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Aymeric Galan

Publications and source records attributed to Aymeric Galan.

At least 19 recordsLinked to original sources

TDCOSMO XXXI: New techniques in line-of-sight studies of time delay lenses

The distribution of matter along the same line of sight but external to the main lens, quantified by $\kappa_\mathrm{ext}$, is a key source of uncertainty in time delay cosmography and other applications of strong lensing, and must be independently estimated to avoid biasing the inferred value of the Hubble constant ($H_0$). We present advancements and standardisations in the weighted number counts techniques used to constrain $\kappa_\mathrm{ext}$, in particular the use of the Euclid Flagship Simulation and a breakdown of the line-of-sight contributions between the observer, lens and source. As part of the TDCOSMO 2026 milestone analysis, we apply this updated method to the sample of 11 time delay lenses used in that study. Our estimates for certain systems are sensitive to these methodological changes, but are nonetheless consistent within $1\sigma$ for all but one of the systems which had been studied previously, with the median $\kappa_\mathrm{ext}$ across those systems changing from $-0.002$ to $-0.006$ following this new analysis. This work represents the first estimate of $\kappa_\mathrm{ext}$ values which includes the contribution of the observer-lens and lens-source terms, and largest standardised analysis of time delay lens environments to date.

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TDCOSMO XXX: Spatially resolved kinematics of the deflectors in time-delay lens systems B1608+656 and SDSSJ1206+4332 from JWST-NIRSpec observation

We present spatially resolved stellar kinematics of the deflector galaxies in two time-delay lens systems, B1608+656 and SDSS J1206+4332, measured from JWST NIRSpec integral field spectroscopy. B1608+656 is a quadruply imaged quasar lensed by a pair of tidally interacting galaxies, the primary deflector (G1) and a close companion (G2), for which we quantified G2's tidal influence on G1 and define a criterion identifying the region where this perturbation is negligible and standard Jeans-equation modeling remains valid. SDSS J1206+4332 is a doubly imaged quasar whose extended host galaxy is itself quadruply imaged by a foreground massive galaxy (G0), accompanied by a fainter perturber (G1). For both systems, we report the first spectroscopic redshift and velocity dispersion measurements of the perturber galaxies. For G2 in B1608+656, we measure {\sigma}_v = 188 +/- 7 (stat.) +/- 2 (sys.) km/s at z = 0.6307, identical to the redshift of the main deflector G1. For G1 in SDSS J1206+4332, we find {\sigma}_v = 100 +/- 21 (stat.) +/- 2 (sys.) km/s at z = 0.7403, slightly lower than the primary deflector G0. The resolved velocity maps of both primary deflectors show clear rotation signatures and we visually classify them as fast or regular rotators. The data products will be made publicly available and will be combined with time delays, lens models, and line-of-sight convergence to measure cosmological parameters in the upcoming TDCOSMO 2026 milestone publication.

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ALPACA I: Controlling source and PSF systematics in JWST time-delay cosmography with differentiable lens modeling

Time-delay cosmography provides an independent probe of the Hubble constant ($H_0$), but exploiting the high-resolution imaging of the James Webb Space Telescope (JWST) requires strong-lens models that capture complex source-galaxy morphologies and the point spread function (PSF) with high fidelity. Flexible source and joint PSF reconstruction are computationally expensive, motivating more rigid parameterizations that can bias the inferred time-delay distance $D_{\Delta t}$. We present ALPACA, a semi-automatic, GPU-accelerated, differentiable strong-lens modeling pipeline built on Herculens, Starred, and JAX. ALPACA jointly reconstructs a pixelated correlated-field source, the lens mass, and the PSF within a single likelihood, reducing per-system runtimes from weeks to less than 15 hours. We validate the pipeline on realistic mock JWST/NIRCam observations built from morphologically complex COSMOS galaxy sources. Holding an imperfect PSF (RMSE = 12% from truth) fixed biases $D_{\Delta t}$ by up to 14%, whereas joint PSF reconstruction reduces this bias to the percent level. We further show that the reconstructed PSF core can be marginalized over jointly with the lens parameters during posterior sampling. Propagating the per-lens posteriors of a ten-lens mock sample through hierarchical inference, we recover the input cosmology without significant bias. ALPACA shows that rigorous control of source and PSF systematics is computationally affordable, supporting reliable percent-level precision in the JWST era.

