Mass-sheet degeneracy, power-law models and external convergence: Impact on the determination of the Hubble constant from gravitational lensing
Peter Schneider, Dominique Sluse
备注：
Accepted for publication in Astronomy and Astrophysics. Discussion expanded (MSD and velocity dispersion, MSD and free form lens models, MSD and multiple source redshifts)
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 analytical arguments and numerical experiments reveal that mass-sheet like transformations yield significant freedom in choosing the mass profile, even when exquisite Einstein rings are observed. Furthermore, the characterization of the environment of the lens does not break that degeneracy which is not physically linked to extrinsic convergence. We present an illustrative example where the multiple imaging properties of a composite (baryons + dark matter) lens can be extremely well reproduced by a power-law model having the same velocity dispersion, but with predictions for the Hubble constant that deviate by $\sim 20%$. Hence we conclude that the impact of degeneracies between parametrized models have been underestimated in current $H_0$ measurements from lensing, and need to be carefully reconsidered.
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…
Massive neutrinos, due to their free streaming, produce a suppression in the matter power spectrum at intermediate and small scales which could be probed by galaxy clustering and/or weak lensing observables. This effect happens at scales that are also influenced by baryon feedback, i.e. galactic winds or Active Galactic Nuclei (AGN) feedback, which in realistic hydrodynamic simulations has also been shown to produce a suppression of power. Leaving aside, for the moment, the complex issue of galaxy bias, we focus here on matter clustering and tomographic weak lensing, we investigate the possibl…
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…
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）
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…
We study the optimal use of third order statistics in the analysis of weak lensing by large-scale structure. These higher order statistics have long been advocated as a powerful tool to break measured degeneracies between cosmological parameters. Using ray-tracing simulations, incorporating important survey features such as a realistic depth-dependent redshift distribution, we find that a joint two- and three-point correlation function analysis is a much stronger probe of cosmology than the skewness statistic. We compare different observing strategies, showing that for a limited survey time th…
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 tr…
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…
We consider several aspects of the generalized multi-plane gravitational lens theory, in which light rays from a distant source are affected by several main deflectors, and in addition by the tidal gravitational field of the large-scale matter distribution in the Universe when propagating between the main deflectors. Specifically, we derive a simple expression for the time-delay function in this case, making use of the general formalism for treating light propagation in inhomogeneous spacetimes which leads to the characterization of distance matrices between main lens planes. Applying Fermat's…
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…
Context. Time delay lensing is a powerful tool to measure the Hubble constant $H_0$. In order to obtain an accurate estimate of $H_0$ from a sample of time delay strong lenses, however, it is necessary to have a very good knowledge of the mass structure of the lens galaxies. Strong lensing data on their own are not sufficient to break the degeneracy between $H_0$ and the lens model parameters, on a single object basis. Aims. The goal of this study is to determine whether it is possible to break the $H_0$-lens structure degeneracy with the statistical combination of a large sample of time-delay…
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…
We constrain properties of cluster haloes by performing likelihood analysis using lensing shear and flexion data. We test our analysis using two mock cluster haloes: an isothermal ellipsoid (SIE) model and a more realistic elliptical Navarro-Frenk-White (eNFW) model. For both haloes, we find that flexion is more sensitive to the halo ellipticity than shear. The introduction of flexion information significantly improves the constraints on halo ellipticity, orientation and mass. We also point out that there is a degeneracy between the mass and the ellipticity of SIE models in the lensing signal.
