Gravitational Lens Models Research Articles

Gravity.jl: Fast and accurate gravitational lens modeling in Julia

We present Gravity.jl, a new proprietary software for the modeling of gravitational lens systems. Gravity.jl is written in the Julia programming language, and is designed to be fast, accurate, and flexible. It can be used to model gravitational lens systems composed of multiple lensing planes, and to perform Bayesian inference on the lens model parameters. In this paper we present the theoretical and statistical ideas behind the code, and we describe its main features. In this first paper of the series, we focus on the modeling of point-like and small extended sources, for which we can linearize the lens equation. We show a practical use of Gravity.jl on a galaxy-scale lens, and we compare the results with those obtained with other codes. We also show how Gravity.jl can be used to perform Bayesian inference on cosmological parameters.

Weak gravitational lensing around dyonic ModMax black hole in plasma

In this study, we examine the effects of weak gravitational lensing and determine the shadow radius around black holes within the Dyonic ModMax (DM) spacetime, also accounting for models with nonuniform plasma distributions. By analyzing various gravitational lens models, we compare corrections to vacuum lensing due to gravitational effects within plasma and plasma inhomogeneity, finding that these effects could be observed in hot gas within galaxy clusters. Starting with the orbits of photons around a black hole in DM, we investigate the shadow and weak gravitational lensing phenomena. Utilizing observational data from the Event Horizon Telescope for M87* and SgrA*, we constrain parameters within DM gravity. To connect our findings to observations, we examine the magnification and positioning of lensed images, along with the weak deflection angle and magnification for sources near different galaxies.

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.

Gravitational imaging through a triple source plane lens: revisiting the ΛCDM-defying dark subhalo in SDSSJ0946+1006

ABSTRACT The ΛCDM paradigm successfully explains the large-scale structure of the Universe, but is less well constrained on subgalactic scales. Gravitational lens modelling has been used to measure the imprints of dark substructures on lensed arcs, testing the small-scale predictions of ΛCDM. However, the methods required for these tests are subject to degeneracies among the lens mass model and the source light profile. We present a case study of the unique compound gravitational lens SDSSJ0946+1006, wherein a dark, massive substructure has been detected, whose reported high concentration would be unlikely in a ΛCDM universe. For the first time, we model the first two background sources in both I- and U-band HST imaging, as well as VLT-MUSE emission line data for the most distant source. We recover a lensing perturber at a 5.9σ confidence level with mass $\log _{10}(M_\mathrm{sub}/{\rm M}_{\odot })=9.2^{+0.4}_{-0.1}$ and concentration $\log _{10}c=2.4^{+0.5}_{-0.3}$. The concentration is more consistent with CDM subhaloes than previously reported, and the mass is compatible with that of a dwarf satellite galaxy whose flux is undetectable in the data at the location of the perturber. A wandering black hole with mass $\log _{10}(M_\mathrm{BH}/{\rm M}_{\odot })=8.9^{+0.2}_{-0.1}$ is a viable alternative model. We systematically investigate alternative assumptions about the complexity of the mass distribution and source reconstruction; in all cases the subhalo is detected at around the ≥5σ level. However, the detection significance can be altered substantially (up to 11.3σ) by alternative choices for the source regularization scheme.

GA-NIFS: JWST/NIRSpec integral field unit observations of HFLS3 reveal a dense galaxy group at z ∼ 6.3

