Lens Inversion Research Articles

Extrapolating the projected potential of gravitational lens models: property-preserving degeneracies

ABSTRACT While gravitational lens inversion holds great promise to reveal the structure of the light-deflecting mass distribution, both light and dark, the existence of various kinds of degeneracies implies that care must be taken when interpreting the resulting lens models. This article illustrates how thinking in terms of the projected potential helps to gain insight into these matters. Additionally it is shown explicitly how, when starting from a discretized version of the projected potential of one particular lens model, the technique of quadratic programming can be used to create a multitude of equivalent lens models that preserve all or a subset of lens properties. This method is applied to a number of scenarios, showing the lack of grasp on the mass outside the strong lensing region, revisiting mass redistribution in between images, and applying this to a recent model of the SDSS J1004+4112 cluster, as well as illustrating the generalized mass sheet degeneracy and source-position transformation. In the case of J1004, we show that this mass redistribution did not succeed at completely eliminating a dark mass clump recovered by grale near one of the quasar images.

Assessing the potential of using a virtual Veselago lens in quantitative microwave imaging

This study explores the potential of implementing the focusing properties of a virtual ideal Veselago lens within a standard free-space microwave imaging scenario. To achieve this, the virtual lens is introduced as an inhomogeneous numerical background for the inverse source problem. This numerical Vesealgo lens is incorporated into the incident and scattered field decomposition, resulting in a new data equation that involves the Veselago lens Green’s function. In addition to the contrast sources within the object-of-interest, the lens introduces virtual contrast sources along the lens boundaries that depend on the total tangential magnetic field. It is shown that a surface integral contribution that takes into account these surface contrast sources must be added to the collected free-space data before one can invert using the well-conditioned Veselago lens inversion operator. A preliminary investigation of the accuracy to which this surface integral contribution must be computed is performed using additive Gaussian noise. Results show that an error of less than one percent is required to achieve imaging performance similar to utilizing an actual Veselago lens. All results are performed within a 2D simulation environment.

Free-form and hybrid lens models for SDSS J1004+4112: substructure and central image time delay constraints

ABSTRACT SDSS J1004+4112 is a well-studied gravitational lens with a recently measured time delay between its first and fourth arriving quasar images. Using this new constraint, we present updated free-form lens reconstructions using the lens inversion method grale, which only uses multiple image and time delay data as inputs. In addition, we obtain hybrid lens reconstructions by including a model of the brightest cluster galaxy as a Sersic lens. For both reconstructions, we use two sets of images as input: one with all identified images, and the other a revised set leaving out images that have been potentially misidentified. We also develop a source position optimization Markov chain Monte Carlo (MCMC) routine, performed on completed grale runs, that allows each model to better match observed image positions and time delays. All the reconstructions produce similar mass distributions, with the hybrid models finding a steeper profile in the centre. Similarly, all the mass distributions are fitted by the Navarro–Frenk–White profile, finding results consistent with previous parametric reconstructions and those derived from Chandra X-ray observations. We identify an ∼5 × 1011 M⊙ substructure apparently unaffiliated with any cluster member galaxy and present in all our models, and study its reality. Using our free-form and hybrid models, we predict a central quasar image time delay of ∼2980 ± 270 and ∼3280 ± 215 d, respectively. A potential future measurement of this time delay will, while being an observational challenge, further constrain the steepness of the central density profile.

What multiple images say about the large-scale mass maps of galaxy clusters

ABSTRACT All lens modelling methods, simply parametrized, hybrid, and free-form, use assumptions to reconstruct galaxy clusters with multiply imaged sources, though the nature of these assumptions (priors) can differ considerably between methods. This raises an important question in strong lens modelling: how much information about the mass model comes from the lensed images themselves, and how much is a consequence of model priors. One way to assess the relative contributions of the lensing data versus model priors is to estimate global lens properties through images alone, without any prior assumptions about the mass distribution. This is our approach. We use 200 mock cluster lenses, half of which have substructures, which vary from clumpy and compact to smooth and extended, a simulated cluster Ares, and real clusters Abell 1689 and RX J1347.5−1145 to show that the centre, ellipticity, and position angle can be estimated quite well, and nearly perfectly for weakly substructured clusters, implying that the recovery of these properties is largely driven by the images, not priors. However, the correlation between the true and image-estimated amount of substructure has a lot of scatter, suggesting that multiple images do not uniquely constrain substructure. Therefore, in general, lens model priors have a stronger effect on smaller scales. Our analysis partly explains why reconstructions using different methodologies can produce qualitatively different mass maps on substructure scales. Our analysis is not meant to aide or replace lens inversion methods, but only to investigate what cluster properties are constrained with multiple images.

