We make a number of comments regarding modeling degeneracies in strong lensing measurements of the Hubble parameter $H_0$. The first point concerns the impact of weak lensing associated with different segments of the line of sight. We show that external convergence terms associated with the lens-source and observer-lens segments need to be included in cosmographic modeling, in addition to the usual observer-source term, to avoid systematic bias in the inferred value of $H_0$. Specifically, we show how an incomplete account of some line of sight terms biases stellar kinematics as well as ray tracing simulation methods to alleviate the mass sheet degeneracy. The second point concerns the use of imaging data for multiple strongly-lensed sources in a given system. We show that the mass sheet degeneracy is not fully resolved by the availability of multiple sources: some degeneracy remains because of differential external convergence between the different sources. Similarly, differential external convergence also complicates the use of multiple sources in addressing the approximate mass sheet degeneracy associated with a local ("internal") core component in lens galaxies. This internal-external degeneracy is amplified by the non-monotonicity of the angular diameter distance as a function of redshift. For a rough assessment of the weak lensing effects, we provide estimates of external convergence using the nonlinear matter power spectrum, paying attention to non-equal time correlators.
Time-delay cosmography with gravitationally lensed quasars plays an important role in anchoring the absolute distance scale and hence measuring the Hubble constant, $H_{0}$, independent of traditional distance ladder methodology. A current potential limitation of time delay distance measurements is the "mass-sheet transformation" (MST) which leaves the lensed imaging unchanged but changes the distance measurements and the derived value of $H_0$. In this work we show that the standard method of addressing the MST in time delay cosmography, through a combination of high-resolution imaging and th…
The time delays between the multiple images of a strong lens system, together with a model of the lens mass distribution, allow a one-step measurement of a cosmological distance, namely, the "time-delay distance" of the lens (D_dt) that encodes cosmological information. The time-delay distance depends sensitively on the radial profile slope of the lens mass distribution; consequently, the lens slope must be accurately constrained for cosmological studies. We show that the slope cannot be constrained in two-image systems with single-component compact sources, whereas it can be constrained in sy…
We present extended modeling of the strong lens system RXJ1131-1231 with archival data in two HST bands in combination with existing line-of-sight contribution and velocity dispersion estimates. Our focus is on source size and its influence on time-delay cosmography. We therefore examine the impact of mass-sheet degeneracy and especially the degeneracy pointed out by Schneider & Sluse (2013) using the source reconstruction scale. We also extend on previous work by further exploring the effects of priors on the kinematics of the lens and the external convergence in the environment of the lensin…
Mass determinations from gravitational lensing shear and the higher order estimator flexion are both subject to the mass sheet degeneracy. Mass sheet degeneracy refers to a transformation that leaves the reduced shear and flexion invariant. In general, this transformation can be approximated by the addition of a constant surface mass density sheet. We propose a new technique to break the mass sheet degeneracy. The method uses mass moments of the shear or flexion fields in combination with convergence information derived from number counts which exploit the magnification bias. The difference be…
The mass-sheet degeneracy is a well-known problem in gravitational lensing which limits our capability to infer astrophysical lens properties or cosmological parameters from observations. As the number of gravitational wave observations grows, detecting lensed events will become more likely, and to assess how the mass-sheet degeneracy may affect them is crucial. Here we study both analytically and numerically how the lensed waveforms are affected by the mass-sheet degeneracy computing the amplification factor from the diffraction integral. In particular, we differentiate between the geometrica…
Time delay lensing is a mature and competitive cosmological probe. However, it is limited in accuracy by the well-known problem of the mass-sheet degeneracy: too rigid assumptions on the density profile of the lens can potentially bias the inference on cosmological parameters. I investigate the degeneracy between the choice of the lens density profile and the inference on the Hubble constant, focusing on double image systems. By expanding lensing observables in terms of the local derivatives of the lens potential around the Einstein radius, and assuming circular symmetry, I show that three deg…
Degeneracies and modelling choices in double-plane time-delay cosmography
