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 et al. (2005), this paper demonstrates that perturbation theory plays an important role in theoretical lensing studies.
We present here an alternative method for calculating magnifications in gravitational lensing calculations -- the Ray Bundle method. We provide a detailed comparison between the distribution of magnifications obtained compared with analytic results and conventional ray-shooting methods. The Ray Bundle method provides high accuracy in the weak lensing limit, and is computationally much faster than (non-hierarchical) ray shooting methods to a comparable accuracy. The Ray Bundle method is a powerful and efficient technique with which to study gravitational lensing within realistic cosmological mo…
To determine the magnification of an extended source caused by gravitational lensing one has to perform a two-dimensional integral over point-source magnifications in general. Since the point-source magnification jumps to an infinite value on caustics, special care is required. For a uniformly bright source, it has been shown earlier that the calculation simplifies if one determines the magnification from the area of the images of the extended source by applying Green's theorem so that one ends up with a one-dimensional integration over the image boundaries. This approach is discussed here in …
We present a generalization of the concept of magnification bias for gravitationally-lensed quasars, in which the quasars are selected by flux in more than one wavelength band. To illustrate the principle, we consider the case of two-band selection, in which the fluxes in the two bands are uncorrelated, perfectly correlated, or correlated with scatter. For uncorrelated fluxes, we show that the previously-held result - that the bias is the product of the single-band biases - is generally false. We demonstrate some important properties of the multi-band magnification bias using model luminosity …
We prove that, independent of the choice of a lens model, the total signed magnification always sums to zero for a source anywhere in the four-image regions of swallowtail, elliptic umbilic, and hyperbolic umbilic caustics. This is a more global and higher-order analog of the well-known fold and cusp magnification relations, in which the total signed magnification in the two-image region of the fold, and the three-image region of the cusp, are both always zero. As an application, we construct a lensing observable for the hyperbolic umbilic magnification relation and compare it with the corresp…
We present a new method of studying quadruple lenses in elliptical power-law potentials parameterized by $ψ(x,y) \propto (x^2+y^2/q^2)^{β/2}/β(0 \leq β< 2)$. For this potential, the moments of the four image positions weighted by signed magnifications (magnification times parity) have very simple properties. In particular, we find that the zeroth moment -- the sum of four signed magnifications satisfies $\simeq 2/(2-β)$; the relation is exact for $β=0$ (point-lens) and $β=1$ (isothermal potential), independent of the axial ratio. Similar relations can be derived when a shear is present along t…
We present a rigorous, detailed study of the generic, quantitative properties of gravitational microlensing near cusp catastrophes. We derive explicit formulas for the total magnification and centroid of the images created for sources outside, on, and inside the cusped caustic. We obtain results on how the image magnifications scale with respect to separation from the cusped caustic for arbitrary source positions. Along the axis of symmetry of the cusp, the total magnification $μ$ scales as $μ\propto u^{-1}$, where $u$ is the distance of the source from the cusp, whereas perpendicular to this …
Recent observations of galaxy luminosity profiles and dark matter simulations find luminosity and mass distributions characterized by central cusps rather than finite core radii. We introduce and implement a set of cusped ellipsoidal lens models which include limits similar to the Jaffe, Hernquist, ηand NFW models and apply them to the gravitational lenses APM 08279+5255 and B 1933+503. A successful model of APM 08279+5255 with its central, odd image requires a very shallow cusp, $γ\ltorder 0.4$ where $ρ\propto r^{-γ}$ as $r\to 0$, which is similar to a core rather than the favored $1 \ltorder…
The cusps of the caustics of any gravitational lens model can be classified into positive and negative ones. This distinction lies on the parity of the images involved in the creation/destruction of pairs occurring when a source crosses a caustic in a cusp. In this paper, we generalize the former definition of the sign of the cusps. Then we apply it to the binary lens. We demonstrate that the cusps on the axis joining the two lenses are positive while the others are negative. To achieve our objective, we combine catastrophe theory, usually employed in the derivation of the properties of causti…
We show that the gravitational magnification factor averaged over all configurations of lenses in a locally inhomogeneous universe satisfy a second order differential equation with redshift $z$ by taking the continuous limit of multi-plane gravitational lens equation (the number $N$ of lenses $\to\infty$) and that the gravitationally magnified Dyer-Roeder distance in a clumpy universe becomes to that of the Friedmann-Lema\^ıtre universe for arbitrary values of the density parameter $Ω_{0}$ and of a mass fraction $\barα$ (smoothness parameter).