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…
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…
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…
This paper is the first of a series of papers constraining cosmological parameters with weak lensing peak statistics using $\sim 450~\rm deg^2$ of imaging data from the Kilo Degree Survey (KiDS-450). We measure high signal-to-noise ratio (SNR: $ν$) weak lensing convergence peaks in the range of $3<ν<5$, and employ theoretical models to derive expected values. These models are validated using a suite of simulations. We take into account two major systematic effects, the boost factor and the effect of baryons on the mass-concentration relation of dark matter haloes. In addition, we investigate t…
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 perform a thorough examination of the neutrino mass ($M_ν$) constraints achievable by combining future spectroscopic galaxy surveys with cosmic microwave background (CMB) experiments, focusing on the contribution of CMB lensing and galaxy-CMB lensing. CMB lensing can help by breaking the $M_ν$-curvature degeneracy when combined with baryon acoustic oscillation (BAO)-only measurements, but we demonstrate this combination wastes a great deal of constraining power, as the broadband shape of the power spectrum contributes significantly to constraints. We also expand on previous work to demonstr…
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…
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）
We aim to constrain mass-varying neutrino models using large scale structure observations and produce forecast for the Euclid survey. We investigate two models with different scalar field potential and both positive and negative coupling parameters β. These parameters correspond to growing or decreasing neutrino mass, respectively. We explore couplings up to |β|<5. In the case of the exponential potential, we find an upper limit on $Ω_νh^2$<0.004 at 2-$σ$ level. In the case of the inverse power law potential the null coupling can be excluded with more than 2-σsignificance; the limits on the co…
We investigate how the cosmological constraints from SNe Ia are improved by including the effects of weak-lensing convergence. To do so, we introduce the lognormal function as the convergence PDF modeling the lensing scatter of SN Ia magnitude, and apply a sample selection for SNe Ia to avoid strongly lensed samples. Comparing with the contribution of other uncertainties (e.g., the intrinsic magnitude scatter), we find that the lensing effect is dominant at $z > 1$. Then forecasting the parameter constraints for the Wide-Field InfraRed Survey Telescope survey, we show that considering the weak…
We explore the Minkowski functionals of weak lensing convergence map to distinguish between $f(R)$ gravity and the general relativity (GR). The mock weak lensing convergence maps are constructed with a set of high-resolution simulations assuming different gravity models. It is shown that the lensing MFs of $f(R)$ gravity can be considerably different from that of GR because of the environmentally dependent enhancement of structure formation. We also investigate the effect of lensing noise on our results, and find that it is likely to distinguish F5, F6 and GR gravity models with a galaxy surve…
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…
Uncertain feedback processes in galaxies affect the distribution of matter, currently limiting the power of weak lensing surveys. If we can identify cosmological statistics that are robust against these uncertainties, or constrain these effects by other means, then we can enhance the power of current and upcoming observations from weak lensing surveys such as DES, Euclid, the Rubin Observatory, and the Roman Space Telescope. In this work, we investigate the potential of the electron density auto-power spectrum as a robust probe of cosmology and baryonic feedback. We use a suite of (magneto-)hy…
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）
Weak gravitational lensing has become a common tool to constrain the cosmological model. The majority of the methods to derive constraints on cosmological parameters use second-order statistics of the cosmic shear. Despite their success, second-order statistics are not optimal and degeneracies between some parameters remain. Tighter constraints can be obtained if second-order statistics are combined with a statistic that is efficient to capture non-Gaussianity. In this paper, we search for such a statistical tool and we show that there is additional information to be extracted from statistical…
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…
We examine the dark energy and matter densities allowed by precision measurements of distances out to various redshifts, in the presence of spatial curvature and (near) arbitrary behavior of the dark energy equation of state. Degeneracies among the parameters permit a remarkably large variation in their values when using only distance measurements of the late time universe and making no assumptions about the dark energy or curvature. Going beyond distance measurements to a lower limit on the growth of structure bounds the allowed region significantly but still leaves considerable freedom to ma…
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 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。
Measuring dark matter substructure within galaxy cluster haloes is a fundamental probe of the Lambda-CDM model of structure formation. Gravitational lensing is a technique for measuring the total mass distribution which is independent of the nature of the gravitating matter, making it a vital tool for studying these dark-matter dominated objects. We present a new method for measuring weak gravitational lensing flexions, the gradients of the lensing shear field, to measure mass distributions on small angular scales. While previously published methods for measuring flexions focus on measuring de…
Matter near a gravitational lens galaxy or projected along the line of sight (LOS) can affect strong lensing observables by more than contemporary measurement errors. We simulate lens fields with realistic three-dimensional mass configurations (self-consistently including voids), and then fit mock lensing observables with increasingly complex lens models to quantify biases and uncertainties associated with different ways of treating the lens environment (ENV) and LOS. We identify the combination of mass, projected offset, and redshift that determines the importance of a perturbing galaxy for l…
