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We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point masses all reside in a single lens plane), but this problem remains open in the multiplane setting. We construct examples of K-plane point-mass gravitational lens ensembles that produce ∏ i = 1 K (5 g i − 5) images of a single background source, where gi is the number of point masses in the ith plane. This gives asymptotically (for large gi with K fixed) 5K times the minimal number of lensed images. Our construction uses Rhie's single-plane examples and a structured parameter-rescaling algorithm to produce preliminary systems of equations with the desired number of solutions. Utilizing the stability principle from the differential topology, we then show that preliminary (nonphysical) examples can be perturbed to produce physically meaningful examples while preserving the number of solutions. We provide numerical simulations illustrating the result of our construction, including positions of lensed images and the structure of critical curves and caustics. We observe an interesting "caustic of multiplicity" phenomenon that occurs in the nonphysical case and has a noticeable effect on the caustic structure in the physically meaningful perturbative case. [ABSTRACT FROM AUTHOR]

The Hubble Frontier Fields data, along with multiple data sets obtained by other telescopes, have provided some of the most extensive constraints on cluster lenses to date. Multiple lens modelling teams analyzed the fields and made public a number of deliverables. By comparing these results, we can then undertake a unique and vital test of the state of cluster lens modelling. Specifically, we see how well the different teams can reproduce similar magnifications and mass profiles. We find that the circularly averaged mass profiles of the fields are remarkably constrained (scatter |$\lt 5{{\ \rm per\ cent}}$|⁠) at distances of 1 arcmin from the cluster core, yet magnifications can vary significantly. Averaged across the six fields, we find a bias of −6 per  cent (−17 per cent) and a scatter of ∼40 per cent (∼65 per cent) at a modest magnification of 3 (10). Statistical errors reported by individual teams are often significantly smaller than the differences among all the teams, indicating the importance of continued systematics studies in cluster lensing. [ABSTRACT FROM AUTHOR]

Cluster lensing has become an important tool in the search for high-redshift galaxies through its ability to magnify sources. In order to determine the intrinsic properties of these galaxies, lensing mass models must be constructed to determine the magnification of the images. These models are traditionally 2D, focusing on the mass within the cluster and either ignoring or approximating any contribution from line-of-sight galaxies. In this paper, we present the first full set of 3D mass models of the six Hubble Frontier Fields and use them to test for systematic biases in magnifications due to using the traditional 2D approach. We find that omitting foreground or background galaxies causes image position offsets between 0.1 and 0.4 arcsec, a non-negligible fraction of the typical 0.3–0.7 arcsec residuals of current state-of-the-art models. We also find that median image magnifications can shift by up to 6 per cent, though it is dependent on the field. This can be alleviated in some cases by approximating the mass in the lensing plane, but a 5 per cent magnification bias still exists in other cases; image position offsets are also improved, but are still present at 0.10 arcsec. [ABSTRACT FROM AUTHOR]

Gravitational lens modelling 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 regularization, 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 regularization schemes that minimize derivatives of the source (using two finite difference methods) and introduce a scheme that minimizes deviations from an analytic source profile. Careful choice of gridding and regularization can reduce ‘discreteness noise’ in the χ2 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 realizations 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 per cent for a peak signal-to-noise ratio of 10, which decreases as the data quality improves. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Gravitational lensing provides a unique and powerful probe of the mass distributions of distant galaxies. Four-image lens systems with fold and cusp configurations have two or three bright images near a critical point. Within the framework of singularity theory, we derive analytic relations that are satisfied for a light source that lies a small but finite distance from the astroid caustic of a four-image lens. Using a perturbative expansion of the image positions, we show that the time delay between the close pair of images in a fold lens scales with the cube of the image separation, with a constant of proportionality that depends on a particular third derivative of the lens potential. We also apply our formalism to cusp lenses, where we develop perturbative expressions for the image positions, magnifications and time delays of the images in a cusp triplet. Some of these results were derived previously for a source asymptotically close to a cusp point, but using a simplified form of the lens equation whose validity may be in doubt for sources that lie at astrophysically relevant distances from the caustic. Along with the work of Keeton, Gaudi & Petters, this paper demonstrates that perturbation theory plays an important role in theoretical lensing studies. [ABSTRACT FROM AUTHOR]

We study microlensing of the faint images that form close to the centres 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 overfocused (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 mag for , or 0.3 mag for . For comparison, the dispersions are 0.5–0.6 mag for image B and 0.05–0.1 mag for image A, if ; and just 0.1 mag for B and 0.008 mag for A if . (The dispersions can be extrapolated to larger sources sizes as .) Thus, central images are more susceptible than other lensed images to microlensing and hence good probes for measuring source sizes. [ABSTRACT FROM AUTHOR]