Massive, starbursting galaxies in the early Universe represent some of the most extreme objects in the study of galaxy evolution. One such source is HFLS3 (z ∼ 6.34), which was originally identified as an extreme starburst galaxy with mild gravitational magnification (μ ∼ 2.2). Here, we present new observations of HFLS3 with the JWST/NIRSpec integral field unit in both low (PRISM/CLEAR; R ∼ 100) and high spectral resolution (G395H/290LP; R ∼ 2700), with high spatial resolution (∼0.1″) and sensitivity. Using a combination of the NIRSpec data and a new lensing model with accurate spectroscopic redshifts, we find that the 3″ × 3″ field is crowded, with a lensed arc (C, z = 6.3425 ± 0.0002), two galaxies to the south (S1 and S2, z = 6.3592 ± 0.0001), two galaxies to the west (W1, z = 6.3550 ± 0.0001; W2, z = 6.3628 ± 0.0001), and two low-redshift interlopers (G1, z = 3.4806 ± 0.0001; G2, z = 2.00 ± 0.01). We present spectral fits and morpho-kinematic maps for each bright emission line (e.g. [OIII]λ5007, Hα, and [NII]λ6584) from the R2700 data for all sources except G2 (whose spectral lines fall outside the observed wavelengths of the R2700 data). From a line ratio analysis, we find that the galaxies in component C are likely powered by star formation, though we cannot rule out or confirm the presence of active galactic nuclei in the other high-redshift sources. We performed gravitational lens modelling, finding evidence for a two-source composition of the lensed central object and a magnification factor (μ = 2.1 − 2.4) comparable to findings of previous work. The projected distances and velocity offsets of each galaxy suggest that they will merge within the next ∼1 Gyr. Finally, we examined the dust extinction-corrected SFRHα of each z > 6 source, finding that the total star formation (510 ± 140 M⊙ yr−1, magnification-corrected) is distributed across the six z ∼ 6.34 − 6.36 objects over a region of diameter ∼11 kpc. Altogether, this suggests that HFLS3 is not a single starburst galaxy, but instead a merging system of star-forming galaxies in the epoch of reionisation.

COOL-LAMPS. Discovery of COOL J0335−1927, a Gravitationally Lensed Quasar at z = 3.27 with an Image Separation of 23.″3

We report the discovery of COOL J0335−1927, a quasar at z = 3.27 lensed into three images with a maximum separation of 23.″3 by a galaxy cluster at z = 0.4178. To date, this is the highest redshift wide-separation lensed quasar known. In addition, COOL J0335−1927 shows several strong intervening absorbers visible in the spectra of all three quasar images with varying equivalent widths. The quasar also shows mini-broad line absorption. We construct a parametric strong gravitational lens model using ground-based imaging, constrained by the redshift and positions of the quasar images as well as the positions of three other multiply imaged background galaxies. Using our best-fit lens model, we calculate the predicted time delays between the three quasar images to be Δt AB = (stat) and Δt AC = (stat) days. Folding in systematic uncertainties, the model-predicted time delays are within the ranges 240 < Δt AB < 700 and −300 < Δt AC < −30. We also present g-band photometry from archival Dark Energy Camera Legacy Survey and Pan-STARRS imaging, and new multi-epoch observations obtained between 2022 September 18 UT and 2023 February 22 UT, which demonstrate significant variability in the quasar and will eventually enable the measurement of the time delay between the three quasar images. The currently available light curves are consistent with the model-predicted time delays. This is the fifth paper from the COOL-LAMPS collaboration.

HOLISMOKES

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

New Tests of Milli-lensing in the Blazar PKS 1413 + 135

Abstract Symmetric achromatic variability (SAV) is a rare form of radio variability in blazars that has been attributed to gravitational milli-lensing by a ∼102–105 M ⊙ mass condensate. Four SAVs have been identified between 1980 and 2020 in the long-term radio monitoring data of the blazar PKS 1413 + 135. We show that all four can be fitted with the same, unchanging, gravitational lens model. If SAV is due to gravitational milli-lensing, PKS 1413 + 135 provides a unique system for studying active galactic nuclei with unprecedented microarcsecond resolution, as well as for studying the nature of the milli-lens itself. We discuss two possible candidates for the putative milli-lens: a giant molecular cloud hosted in the intervening edge-on spiral galaxy, and an undetected dwarf galaxy with a massive black hole. We find a significant dependence of SAV crossing time on frequency, which could indicate a fast shock moving in a slower underlying flow. We also find tentative evidence for a 989 day periodicity in the SAVs, which, if real, makes possible the prediction of future SAVs: the next three windows for possible SAVs begin in 2022 August, 2025 May, and 2028 February.

Large-scale Gravitational Lens Modeling with Bayesian Neural Networks for Accurate and Precise Inference of the Hubble Constant

Abstract We investigate the use of approximate Bayesian neural networks (BNNs) in modeling hundreds of time delay gravitational lenses for Hubble constant (H 0) determination. Our BNN was trained on synthetic Hubble Space Telescope quality images of strongly lensed active galactic nuclei with lens galaxy light included. The BNN can accurately characterize the posterior probability density functions (PDFs) of model parameters governing the elliptical power-law mass profile in an external shear field. We then propagate the BNN-inferred posterior PDFs into an ensemble H 0 inference, using simulated time delay measurements from a plausible dedicated monitoring campaign. Assuming well-measured time delays and a reasonable set of priors on the environment of the lens, we achieve a median precision of 9.3% per lens in the inferred H 0. A simple combination of a set of 200 test lenses results in a precision of 0.5 km s−1 Mpc−1 (0.7%), with no detectable bias in this H 0 recovery test. The computation time for the entire pipeline—including the generation of the training set, BNN training and H 0 inference—translates to 9 minutes per lens on average for 200 lenses and converges to 6 minutes per lens as the sample size is increased. Being fully automated and efficient, our pipeline is a promising tool for exploring ensemble-level systematics in lens modeling for H 0 inference.