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 dif…
Weak lensing of high-redshift Type Ia supernovae induces an external dispersion in their observed standard candle brightnesses, comparable in magnitude to the intrinsic dispersion for redshifts z>1. The same matter fluctuations responsible for the magnification of distant supernovae also generate shear in the images of background galaxies. We investigate the possibility of using lensing shear maps constructed from galaxies surrounding the supernovae as a means of correcting the lensing-induced magnification dispersion. We find that a considerable fraction of the lensing dispersion derives from…
We demonstrate that the maximum-entropy method for gravitational lens reconstruction presented in Bridle et al. (1998) may be applied even when only shear \emph{or} magnification information is present. We also demonstrate that the method can easily handle irregularly shaped observing fields and, because shear is a non-local function of the lensing mass, reconstructions that use shear information can successfully bridge small gaps in observations. For our simulations we use a mass density distribution that is realistic for a z=0.4 cluster of total mass around 10^15 h^-1 M_solar. Using HST-qual…
The inability of standard models to explain the flux ratios in many 4-image gravitational lenses has been cited 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 associated with a source near a cusp caustic, the sum of the signed magnifications should approximately vanish. We derive realistic upper bounds on the sum, and argue th…
We demonstrate that for several of the gravitational lens models used to describe galaxies, there exists a quantity we dub the magnification invariant, equaling the sum of the signed magnifications of the images, that is a constant when the image multiplicity is a maximum. This invariant is independent of most of the model parameters, and is independent of the source position as long as the source lies inside the caustic. It is not necessary to solve the lens equations to compute this invariant. For quad lenses, it is usually easy to assign image parities and thus one can compute the sum of th…
We consider the influence of (milli/micro)lensing on the spectra of lensed QSOs. We propose a method for the observational detection of microlensing in the spectra of lensed QSOs and apply it to the spectra of the three lensed QSOs (PG 1115+080, QSO 1413+117 and QSO 0957+561) observed with Hubble Space Telescope (HST). We find that the flux ratio between images A1 and A2 of PG 1115+080 is wavelength-dependent and shows differential magnification between the emission lines and the continuum. We interpret this magnification as arising from millilensing. We also find that the temporal variations …
Galactic sized gravitational lenses are simulated by combining a cosmological N-body simulation and models for the baryonic component of the galaxy. The lens caustics, critical curves, image locations and magnification ratios are calculated by ray-shooting on an adaptive grid. When the source is near a cusp in a smooth lens' caustic the sum of the magnifications of the three closest images should be close to zero. It is found that in the observed cases this sum is generally too large to be consistent with the simulations implying that there is not enough substructure in the simulations. This s…
When a gravitationally lensed source crosses a caustic, a pair of images is created or destroyed. We calculate the mean number of such pairs of micro-images $<n>$ for a given macro-image of a gravitationally lensed point source, due to microlensing by the stars of the lensing galaxy. This quantity was calculated by Wambsganss, Witt & Schneider (1992) for the case of zero external shear, $γ=0$, at the location of the macro-image. Since in realistic lens models a non-zero shear is expected to be induced by the lensing galaxy, we extend this calculation to a general value of $γ$. We find a comple…
We discuss the classic theorem according to which a gravitational lens always produces a total magnification greater than unity. This theorem seems to contradict the conservation of total flux from a lensed source. The standard solution to this paradox is based on the exact definition of the reference 'unlensed' situation. We calculate magnifications and amplifications for general lensing scenarios not limited to regions close to the optical axis. In this way the formalism is naturally extended from tangential planes for the source and lensed images to complete spheres. We derive the lensing p…
Several gravitationally lensed quasars are observed with anomalous magnifications in pairs of images that straddle a critical curve. Simple theoretical arguments suggest that the magnification of these images should be approximately equivalent, whereas one image is observed to be significantly demagnified. Microlensing provides a possible explanation for this discrepancy. There are two key parameters when modelling this effect. The first, the fraction of smooth matter in the lens at the image positions, has been explored by Schechter and Wambsganss (2002). They have shown that the anomalous fl…
We propose a new, two-stage algorithm for inverting strong gravitational lenses. The key to the algorithm is decoupling the effects of lens magnification and intrinsic structure in the background source in the appearance of lensed arcs. First, the distribution of mass on the deflector plane and the geometry of the source-deflector-observer optical system are established. This is done by numerically simulating the lensing of light past a parametric mass model, and adjusting the handful of model parameters to match the positions and shapes of the observed and simulated lensed arcs. At the same t…
