Non-parametric strong lens inversion of Cl 0024+1654: illustrating the monopole degeneracy

The cluster lens Cl 0024+1654 is undoubtedly one of the most beautiful examples of strong gravitational lensing, providing five large images of a single source with well-resolved substructure. Using the information contained in the positions and the shapes of the images, combined with the null space information, a non-parametric technique is used to infer the strong lensing mass map of the central region of this cluster. This yields a strong lensing mass of  1.60 × 10¹⁴ M_⊙  within a 0.5  arcmin radius around the cluster centre. This mass distribution is then used as a case study of the monopole degeneracy, which may be one of the most important degeneracies in gravitational lensing studies and which is extremely hard to break. We illustrate the monopole degeneracy by adding circularly symmetric density distributions with zero total mass to the original mass map of Cl 0024+1654. These redistribute mass in certain areas of the mass map without affecting the observed images in any way. We show that the monopole degeneracy and the mass-sheet degeneracy together lie at the heart of the discrepancies between different gravitational lens reconstructions that can be found in the literature for a given object, and that many images/sources, with an overall high image density in the lens plane, are required to construct an accurate, high-resolution mass map based on strong lensing data.     

A genetic algorithm for the non-parametric inversion of strong lensing systems

We present a non-parametric technique to infer the projected mass distribution of a gravitational lens system with multiple strong-lensed images. The technique involves a dynamic grid in the lens plane on which the mass distribution of the lens is approximated by a sum of basis functions, one per grid cell. We used the projected mass densities of Plummer spheres as basis functions. A genetic algorithm then determines the mass distribution of the lens by forcing images of a single source, projected back on to the source plane, to coincide as well as possible. Averaging several tens of solutions removes the random fluctuations that are introduced by the reproduction process of genomes in the genetic algorithm and highlights those features common to all solutions. Given the positions of the images and the redshifts of the sources and the lens, we show that the mass of a gravitational lens can be retrieved with an accuracy of a few percent and that, if the sources sufficiently cover the caustics, the mass distribution of the gravitational lens can also be reliably retrieved. A major advantage of the algorithm is that it makes full use of the information contained in the radial images, unlike methods that minimize the residuals of the lens equation, and is thus able to accurately reconstruct also the inner parts of the lens.     

基于Shapelets基函数的引力透镜质量重构方法

引力透镜效应是探测星系团物质分布的有效方法之一.目前,利用引力透镜数据重构星系团质量分布的主流方法可以分为两大类,即参数法和非参数法.在实际研究工作中,受限于质量模型假设和计算分辨率等方面的影响,现有的重构算法仍有诸多亟需解决的问题.基于Shapelets基函数的引力透镜质量重构方法通过基函数来实现引力透镜质量重构,使用Shapelets基函数分解引力透镜势,以引力透镜中多重像的位置和背景星系椭率畸变为限制条件来迭代求解基函数系数从而得到透镜体的质量分布.通过拟合一个模拟的NFW (Navarro,Frenk and White)透镜系统测试了新方法的可行性,结果表明新方法可以在整体上重构出透镜体的质量分布,并拟合出接近真实的源位置,能够为星系团质量测量提供一套灵活且高效的重构算法.     

The anatomy of a quadruply imaged gravitational lens system

The key to using a strong gravitational lens system to measure the Hubble constant is to obtain an accurate model of the lens potential. In this paper, we investigate the properties of gravitational lens B1608+656, a quadruply imaged lens system with an extended source intensity distribution. Our analysis is valid for generic quadruply lensed systems. Limit curves and isophotal separatrices are defined for such systems, and we show that the isophotal separatrices must intersect at the critical curves and the satellite isophotes must be tangent to the limit curves. The most recent model of B1608+656 by Koopmans et al. satisfies these criteria for some, but not all, of the isophotal separatrices within the observational uncertainty. We study a non-parametric method of potential reconstruction proposed by Blandford, Surpi & Kundic and demonstrate that although the method works in principle and elucidates image formation, the initial potential only converges to the true model when it is within ∼1 per cent of the true model.     

