spectroscopically confirmed quasars in the SDSS Data Release 5. We then define a complete sample of 19 lenses, including 11 from our previous search in the SDSS Data Release 3, from the sample of 36,287 quasars with i, Comment: 37 pages, 2 figures and 5 tables, accepted to AJ

Motivated by debris disk studies, we investigate the gravitational microlensing of background starlight by a planetesimal disk around a foreground star. We use dynamical survival models to construct a plausible example of a planetesimal disk and study its microlensing properties using established ideas of microlensing by small bodies. When a solar-type source star passes behind a planetesimal disk, the microlensing light curve may exhibit short-term, low-amplitude residuals caused by planetesimals several orders of magnitude below Earth mass. The minimum planetesimal mass probed depends on the photometric sensitivity and the size of the source star, and is lower when the planetesimal lens is located closer to us. Planetesimal lenses may be found more nearby than stellar lenses because the steepness of the planetesimal mass distribution changes how the microlensing signal depends on the lens/source distance ratio. Microlensing searches for planetesimals require essentially continuous monitoring programs that are already feasible and can potentially set constraints on models of debris disks, the supposed extrasolar analogues of Kuiper belts., 15 pages, 8 figures (emulateapj). Accepted by ApJ

We examine the ability of gravitational lens time delays to reveal complex structure in lens potentials. In Congdon, Keeton & Nordgren (2008), we predicted how the time delay between the bright pair of images in a "fold" lens scales with the image separation, for smooth lens potentials. Here we show that the proportionality constant increases with the quadrupole moment of the lens potential, and depends only weakly on the position of the source along the caustic. We use Monte Carlo simulations to determine the range of time delays that can be produced by realistic smooth lens models consisting of isothermal ellipsoid galaxies with tidal shear. We can then identify outliers as "time delay anomalies". We find evidence for anomalies in close image pairs in the cusp lenses RX J1131$-$1231 and B1422+231. The anomalies in RX J1131$-$1231 provide strong evidence for substructure in the lens potential, while at this point the apparent anomalies in B1422+231 mainly indicate that the time delay measurements need to be improved. We also find evidence for time delay anomalies in larger-separation image pairs in the fold lenses, B1608+656 and WFI 2033$-$4723, and the cusp lens RX J0911+0551. We suggest that these anomalies are caused by some combination of substructure and a complex lens environment. Finally, to assist future monitoring campaigns we use our smooth models with shear to predict the time delays for all known four-image lenses., 24 pages, 10 figures, 5 tables, accepted by ApJ, minor edits to match text of published version

We present a detailed strong lensing analysis of an HST/ACS legacy dataset for the first gravitational lens, Q0957+561. With deep imaging we identify 24 new strongly lensed features, which we use to constrain mass models. We model the stellar component of the lens galaxy using the observed luminosity distribution, and the dark matter halo using several different density profiles. We draw on the weak lensing analysis by Nakajima et al. (2009) to constrain the mass sheet and environmental terms in the lens potential. Adopting the well-measured time delay, we find H_0 = 85 (+14/-13) km/s/Mpc (68% CL) using lensing constraints alone. The principal uncertainties in H_0 are tied to the stellar mass-to-light ratio (a variant of the radial profile degeneracy in lens models). Adding constraints from stellar population synthesis models, we obtain H_0 = 79.3 (+6.7/-8.5) km/s/Mpc (68% CL). We infer that the lens galaxy has a rising rotation curve and a dark matter distribution with an inner core. Intriguingly, we find the quasar flux ratios predicted by our models to be inconsistent with existing radio measurements, suggesting the presence of substructure in the lens., Comment: 23 pages, 19 figures, 7 tables; ApJ accepted. Some additional comments and minor changes from v1

We model the massive dark object at the center of the Galaxy as a Schwarzschild black hole as well as Janis-Newman-Winicour naked singularities, characterized by the mass and scalar charge parameters, and study gravitational lensing (particularly time delay, magnification centroid, and total magnification) by them. We find that the lensing features are qualitatively similar (though quantitatively different) for the Schwarzschild black holes, weakly naked, and marginally strongly naked singularities. However, the lensing characteristics of strongly naked singularities are qualitatively very different from those due the Schwarzschild black holes. The images produced by Schwarzschild black hole lenses and weakly naked and marginally strongly naked singularity lenses always have positive time delays. On the other hand, the strongly naked singularity lenses can give rise to images with positive, zero, or negative time delays. In particular, for a large angular source position the direct image (the outermost image on the same side as the source) due to strongly naked singularity lensing always has negative time delay. We also found that the scalar field decreases the time delay and increases the magnitude of magnifications of images; this result could have important implications for cosmology. As the Janis-Newman-Winicour metric also describes the exterior gravitational field of a scalar star, naked singularities as well as scalar star lenses, if these exist in nature, will serve as more efficient cosmic telescopes than regular gravitational lenses., RevTex, 14 pages, 5 figures

(abridged) The Observatory for Multi-Epoch Gravitational Lens Astrophysics (OMEGA) is a mission concept for a 1.5-m near-UV through near-IR space observatory that will be dedicated to frequent imaging and spectroscopic monitoring of ~100 multiply-imaged active galactic nuclei over the whole sky. Using wavelength-tailored dichroics with extremely high transmittance, efficient imaging in six channels will be done simultaneously during each visit to each target. The separate spectroscopic mode, engaged through a flip-in mirror, uses an image slicer spectrograph. After a period of many visits to all targets, the resulting multidimensional movies can then be analyzed to a) measure the mass function of dark matter substructure; b) measure precise masses of the accreting black holes as well as the structure of their accretion disks and their environments over several decades of physical scale; and c) measure a combination of the local Hubble expansion and cosmological distances to unprecedented precision., Comment: Proceedings paper for the SPIE Astronomical Telescopes and Instrumentation 2008