We construct a filament catalogue using an extension of the halo based filament finder of Zhang et al.(2009), in a 250 Mpc/h side N-body simulation, and study the properties of filaments ending upon or surrounding galaxy clusters (within 10 Mpc/h). In this region, the majority of filamentary mass, halo mass, and galaxy richness centered upon the cluster tends to lie in sheets, which are not always coincident. Fixing a sheet width of 3 Mpc/h for definiteness, we find the sheet orientations and (connected) filamentary mass, halo mass and richness fractions relative to the surrounding sphere. Fil…
We present an analytic approach to lift the mass-anisotropy degeneracy in clusters of galaxies by utilizing the line-of-sight velocity dispersion of clustered galaxies jointly with weak lensing inferred masses. More specifically, we solve the spherical Jeans equation by assuming a simple relation between the line-of-sight velocity dispersion and the radial velocity dispersion and recast the Jeans equation as a Bernoulli differential equation that has a well-known analytic solution. We first test our method in cosmological N-body simulations and then derive the anisotropy profiles for 35 archiv…
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%…
Minkowski functionals (MFs) quantify the topological properties of a given field probing its departure from Gaussianity. We investigate their use on lensing convergence maps in order to see whether they can provide further insights on the underlying cosmology with respect to the standard second-order statistics, i.e., cosmic shear tomography. To this end, we first present a method to match theoretical predictions with measured MFs taking care of the shape noise, imperfections in the map reconstruction, and inaccurate description of the nonlinearities in the matter power spectrum and bispectrum…
We study the constraints which the next generation of radio telescopes could place on the mass and number of neutrino species by studying the gravitational lensing of high redshift 21 cm emission in combination with wide-angle surveys of galaxy lensing. We use simple characterizations of reionization history and of proposed telescope designs to forecast the constraints and detectability threshold for neutrinos. It is found that the degeneracy between neutrino parameters and dark energy parameters is significantly reduced by incorporating 21 cm lensing. The combination of galaxy and 21 cm lensi…
In recent years, it has become possible to detect individual dark matter subhalos near images of strongly lensed extended background galaxies. Typically, only the most massive subhalos in the strong lensing region may be detected this way. In this work, we show that strong lenses may also be used to constrain the much more numerous population of lower mass subhalos that are too small to be detected individually. In particular, we show that the power spectrum of projected density fluctuations in galaxy halos can be measured using strong gravitational lensing. We develop the mathematical framewo…
We revise the cosmological bounds on Hořava gravity taking into accounts the stringent constraint on the speed of propagation of gravitational waves from GW170817 and GRB170817A. In light of this we also investigate the degeneracy between massive neutrinos and Hořava gravity. We show that a luminal propagation of gravitational waves suppresses the large-scale Cosmic Microwave Background (CMB) radiation temperature anisotropies and the presence of massive neutrinos increases this effect. On the contrary large neutrinos mass can compensate the modifications induced by Hořava gravity in the lensi…
For the first time, measurements of the cosmic microwave background radiation (CMB) alone favor cosmologies with $w=-1$ dark energy over models without dark energy at a 3.2-sigma level. We demonstrate this by combining the CMB lensing deflection power spectrum from the Atacama Cosmology Telescope with temperature and polarization power spectra from the Wilkinson Microwave Anisotropy Probe. The lensing data break the geometric degeneracy of different cosmological models with similar CMB temperature power spectra. Our CMB-only measurement of the dark energy density $Ω_Λ$ confirms other measureme…
As weak lensing surveys are becoming deeper and cover larger areas, information will be available on small angular scales down to the arcmin level. To extract this extra information, accurate modelling of baryonic effects is necessary. In this work, we adopt a baryonic correction model, which includes gas both bound inside and ejected from dark matter (DM) haloes, a central galaxy, and changes in the DM profile induced by baryons. We use this model to incorporate baryons into a large suite of DM-only $N$-body simulations, covering a grid of 75 cosmologies in the $Ω_\mathrm{m}-σ_8$ parameter sp…
Based on a mass-selected sample of galaxy-scale strong gravitational lenses from the SLACS, BELLS, LSD and SL2S surveys and using a well-motivated fiducial set of lens-galaxy parameters we tested the weak-field metric on kiloparsec scales and found a constraint on the post-Newtonian parameter $γ= 0.995^{+0.037}_{-0.047}$ under the assumption of a flat $Λ$CDM universe with parameters taken from \textit{Planck} observations. General relativity (GR) predicts exactly $γ=1$. Uncertainties concerning the total mass density profile, anisotropy of the velocity dispersion and the shape of the light-pro…
Massive neutrinos and interacting dark matter look alike through the lens of lensing
Lehman College, City University of New York（纽约市立大学莱曼学院）；Graduate Center, City University of New York（纽约市立大学研究生院）；American Museum of Natural History（美国自然历史博物馆）；University of Hawaii at Manoa（夏威夷大学马诺阿分校）
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…
We present a revised strong lensing mass reconstruction of the galaxy cluster RX J1347.5-1145. The X-ray luminous cluster at redshift z=0.451 has already been studied intensively in the past. Based on information of two such previous (strong-)lensing studies by Halkola et al. (2008) and Bradac et al. (2008), as well as by incorporating newly available data from the Cluster Lensing And Supernovae survey with Hubble (CLASH, Postman et al. 2012), we identified four systems of multiply lensed images (anew) in the redshift range 1.75 <= z <= 4.19. One multiple image system consists of in total eigh…
