Singular Isothermal Ellipsoid Research Articles

HOLISMOKES

Modeling of strongly gravitationally lensed galaxies is often required in order to use them as astrophysical or cosmological probes. With current and upcoming wide-field imaging surveys, the number of detected lenses is increasing significantly such that automated and fast modeling procedures for ground-based data are urgently needed. This is especially pertinent to short-lived lensed transients in order to plan follow-up observations. Therefore, we present in a companion paper a neural network predicting the parameter values with corresponding uncertainties of a singular isothermal ellipsoid (SIE) mass profile with external shear. In this work, we also present a newly developed pipeline glee_auto.py that can be used to model any galaxy-scale lensing system consistently. In contrast to previous automated modeling pipelines that require high-resolution space-based images, glee_auto.py is optimized to work well on ground-based images such as those from the Hyper-Suprime-Cam (HSC) Subaru Strategic Program or the upcoming Rubin Observatory Legacy Survey of Space and Time. We further present glee_tools.py, a flexible automation code for individual modeling that has no direct decisions and assumptions implemented on the lens system setup or image resolution. Both pipelines, in addition to our modeling network, minimize the user input time drastically and thus are important for future modeling efforts. We applied the network to 31 real galaxy-scale lenses of HSC and compare the results to traditional, Markov chain Monte Carlo sampling-based models obtained from our semi-autonomous pipelines. In the direct comparison, we find a very good match for the Einstein radius. The lens mass center and ellipticity show reasonable agreement. The main discrepancies pretrain to the external shear, as is expected from our tests on mock systems where the neural network always predicts values close to zero for the complex components of the shear. In general, our study demonstrates that neural networks are a viable and ultra fast approach for measuring the lens-galaxy masses from ground-based data in the upcoming era with ~105 lenses expected.

HOLISMOKES

Modeling of strong gravitational lenses is a necessity for further applications in astrophysics and cosmology. With the large number of detections in current and upcoming surveys, such as the Rubin Legacy Survey of Space and Time (LSST), it is pertinent to investigate automated and fast analysis techniques beyond the traditional and time-consuming Markov chain Monte Carlo sampling methods. Building upon our (simple) convolutional neural network (CNN), we present here another CNN, specifically a residual neural network (ResNet), that predicts the five mass parameters of a singular isothermal ellipsoid (SIE) profile (lens center x and y, ellipticity ex and ey, Einstein radius θE) and the external shear (γext, 1, γext, 2) from ground-based imaging data. In contrast to our previous CNN, this ResNet further predicts the 1σ uncertainty for each parameter. To train our network, we use our improved pipeline to simulate lens images using real images of galaxies from the Hyper Suprime-Cam Survey (HSC) and from the Hubble Ultra Deep Field as lens galaxies and background sources, respectively. We find very good recoveries overall for the SIE parameters, especially for the lens center in comparison to our previous CNN, while significant differences remain in predicting the external shear. From our multiple tests, it appears that most likely the low ground-based image resolution is the limiting factor in predicting the external shear. Given the run time of milli-seconds per system, our network is perfectly suited to quickly predict the next appearing image and time delays of lensed transients. Therefore, we use the network-predicted mass model to estimate these quantities and compare to those values obtained from our simulations. Unfortunately, the achieved precision allows only a first-order estimate of time delays on real lens systems and requires further refinement through follow-up modeling. Nonetheless, our ResNet is able to predict the SIE and shear parameter values in fractions of a second on a single CPU, meaning that we are able to efficiently process the huge amount of galaxy-scale lenses expected in the near future.