Laboratoire Univers et Particules de Montpellier (LUPM), CNRS & Université de Montpellier（蒙彼利埃宇宙与粒子实验室（LUPM），法国国家科学研究中心与蒙彼利埃大学）；Université de Genève（日内瓦大学）；Institute for Particle Physics and Astrophysics, ETH Zurich（苏黎世联邦理工学院粒子物理与天体物理研究所）
TDCOSMO XXIV. First spatially resolved kinematics of the lens galaxy obtained using JWST-NIRSpec to improve time-delay cosmography
Kavli Institute for Cosmological Physics, University of Chicago（芝加哥大学卡弗里宇宙物理学研究所）；Center for Astronomy, Space Science and Astrophysics, Independent University, Bangladesh（孟加拉独立大学天文学、空间科学与天体物理中心）；Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理与天文系）；Technical University of Munich, TUM School of Natural Sciences, Physics Department（慕尼黑工业大学自然科学院物理系）；Max Planck Institute for Astrophysics（马克斯·普朗克天体物理学研究所）；Space Telescope Science Institute（太空望远镜科学研究所）；Sub-Department of Astrophysics, Department of Physics, University of Oxford（牛津大学物理系天体物理子部门）
Strong lensing time delays can measure the Hubble constant H$_0$ independent of any other probe. Assuming commonly used forms for the radial mass density profile of the lenses, a 2\% precision has been achieved with 7 Time-Delay Cosmography (TDCOSMO) lenses, in tension with the H$_0$ from the cosmic microwave background. However, without assumptions on the radial mass density profile -- and relying exclusively on stellar kinematics to break the mass-sheet degeneracy -- the precision drops to 8\% with the current data of the 7 TDCOSMO lenses, insufficient to resolve the H$_0$ tension. With the …
LISA might detect gravitational waves from mergers of massive black hole binaries strongly lensed by intervening galaxies (Sereno et al. 2010). The detection of multiple gravitational lensing events would provide a new tool for cosmography. Constraints on cosmological parameters could be placed by exploiting either lensing statistics of strongly lensed sources or time delay measurements of lensed gravitational wave signals. These lensing methods do not need the measurement of the redshifts of the sources and the identification of their electromagnetic counterparts. They would extend cosmologic…
Gravitational lensing offers a competitive method to measure $H_0$ with the goal of 1% precision. A major obstacle comes in the form of lensing degeneracies, such as the mass sheet degeneracy (MSD), which make it possible for a family of density profiles to reproduce the same lensing observables but return different values of $H_0$. The modeling process artificially selects one choice from this family, potentially biasing $H_0$. The effect is more pronounced when the profile of a given lens is not perfectly described by the lens model, which will always be the case to some extent. To explore t…
Breaking the Mass-sheet Degeneracy in Time-delay Cosmology with Lensed and Unlensed Type Ia Supernovae
College of Physics, Hebei Normal University（河北师范大学物理学院）；Shijiazhuang Key Laboratory of Astronomy and Space Science, Hebei Normal University（河北师范大学石家庄天文与空间科学重点实验室）；School of Physics and Technology, Wuhan University（武汉大学物理科学与技术学院）
The light travel time differences in strong gravitational lensing systems allows an independent determination of the Hubble constant. This method has been successfully applied to several lens systems. The formally most precise measurements are, however, in tension with the recent determination of $H_0$ from the Planck satellite for a spatially flat six-parameters $ΛCDM$ cosmology. We reconsider the uncertainties of the method, concerning the mass profile of the lens galaxies, and show that the formal precision relies on the assumption that the mass profile is a perfect power law. Simple analyt…
Precision cosmology is crucial to understand the different energy components in the Universe and their evolution through cosmic time. Gravitational wave sources are standard sirens that can accurately map out distances in the Universe. Together with the source redshift information, we can then probe the expansion history of the Universe. We explore the capabilities of various gravitational-wave detector networks to constrain different cosmological models while employing separate waveform models for inspiral and post-merger part of the gravitational wave signal from equal mass binary neutron st…
The H0LiCOW collaboration inferred via gravitational lensing time delays a Hubble constant $H_0=73.3^{+1.7}_{-1.8}$ km s$^{-1}{\rm Mpc}^{-1}$, describing deflector mass density profiles by either a power-law or stars plus standard dark matter halos. The mass-sheet transform (MST) that leaves the lensing observables unchanged is considered the dominant source of residual uncertainty in $H_0$. We quantify any potential effect of the MST with flexible mass models that are maximally degenerate with H0. Our calculation is based on a new hierarchical approach in which the MST is only constrained by …
Time delay measurements with Broken Power Law model
Institute for Frontier in Astronomy and Astrophysics, Beijing Normal University（北京师范大学前沿天文与天体物理研究所）；Shanghai Key Lab for Astrophysics, Shanghai Normal University（上海师范大学天体物理上海市重点实验室）；School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学院）
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 scal…