Flux ratio ‘anomalies’ in quadruply imaged gravitational lenses can be explained with galactic substructure of the sort predicted by ΛCDM, but the strength and uniqueness of that hypothesis need to be further assessed. A good way to do that is to use the physical scale associated with the size of the source quasar, and its dependence on wavelength. We develop a toy model to study finite source effects in substructure lensing. Treating substructure as a singular isothermal sphere allows us to compute the images of a finite source analytically, and then to explore how the image configurations and magnifications depend on source position and size. Although simplified, our model yields instructive general principles: image positions and magnifications are basically independent of source size until the source is large enough to intersect a substructure caustic; even sources that are much larger than the substructure Einstein radius can be perturbed at a detectable level; and, most importantly, there is a tremendous amount to be learned from comparing image positions and magnifications at wavelengths that correspond to different source sizes. In a separate analysis, we carefully study four observed radio lenses to determine which of the images are anomalous. In B0712+472, the evidence for a radio flux ratio anomaly is marginal, but if the anomaly is real then image C is probably the culprit. In B1422+231, the anomaly is in image A. Interestingly, B2045+265 and B1555+375 both appear to have two anomalous images. Coincidentally, in each system one of the anomalies is in image C, and the other is in either image A or image B (both possibilities lead to acceptable models). It remains to be seen whether ΛCDM predicts enough substructure to explain multiple anomalies in multiple lenses. When we finally join our modelling results and substructure theory, we obtain lower bounds on the masses of the substructures responsible for the observed anomalies. The mass bounds are broadly consistent with expectations for ΛCDM. Perhaps more importantly, we outline various systematic effects in the mass bounds; poor knowledge of whether the substructure lies within the main lens galaxy or elsewhere along the line of sight appears to be the dominant systematic. [ABSTRACT FROM AUTHOR]

We study the impact of lens galaxy environments on the image separation distribution of lensed quasars. We account for both environmental convergence and shear, using a joint distribution derived from galaxy formation models calibrated by galaxy–galaxy lensing data and number counts of massive elliptical galaxies. We find that the external field enhances lensing probabilities, particularly, at large image separations; the increase is ∼30 per cent at and ∼200 per cent at , when we adopt a power-law source luminosity function . The enhancement is mainly driven by convergence, which boosts both the image separation and magnification bias (for a fixed lens galaxy mass). These effects have been neglected in previous studies of lens statistics. Turning the problem around, we derive the posterior convergence and shear distributions and point out that they are strong functions of image separation; lens systems with larger image separations are more likely to lie in dense environments. [ABSTRACT FROM AUTHOR]

It has been known for over a decade that many four-image gravitational lenses exhibit anomalous radio flux ratios. These anomalies can be explained by adding a clumpy cold dark matter (CDM) component to the background galactic potential of the lens. As an alternative, Evans & Witt (2003) recently suggested that smooth multipole perturbations provide a reasonable alternative to CDM substructure in some but not all cases. We generalize their method in two ways so as to determine whether multipole models can explain highly anomalous systems. We carry the multipole expansion to higher order, and also include external tidal shear as a free parameter. Fitting for the shear proves crucial to finding a physical (positive-definite density) model. For B1422+231, working to order (and including shear) yields a model that is physical but implausible. Going to higher order reduces global departures from ellipticity, but at the cost of introducing small-scale wiggles in proximity to the bright images. These localized undulations are more pronounced in B2045+265, where multipoles are required to smooth out large-scale deviations from elliptical symmetry. Such modes surely cannot be taken at face value; they must indicate that the models are trying to reproduce some other sort of structure. Our formalism naturally finds models that fit the data exactly, but we use B0712+472 to show that measurement uncertainties have little effect on our results. Finally, we consider the system B1933+503, where two sources are lensed by the same foreground galaxy. The additional constraints provided by the images of the second source render the multipole model unphysical. We conclude that external shear must be taken into account to obtain plausible models, and that a purely smooth angular structure for the lens galaxy does not provide a viable alternative to the prevailing CDM clump hypothesis. [ABSTRACT FROM AUTHOR]

We report Atacama Large Millimeter/Submillimeter Array (ALMA) observations of the dust continuum and [C ii ] emission of the host galaxy of J0439+1634, a gravitationally lensed quasar at z = 6.5. Gravitational lensing boosts the source-plane resolution to ∼0.″15 (∼0.8 kpc). The lensing model derived from the ALMA data is consistent with the fiducial model in Fan et al. based on HST imaging. The host galaxy of J0439+1634 can be well-fitted by a Sérsic profile consistent with an exponential disk, both in the far-infrared (FIR) continuum and the [C ii ] emission. The overall magnification is 4.53 ± 0.05 for the continuum and 3.44 ± 0.05 for the [C ii ] line. The host galaxy of J0439+1634 is a compact ultraluminous infrared galaxy, with a total star formation rate (SFR) of 1.56 × 103M⊙ yr−1 after correcting for lensing and an effective radius of 0.74 kpc. The resolved regions in J0439+1634 follow the "[C ii ] deficit," where the [C ii ]-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 is 130 km s−1 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. 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. [ABSTRACT FROM AUTHOR]