We present 426 epochs of optical monitoring data spanning 1000 days from 2003 December to 2006 June for the gravitationally lensed quasar SDSS J1004+4112. The time delay between the A and B images is ΔtBA = 38.4 ± 2.0 days (Δχ2 = 4) in the expected sense that B leads A and the overall time ordering is C-B-A-D-E. The measured delay invalidates all published models. The models probably failed because they neglected the perturbations from cluster member galaxies. Models including the galaxies can fit the data well, but conclusions about the cluster mass distribution should await the measurement of the longer, and less substructure sensitive, delays of the C and D images. For these images, a delay of ΔtCB ≃ 681 ± 15 days is plausible but requires confirmation, while delays of ΔtCB 560 days and Δt AD > 800 days are required. We clearly detect microlensing of the A/B images, with the delay-corrected flux ratios changing from mB - mA -0-44 ± 0.01 mag in the first season to 0.29 ± 0.01 mag in the second season and 0.32 ± 0.01 mag in the third season. © 2007. The American Astronomical Society. All rights reserved.

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 ratio of the angular gravitational radius to 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., Comment: submitted

By directly probing mass distributions, gravitational lensing offers several new tests of the CDM paradigm. Lens statistics place upper limits on the dark matter content of elliptical galaxies. Galaxies built from CDM mass distributions are too concentrated to satisfy these limits, so lensing extends the ``concentration problem'' in CDM to elliptical galaxies. The central densities of the model galaxies are too low on ~10 pc scales to agree with the lack of central images in observed lenses. The flux ratios of four-image lenses imply a substantial population of dark matter clumps with a typical mass ~10^6 Msun. Thus, lensing implies the need for a mechanism that reduces dark matter densities on kiloparsec scales without erasing structure on smaller scales.

Many strong gravitational lenses lie in complex environments, such as poor groups of galaxies, that significantly bias conclusions from lens analyses. We are undertaking a photometric survey of all known galaxy-mass strong lenses to characterize their environments and include them in careful lens modeling, and to build a large, uniform sample of galaxy groups at intermediate redshifts for evolutionary studies. In this paper we present wide-field photometry of the environments of twelve lens systems with 0.24 < z_lens < 0.5. Using a red-sequence identifying technique, we find that eight of the twelve lenses lie in groups, and that ten group-like structures are projected along the line of sight towards seven of these lenses. Follow-up spectroscopy of a subset of these fields confirms these results. For lenses in groups, the group centroid position is consistent with the direction of the external tidal shear required by lens models. Lens galaxies are not all super-L_* ellipticals; the median lens luminosity is < L_*, and the distribution of lens luminosities extends 3 magnitudes below L_* (in agreement with theoretical models). Only two of the lenses in groups are the brightest group galaxy, in qualitative agreement with theoretical predictions. As in the local Universe, the highest velocity-dispersion groups contain a brightest member spatially coincident with the group centroid, whereas lower-dispersion groups tend to have an offset brightest group galaxy. This suggests that higher-dispersion groups are more dynamically relaxed than lower-dispersion groups and that at least some evolved groups exist by z ~ 0.5., Accepted for publication to the Astrophysical Journal. Figure 1 reduced in resolution. Requires emulateapj.sty. Table 6 to be published electronically. Revised version includes moderate changes to text, minor changes to conclusions, addition of one subsection

We study microlensing of the faint images that form close to the centers of strong gravitational lens galaxies. These central images, which have finally begun to yield to observations, naturally appear in dense stellar fields and may be particularly sensitive to fine granularity in the mass distribution. The microlensing magnification maps for overfocussed (i.e., demagnified) images differ strikingly from those for magnified images. In particular, the familiar "fold" and "cusp" features of maps for magnified images are only present for certain values of the fraction, f, of the surface mass density contained in stars. For central images, the dispersion in microlensing magnifications is generally larger than for normal (minimum and saddle) images, especially when the source is comparable to or larger than the stellar Einstein radius. The dispersion depends in a complicated way on f; this behaviour may hold the key to using microlensing as a probe of the relative densities of stars and dark matter in the cores of distant galaxies. Quantitatively, we predict that the central image C in PMN J1632-0033 has a magnification dispersion of 0.6 magnitudes for Rsrc/Rein, Comment: 12 pages; accepted in MNRAS; many new magnification maps and significantly expanded analysis of magnification map structure

SDSS J0924+0219 is the most glaring example of a gravitational lens with anomalous flux ratios: optical broad-band photometry shows image D to be a factor of 12 fainter than expected for smooth lens potentials. We report spectroscopy showing that the anomaly is present in the broad emission line flux ratios as well. There are differences between the emission line and continuum flux ratios: the A/D ratio is 10 in the broad Lyman-alpha line and 19 in the associated continuum. Known variability argues for the presence of microlensing. We show that microlensing can account for both the continuum and emission line flux ratios, if the broad emission line region is comparable in size to the Einstein radii of the microlenses. Specifically, we need the half-light radius of the broad-line region to be R_BLR, Comment: accepted in ApJ; two figures have been added, and the statistical analysis has been improved

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 first- and second-order corrections to the image positions, magnifications, and time delays. 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 GR in the weak-deflection limit. For the Reissner-Nordstrom 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., Comment: accepted in PRD