An excursion into the core of the cluster lens Abell 1689

ABSTRACT Abell 1689 is a well-studied cluster of galaxies and one of the largest gravitational lens systems ever observed. We have obtained a reconstruction of the cluster Abell 1689 using grale, a free-form lens inversion method that relies exclusively on the multiple image data. Non-inclusion of any data related to cluster member galaxies ensures an unbiased measure of the mass distribution, which is the most notable feature of free-form methods like grale. We used two different sets of multiple image systems from the available strong lensing data – one containing only the secure systems (107 images) and the other containing all available systems, only excluding some very non-secure systems (151 images). For the very well-constrained central ∼100 kpc region of the cluster, we made a detailed comparison of the grale reconstructed lensing mass and stellar mass retrieved by the Spectral Energy Distribution (SED) fitting software fast++. We found a light-unaccompanied mass peak in this region, whose existence, while tentative, is favoured by the distribution of nearby images that are local maxima in the Fermat potential. However, further tests, using different methodologies are needed to confirm the reality of this feature. If it is shown to be real, this light-unaccompanied mass peak is consistent with dark matter self-interaction cross-section σ ≲ 1 cm2 g−1, while being in tension with larger cross-sections.

Further support for a trio of mass-to-light deviations in Abell 370: free-form grale lens inversion using BUFFALO strong lensing data

ABSTRACT We use the Beyond Ultra-deep Frontier Fields and Legacy Observations (BUFFALO) strong lensing image catalogue of the merging galaxy cluster Abell 370 to obtain a mass model using free-form lens inversion algorithm grale. The improvement of the strong lensing data quality results in a lens plane rms of only 0.45 arcsec, about a factor of two lower than that of our existing HFF v4 reconstruction. We attribute the improvement to spectroscopic data and use of the full reprocessed HST mosaics. In our reconstructed mass model, we found indications of three distinct mass features in Abell 370: (i) a ∼35 kpc offset between the northern BCG and the nearest mass peak, (ii) a ∼100 kpc mass concentration of roughly critical density ∼250 kpc east of the main cluster, and (iii) a probable filament-like structure passing N-S through the cluster. While (i) is present in some form in most publicly available reconstructions spanning the range of modelling techniques: parametric, hybrid, and free-form, (ii) and (iii) are recovered by only about half of the reconstructions. We tested our hypothesis on the presence of the filament-like structure by creating a synthetic cluster – Irtysh IIIc – mocking the situation of a cluster with external mass. We also computed the source plane magnification distributions. Using them, we estimated the probabilities of magnifications in the source plane, and scrutinized their redshift dependence. Finally, we explored the lensing effects of Abell 370 on the luminosity functions of sources at zs = 9.0, finding it consistent with published results.