Auto-identification of unphysical source reconstructions in strong gravitational lens modelling

ABSTRACT With the advent of next-generation surveys and the expectation of discovering huge numbers of strong gravitational lens systems, much effort is being invested into developing automated procedures for handling the data. The several orders of magnitude increase in the number of strong galaxy–galaxy lens systems is an insurmountable challenge for traditional modelling techniques. Whilst machine learning techniques have dramatically improved the efficiency of lens modelling, parametric modelling of the lens mass profile remains an important tool for dealing with complex lensing systems. In particular, source reconstruction methods are necessary to cope with the irregular structure of high-redshift sources. In this paper, we consider a convolutional neural network (CNN) that analyses the outputs of semi-analytic methods that parametrically model the lens mass and linearly reconstruct the source surface brightness distribution. We show the unphysical source reconstructions that arise as a result of incorrectly initialized lens models can be effectively caught by our CNN. Furthermore, the CNN predictions can be used to automatically reinitialize the parametric lens model, avoiding unphysical source reconstructions. The CNN, trained on reconstructions of lensed Sérsic sources, accurately classifies source reconstructions of the same type with a precision P &amp;gt; 0.99 and recall R &amp;gt; 0.99. The same CNN, without retraining, achieves P = 0.89 and R = 0.89 when classifying source reconstructions of more complex lensed Hubble Ultra-Deep Field (HUDF) sources. Using the CNN predictions to reinitialize the lens modelling procedure, we achieve a 69 per cent decrease in the occurrence of unphysical source reconstructions. This combined CNN and parametric modelling approach can greatly improve the automation of lens modelling.

A novel approach to visibility-space modelling of interferometric gravitational lens observations at high angular resolution

ABSTRACT We present a new gravitational lens modelling technique designed to model high-resolution interferometric observations with large numbers of visibilities without the need to pre-average the data in time or frequency. We demonstrate the accuracy of the method using validation tests on mock observations. Using small data sets with ∼103 visibilities, we first compare our approach with the more traditional direct Fourier transform (DFT) implementation and direct linear solver. Our tests indicate that our source inversion is indistinguishable from that of the DFT. Our method also infers lens parameters to within 1 to 2 per cent of both the ground truth and DFT, given sufficiently high signal-to-noise ratio (SNR). When the SNR is as low as 5, both approaches lead to errors of several tens of per cent in the lens parameters and a severely disrupted source structure, indicating that this is related to the SNR and choice of priors rather than the modelling technique itself. We then analyse a large data set with ∼108 visibilities and a SNR matching real global Very Long Baseline Interferometry observations of the gravitational lens system MG J0751+2716. The size of the data is such that it cannot be modelled with traditional implementations. Using our novel technique, we find that we can infer the lens parameters and the source brightness distribution, respectively, with an RMS error of 0.25 and 0.97 per cent relative to the ground truth.

Overconstrained gravitational lens models and the Hubble constant

ABSTRACT It is well known that measurements of H0 from gravitational lens time delays scale as H0 ∝ 1 − κE, where κE is the mean convergence at the Einstein radius RE but that all available lens data other than the delays provide no direct constraints on κE. The properties of the radial mass distribution constrained by lens data are RE and the dimensionless quantity ξ = REα″(RE)/(1 − κE), where α″(RE) is the second derivative of the deflection profile at RE. Lens models with too few degrees of freedom, like power-law models with densities ρ ∝ r−n, have a one-to-one correspondence between ξ and κE (for a power-law model, ξ = 2(n − 2) and κE = (3 − n)/2 = (2 − ξ)/4). This means that highly constrained lens models with few parameters quickly lead to very precise but inaccurate estimates of κE and hence H0. Based on experiments with a broad range of plausible dark matter halo models, it is unlikely that any current estimates of H0 from gravitational lens time delays are more accurate than ${\sim} 10{{\ \rm per\ cent}}$, regardless of the reported precision.