Gravitational lensing by the dual cusp catastrophes of the cold dark matter (CDM) caustic rings at cosmological distances may provide the tantalizing opportunity to detect CDM indirectly, and discriminate between axions and weakly interacting massive particles (WIMPs). Caustics are places where the CDM particles are naturally focussed. Our focus is upon the caustic rings which are closed tubes whose cross-section is an elliptic umbilic catastrophe with three dual cusps. A caustic ring has a specific density profile, a specific geometry and, therefore, precisely calculable gravitational lensing…
We study the statistics of large-separation gravitational lens systems produced by non-spherical halos in the Cold Dark Matter (CDM) model. Specifically, we examine how the triaxiality of CDM halos affects the overall lensing probabilities and the relative numbers of different image configurations (double, quadruple, and naked cusp lenses). We find that triaxiality significantly enhances lensing probabilities by a factor of ~2-4, so it cannot be ignored. If CDM halos have central density slopes alpha<~1.5, we predict that a significant fraction (>~20%) of large-separation lenses should have na…
For a wide variety of cosmological models characterized by the cosmic mass density $Ω_M$ and the normalized cosmological constant $Ω_Λ$, we derive an analytic expression for the estimate of magnification cross-sections by an ensemble of isothermal spheres as models of galactic mass distributions. This provides a simple approach to demonstrate how the lensing probability by galaxies depends on the cosmological parameters. An immediate consequence is that, while a non-zero cosmological constant indeed leads to a significant increase of the lensing probability as it has been shown in the literatu…
We present detailed calculations of the magnification distribution, including both weak and strong lensing, using very recent solutions of the Dyer-Roeder (1973) equation for light propagation in a inhomogeneous universe with a cosmological constant and up-to-date models for the evolving cosmological distribution of dark matter halos. We address in particular the effect of a non-zero cosmological constant, of different density profiles of lenses, and of finite sizes of lensed sources. We show that, if dark matter fluctuations are normalized to the local cluster abundance, in the presence of a …
Image flux ratio anomalies have been attributed to substructures within the gravitational lens and to small mass halos (M <~ 10^10 Msun) in intergalactic space. In this paper, analytic calculations are presented that help in the understanding of how intergalactic halos affect magnification ratios. It is found that intergalactic halos can produce anomalies at a similar level to those that are observed. Intergalactic halos with masses <10^10 Msun are expected to cause relative deflections between images of order 10 milliarcseconds, which are then magnified by the primary lens. They will also cau…
CONTEXT: Gravitationally lensed quasars constitute an independent tool to derive H0 through time-delays; they offer as well the opportunity to study the mass distribution and interstellar medium of their lensing galaxies and, through microlensing they also allow one to study details of the emitting source. AIMS: For such studies, one needs to have an excellent knowledge of the close environment of the lensed images in order to model the lensing potential: this means observational data over a large field-of-view and spectroscopy at high spatial resolution. METHODS: We present VIMOS integral fie…
We present flux-ratio curves of the fold and cusp (i.e. close multiple) images of six JVAS/CLASS gravitational lens systems. The data were obtained over a period of 8.5 months in 2001 with the Multi-Element Radio-Linked Interferometer Network (MERLIN) at 5-GHz with 50 mas resolution, as part of a MERLIN Key-Project. Even though the time delays between the fold and cusp images are small (<~1 day) compared to the time-scale of intrinsic source variability, all six lens systems show evidence that suggests the presence of extrinsic variability. In particular, the cusp images of B2045+265 -- regard…
A method is developed to evaluate the magnifications of the images of galaxies with lensing potentials stratified on similar concentric ellipses. A simple contour integral is provided which enables the sums of the magnifications of even parity or odd parity or the central image to be easily calculated. The sums for pairs of images vary considerably with source position, while the signed sums can be remarkably uniform inside the tangential caustic in the absence of naked cusps. For a family of models in which the potential is a power-law of the elliptic radius, the number of visible images is f…
We report the discovery of a new quadruply imaged quasar surrounded by an optical Einstein ring candidate. Spectra of the different components of 1RXS J113155.4-123155 reveal a source at z=0.658. Up to now, this object is the closest known gravitationally lensed quasar. The lensing galaxy is clearly detected. Its redshift is measured to be z=0.295. Additionally, the total V magnitude of the system has varied by 0.3 mag between two epochs separated by 33 weeks. The measured relative astrometry of the lensed images is best fitted with an SIS model plus shear. This modeling suggests very high mag…