SLITRONOMY: Towards a fully wavelet-based strong lensing inversion technique

Strong gravitational lensing provides a wealth of astrophysical information on the baryonic and dark matter content of galaxies. It also serves as a valuable cosmological probe by allowing us to measure the Hubble constant independently of other methods. These applications all require the difficult task of inverting the lens equation and simultaneously reconstructing the mass profile of the lens along with the original light profile of the unlensed source. As there is no reason for either the lens or the source to be simple, we need methods that both invert the lens equation with a large number of degrees of freedom and also enforce a well-controlled regularisation that avoids the appearance of spurious structures. This can be beautifully accomplished by representing signals in wavelet space. Building on the Sparse Lens Inversion Technique (SLIT), we present an improved sparsity-based method that describes lensed sources using wavelets and optimises over the parameters given an analytical lens mass profile. We applied our technique on simulated HST and E-ELT data, as well as on real HST images of lenses from the Sloan Lens ACS sample, assuming a lens model. We show that wavelets allowed us to reconstruct lensed sources containing detailed substructures when using both present-day data and very high-resolution images expected from future thirty-metre-class telescopes. In the latter case, wavelets moreover provide a much more tractable solution in terms of quality and computation time compared to using a source model that combines smooth analytical profiles and shapelets. Requiring very little human interaction, our flexible pixel-based technique fits into the ongoing effort to devise automated modelling schemes. It can be incorporated in the standard workflow of sampling analytical lens model parameters while modelling the source on a pixelated grid. The method, which we call SLITRONOMY, is freely available as a new plug-in to the modelling software LENSTRONOMY.

Free-form grale lens inversion of galaxy clusters with up to 1000 multiple images

ABSTRACT In the near future, ultra deep observations of galaxy clusters with Hubble Space Telescope or James Webb Space Telescope will uncover 300–1000 lensed multiple images, increasing the current count per cluster by up to an order of magnitude. This will further refine our view of clusters, leading to a more accurate and precise mapping of the total and dark matter distribution in clusters, and enabling a better understanding of background galaxy population and their luminosity functions. However, to effectively use that many images as input to lens inversion will require a re-evaluation of, and possibly upgrades to the existing methods. In this paper, we scrutinize the performance of the free-form lens inversion method grale in the regime of 150–1000 input images, using synthetic massive galaxy clusters. Our results show that with an increasing number of input images, grale produces improved reconstructed mass distributions, with the fraction of the lens plane recovered at better than $10{{\ \rm per\ cent}}$ accuracy increasing from $40\!-\!50{{\ \rm per\ cent}}$ for ∼150 images to $65{{\ \rm per\ cent}}$ for ∼1000 images. The reconstructed time delays imply a more precise measurement of H0, with $\lesssim 1{{\ \rm per\ cent}}$ bias. While the fidelity of the reconstruction improves with the increasing number of multiple images used as model constraints, ∼150 to ∼1000, the lens plane rms deteriorates from ∼0.11 to ∼0.28 arcsec. Since lens plane rms is not necessarily the best indicator of the quality of the mass reconstructions, looking for an alternative indicator is warranted.

A centroid-based modelling approach to lens inversion for gravitationally lensed systems

In this research, we propose a novel method for determining the coordinate of a gravitational lens in systems where the lens has not yet been directly observed. Our technique uses image processing software to locate the optical centroid of strongly lensed systems and then applies a geometric analysis to derive the coordinates of the lens from the coordinates of the centroid and the arrangement of the lensed images. We demonstrate this method on gravitationally lensed quasar systems in which the lens has been observed to empirically validate our model, and then apply it to the GraL group's list of lensing candidates derived from Gaia DR2 to propose lens coordinates in these candidate systems.

Sparse Lens Inversion Technique (SLIT): lens and source separability from linear inversion of the source reconstruction problem

Strong gravitational lensing offers a wealth of astrophysical information on the background source it affects, provided the lensed source can be reconstructed as if it was seen in the absence of lensing. In the present work, we illustrate how sparse optimisation can address the problem. As a first step towards a full free-form-lens-modelling technique, we consider linear inversion of the lensed source under sparse regularisation and joint deblending from the lens light profile. The method is based on morphological component analysis, assuming a known mass model. We show with numerical experiments that representing the lens and source light using an undecimated wavelet basis allows us to reconstruct the source and to separate it from the foreground lens at the same time. Both the source and lens light have a non-analytic form, allowing for the flexibility needed in the inversion to represent arbitrarily small and complex luminous structures in the lens and source. In addition, sparse regularisation avoids over-fitting the data and does not require the use of an adaptive mesh or pixel grid. As a consequence, our reconstructed sources can be represented on a grid of very small pixels. Sparse regularisation in the wavelet domain also allows for automated computation of the regularisation parameter, thus minimising the impact of the arbitrary choice of initial parameters. Our inversion technique for a fixed mass distribution can be incorporated into future lens-modelling techniques iterating over the lens mass parameters.