Lenstool-HPC: A High Performance Computing based mass modelling tool for cluster-scale gravitational lenses

With the upcoming generation of telescopes, cluster scale strong gravitational lenses will act as an increasingly relevant probe of cosmology and dark matter. The better resolved data produced by current and future facilities requires faster and more efficient lens modeling software. Consequently, we present Lenstool-HPC, a strong gravitational lens modeling and map generation tool based on High Performance Computing (HPC) techniques and the renowned Lenstool software. We also showcase the HPC concepts needed for astronomers to increase computation speed through massively parallel execution on supercomputers. Lenstool-HPC was developed using lens modelling algorithms with high amounts of parallelism. Each algorithm was implemented as a highly optimised CPU, GPU and Hybrid CPU-GPU version. The software was deployed and tested on the Piz Daint cluster of the Swiss National Supercomputing Centre (CSCS). Lenstool-HPC perfectly parallel lens map generation and derivative computation achieves a factor 30 speed-up using only 1 GPUs compared to Lenstool. Lenstool-HPC hybrid Lens-model fit generation tested at Hubble Space Telescope precision is scalable up to 200 CPU-GPU nodes and is faster than Lenstool using only 4 CPU-GPU nodes.

Gravitational lensing probability for the Konus-Wind gamma-ray bursts detected in the triggered mode

Although half a century has passed since the discovery of Gamma-ray bursts (GRBs), there is no evidence for any strongly lensed event. Using GRB luminosity function and GRB formation rate estimates for the Konus-Wind burst sample with known redshifts, and a Singular Isothermal Sphere as a gravitational lens model, we predict the detection rate of both GRB images being ≈ 0.02–0.05 events per year. The result is consistent with the non-detection of lensed bursts in the KW triggered GRB sample.

The Atacama Cosmology Telescope: CO(J = 3 – 2) Mapping and Lens Modeling of an ACT-selected Dusty Star-forming Galaxy

Abstract We report Northern Extended Millimeter Array CO(J = 3 − 2) observations of the dusty star-forming galaxy ACT-S J020941+001557 at z = 2.5528, which was detected as an unresolved source in the Atacama Cosmology Telescope (ACT) equatorial survey. Our spatially resolved spectral line data support the derivation of a gravitational lens model from 37 independent velocity channel maps using a pixel-based algorithm, from which we infer a velocity-dependent magnification factor μ ≈ 7–22 with a luminosity-weighted mean . The resulting source-plane reconstruction is consistent with a rotating disk, although other scenarios cannot be ruled out by our data. After correction for lensing, we derive a line luminosity , a cold gas mass , a dynamical mass , and a gas mass fraction . The line brightness temperature ratio of r 3,1 ≈ 1.6 relative to a Green Bank Telescope CO(J = 1 − 0) detection may be elevated by a combination of external heating of molecular clouds, differential lensing, and/or pointing errors.

Searching for the 380 GHz H2O emission from the high-z lensed quasi-stellar object MG J0414+0534

Abstract We report the results of our search for the 380 GHz H2O line emissions from the quadruply lensed quasi-stellar object (QSO) MG J0104+0534 at z = 2.639 with the Atacama Large Millimeter/submillimeter Array. Our observation shows a tentative detection of the 380 GHz line from the lensed QSO, and the line spectrum shows a broad spectral distribution that has a velocity width of ∼290 km s−1 and a peak flux of ∼0.8 mJy. The integrated intensity map of the H2O line shows lensed emissions at the A1 and A2 component of the QSO, with the A2 component slightly resolved. The integrated line flux ratio between the A1 and A2 components shows an unexpected difference from the continuum flux ratio. Based on our gravitational lens modeling, assuming our tentative detection is real, this flux ratio anomaly would suggest that the 380 GHz line emissions come from two or three spatially displaced locations in the QSO, with the dominant one located at the position of the continuum emission from the QSO, and the other one(s) displaced from the continuum by ∼1.5 kpc on the source plane.