State Key Laboratory of Dark Matter Physics, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院暗物质物理全国重点实验室）；Key Laboratory for Particle Astrophysics and Cosmology (MOE) / Shanghai Key Laboratory for Particle Physics and Cosmology（粒子天体物理学与宇宙学教育部重点实验室/上海粒子物理与宇宙学重点实验室）；INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna（意大利国家天体物理研究所博洛尼亚空间天体物理与科学观测站）；INFN – Sezione di Bologna（意大利国家核物理研究所博洛尼部分所）
TDCOSMO XXV: Measuring H0 to 6.5% precision with quasar strong lensing and maximally flexible mass-sheet dynamics
University of California, Los Angeles（加州大学洛杉矶分校）；ETH Zurich（苏黎世联邦理工学院）；Université de Genève（日内瓦大学）；European Southern Observatory（欧洲南方天文台）；Ecole Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Stony Brook University（石溪大学）；Université de Montpellier（蒙彼利埃大学）
基于四重成像类星体SDSSJ1433+6007，结合哈勃空间望远镜成像、时间延迟监测和空间分辨恒星运动学，通过联合透镜和动力学模型并考虑质量片变换的最大灵活性，在平坦ΛCDM宇宙学下以6.5%精度测得哈勃常数$H_0 = 73.2^{+4.8}_{-4.7}$ km s$^{-1}$ Mpc$^{-1}$，并发现内部质量片参数$λ_{\rm int}=1.12^{+0.05}_{-0.06}$与1有2σ偏离。
Gravitational lensing time delays offer an avenue to measure the Hubble parameter $H_0$, with some analyses suggesting a tension with early-type probes of $H_0$. The lensing measurements must mitigate systematic uncertainties due to the mass modelling of lens galaxies. In particular, a core component in the lens density profile would form an approximate local mass sheet degeneracy and could bias $H_0$ in the right direction to solve the lensing tension. We consider ultralight dark matter as a possible mechanism to generate such galactic cores. We show that cores of roughly the required propert…
The central ambition of the modern time delay cosmography consists in determining the Hubble constant $H_0$ with a competitive precision. However, the tension with $H_0$ obtained from the Planck satellite for a spatially-flat $Λ$CDM cosmology suggests that systematic errors may have been underestimated. The most critical one probably comes from the degeneracy existing between lens models that was first formalized by the well-known mass-sheet transformation (MST). In this paper, we assess to what extent the source position transformation (SPT), a more general invariance transformation which con…
Inspiralling compact binaries as standard sirens will soon become an invaluable tool for cosmology when advanced interferometric gravitational-wave detectors begin their observations in the coming years. However, a degeneracy in the information carried by gravitational waves between the total rest-frame mass $M$ and the redshift $z$ of the source implies that neither can be directly extracted from the signal, but only the combination $M(1+z)$, the redshifted mass. Recent work has shown that for binary neutron star systems, a tidal correction to the gravitational-wave phase in the late-inspiral…
We develop a novel statistical strong lensing approach to probe the cosmological parameters by exploiting multiple redshift image systems behind galaxies or galaxy clusters. The method relies on free-form mass inversion of strong lenses and does not need any additional information other than gravitational lensing. Since in free-form lensing the solution space is a high-dimensional convex polytope, we consider Bayesian model comparison analysis to infer the cosmological parameters. The volume of the solution space is taken as a tracer of the probability of the underlying cosmological assumption…
Strong gravitational lensing of sources with different redshifts has been used to determine cosmological distance ratios, which in turn depend on the expansion history. Hence, such systems are viewed as potential tools for constraining cosmological parameters. Here we show that in lens systems with two distinct source redshifts, of which the nearest one contributes to the light deflection towards the more distant one, there exists an invariance transformation which leaves all strong lensing observables unchanged (except the product of time delay and Hubble constant), generalizing the well-know…
Attempts to constrain the Hubble constant using the strong gravitational lens system Q0957+561 are limited by systematic uncertainties in the mass model, since the time delay is known very precisely. One important systematic effect is the mass sheet degeneracy, which arises because strong lens modeling cannot constrain the presence or absence of a uniform mass sheet $κ$, which rescales $H_0$ by the factor $(1-κ)$. In this paper we present new constraints on the mass sheet derived from a weak-lensing analysis of the Hubble Space Telescope imaging of a 6 arcmin square region surrounding the lens…
Strongly lensed explosive transients such as supernovae, gamma-ray bursts, fast radio bursts, and gravitational waves are very promising tools to determine the Hubble constant ($H_0$) in the near future in addition to strongly lensed quasars. In this work, we show that the transient nature of the point source provides an advantage over quasars: the lensed host galaxy can be observed before or after the transient's appearance. Therefore, the lens model can be derived from images free of contamination from bright point sources. We quantify this advantage by comparing the precision of a lens mode…