HOLISMOKES

Modeling the mass distributions of strong gravitational lenses is often necessary in order to use them as astrophysical and cosmological probes. With the large number of lens systems (≳105) expected from upcoming surveys, it is timely to explore efficient modeling approaches beyond traditional Markov chain Monte Carlo techniques that are time consuming. We train a convolutional neural network (CNN) on images of galaxy-scale lens systems to predict the five parameters of the singular isothermal ellipsoid (SIE) mass model (lens center x and y, complex ellipticity ex and ey, and Einstein radius θE). To train the network we simulate images based on real observations from the Hyper Suprime-Cam Survey for the lens galaxies and from the Hubble Ultra Deep Field as lensed galaxies. We tested different network architectures and the effect of different data sets, such as using only double or quad systems defined based on the source center and using different input distributions of θE. We find that the CNN performs well, and with the network trained on both doubles and quads with a uniform distribution of θE > 0.5″ we obtain the following median values with 1σ scatter: Δx = (0.00−0.30+0.30)″, Δy = (0.00−0.29+0.30)″, ΔθE = (0.07−0.12+0.29)″, Δex = −0.01−0.09+0.08, and Δey = 0.00−0.09+0.08. The bias in θE is driven by systems with small θE. Therefore, when we further predict the multiple lensed image positions and time-delays based on the network output, we apply the network to the sample limited to θE > 0.8″. In this case the offset between the predicted and input lensed image positions is (0.00−0.29+0.29)″ and (0.00−0.31+0.32)″ for the x and y coordinates, respectively. For the fractional difference between the predicted and true time-delay, we obtain 0.04−0.05+0.27. Our CNN model is able to predict the SIE parameter values in fractions of a second on a single CPU, and with the output we can predict the image positions and time-delays in an automated way, such that we are able to process efficiently the huge amount of expected galaxy-scale lens detections in the near future.

Discovery of Two Einstein Crosses from Massive Post-blue Nugget Galaxies at z > 1 in KiDS*

Abstract We report the discovery of two Einstein Crosses (ECs) in the footprint of the Kilo-Degree Survey (KiDS): KIDS J232940-340922 and KIDS J122456+005048. Using integral field spectroscopy from the Multi Unit Spectroscopic Explorer at the Very Large Telescope, we confirm their gravitational-lens nature. In both cases, the four spectra of the source clearly show a prominence of absorption features, hence revealing an evolved stellar population with little star formation. The lensing model of the two systems, assuming a singular isothermal ellipsoid (SIE) with external shear, shows that: (1) the two crosses, located at redshift z = 0.38 and 0.24, have Einstein radius R E = 5.2 kpc and 5.4 kpc, respectively; (2) their projected dark matter fractions inside the half effective radius are 0.60 and 0.56 (Chabrier initial mass function); (3) the sources are ultra-compact galaxies, R e ∼ 0.9 kpc (at redshift, z s = 1.59) and R e ∼ 0.5 kpc (z s = 1.10), respectively. These results are unaffected by the underlying mass density assumption. Due to size, blue color, and absorption-dominated spectra, corroborated by low specific star formation rates derived from optical–near-infrared spectral energy distribution fitting, we argue that the two lensed sources in these ECs are blue nuggets migrating toward their quenching phase.

A New Einstein Cross Gravitational Lens of a Lyman-break Galaxy

Abstract We report the study of an “Einstein Cross” configuration first identified in a set of HST images by Cerny et al. Deep spectroscopic observations obtained at the Spanish 10.4 m Gran Telescopio Canarias telescope, allowed us to demonstrate the lens nature of the system, that consists of a Lyman-break galaxy (LBG), not a quasi-stellar object as is usually the case, at z = 3.03 lensed by a galaxy at z = 0.556. Combining the new spectroscopy with the archival HST data, it turns out that the lens is an elliptical galaxy with M V = −21.0, effective radius 2.8 kpc, and stellar velocity dispersion σ = 208 ± 39 km s−1. The source is an LBG with Lyα luminosity ∼L* at that redshift. From the modeling of the system, performed by assuming a singular isothermal ellipsoid (SIE) with external shear, we estimate that the flux source is magnified about 4.5 times, and the velocity dispersion of the lens is km s−1, in good agreement with the value derived spectroscopically. This is the second case known of an Einstein cross of an LBG.