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Disentangling the dark and stellar mass through precise lens modelling of the JWST observation of lensed quasar WFI2033--4723

We use high-resolution JWST/NIRCam imaging and measured time delays to model the quadruply imaged quasar WFI2033--4723 with a composite stellar plus dark-matter mass model. We first construct an elliptical power-law baseline model and recover Fermat-potential differences (fpd) consistent with previous HST-based and JWST-based analyses, providing a reference scale for composite modelling. We then replace the total mass profile with a physically motivated decomposition in which the stellar mass follows a multi-Gaussian expansion of the lens light, with a free radial mass-to-light gradient, and the dark matter is described by a generalized Navarro--Frenk--White (gNFW) halo. Using two external cosmological priors, Planck+DESI and Pantheon+SH0ES, the measured time delays constrain the mass-sheet-transformation freedom that would otherwise damage the stellar--dark-matter decomposition. In both cosmological cases, the stellar normalization lies between the expectations for Chabrier and Salpeter initial mass functions, while the radial mass-to-light gradient is not strongly required by the data (mildly positive). The dark matter halo has an inner slope $\gamma_{\rm in}\simeq1.3$, steeper than a standard NFW cusp, and the main astrophysical conclusions are insensitive to the adopted cosmological prior. This work shows that composite time-delay lens modelling can effectively separate baryons from dark matter. As a qualitative check, we reverse the logic and use our composite lens model without kinematic information to infer the cosmology instead. However, the strong degeneracy between $H_0$ and the halo scale radius $R_s$ prevents a robust standalone constraint.

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JAXtronomy: A JAX port of lenstronomy

Gravitational lensing is a phenomenon where light bends around massive objects, resulting in distorted images seen by an observer. Studying gravitationally lensed systems provides insights into cosmology and astrophysics, including constraints of the expansion rate of the Universe and the distribution of dark matter. Thus, we introduce JAXtronomy, a re-implementation of the gravitational lensing software package lenstronomy (Birrer, 2021; Birrer & Amara, 2018) using JAX (Bradbury et al., 2018). JAX is a Python library that uses an accelerated linear algebra (XLA) compiler to improve the performance of computing software. Our core design principle of JAXtronomy is to maintain an identical API to that of lenstronomy. The main JAX features utilized in JAXtronomy are just-in-time compilation, which can lead to significant reductions in execution time, and automatic differentiation, which allows for the implementation of gradient-based algorithms that were previously impossible. Additionally, JAX allows code to be run on GPUs or parallelized across CPU cores, further boosting the performance of JAXtronomy.

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The Stellar IMF and Dark Matter Halo of ESO0286: Constraints from Strong Lensing and Dynamics

The internal mass structure of elliptical galaxies offers critical insights into galaxy formation, yet disentangling stellar mass from dark matter and determining the stellar initial mass function (IMF) remains challenging. We present a detailed analysis of ESO0286-G022 ($z=0.0312$), a rare nearby strong-lens system with a fast-rotating elliptical galaxy, combining high-resolution Hubble Space Telescope (HST) imaging with VLT/MUSE integral-field stellar kinematics. We construct axisymmetric and triaxial Schwarzschild orbit-superposition models to reconstruct its intrinsic shape and mass distribution. Despite being a fast rotator, ESO0286 exhibits clear kinematic signatures of intrinsic triaxiality, characterized by rotation along both the major and minor axes, making it only the second such confirmed case. By incorporating the mass enclosed within the Einstein radius from strong lensing as a complementary constraint, we tightly anchor the total mass at large radii. This significantly reduces the uncertainty on the outer mass profile and orbital structure, demonstrating that only models with strong radial anisotropy beyond the IFU field of view are compatible with the data. In the inner regions, we robustly constrain an upper limit for the stellar mass around $r \sim 0.7$ kpc, ruling out an IMF more bottom-heavy than Kroupa, though a gentle gradient toward a slightly heavier central IMF is permitted. This aligns with recent dynamical studies of local massive early-type galaxies but contrasts with heavier IMFs reported for lenses at $z>0.1$. Our work demonstrates the power of combining lensing and dynamical modeling to resolve the detailed inner structure of massive galaxies.