We identify a third image in the unique quasar lens SDSS J1029+2623, the second known quasar lens produced by a massive cluster of galaxies. The spectrum of the third image shows similar emission and absorption features, but has a redder continuum than the other two images which can be explained by differential extinction or microlensing. We also identify several lensed arcs. Our observations suggest a complicated structure of the lens cluster at z~0.6. We argue that the three lensed images are produced by a naked cusp on the basis of successful mass models, the distribution of cluster member …
A contour plot of positive iso-J curves is shown for a gravitational binary lens epsilon_2 = 0.1883 and ell = 0.687. The caustic curve is made of 4-cusped central caustic (so-called "stealth bomber") on the lens axis and two triangular caustics off the lens axis. The cusps of the trioids are all negative cusps. The positive image magnification contours outside the positive cusps on the lens axis are elongated along the symmetry axis. It is in contrast with figure 4 of astro-ph/0206162 (Gaudi and Petters) where the contours appear to contract along the symmetry axis. We apologize if our citatio…
Criticality in graviational microlensing is an everyday issue because that is what generates microlensing signals which may be of photon-challenged compact objects such as black holes or planetary systems ET calls home. The criticality of these quasi-analytic lenses is intrinsically quadratic, and the critical curve behaves as a mirror generating two mirror images along the image line (parallel +/- E_-) at the same distances from the critical curve in the opposite sides. At the (pre)cusps where the caustic curve "reflects" and develops cusps, however, "would-be" two pairs of quadratic images "…
We discuss new observational constraints on the abundance of faint high redshift Lyman-alpha emitters secured from a deep Keck near-infrared spectroscopic survey which utilizes the strong magnification provided by lensing galaxy clusters at intermediate redshift (z=0.2-0.5). In each of 9 clusters we have undertaken a systematic `blind' search for line emission with NIRSPEC in the J-band within carefully-selected regions which offer very high magnifications (>10-50x) for background sources with redshifts z~10. The high magnification enables the detection of emission at unprecedented limits (10^…
Strong gravitational lensing of distant supernovae (SNe), particularly Type Ia's, has some exploitable properties not available when other sorts of cosmologically distant sources are lensed. One such property is that the ``standard candle'' nature of the SN at peak brightness allows a direct determination of the lensing magnification factor for each well observed image. Another is that the duration of an SN event is of the same order as the differential time delays between the various lens images. The comparable time scales of the photometric event and the time delay invite a variety of applic…
When the logarithmic slope of the galaxy counts is lower than 0.4 (this is the case in all filters at high magnitude), the magnification bias due to the lens makes the number density of objects decrease. Consequently, the radial distribution shows a typical depletion curve. We present simulations of depletion curves obtained for a variety of different lens models and we show how the model parameters affect the depletion area.
We study the problem of gravitational lensing by an isothermal elliptical density galaxy in the presence of a tidal perturbation. When the perturbation is fairly strong and oriented near the galaxy's minor axis, the lens can produce image configurations with six or even eight highly magnified images lying approximately on a circle. We classify the caustic structures in the model and identify the range of models that can produce such lenses. Sextuple and octuple lenses are likely to be rare because they require special lens configurations, but a full calculation of the likelihood will have to i…
Using HST WFPC2 V- and R-band data of the z=0.37 cluster MS1512+36 we show that the z=2.72 `protogalaxy' cB58 is not extraordinarily luminous intrinsically but lensed into a gravitational fold arc by the cluster. The arc has a surface brightness weighted axis ratio of 1:7, is marginally resolved in width and about 3 arcsec long. Its counterimage was identified and found to be very compact r(1/2) = 2.4-4.0 / h50 kpc in a q0=0.05 cosmology). In addition, we found three further multiple image systems, one with five and two with three images each. The positions of the multiple images is explained …
Gravitational lensing of a background source by a foreground galaxy lens occasionally produces four images of the source. The cusp and the fold relations impose conditions on the ratios of magnifications of these four-image lenses. In this theoretical investigation, we explore the sensitivity of these relations to the presence of substructure in the lens. Starting with a smooth lens potential, we add varying amounts of substructure, while keeping the source position fixed, and find that the fold relation is a more robust indicator of substructure than the cusp relation for the images. This rob…
We discuss a method to build up and study a sample of distant galaxies, with 2<z<5, using the gravitational amplification effect in cluster-lenses for which the mass distribution is well known. The candidates are selected close to the critical lines at high-z, through photometric redshift estimates on a large wavelength range. With respect to other methods, this procedure allows to reduce the selection biases in luminosity, thanks to the cluster magnification of ~1 magnitude, and towards active SFRs when the near-IR domain is included. Some new results obtained with this technique are presente…