Uncertainties in Parameters Estimated with Neural Networks: Application to Strong Gravitational Lensing

Abstract In Hezaveh et al. we showed that deep learning can be used for model parameter estimation and trained convolutional neural networks to determine the parameters of strong gravitational-lensing systems. Here we demonstrate a method for obtaining the uncertainties of these parameters. We review the framework of variational inference to obtain approximate posteriors of Bayesian neural networks and apply it to a network trained to estimate the parameters of the Singular Isothermal Ellipsoid plus external shear and total flux magnification. We show that the method can capture the uncertainties due to different levels of noise in the input data, as well as training and architecture-related errors made by the network. To evaluate the accuracy of the resulting uncertainties, we calculate the coverage probabilities of marginalized distributions for each lensing parameter. By tuning a single variational parameter, the dropout rate, we obtain coverage probabilities approximately equal to the confidence levels for which they were calculated, resulting in accurate and precise uncertainty estimates. Our results suggest that the application of approximate Bayesian neural networks to astrophysical modeling problems can be a fast alternative to Monte Carlo Markov Chains, allowing orders of magnitude improvement in speed.

The discovery of a five-image lensed quasar at z = 3.34 using PanSTARRS1 and Gaia

Abstract We report the discovery, spectroscopic confirmation and mass modelling of the gravitationally lensed quasar system PS J0630−1201. The lens was discovered by matching a photometric quasar catalogue compiled from Pan-STARRS1 and Wide-field Infrared Survey Explorer photometry to the Gaia data release 1 catalogue, exploiting the high spatial resolution of the latter (full width at half-maximum ∼0.1 arcsec) to identify the three brightest components of the lensed quasar system. Follow-up spectroscopic observations with the William Herschel Telescope confirm the multiple objects are quasars at redshift zq = 3.34. Further follow-up with Keck adaptive optics high-resolution imaging reveals that the system is composed of two lensing galaxies and the quasar is lensed into an ∼2.8 arcsec separation four-image cusp configuration with a fifth image clearly visible, and a 1.0 arcsec arc due to the lensed quasar host galaxy. The system is well modelled with two singular isothermal ellipsoids, reproducing the position of the fifth image. We discuss future prospects for measuring time delays between the images and constraining any offset between mass and light using the faintly detected Einstein arcs associated with the quasar host galaxy.

SDSS J1640+1932: a spectacular galaxy–quasar strong lens system

We present Canada-France-Hawaii Telescope (CFHT) MegaCam observations of a galaxy-quasar strong gravitational lens system, SDSS J1640+1932. This system, located at z=0.195 (foreground elliptical galaxy) and z=0.778 (background quasar), was first visually identified by us in the Sloan Digital Sky Survey (SDSS) database. Our CFHT imaging with an angular resolution of 0.7$^{\prime \prime}$ clearly resolves 4 lensed images and a nearly complete Einstein ring. Modeling the system with a singular isothermal ellipsoid (SIE) total mass distribution, we find an Einstein radius of ${2.49^{\prime \prime}}_{-0.049}^{+0.063}$ enclosing a inferred mass of $7.25_{-0.29}^{+0.37}\times10^{11} M_{\odot}$. The quasar and its host galaxy have been magnified by a factor of 23, and the time delay relative to the leading image is determined to be 23.4-25.2 days. These parameters vary minimally when our model is fitted to the {\it{g}}-, {\it{r}}- or {\it{i}}-band images.

REVISITING STUDIES OF THE STATISTICAL PROPERTY OF A STRONG GRAVITATIONAL LENS SYSTEM AND MODEL-INDEPENDENT CONSTRAINT ON THE CURVATURE OF THE UNIVERSE