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TDCOSMO. XXIII. Measurement of the Hubble constant from the doubly lensed quasar HE1104-1805

Time-delay cosmography leverages strongly lensed quasars to measure the Universe's current expansion rate, H_0, independently from other methods. While the latest TDCOSMO results relied mainly on quadruply lensed quasars, doubly lensed systems are far more common and offer precise time delays, potentially enlarging the usable sample by a factor of five and enabling percent-level constraints on H_0. We present the first TDCOSMO analysis of a doubly imaged source, HE1104-1805, including the measurement of the four necessary ingredients. First, by combining 17 years of data from the SMARTS, Euler and WFI telescopes, we measure a time delay of 176.3\pm 10.8 days. Second, using MUSE data, we extract stellar velocity dispersion measurements in three radial bins with up to 5% precision. Third, employing F160W HST imaging for lens modelling and marginalising over various modelling choices, we measure the Fermat potential difference between the images. Fourth, using wide-field imaging, we measure the convergence added by objects not included in the lens modelling. Hence, we measure the time delay distance and the angular diameter distance to the deflector, favouring a power-law mass model over a baryonic and dark matter composite model. The measurement was performed blindly and yielded H_0 = 64.2^{+5.8}_{-5.0} x $\lambda_{int} km s^{-1} Mpc^{-1}, where \lambda_{int} is the internal mass sheet degeneracy parameter. This is in agreement with the TDCOSMO-2025 milestone and its precision for \lambda_{int}=1 is comparable to that obtained with the best-observed quadruply lensed quasars (4-6%). This work is a stepping stone towards a precise measurement of H_0 using a large sample of doubly lensed quasars, supplementing the current sample. The next TDCOSMO milestone paper will include this system in its hierarchical analysis, constraining \lambda_{int} and H_0 jointly with multiple lenses.

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HOLISMOKES XIX: SN 2025wny at $z=2$, the first strongly lensed superluminous supernova

We present imaging and spectroscopic observations of supernova SN 2025wny, associated with the lens candidate PS1 J0716+3821. Photometric monitoring from the Lulin and Maidanak observatories confirms multiple point-like images, consistent with SN 2025wny being strongly lensed by two foreground galaxies. Optical spectroscopy of the brightest image with the Nordic Optical Telescope and the University of Hawaii 88-inch Telescope allows us to determine the redshift to be z_s = 2.008 +- 0.001, based on narrow absorption lines originating in the interstellar medium of the supernova host galaxy. At this redshift, the spectra of SN 2025wny are consistent with those of superluminous supernovae of Type I. We find a high ejecta temperature and depressed spectral lines compared to other similar objects. We also measure, for the first time, the redshift of the fainter of the two lens galaxies (the "perturber") to be z_p = 0.375 +- 0.001, fully consistent with the DESI spectroscopic redshift of the main deflector at z_d = 0.3754. SN 2025wny thus represents the first confirmed galaxy-scale strongly lensed supernova with time delays likely in the range of days to weeks, as judged from the image separations. This makes SN 2025wny suitable for cosmography, offering a promising new system for independent measurements of the Hubble constant. Following a tradition in the field of strongly-lensed SNe, we give SN 2025wny the nickname SN Winny.