We use a new non-parametric gravitational modelling tool -- \Glass{} -- to determine what quality of data (strong lensing, stellar kinematics, and/or stellar masses) are required to measure the circularly averaged mass profile of a lens and its shape. \Glass{} uses an under-constrained adaptive grid of mass pixels to model the lens, searching through thousands of models to marginalise over model uncertainties. Our key findings are as follows: (i) for pure lens data, multiple sources with wide redshift separation give the strongest constraints as this breaks the well-known mass-sheet or steepne…
Project Dinos II: Redshift evolution of dark and luminous matter density profiles in strong-lensing elliptical galaxies across 0.1 < z < 0.9
University of California, Los Angeles（加州大学洛杉矶分校）；Kavli Institute for Cosmological Physics, University of Chicago（芝加哥大学卡弗里宇宙物理研究所）；Department of Astronomy and Astrophysics, University of Chicago（芝加哥大学天文与天体物理系）；Shanghai Jiao Tong University（上海交通大学）；Stony Brook University（石溪大学）；University of Oxford（牛津大学）；Institute of Cosmology and Gravitation, University of Portsmouth（朴茨茅斯大学宇宙学与引力研究所）；Department of Physics, University of Chicago（芝加哥大学物理系）
The modern time-delay cosmography aims to infer the cosmological parameters with a competitive precision from observing a multiply imaged quasar. The success of this technique relies upon a robust modeling of the lens mass distribution. Unfortunately strong degeneracies between density profiles that lead to almost the same lensing observables may bias precise estimate for the Hubble constant. The source position transformation (SPT), which covers the well-known mass sheet transformation (MST) as a special case, defines a new framework to investigate these degeneracies. In this paper, we presen…
Gravitational lensing has long been considered as a valuable tool to determine the total mass of galaxy clusters. The shear profile as inferred from the statistics of ellipticity of background galaxies allows to probe the cluster intermediate and outer regions thus determining the virial mass estimate. However, the mass sheet degeneracy and the need for a large number of background galaxies motivate the search for alternative tracers which can break the degeneracy among model parameters and hence improve the accuracy of the mass estimate. Lensing flexion, i.e. the third derivative of the lensi…
Cosmological parameter constraints from observations of time-delay lenses are becoming increasingly precise. However, there may be significant bias and scatter in these measurements due to, among other things, the so-called mass-sheet degeneracy. To estimate these uncertainties, we analyze strong lenses from the largest EAGLE hydrodynamical simulation. We apply a mass-sheet transformation to the radial density profiles of lenses, and by selecting lenses near isothermality, we find that the bias on H0 can be reduced to 5% with an intrinsic scatter of 10%, confirming previous results performed o…
The main obstacle for gravitational lensing to determine accurate masses of deflectors, or to determine precise estimates for the Hubble constant, is the degeneracy of lensing observables with respect to the mass-sheet transformation (MST). The MST is a global modification of the mass distribution which leaves all image positions, shapes and flux ratios invariant, but which changes the time delay. Here we show that another global transformation of lensing mass distributions exists which almost leaves image positions and flux ratios invariant, and of which the MST is a special case. Whereas for…
We introduce a novel method for reconstructing the projected matter distributions of galaxy clusters with weak-lensing (WL) data based on convolutional neural network (CNN). Training datasets are generated with ray-tracing through cosmological simulations. We control the noise level of the galaxy shear catalog such that it mimics the typical properties of the existing ground-based WL observations of galaxy clusters. We find that the mass reconstruction by our multi-layered CNN with the architecture of alternating convolution and trans-convolution filters significantly outperforms the tradition…
In strong gravitational lens systems, the light bending is usually dominated by one main galaxy, but may be affected by other mass along the line of sight (LOS). Shear and convergence can be used to approximate the contributions from less significant perturbers (e.g. those that are projected far from the lens or have a small mass), but higher order effects need to be included for objects that are closer or more massive. We develop a framework for multiplane lensing that can handle an arbitrary combination of tidal planes treated with shear and convergence and planes treated exactly (i.e., incl…
A power-law density model, i.e., $ρ(r) \propto r^{-γ'}$ has been commonly employed in strong gravitational lensing studies, including the so-called time-delay technique used to infer the Hubble constant $H_0$. However, since the radial scale at which strong lensing features are formed corresponds to the transition from the dominance of baryonic matter to dark matter, there is no known reason why galaxies should follow a power law in density. The assumption of a power law artificially breaks the mass-sheet degeneracy, a well-known invariance transformation in gravitational lensing which affects…