Dual-component plasma lens models

In contrast to the converging, achromatic behaviour of axisymmetric gravitational lenses, diverging frequency-dependent lensing occurs from refraction due to a distribution of over-dense axisymmetric plasma along an observer's line of sight. Such plasma lenses are particularly interesting from the point of view of astronomical observations because they can both magnify and dim the appearance of background sources as a function of frequency. Plasma lensing is believed to be involved in a number of separate phenomena involving the scintillation of radio pulsars, extreme scattering events of background radio sources and may also play a role in the generation of fast radio bursts. These lensing phenomena are believed to occur in dense environments, in which there may be many density perturbations between an observer and background source. In this work we generalize individual plasma lens models to produce dual component lenses using families of plasma lens models previously studied in the literature, namely the exponential and softened power-law lenses. Similar to binary gravitational lens models, these dual component plasma lenses feature a rich and complex critical and caustic morphology, as well as generate more complicated light curves. We map the number of criticals formed for a given component separation and angular size, and highlight a relevant degeneracy between two particular models. This work provides an argument in favor of close monitoring of extreme scattering events in progress in order to break such model degeneracies.

Mass and Light of Abell 370: A Strong and Weak Lensing Analysis

Abstract We present a new gravitational lens model of the Hubble Frontier Fields cluster Abell 370 (z = 0.375) using imaging and spectroscopy from Hubble Space Telescope and ground-based spectroscopy. We combine constraints from a catalog of 909 weakly lensed galaxies and 39 multiply imaged sources comprised of 114 multiple images, including a system of multiply imaged candidates at z = 7.84 ± 0.02, to obtain a best-fit mass distribution using the cluster lens modeling code Strong and Weak Lensing United. As the only analysis of A370 using strong and weak lensing constraints from Hubble Frontier Fields data, our method provides an independent check of assumptions on the mass distribution used in other methods. Convergence, shear, and magnification maps are made publicly available through the HFF website (http://www.stsci.edu/hst/campaigns/frontier-fields). We find that the model we produce is similar to models produced by other groups, with some exceptions due to the differences in lensing code methodology. In an effort to study how our total projected mass distribution traces light, we measure the stellar mass density distribution using Spitzer/Infrared Array Camera imaging. Comparing our total mass density to our stellar mass density in a radius of 0.3 Mpc, we find a mean projected stellar to total mass ratio of (stat.) using the diet Salpeter initial mass function. This value is in general agreement with independent measurements of in clusters of similar total mass and redshift.

Lenstronomy: Multi-purpose gravitational lens modelling software package

We present lenstronomy, a multi-purpose open-source gravitational lens modelling pythonpackage. lenstronomy is able to reconstruct the lens mass and surface brightness distributions of strong lensing systems using forward modelling. lenstronomy supports a wide range of analytic lens and light models in arbitrary combination. The software is also able to reconstruct complex extended sources (Birrer et. al 2015) as well as being able to model point sources. We designed lenstronomy to be stable, flexible and numerically accurate, with a clear user interface that could be deployed across different platforms. Throughout its development, we have actively used lenstronomy to make several measurements including deriving constraints on dark matter properties in strong lenses, measuring the expansion history of the universe with time-delay cosmography, measuring cosmic shear with Einstein rings and decomposing quasar and host galaxy light. The software is distributed under the MIT license. The documentation, starter guide, example notebooks, source code and installation guidelines can be found at https://lenstronomy.readthedocs.io.

Ambiguities in gravitational lens models: impact on time delays of the source position transformation

The central ambition of the modern time delay cosmography consists in determining the Hubble constant H0 with a competitive precision. However, the tension with H0 obtained from the Planck satellite for a spatially flat ΛCDM cosmology suggests that systematic errors may have been underestimated. The most critical of these errors 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 contains the MST as a special case, may affect the time delays predicted by a model. To this aim, we have used pySPT, a new open-source python package fully dedicated to the SPT that we present in a companion paper. For axisymmetric lenses, we find that the time delay ratios between a model and its SPT-modified counterpart simply scale like the corresponding source position ratios, Δtˆ/Δt ≈ βˆ/β, regardless of the mass profile and the isotropic SPT. Similar behavior (almost) holds for nonaxisymmetric lenses in the double image regime and for opposite image pairs in the quadruple image regime. In the latter regime, we also confirm that the time delay ratios are not conserved. In addition to the MST effects, the SPT-modified time delays deviate in general no more than a few percent for particular image pairs, suggesting that its impact on time delay cosmography seems not be as crucial as initially suspected. We also reflected upon the relevance of the SPT validity criterion and present arguments suggesting that it should be reconsidered. Even though a new validity criterion would affect the time delays in a different way, we expect from numerical simulations that our conclusions will remain unchanged.