Non-parametric reconstruction of cluster mass distribution from strong lensing: modelling Abell 370

We describe a new non-parametric technique for reconstructing the mass distribution in galaxy clusters with strong lensing, i.e. from multiple images of background galaxies. The observed positions and redshifts of the images are considered as rigid constraints, and through the lens (ray-trace) equation they provide us with linear constraint equations. These constraints confine the mass distribution to some allowed region, which is then found by linear programming. Within this allowed region we study in detail the mass distribution with minimum mass-to-light variation, and also some other distributions, such as the smoothest mass distribution.
The method is applied to the extensively studied cluster Abell 370, which hosts a giant luminous arc and several other multiply imaged background galaxies. Our mass maps are constrained by the observed positions and redshifts (spectroscopic or model-inferred by previous authors) of the giant arc and multiple-image systems. The reconstructed maps obtained for Abell 370 reveal a detailed mass distribution, with substructure quite different from the light distribution. The method predicts the bimodal nature of the cluster, and that the projected mass distribution is indeed elongated along the axis defined by the two dominant cD galaxies. However, the peaks in the mass distribution appear to be offset from the centres of the cDs.
We also present an estimate for the total mass of the central region of the cluster. This is in good agreement with previous mass determinations. The total mass of the central region is M =(2.0–2.7)×10¹⁴ M⊙ h ⁻¹₅₀, depending on the solution chosen.     

The lens and source of the optical Einstein ring gravitational lens ER 0047–2808

We perform a detailed analysis of the optical gravitational lens ER 0047–2808 imaged with the Wide Field Planetary Camera 2 on the Hubble Space Telescope . Using software specifically designed for the analysis of resolved gravitational lens systems, we focus on how the image alone can constrain the mass distribution in the lens galaxy. We find that the data are of sufficient quality to strongly constrain the lens model with no a priori assumptions about the source. Using a variety of mass models, we find statistically acceptable results for elliptical isothermal-like models with an Einstein radius of 1.17 arcsec. An elliptical power-law model  (Σ∝ R ^−β)  for the surface mass density favours a slope slightly steeper than isothermal with  β= 1.08 ± 0.03  . Other models including a constant mass-to-light ratio (M/L), pure Navarro, Frenk & White halo and (surprisingly) an isothermal sphere with external shear are ruled out by the data. We find the galaxy light profile can only be fit with a Sérsic plus point-source model. The resulting total  M/L _B   contained within the images is  4.7  h ₆₅± 0.3  . In addition, we find the luminous matter is aligned with the total mass distribution within a few degrees. This is the first time a resolved optical gravitational lens image has been quantitatively reproduced using a non-parametric source.
The source, reconstructed by the software, is revealed to have two bright regions, with an unresolved component inside the caustic and a resolved component straddling a fold caustic. The angular size of the entire source is ∼0.1 arcsec and its (unlensed) Lyα flux is  3 × 10⁻¹⁷ erg s⁻¹ cm⁻²  .     

NICMOS images of JVAS/CLASS gravitational lens systems

We present Hubble Space Telescope ( HST ) infrared images of four gravitational lens systems from the JVAS/CLASS gravitational lens survey and compare the new infrared HST pictures with previously published WFPC2 HST optical images and radio maps. Apart from the wealth of information that we get from the flux ratios and accurate positions and separations of the components of the lens systems, which we can use as inputs for better constraints on the lens models, we are able to discriminate between reddening and optical/radio microlensing as the possible cause of differences observed in the flux ratios of the components across the three wavelength bands. Substantial reddening has been known to be present in the lens system B1600+434 and has been further confirmed by the present infrared data. In the two systems B0712+472 and B1030+074 microlensing has been pinpointed as the main cause of the flux ratio discrepancy both in the optical/infrared and in the radio, the radio possibly caused by the substructure revealed in the lensing galaxies. In B0218+357, however, the results are still not conclusive. If we are actually seeing the two 'true' components of the lens system then the flux ratio differences are attributed to a combination of microlensing and reddening or are alternatively the result of some variability in at least one of the images. Otherwise the second 'true' component of B0218+357 may be completely absorbed by a molecular cloud and the anomalous flux density ratios and large difference in separation between the optical/infrared and radio that we see can be explained by emission either from a foreground object or from part of the lensing galaxy.     