The role of multiple images and model priors in measuringH0 from supernova Refsdal in galaxy cluster MACS J1149.5+2223

Multiple image gravitational lensing systems with measured time delays provide a promising one-step method for determining $H_0$. MACS J1149, which lenses SN Refsdal into a quad S1-S4, and two other widely separated images, SX and SY, is a perfect candidate. If time delays are pinned down, the remaining uncertainty arises from the mass distribution in the lens. In MACS J1149, the mass in the relevant lens plane region can be constrained by (i) many multiple images, (ii) the mass of the galaxy splitting S1-S4 (which, we show, is correlated with $H_0$), (iii) magnification of SX (also correlated with $H_0$), and (iv) prior assumptions on the mass distribution. Our goal is not to estimate $H_0$, but to understand its error budget, i.e., estimate uncertainties associated with each of these constraints. Using multiple image positions alone, yields very large uncertainty, despite the fact that the position of SX is recovered to within $\!\le\!0.036$" (rms $\!\le\!0.36$") by GRALE lens inversion. Fixing the mass of the galaxy that splits S1-S4 reduces $1\sigma$ uncertainties to $\sim 23\%$, while fixing the magnification of SX yields $1\sigma$ uncertainties of $32\%$. We conclude that smaller uncertainties, of order few percent, are a consequence of imposing prior assumptions on the shapes of the galaxy and cluster mass distributions, which may or may not apply in a highly non-equilibrium environment of a merging cluster. We propose that if a measurement of $H_0$ is to be considered reliable, it must be supported by a wide range of lens inversion methods.

Lens models under the microscope: comparison ofHubble Frontier Fieldcluster magnification maps

Using the power of gravitational lensing magnification by massive galaxy clusters, the Hubble Frontier Fields provide deep views of six patches of the high redshift Universe. The combination of deep Hubble imaging and exceptional lensing strength has revealed the greatest numbers of multiply-imaged galaxies available to constrain models of cluster mass distributions. However, even with $\mathcal{O}(100)$ images per cluster, the uncertainties associated with the reconstructions are not negligible. The goal of this paper is to show the diversity of model magnification predictions. We examine 7 and 9 mass models of Abell 2744 and MACS J0416, respectively, submitted to the Mikulski Archive for Space Telescopes for public distribution in September 2015. The dispersion between model predictions increases from 30% at common low magnifications ($\mu\sim2$) to 70% at rare high magnifications ($\mu\sim40$). MACS J0416 exhibits smaller dispersions than Abell 2744 for $2<\mu<10$. We show that magnification maps based on different lens inversion techniques typically differ from each other by more than their quoted statistical errors. This suggests that some models underestimate the true uncertainties, which are primarily due to various lensing degeneracies. Though the exact mass sheet degeneracy is broken, its generalized counterpart is not broken at least in Abell 2744. Other, local degeneracies are also present in both clusters. Our comparison of models is complementary to the comparison of reconstructions of known synthetic mass distributions. By focusing on observed clusters, we can identify those that are best constrained, and therefore provide the clearest view of the distant Universe.

Herschel ★ -ATLAS: modelling the first strong gravitational lenses

We have determined the mass-density radial profiles of the first five strong gravitational lens systems discovered by the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). We present an enhancement of the semi-linear lens inversion method of Warren & Dye which allows simultaneous reconstruction of several different wavebands and apply this to dual-band imaging of the lenses acquired with the Hubble Space Telescope. The five systems analysed here have lens redshifts which span a range, 0.22<z<0.94. Our findings are consistent with other studies by concluding that: 1) the logarithmic slope of the total mass density profile steepens with decreasing redshift; 2) the slope is positively correlated with the average total projected mass density of the lens contained within half the effective radius and negatively correlated with the effective radius; 3) the fraction of dark matter contained within half the effective radius increases with increasing effective radius and increases with redshift.