ABSTRACT In this paper, we use a recently compiled data set, which comprises 118 galactic-scale strong gravitational lensing (SGL) systems to constrain the statistical property of the SGL system as well as the curvature of the universe without assuming any fiducial cosmological model. Based on the singular isothermal ellipsoid (SIE) model of the SGL system, we obtain that the constrained curvature parameter is close to zero from the SGL data, which is consistent with the latest result of Planck measurement. More interestingly, we find that the parameter f in the SIE model is strongly correlated with the curvature . Neglecting this correlation in the analysis will significantly overestimate the constraining power of SGL data on the curvature. Furthermore, the obtained constraint on f is different from previous results: (68% confidence level [C.L.]), which means that the standard singular isothermal sphere (SIS) model (f = 1) is disfavored by the current SGL data at more than a C.L. We also divide all of the SGL data into two parts according to the centric stellar velocity dispersion and find that the larger the value of for the subsample, the more favored the standard SIS model is. Finally, we extend the SIE model by assuming the power-law density profiles for the total mass density, , and luminosity density, , and obtain the constraints on the power-law indices: and at a 68% C.L. When assuming the power-law index , this scenario is totally disfavored by the current SGL data, .

THE BOSS EMISSION-LINE LENS SURVEY. IV. SMOOTH LENS MODELS FOR THE BELLS GALLERY SAMPLE*

ABSTRACT We present Hubble Space Telescope F606W-band imaging observations of 21 galaxy-Lyα emitter lens candidates in the Baryon Oscillation Spectroscopic Survey Emission-Line Lens Survey (BELLS) for the GALaxy-Lyα EmitteR sYstems (BELLS GALLERY) survey. Seventeen systems are confirmed to be definite lenses with unambiguous evidence of multiple imaging. The lenses are primarily massive early-type galaxies (ETGs) at redshifts of approximately 0.55, while the lensed sources are Lyα emitters (LAEs) at redshifts from two to three. Although most of the lens systems are well fit by smooth lens models consisting of singular isothermal ellipsoids in an external shear field, a thorough exploration of dark substructures in the lens galaxies is required. The Einstein radii of the BELLS GALLERY lenses are, on average, 60% larger than those of the BELLS lenses because of the much higher source redshifts. This will allow for a detailed investigation of the radius evolution of the mass profile in ETGs. With the aid of the average ∼13× lensing magnification, the LAEs are frequently resolved into individual star-forming knots with a wide range of properties. They have characteristic sizes from less than 100 pc to several kiloparsecs, rest-frame far-UV apparent AB magnitudes from 29.6 to 24.2, and typical projected separations of 500 pc to 2 kpc.

Multiply imaged quasi-stellar objects in theGaiasurvey

Aims. We report a study on the statistical properties of the multiply imaged quasi-stellar objects (QSOs) to be detected within the Gaia survey. Methods. We considered two types of potential deflectors, the singular isothermal sphere (SIS) and the singular isothermal ellipsoid (SIE), to estimate the number of multiply imaged quasars as well as the normalized distributions of the redshifts of the lensed sources and of their associated deflectors. We also investigated the distribution of the lensing events as a function of their angular size and apparent magnitude. We compared the Gaia survey for multiply imaged quasars to typical ground-based surveys and to an ideal survey that would be carried out with a perfect instrument from space. Results. Of the 6.64 × 10 5 QSOs brighter than G = 20 to be detected by Gaia , we expect the discovery of about 2886 multiply imaged sources, 450 of which are expected to be produced by a late-type galaxy. We expect only ~1600 of these multiply imaged quasars to have an angular separation between their images that is large enough to be resolved from seeing-limited observations, and ~80 of them to have more than two lensed images.