CLASS B2108+213: a new wide-separation gravitational lens system

We present observations of CLASS B2108+213, the widest separation gravitational lens system discovered by the Cosmic Lens All-Sky Survey. Radio imaging using the VLA at 8.46 GHz and MERLIN at 5 GHz shows two compact components separated by 4.56 arcsec with a faint third component in between which we believe is emission from a lensing galaxy. 5-GHz VLBA observations reveal milliarcsecond-scale structure in the two lensed images that is consistent with gravitational lensing. Optical emission from the two lensed images and two lensing galaxies within the Einstein radius is detected in Hubble Space Telescope imaging. Furthermore, an optical gravitational arc, associated with the strongest lensed component, has been detected. Surrounding the system is a number of faint galaxies which may help explain the wide image separation. A plausible mass distribution model for CLASS B2108+213 is also presented.     

Resolved nuclear CO(1–0) emission in APM 08279+5255: gravitational lensing by a naked cusp?

The ultraluminous broad absorption line quasar APM 08279+5255 is one of the most luminous systems known. Here, we present an analysis of its nuclear  CO(1–0)  emission. Its extended distribution suggests that the gravitational lens in this system is highly elliptical, probably a highly inclined disc. The quasar core, however, lies in the vicinity of a naked cusp, indicating that APM 08279+5255 is truly the only odd-image gravitational lens. This source is the second system for which the gravitational lens can be used to study structure on sub-kiloparsec scales in the molecular gas associated with the AGN host galaxy. The observations and lens model require CO distributed on a scale of ∼400 pc. Using this scale, we find that the molecular gas mass makes a significant, and perhaps dominant, contribution to the total mass within a couple of hundred parsecs of the nucleus of APM 08279+5255.     

B0712+472: a new radio four-image gravitational lens

A new four-image gravitational lens system, B0712+472, has been discovered during the Cosmic Lens All-Sky Survey. This system consists of four flat-spectrum radio images that are also seen on a Hubble Space Telescope ( HST ) image, together with the lensing galaxy. We present MERLIN, VLA and VLBA maps and WHT spectra of the system as well as the HST images. The light distribution of the lensing galaxy is highly elongated and so too is the mass distribution deduced from modelling. We suggest a redshift of ∼1.33 for the lensed object; the lens redshift will require further investigation. The discovery of this new system further increases the ratio of four-image to two-image lens systems currently known, exacerbating problems of required ellipticity of matter distributions in lensing galaxies.     

A systematic review of strong gravitational lens modeling software

Despite expanding research activity in gravitational lens modeling, there is no particular software which is considered a standard. Much of the gravitational lens modeling software is written by individual investigators for their own use. Some gravitational lens modeling software is freely available for download but is widely variable with regard to ease of use and quality of documentation. This review of 13 software packages was undertaken to provide a single source of information. Gravitational lens models are classified as parametric models or non-parametric models, and can be further divided into research and educational software. Software used in research includes the GRAVLENS package (with both gravlens and lensmodel), Lenstool, LensPerfect, glafic, PixeLens, SimpLens, Lensview, and GRALE. In this review, GravLensHD, G-Lens, Gravitational Lensing, lens and MOWGLI are categorized as educational programs that are useful for demonstrating various aspects of lensing. Each of the 13 software packages is reviewed with regard to software features (installation, documentation, files provided, etc.) and lensing features (type of model, input data, output data, etc.) as well as a brief review of studies where they have been used. Recent studies have demonstrated the utility of strong gravitational lensing data for mass mapping, and suggest increased use of these techniques in the future. Coupled with the advent of greatly improved imaging, new approaches to modeling of strong gravitational lens systems are needed. This is the first systematic review of strong gravitational lens modeling software, providing investigators with a starting point for future software development to further advance gravitational lens modeling research.     