On the accuracy of the perturbative approach for strong lensing: local distortion for pseudo-elliptical models

The Perturbative Approach (PA) introduced by \citet{alard07} provides analytic solutions for gravitational arcs by solving the lens equation linearized around the Einstein ring solution. This is a powerful method for lens inversion and simulations in that it can be used, in principle, for generic lens models. In this paper we aim to quantify the domain of validity of this method for three quantities derived from the linearized mapping: caustics, critical curves, and the deformation cross section (i.e. the arc cross section in the infinitesimal circular source approximation). We consider lens models with elliptical potentials, in particular the Singular Isothermal Elliptic Potential and Pseudo-Elliptical Navarro--Frenk--White models. We show that the PA is exact for this first model. For the second, we obtain constraints on the model parameter space (given by the potential ellipticity parameter $\varepsilon$ and characteristic convergence $\kappa_s$) such that the PA is accurate for the aforementioned quantities. In this process we obtain analytic expressions for several lensing functions, which are valid for the PA in general. The determination of this domain of validity could have significant implications for the use of the PA, but it still needs to be probed with extended sources.

Lensing degeneracies and mass substructure

The inversion of gravitational lens systems is hindered by the fact that multiple mass distributions are often equally compatible with the observed properties of the images. Besides using clear examples to illustrate the effect of the so-called monopole and mass sheet degeneracies, this article introduces the most general form of said mass sheet degeneracy. While the well known version of this degeneracy rescales a single source plane, this generalization allows any number of sources to be rescaled. Furthermore, it shows how it is possible to rescale each of those sources with a different scale factor. Apart from illustrating that the mass sheet degeneracy is not broken by the presence of multiple sources at different redshifts, it will become apparent that the newly constructed mass distribution necessarily alters the existing mass density precisely at the locations of the images in the lens system, and that this change in mass density is linked to the factors with which the sources were rescaled. Combined with the fact that the monopole degeneracy introduces a large amount of uncertainty about the density in between the images, this means that both degeneracies are in fact closely related to substructure in the mass distribution. An example simulated lensing situation based on an elliptical version of a Navarro-Frenk-White profile explicitly shows that such degeneracies are not easily broken by observational constraints, even when multiple sources are present. Instead, the fact that each lens inversion method makes certain assumptions, implicit or explicit, about the smoothness of the mass distribution means that in practice the degeneracies are broken in an artificial manner rather than by observed properties of the lens system.

THE OPTIMAL GRAVITATIONAL LENS TELESCOPE

Given an observed gravitational lens mirage produced by a foreground deflector (cf. galaxy, quasar, cluster,...), it is possible via numerical lens inversion to retrieve the real source image, taking full advantage of the magnifying power of the cosmic lens. This has been achieved in the past for several remarkable gravitational lens systems. Instead, we propose here to invert an observed multiply imaged source directly at the telescope using an ad-hoc optical instrument which is described in the present paper. Compared to the previous method, this should allow one to detect fainter source features as well as to use such an optimal gravitational lens telescope to explore even fainter objects located behind and near the lens. Laboratory and numerical experiments illustrate this new approach.

Perturbative reconstruction of a gravitational lens: when mass does not follow light

The structure and potential of a complex gravitational lens is reconstructed using the perturbative method presented in Alard 2007, MNRAS, 382L, 58; Alard 2008, MNRAS, 388, 375. This lens is composed of 6 galaxies belonging to a small group. The lens inversion is reduced to the problem of reconstructing non-degenerate quantities: the 2 fields of the perturbative theory of strong gravitational lenses. Since in the perturbative theory the circular source solution is analytical, the general properties of the perturbative solution can be inferred directly from the data. As a consequence, the reconstruction of the perturbative fields is not affected by degeneracy, and finding the best solution is only a matter of numerical refinement. The local shape of the potential and density of the lens are inferred from the perturbative solution, revealing the existence of an independent dark component that does not follow light. The most likely explanation is that the particular shape of the dark halo is due to the merging of cold dark matter halos. This is a new result illustrating the structure of dark halos at the scale of galaxies.