It has been argued that the flux anomalies detected in gravitationally lensed QSOs are evidence for substructures in the foreground lensing haloes. In this paper we investigate this issue in greater detail focusing on the Cusp relation which corresponds to images of a source located to the cusp of the inner caustic curve. We use numerical simulations combined with a Monte Carlo approach to study the effects of the expected power law distribution of substructures within LCDM haloes on the multiple images. Generally, the high number of anomalous flux ratios in the cusp configurations is unlikely…
Some aspects of gravitational lensing by large scale structure (LSS) are investigated. We show that lensing causes the damping tail of the cosmic microwave background (CMB) power spectrum to fall less rapidly with decreasing angular scale than previously expected. This is due to a transfer of power from larger to smaller angular scales which produces a fractional change in power spectrum that increases rapidly beyond $\ell \sim 2000$. We also find that lensing produces a nonzero mean magnification of structures on surfaces of constant redshift if weighted by area on the sky. This is a result o…
In this letter, we exploit the gravitational potential of the rich cluster A2218 as a magnifying glass. We demonstrate that the magnification due to the cluster allows us to observe distant background galaxies at a comparable level of detail to galaxies at z~0.1. Using the GMOS Integral Field Unit on Gemini North we observed the spatially-resolved [OII](3727) emission line spectrum for a lensed disk-galaxy at z=1.034. Using a detailed model for the cluster mass distribution, we are able to correct for the lensing by the cluster and reconstruct the source morphology. We find that the overall ma…
(Abridged) In this paper we refine the theory of microlensing for a planar distribution of point masses. We derive the macroimage magnification distribution P(A) at high magnification (A-1 >> tau^2) for a low optical depth (tau << 1) lens distribution by modeling the illumination pattern as a superposition of the patterns due to individual ``point mass plus weak shear'' lenses. We show that a point mass plus weak shear lens produces an astroid- shaped caustic and that the magnification cross-section obeys a simple scaling property. By convolving this cross-section with the shear distribution, …
We study the statistical properties of magnification perturbations by substructures in strong lensed systems using linear perturbation theory and an analytical substructure model including tidal truncation and a continuous substructure mass spectrum. We demonstrate that magnification perturbations are dominated by perturbers found within a tidal radius of an image, and that sizable magnification perturbations may arise from small, coherent contributions from several substructures within the lens halo. We find that the root-mean-square (rms) fluctuation of the magnification perturbation is 10% …
We discuss boson stars as possible gravitational lenses and study the lensing effect by these objects made of scalar particles. The mass and the size of a boson star may vary from an individual Newtonian object similar to the Sun to the general relativistic size and mass of a galaxy close to its Schwarzschild radius. We assume boson stars to be transparent which allows the light to pass through them though the light is gravitationally deflected. We assume boson stars of the mass $M = 10^{10}M_\odot$ to be on non-cosmological distance from the observer. We discuss the lens equation for these st…
The flux from distant type Ia supernovae (SN) is likely to be amplified or de-amplified by gravitational lensing due to matter distributions along the line-of-sight. A gravitationally lensed SN would appear brighter or fainter than the average SN at a particular redshift. We estimate the magnification of 26 SNe in the GOODS fields and search for a correlation with the residual magnitudes of the SNe. The residual magnitude, i.e. the difference between observed and average magnitude predicted by the "concordance model" of the Universe, indicates the deviation in flux from the average SN. The lin…
To date, the study of high-magnification gravitational lensing effects of galaxy clusters has focused upon the grossly distorted, luminous arc-like features formed in massive, centrally condensed clusters. We investigate the formation of a different type of image, highly magnified yet undistorted, in two widely employed cluster mass density profiles; an isothermal sphere with a core, and a universal dark matter halo profile derived from numerical simulations of Navarro et al. We examine the properties of images of extended sources produced by these two cluster profiles, paying particular atten…
I elaborate on my prediction that an indirect detection of cold dark matter (CDM) may be possible by observing the gravitational lensing effects of the CDM cusp caustics at cosmological distances. Cusps in the distribution of CDM are plentiful once density perturbations enter the nonlinear regime of structure formation. Caustic ring model of galactic halo formation provides a well defined density profile and geometry near the cusps of the caustic rings. I calculate the gravitational lensing effects of the cusps in this model. As a pointlike background source passes behind a cusp of a cosmologi…