Perturbation theory of N point-mass gravitational lens systems without symmetry: small mass-ratio approximation

This paper makes the first systematic attempt to determine using perturbation theory the positions of images by gravitational lensing due to arbitrary number of coplanar masses without any symmetry on a plane, as a function of lens and source parameters. We present a method of Taylor-series expansion to solve the lens equation under a small mass-ratio approximation. First, we investigate perturbative structures of a single-complex-variable polynomial, which has been commonly used. Perturbative roots are found. Some roots represent positions of lensed images, while the others are unphysical because they do not satisfy the lens equation. This is consistent with a fact that the degree of the polynomial, namely the number of zeros, exceeds the maximum number of lensed images if   N = 3  (or more). The theorem never tells which roots are physical (or unphysical). In this paper, unphysical ones are identified. Secondly, to avoid unphysical roots, we re-examine the lens equation. The advantage of our method is that it allows a systematic iterative analysis. We determine image positions for binary lens systems up to the third order in mass ratios and for arbitrary N point masses up to the second order. This clarifies the dependence on parameters. Thirdly, the number of the images that admit a small mass-ratio limit is less than the maximum number. It is suggested that positions of extra images could not be expressed as Maclaurin series in mass ratios. Magnifications are finally discussed.     

A small mass,large image separation gravitational lens model

It is shown that gravitational lensing could produce images of comparable brightness which are separated by an angular distance larger than the angular size of the Einstein ring of the lens distribution. Hence, specific lens configurations allow given image separations to be derived with significantly less mass than a standard single lens model. A specific example is supplied, a simple case of two properly positioned lenses acting instead of one. Observational consequences are discussed that would result if such a model was used to explain the candidate gravitational lens systems Hazard 1146+111 B, C and PKS 1145–071. Such a lens distribution might, in many cases, be verifiable with VLBI techniques.     

A new gravitational lens from the MUSCLES survey: ULAS J082016.1+081216

We present observations of a new double-image gravitational lens system, ULAS J082016.1+081216, of image separation 2.3 arcsec and high (∼6) flux ratio. The system is selected from the Sloan Digital Sky Survey (SDSS) spectroscopic quasar list using new high-quality images from the UKIRT (United Kingdom Infrared Telescope) Deep Sky Survey (UKIDSS). The lensed quasar has a source redshift of 2.024, and we identify the lens galaxy as a faint red object of redshift  0.803 ± 0.001  . Three other objects from the UKIDSS survey, selected in the same way, were found not to be lens systems. Together with the earlier lens found using this method, the SDSS–UKIDSS lenses have the potential to significantly increase the number of quasar lenses found in SDSS, to extend the survey to higher flux ratios and lower separations, and to give greater completeness which is important for statistical purposes.     

Galaxy density profiles and shapes – II. Selection biases in strong lensing surveys

Many current and future astronomical surveys will rely on samples of strong gravitational lens systems to draw conclusions about galaxy mass distributions. We use a new strong lensing pipeline (presented in Paper I of this series) to explore selection biases that may cause the population of strong lensing systems to differ from the general galaxy population. Our focus is on point-source lensing by early-type galaxies with two mass components (stellar and dark matter) that have a variety of density profiles and shapes motivated by observational and theoretical studies of galaxy properties. We seek not only to quantify but also to understand the physics behind selection biases related to: galaxy mass, orientation and shape; dark matter profile parameters such as inner slope and concentration; and adiabatic contraction. We study how all of these properties affect the lensing Einstein radius, total cross-section, quad/double ratio and image separation distribution, with a flexible treatment of magnification bias to mimic different survey strategies. We present our results for two families of density profiles: cusped and deprojected Sérsic models. While we use fixed lens and source redshifts for most of the analysis, we show that the results are applicable to other redshift combinations, and we also explore the physics of how our results change for very different redshifts. We find significant (factors of several) selection biases with mass; orientation, for a given galaxy shape at fixed mass; cusped dark matter profile inner slope and concentration; concentration of the stellar and dark matter deprojected Sérsic models. Interestingly, the intrinsic shape of a galaxy does not strongly influence its lensing cross-section when we average over viewing angles. Our results are an important first step towards understanding how strong lens systems relate to the general galaxy population.     