Non-parametric strong lens inversion of SDSS J1004+4112

In this article we study the well-known strong lensing system SDSS J1004+4112. Not only does it host a large-separation lensed quasar with measured time-delay information, but several other lensed galaxies have been identified as well. A previously developed strong lens inversion procedure that is designed to handle a wide variety of constraints, is applied to this lensing system and compared to results reported in other works. Without the inclusion of a tentative central image of one of the galaxies as a constraint, we find that the model recovered by the other constraints indeed predicts an image at that location. An inversion which includes the central image provides tighter constraints on the shape of the central part of the mass map. The resulting model also predicts a central image of a second galaxy where indeed an object is visible in the available ACS images. We find masses of 2.5x10^13 M_O and 6.1x10^13 M_O within a radius of 60 kpc and 110 kpc respectively, confirming the results from other authors. The resulting mass map is compatible with an elliptical generalization of a projected NFW profile, with r_s = 58_{-13}^{+21} arcsec and c_vir = 3.91 +/- 0.74. The orientation of the elliptical NFW profile follows closely the orientation of the central cluster galaxy and the overall distribution of cluster members.

A Molecular Einstein Ring atz= 4.12: Imaging the Dynamics of a Quasar Host Galaxy Through a Cosmic Lens

We present high-resolution (0.3") Very Large Array (VLA) imaging of the molecular gas in the host galaxy of the high redshift quasar PSS J2322+1944 (z=4.12). These observations confirm that the molecular gas (CO) in the host galaxy of this quasar is lensed into a full Einstein ring, and reveal the internal dynamics of the molecular gas in this system. The ring has a diameter of ~1.5", and thus is sampled over ~20 resolution elements by our observations. Through a model-based lens inversion, we recover the velocity gradient of the molecular reservoir in the quasar host galaxy of PSS J2322+1944. The Einstein ring lens configuration enables us to zoom in on the emission and to resolve scales down to ~1 kpc. From the model-reconstructed source, we find that the molecular gas is distributed on a scale of 5 kpc, and has a total mass of M(H2)=1.7 x 10^10 M_sun. A basic estimate of the dynamical mass gives M_dyn = 4.4 x 10^10 (sin i)^-2 M_sun, that is, only ~2.5 times the molecular gas mass, and ~30 times the black hole mass (assuming that the dynamical structure is highly inclined). The lens configuration also allows us to tie the optical emission to the molecular gas emission, which suggests that the active galactic nucleus (AGN) does reside within, but not close to the center of the molecular reservoir. Together with the (at least partially) disturbed structure of the CO, this suggests that the system is interacting. Such an interaction, possibly caused by a major `wet' merger, may be responsible for both feeding the quasar and fueling the massive starburst of 680 M_sun/yr in this system, in agreement with recently suggested scenarios of quasar activity and galaxy assembly in the early universe.

Separation of the visible and dark matter in the Einstein ring LBG J213512.73-010143

We model the mass distribution in the recently discovered Einstein ring LBGJ213512.73-010143 (the 'Cosmic Eye') using archival Hubble Space Telescope imaging. We reconstruct the mass density profile of the z= 0.73 lens and the surface brightness distribution of the z = 3.07 source and find that the observed ring is best fitted with a dual-component lens model consisting of a baryonic Sersic component nested within a dark matter halo. The dark matter halo has an inner slope of 1.42 +0.24 -0.22 , consistent with cold dark matter simulations after allowing for baryon contraction. The baryonic component has a mass-to-light ratio of 1.71 +0.28 0.38 ( M⊙/L B ⊙ which when evolved to the present day is in agreement with local ellipticals. Within the Einstein radius of 0.77 arcsec (5.6 kpc), the baryons account for 46 ± 11 per cent of the projected lens mass. External shear from a nearby foreground cluster is accurately predicted by the model. The reconstructed surface brightness distribution in the source plane clearly shows two peaks. Through a generalization of our lens inversion method, we conclude that the redshifts of both peaks are consistent with each other, suggesting that we are seeing structure within a single galaxy.