Extensive light profile fitting of galaxy-scale strong lenses

We investigate the merits of a massive forward modeling of ground-based optical imaging as a diagnostic for the strong lensing nature of Early-Type Galaxies, in the light of which blurred and faint Einstein rings can hide. We simulate several thousand mock strong lenses under ground- and space-based conditions as arising from the deflection of an exponential disk by a foreground de Vaucouleurs light profile whose lensing potential is described by a Singular Isothermal Ellipsoid. We then fit for the lensed light distribution with sl_fit after having subtracted the foreground light emission off (ideal case) and also after having fitted the deflector's light with galfit. By setting thresholds in the output parameter space, we can decide the lens/not-a-lens status of each system. We finally apply our strategy to a sample of 517 lens candidates present in the CFHTLS data to test the consistency of our selection approach. The efficiency of the fast modeling method at recovering the main lens parameters like Einstein radius, total magnification or total lensed flux, is quite comparable under CFHT and HST conditions when the deflector is perfectly subtracted off (only possible in simulations), fostering a sharp distinction between the good and the bad candidates. Conversely, for a more realistic subtraction, a substantial fraction of the lensed light is absorbed into the deflector's model, which biases the subsequent fitting of the rings and then disturbs the selection process. We quantify completeness and purity of the lens finding method in both cases. This suggests that the main limitation currently resides in the subtraction of the foreground light. Provided further enhancement of the latter, the direct forward modeling of large numbers of galaxy-galaxy strong lenses thus appears tractable and could constitute a competitive lens finder in the next generation of wide-field imaging surveys.

Constraints on warm dark matter from weak lensing in anomalous quadruple lenses

We investigate the weak lensing effect by line-of-sight structures with a surface mass density of <~10^8 solar mass/arcsec^2 in QSO-galaxy quadruple lens systems. Using high-resolution N-body simulations in warm dark matter (WDM) models and observed four quadruple lenses that show anomalies in the flux ratios, we obtain constraints on the mass of thermal WDM, m_WDM>= 1.3keV(95%CL) assuming that the density of the primary lens is described by a singular isothermal ellipsoid (SIE). The obtained constraint is consistent with those from Lyman-$\alpha$ forests and the number counts of high-redshift galaxies at z>4. Our results show that WDM with a free-streaming comoving wavenumber k_{fs} <= 27 h/Mpc is disfavored as the major component of cosmological density at redshifts 0.5 <~ z <~ 4 provided that the SIE models describe the gravitational potentials of the primary lenses correctly.

Asymptotic solutions for the case of nearly symmetric gravitational lens systems

Gravitational lensing provides a powerful tool to determine the Hubble parameter H0 from the measurement of the time delay Δt between two lensed images of a background variable source. Nevertheless, knowledge of the deflector mass distribution constitutes a hurdle. We propose in the present work interesting solutions for the case of nearly symmetric gravitational lens systems. For the case of a small misalignment between the source, the deflector and the observer, we first consider power-law (ɛ) axially symmetric models for which we derive an analytical relation between the amplification ratio and source position which is independent of the power-law slope ɛ. According to this relation, we deduce an expression for H0 also irrespective of the value ɛ. Secondly, we consider the power-law axially symmetric lens models with an external large-scale gravitational field, the shear γ, resulting in the so-called ɛ−γ models, for which we deduce simple first-order equations linking the model parameters and the lensed image positions, the latter being observable quantities. We also deduce simple relations between H0 and observables quantities only. From these equations, we may estimate the value of the Hubble parameter in a robust way. Nevertheless, comparison between the ɛ−γ and singular isothermal ellipsoid (SIE) models leads to the conclusion that these models remain most often distinct. Therefore, even for the case of a small misalignment, use of the first-order equations and precise astrometric measurements of the positions of the lensed images with respect to the centre of the deflector enables one to discriminate between these two families of models. Finally, we confront the models with numerical simulations to evaluate the intrinsic error of the first-order expressions used when deriving the model parameters under the assumption of a quasi-alignment between the source, the deflector and the observer. From these same simulations, we estimate for the case of the ɛ−γ family of models that the standard deviation affecting H0 is which merely reflects the adopted astrometric uncertainties on the relative image positions, typically arcsec. In conclusions, we stress the importance of getting very accurate measurements of the relative positions of the multiple lensed images and of the time delays for the case of nearly symmetric gravitational lens systems, in order to derive robust and precise values of the Hubble parameter.