A search for the third lensed image in JVAS B1030+074

Central gravitational image detection is very important for the study of the mass distribution of the inner parts (∼100 pc) of lens galaxies. However, the detection of such images is extremely rare and difficult. We present a 1.7-GHz High Sensitivity Array (HSA) observation of the double-image radio lens system B1030+074. The data are combined with archive Very Long Baseline Array and global very long baseline interferometry (VLBI) observations, and careful consideration is given to the effects of noise, clean ing and self-calibration. An upper limit is derived for the strength of the central image of 180 μJy (90 per cent confidence level), considerably greater than would have been expected on the basis of a simple analysis. This gives a lower limit of ∼10³ for the ratio of the brightest image to the central image. For cusped models of lens mass distributions, we have made use of this non-detection to constrain the relation between inner power-law slope β of the lensing galaxy mass profile, and its break radius r _b. For   r _b > 130 pc  the power-law slope is required to be close to isothermal  (β > 1.8)  . A flatter inner slope is allowed if a massive black hole is present at the centre of the lensing galaxy, but the effect of the black hole is small unless it is ∼10 times more massive than that implied by the relation between black hole mass and stellar velocity dispersion. By comparing four epochs of VLBI observations, we also detected possible superluminal motion in the jet in the brighter image A. The B jet remains unresolved, as expected from a simple lens model of the system.     

A new quadruple gravitational lens system: CLASS B0128+437

High-resolution MERLIN observations of a newly discovered four-image gravitational lens system, B0128+437, are presented. The system was found after a careful re-analysis of the entire CLASS data set. The MERLIN observations resolve four components in a characteristic quadruple-image configuration; the maximum image separation is 542 mas and the total flux density is 48 mJy at 5 GHz. A best-fitting lens model with a singular isothermal ellipsoid results in large errors in the image positions. A significantly improved fit is obtained after the addition of a shear component, suggesting that the lensing system is more complex and may consist of multiple deflectors. The integrated radio spectrum of the background source indicates that it is a gigahertz peaked spectrum source. It may therefore be possible to resolve structure within the radio images with deep VLBI observations and thus to constrain the lensing mass distribution better.     

Degeneracies and scaling relations in general power-law models for gravitational lenses

The time-delay in gravitational lenses can be used to derive the Hubble constant in a relatively simple way. The results of this method are less dependent on astrophysical assumptions than in many other methods. For systems with accurately measured positions and time-delays, the most important uncertainty is related to the mass model used. Simple parametric models like isothermal ellipsoidal mass distributions seem to provide consistent results with a reasonably small scatter when applied to several lens systems. We discuss a family of models with a separable radial power law and an arbitrary angular dependence for the potential   ψ = r ^β F ( θ )  . Isothermal potentials are a special case of these models with   β =1  . An additional external shear is used to take into account perturbations from other galaxies. Using a simple linear formalism for quadruple lenses, we can derive H ₀ as a function of the observables and the shear. If the latter is fixed, the result depends on the assumed power-law exponent according to   H ₀∝(2- β )/ β   . The effect of external shear is quantified by introducing a 'critical shear' γ _c as a measure for the amount of shear that changes the result significantly. The analysis shows that in the general case H ₀ and γ _c do not depend on the position of the lens galaxy. Spherical lens models with images close to the Einstein radius with fitted external shear differ by a factor of   β /2  from shearless models, leading to   H ₀∝2- β   in this case. We discuss these results and compare them with numerical models for a number of real lens systems.     

临界曲线附近的微引力透镜效应模拟

在光滑物质分布模型下,临界曲线是强引力透镜系统中像平面上一条放大率为无穷的线,而考虑少量离散质量的微透镜效应后,源平面上的放大率分布会出现复杂的结构,为暗物质成分的探测提供了一种有效途径.模拟临界曲线附近微透镜效应存在临界曲线上放大率无穷大和计算量巨大的困难.要达到所需的模拟精度,直接使用传统的光线追踪算法需要巨大的计算资源.为此发展了一个能实现海量计算的Graphics Processing Unit (GPU)并行方法来模拟临界曲线附近的微引力透镜效应.在型号为NVIDIA Tesla V100S PCIe 32 GB的GPU上,对于需要处理13000多个微透镜天体、发射1013量级光线的模拟,耗时在7000 s左右.在GPU并行的基础上,与直接的光线追踪算法相比,插值近似的引入使计算速度提升约两个数量级.利用该方法生成80个放大率分布图,并从中抽取800条光变曲线,进行了微焦散线数密度和峰值放大率的统计.