We present a detailed strong lensing analysis of an HST/ACS legacy dataset for the first gravitational lens, Q0957+561. With deep imaging we identify 24 new strongly lensed features, which we use to constrain mass models. We model the stellar component of the lens galaxy using the observed luminosity distribution, and the dark matter halo using several different density profiles. We draw on the weak lensing analysis by Nakajima et al. (2009) to constrain the mass sheet and environmental terms in the lens potential. Adopting the well-measured time delay, we find H_0 = 85 (+14/-13) km/s/Mpc (68%…
Lens magnification by galaxy clusters induces characteristic spatial variations in the number counts of background sources, amplifying their observed fluxes and expanding the area of sky, the net effect of which, known as magnification bias, depends on the intrinsic faint-end slope of the source luminosity function. The bias is strongly negative for red galaxies, dominated by the geometric area distortion, whereas it is mildly positive for blue galaxies, enhancing the blue counts toward the cluster center. We generalize the Bayesian approach of Umetsu et al. for reconstructing projected cluste…
TDCOSMO XXVIII. The Hubble constant from the quadruply lensed quasar J1537$-$3010 with precise time delays
University of Geneva（日内瓦大学）；Technical University of Munich（慕尼黑工业大学）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；Laboratoire Univers et Particules de Montpellier (LUPM), CNRS & Université Montpellier（蒙彼利埃宇宙与粒子实验室（LUPM），法国国家科学研究中心与蒙彼利埃大学）；STAR Institute, University of Liège（列日大学STAR研究所）；Armagh Observatory and Planetarium（阿马天文台和天文馆）；Fermi National Accelerator Laboratory（费米国家加速器实验室）
Strong Lensing Tomography: Double and pseudo multi-source plane strong gravitational lensing to constrain dark energy
Stony Brook University（石溪大学）；Stanford University（斯坦福大学）；SLAC National Accelerator Laboratory（SLAC国家加速器实验室）；Universidad Andres Bello（安德斯贝洛大学）；Oxford University（牛津大学）；University of Portsmouth（朴茨茅斯大学）
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 a…
The information about the mass density of galaxy clusters provided by the gravitational lens effect has inspired many inversion techniques. In this article, updates to the previously introduced method in Grale are described, and explored in a number of examples. The first looks into a different way of incorporating time delay information, not requiring the unknown source position. It is found that this avoids a possible bias that leads to "over-focusing" the images, i.e. providing source position estimates that lie in a considerably smaller region than the true positions. The second is inspire…
Strong gravitational lensing is regarded as the most precise technique to measure the mass in the inner region of galaxies or galaxy clusters. In particular, the mass within one Einstein radius can be determined with an accuracy of order of a few percent or better, depending on the image configuration. For other radii, however, degeneracies exist between galaxy density profiles, precluding an accurate determination of the enclosed mass. The source position transformation (SPT), which includes the well-known mass-sheet transformation (MST) as a special case, describes this degeneracy of the len…
We reconstruct the two-dimensional (2D) matter distributions in 20 high-mass galaxy clusters selected from the CLASH survey by using the new approach of performing a joint weak lensing analysis of 2D shear and azimuthally averaged magnification measurements. This combination allows for a complete analysis of the field, effectively breaking the mass-sheet degeneracy. In a Bayesian framework, we simultaneously constrain the mass profile and morphology of each individual cluster assuming an elliptical Navarro-Frenk-White halo characterized by the mass, concentration, projected axis ratio, and pos…
Weak lensing is one of the best available diagnostic tools to measure the total density profiles of distant clusters of galaxies. Unfortunately, it suffers from the well-known mass-sheet degeneracy, so that weak lensing analyses cannot lead to fully reliable determinations of the total mass of the clusters. One possible way to set the relevant scale of the density profile would be to make a direct measurement of the magnification produced by the clusters as gravitational lenses. In this paper we revisit a suggestion made a few years ago for this general purpose, based on the use of the Fundame…
Shape Degeneracies: the likely culprit for the differences between lens mass models of galaxy clusters
University of Minnesota（明尼苏达大学）；UHasselt（荷语哈瑟尔特大学）；Indian Institute of Science Bangalore（印度科学研究所班加罗尔分校）；Aix Marseille Univ（艾克斯-马赛大学）；Max Planck Institute for Extraterrestrial Physics（马克斯·普朗克地外物理研究所）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理研究所）
We perform a 3D multi-probe analysis of the rich galaxy cluster A1689 by combining improved weak-lensing data from new BVRi'z' Subaru/Suprime-Cam observations with strong-lensing, X-ray, and Sunyaev-Zel'dovich effect (SZE) data sets. We reconstruct the projected matter distribution from a joint weak-lensing analysis of 2D shear and azimuthally integrated magnification constraints, the combination of which allows us to break the mass-sheet degeneracy. The resulting mass distribution reveals elongation with axis ratio ~0.7 in projection. When assuming a spherical halo, our full weak-lensing anal…