Constraints on cosmological models from lens redshift data

Strong lensing has developed into an important astrophysical tool for probing both cosmology and galaxies (their structures, formations, and evolutions). Now several hundreds of strong lens systems produced by massive galaxies have been discovered, which may form well-defined samples useful for statistical analyses. To collect a relatively complete lens redshift data from various large systematic surveys of gravitationally lensed quasars and check the possibility to use it as a future complementarity to other cosmological probes. We use the distribution of gravitationally-lensed image separations observed in the Cosmic Lens All-Sky Survey (CLASS), the PMN-NVSS Extragalactic Lens Survey (PANELS), the Sloan Digital Sky Survey (SDSS) and other surveys, considering a singular isothermal ellipsoid (SIE) model for galactic potentials as well as improved new measurements of the velocity dispersion function of galaxies based on the SDSS DR5 data and recent semi-analytical modeling of galaxy formation, to constrain two dark energy models ($\Lambda$CDM and constant $w$) under a flat universe assumption. We find that the current lens redshift data give a relatively weak constraint on the model parameters. However, by combing the redshift data with the baryonic acoustic oscillation peak and the comic macrowave background data, we obtain more stringent results, which show that the flat $\Lambda$ CDM model is still included at 1$\sigma$.

Effects of supermassive binary black holes on gravitational lenses

Recent observations indicate that many if not all galaxies host massive central black holes (BHs). In this paper we explore the influence of supermassive binary black holes (SMBBHs) on their actions as gravitational lenses. When lenses are modelled as singular isothermal ellipsoids, binary black holes change the critical curves and caustics differently as a function of distance. Each black hole can in principle create at least one additional image, which, if observed, provides evidence of black holes. By studying how SMBBHs affect the cumulative distribution of magnification for images created by black holes, we find that the cross section for at least one such additional image to have a magnification larger than $10^{-5}$ is comparable to the cross section for producing multiple-images in singular isothermal lenses. Such additional images may be detectable with high-resolution and large dynamic range maps of multiply-imaged systems from future facilities, such as the SKA. The probability of detecting at least one image (two images) with magnification above $10^{-3}$ is $\sim 0.2 \fBH$ ($\sim 0.05 \fBH$) in a multiply-imaged lens system, where $\fBH$ is the fraction of galaxies housing binary black holes. We also study the effects of SMBBHs on the core images when galaxies have shallower central density profiles (modelled as non-singular isothermal ellipsoids). We find that the cross section of the usually faint core images is further suppressed by SMBBHs. Thus their presence should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.

Effects of galaxy-halo alignment and adiabatic contraction on gravitational lens statistics

We study the strong gravitational lens statistics of triaxial cold dark matter (CDM) halos occupied by central early-type galaxies. We calculate the image separation distribution for double, cusp and quad configurations. The ratios of image multiplicities at large separations are consistent with the triaxial NFW model, and at small separations are consistent with the singular isothermal ellipsoid (SIE) model. At all separations, the total lensing probability is enhanced by adiabatic contraction. If no adiabatic contraction is assumed, naked cusp configurations become dominant at approximately 2.5'', which is inconsistent with the data. We also show that at small-to-moderate separations, the image multiplicities depend sensitively on the alignment of the shapes of the luminous and dark matter projected density profiles. In constrast to other properties that affect these ratios, the degree of alignment does not have a significant effect on the total lensing probability. These correlations may therefore be constrained by comparing the theoretical image separation distribution to a sufficiently large lens sample from future wide and deep sky surveys such as Pan-Starrs, LSST and JDEM. Understanding the correlations in the shapes of galaxies and their dark matter halo is important for future weak lensing surveys.