In the cold dark matter (CDM) picture of structure formation, galaxy mass distributions are predicted to have a considerable amount of structure on small scales. Strong gravitational lensing has proven to be a useful tool for studying this small-scale structure. Much of the attention has been given to detecting individual dark matter subhaloes through lens modelling, but recent work has suggested that the full population of subhaloes could be probed using a power spectrum analysis. In this paper, we quantify the power spectrum of small-scale structure in simulated galaxies, with the goal of understanding theoretical predictions and setting the stage for using measurements of the power spectrum to test dark matter models. We use a sample of simulated galaxies generated from the galacticus semi-analytic model to determine the power spectrum distribution first in the CDM paradigm and then in a warm dark matter scenario. We find that a measurement of the slope and amplitude of the power spectrum on galaxy strong lensing scales (k ∼ 1 kpc−1) could be used to distinguish between CDM and alternate dark matter models, especially if the most massive subhaloes can be directly detected via gravitational imaging. [ABSTRACT FROM AUTHOR]

The free streaming length of dark matter particles determines the abundance of structure on sub-galactic scales. We present a statistical technique, amendable to any parametrization of subhalo density profile and mass function, to probe dark matter on these scales with quadrupole image lenses. We consider a warm dark matter particle with a mass function characterized by a normalization and free streaming scale m hm. We forecast bounds on dark matter warmth for 120–180 lenses, attainable with future surveys, at typical lens (source) redshifts of 0.5 (1.5) in early-type galaxies with velocity dispersions of  220–270 km s-1. We demonstrate that limits on m hm deteriorate rapidly with increasing uncertainty in image fluxes, underscoring the importance of precise measurements and accurate lens models. For our forecasts, we assume the deflectors in the lens sample do not exhibit complex morphologies, so we neglect systematic errors in their modelling. Omitting the additional signal from line-of-sight haloes, our constraints underestimate the true power of the method. Assuming cold dark matter, for a low normalization, corresponding to the destruction of all subhaloes within the host scale radius, we forecast 2σ bounds on m hm (thermal relic mass) of 107.5 (5.0), 108 (3.6), and |$10^{8.5} \ (2.7) \ {\rm M}_{\odot } \left(\rm {keV}\right)$| for flux errors of |$2{{\ \rm per\ cent}}$|⁠, |$4{{\ \rm per\ cent}}$|⁠, and |$8{{\ \rm per\ cent}}$|⁠. With a higher normalization, these constraints improve to 107.2 (6.6), 107.5 (5.3), and |$10^{7.8} \ (4.3) \ {\rm M}_{\odot } \left(\rm {keV}\right)$| with 120 systems. We are also able to measure the normalization of the mass function, which has implications for baryonic feedback models and tidal stripping. [ABSTRACT FROM AUTHOR]

The best gravitational lenses for detecting distant galaxies are those with the largest mass concentrations and the most advantageous configurations of that mass along the line of sight. Our new method for finding such gravitational telescopes uses optical data to identify projected concentrations of luminous red galaxies (LRGs). LRGs are biased tracers of the underlying mass distribution, so lines of sight with the highest total luminosity in LRGs are likely to contain the largest total mass. We apply this selection technique to the Sloan Digital Sky Survey and identify the 200 fields with the highest total LRG luminosities projected within a 3.5 radius over the redshift range 0.1 ⩽ z ⩽ 0.7. The redshift and angular distributions of LRGs in these fields trace the concentrations of non-LRG galaxies. These fields are diverse; 22.5% contain one known galaxy cluster and 56.0% contain multiple known clusters previously identified in the literature. Thus, our results confirm that these LRGs trace massive structures and that our selection technique identifies fields with large total masses. These fields contain two to three times higher total LRG luminosities than most known strong-lensing clusters and will be among the best gravitational lensing fields for the purpose of detecting the highest redshift galaxies. [ABSTRACT FROM AUTHOR]

Abstract: 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 ≲6kpc), and span a range of morphologies consistent with early-types (though smaller in mass) to disky/late-type. The ratio of the black hole mass (, from the virial technique) to the bulge mass (, from the stellar luminosity) at 1≲ z ≲1.7 is broadly consistent with the local value; while at z ≳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 107 yr, with sizable scatter. [Copyright &y& Elsevier]

Certain configurations of massive structures projected along the line of sight (LOS) maximize the number of detections of gravitationally lensed z ∼ 10 galaxies. We characterize such LOSs with the étendue σμ, the area in the source plane magnified over some threshold μ. We use the Millennium I and Millennium XXL cosmological simulations to determine the frequency of high σμ beams on the sky, their properties, and efficient selection criteria. We define the best beams as having σμ > 3 > 2000 arcsec2, for a z ∼ 10 source plane, and predict 477 ± 21 such beams on the sky. The total mass in the beam and σμ > 3 are strongly correlated. After controlling for total mass, we find a significant residual correlation between σμ > 3 and the number of cluster-scale halos (>1014M☼h–1) in the beam. Beams with σμ > 3 > 2000 arcsec2, which should be best at lensing z ∼ 10 galaxies, are 10 times more likely to contain multiple cluster-scale halos than a single cluster-scale halo. Beams containing an A1689-like massive cluster halo often have additional structures along the LOS, including at least one additional cluster-scale (M200 > 1014M☼h–1) halo 28% of the time. Selecting beams with multiple, massive structures will lead to enhanced detection of the most distant and intrinsically faint galaxies. [ABSTRACT FROM AUTHOR]