Constraints on the velocity profiles of galaxies from strong lensing statistics and semi-analytical modelling of galaxy formation

Semi-analytical models of galaxy formation can be used to predict the evolution of the number density of early-type galaxies as a function of the circular velocity at the virial radius, v_{c,vir}. Gravitational lensing probability and separation distribution on the other hand are sensitive to the velocity dispersion (or circular velocity) at about the effective radius. We adopt the singular isothermal ellipsoid (SIE) lens model to estimate the velocity dispersion at the effective radius. We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the velocity dispersions, \sigma_SIE, are related to v_{c,vir}. When we include both the lensing probability and separation distribution as our lensing constraints, we find \sigma_SIE /(200\kms) = [(1.17_{-0.26}^{+0.40}) v_{c,vir}/ (200\kms)]^{0.22^{+0.05}_{-0.04}} for 200\kms \la \sigma_SIE \la 260\kms; at \sigma_SIE = 200\kms, the ratio \sqrt{2} \sigma_SIE / v_{c,vir} is about 1.65^{+0.57}_{-0.37} (68% CL) but decreases to 0.65_{-0.12}^{+0.15} (68% CL) for \sigma_SIE = 260\kms. These results are consistent with those of Seljak (2002) obtained from galaxy-galaxy weak lensing for galaxies of around L_*. However, our results clearly suggest that the ratio must vary significantly as \sigma_SIE is varied and are marginally discrepant with the Seljak results at \sigma_SIE = 260\kms. The scaling \sigma_SIE \propto v_{c,vir}^{0.22\pm 0.05} is broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study. (Abridged)

Three‐dimensional Mapping of CDM Substructure at Submillimeter Wavelengths

The cold dark matter (CDM) structure formation model predicts that about 5-10 percent of a typical galactic halo of mass $\sim 10^{12} \ms$ is in substructures with masses $\lesssim 10^8 \ms$. To directly detect such substructures, we propose to observe dust continuum emission from a strongly lensed QSO-host galaxy using a large submillimeter interferometer array with a high angular resolution of $\sim 0.01$arcsec such as the planned Atacama Large Submillimeter Array (ALMA). To assess their observational feasibility, we numerically simulate millilensing of an extended circular source by a CDM substructure modeled as a tidally truncated singular isothermal sphere (SIS) embedded in a typical QSO-galaxy lens system, B1422+231, modeled as a singular isothermal ellipsoid (SIE) with an external constant shear and a constant convergence. Assuming an angular resolution of 0.01arcsec, we find that the angular positions of $\sim 10^8 \ms$ substructures at several kpc from the center of the macrolens halo can be directly measured if the size of the dust continuum emission region and the gradient of the surface brightness at the position of the perturber are sufficiently large. From the astrometric shift on a scale of a few times $10~$mas of an image perturbed by a subhalo with respect to an unperturbed macrolensed image, we can break the degeneracy between subhalo mass and distance provided that macrolensing parameters are determined from positions and fluxes of multiple images.

Constraints on the Velocity Dispersion Function of Early‐Type Galaxies from the Statistics of Strong Gravitational Lensing

We use the distribution of gravitationally lensed image separations observed in the Cosmic Lens All-Sky Survey (CLASS) and the PMN-NVSS Extragalactic Lens Survey (PANELS), which are (nearly) complete for the image separation range 03 ≤ Δθ ≤ 6'', to constrain a model velocity dispersion function (VF) of early-type galaxies. Assuming a current concordance cosmological model and adopting a singular isothermal ellipsoid (SIE) model for galactic potentials, we consider constraining both a characteristic velocity dispersion (parameter σ*) and the shape of the function (parameters α and β; from Sheth et al.) for 0.3 z 1. If all three parameters are allowed to vary, then none of the parameters can be tightly constrained by the lensing data because of the small size of the sample. If we fix the shape of the function by either the SDSS local stellar VF or an inferred local stellar VF based on the SSRS2 galaxy sample, then the constrained values of σ* are nearly equal to the corresponding stellar values; we have fSIE/center(≡ σ*,SIE/σ*,center) = 0.90 ± 0.18 (SDSS) or 1.04 ± 0.19 (SSRS2), assuming nonevolution of the function between the present epoch and z ~ 0.6. Finally, using only the CLASS statistical sample (from Browne et al.), and thus including an absolute multiple-imaging probability, we find that the SDSS stellar VF may have significantly underestimated the abundance of morphologically early-type galaxies.