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TDCOSMO XXIV. First spatially resolved kinematics of the lens galaxy obtained using JWST-NIRSpec to improve time-delay cosmography

Spatially resolved stellar kinematics has become a key ingredient in time-delay cosmography to break the mass-sheet degeneracy in the mass profile and in turn provide a precise constraint on the Hubble constant and other cosmological parameters. In this paper, we present the first measurements of 2D resolved stellar kinematics for the lens galaxy in the quadruply lensed quasar system RXJ1131$-$1231 using integral field spectroscopy from JWST's Near-Infrared Spectrograph (NIRSpec), marking the first such measurement conducted with JWST. In extracting robust kinematic measurements from this first-of-its-kind dataset, we have made methodological improvements both in the data reduction and kinematic extraction. In our kinematic extraction procedure, we performed joint modeling of the lens galaxy, the quasar, and its host galaxy's contributions in the spectra to deblend the lens galaxy component and robustly constrain its stellar kinematics. Our improved methodological frameworks are released as software pipelines for future use: squirrel, for extracting stellar kinematics, and RegalJumper, for JWST-NIRSpec data reduction. We incorporated additional artifact cleaning beyond the standard JWST pipeline. We compared our measured stellar kinematics from the JWST NIRSpec with previously obtained ground-based measurements from the Keck Cosmic Web Imager integral field unit and find that the two datasets are statistically consistent at a $\sim$1.1$\sigma$ confidence level. Our measured kinematics will be used in a future study to improve the precision of the Hubble constant measurement.

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TDCOSMO 2025: Cosmological constraints from strong lensing time delays

We present cosmological constraints from 8 strongly lensed quasars (hereafter, the TDCOSMO-2025 sample). Building on previous work, our analysis incorporated new deflector stellar velocity dispersions measured from spectra obtained with the James Webb Space Telescope (JWST), the Keck Telescopes, and the Very Large Telescope (VLT), utilizing improved methods. We used integrated JWST stellar kinematics for 5 lenses, VLT-MUSE for 2, and resolved kinematics from Keck and JWST for RXJ1131-1231. We also considered two samples of non-time-delay lenses: 11 from the Sloan Lens ACS (SLACS) sample with Keck-KCWI resolved kinematics; and 4 from the Strong Lenses in the Legacy Survey (SL2S) sample. We improved our analysis of line-of-sight effects, the surface brightness profile of the lens galaxies, and orbital anisotropy, and corrected for projection effects in the dynamics. Our uncertainties are maximally conservative by accounting for the mass-sheet degeneracy in the deflectors' mass density profiles. The analysis was blinded to prevent experimenter bias. Our primary result is based on the TDCOSMO-2025 sample, in combination with $\Omega_{\rm m}$ constraints from the Pantheon+ Type Ia supernovae (SN) dataset. In the flat $\Lambda$ cold dark matter (CDM), we find $H_0=71.6^{+3.9}_{-3.3}$ km s$^{-1}$ Mpc$^{-1}$. The SLACS and SL2S samples are in excellent agreement with the TDCOSMO-2025 sample, improving the precision on $H_0$ in flat $\Lambda$CDM to 4.6%. Using the Dark Energy Survey SN Year-5 dataset (DES-SN5YR) or DESI-DR2 baryonic acoustic oscillations (BAO) likelihoods instead of Pantheon+ yields very similar results. We also present constraints in the open $\Lambda$CDM, $w$CDM, $w_0w_a$CDM, and $w_{\phi}$CDM cosmologies. The TDCOSMO $H_0$ inference is robust and consistent across all presented cosmological models, and our cosmological constraints in them agree with those from the BAO and SN.

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TDCOSMO XVII. New time delays in 22 lensed quasars from optical monitoring with the ESO-VST 2.6m and MPG 2.2m telescopes

We present new time delays, the main ingredient of time delay cosmography, for 22 lensed quasars resulting from high-cadence r-band monitoring on the 2.6 m ESO VLT Survey Telescope and Max-Planck-Gesellschaft 2.2 m telescope. Each lensed quasar was typically monitored for one to four seasons, often shared between the two telescopes to mitigate the interruptions forced by the COVID-19 pandemic. The sample of targets consists of 19 quadruply and 3 doubly imaged quasars, which received a total of 1 918 hours of on-sky time split into 21 581 wide-field frames, each 320 seconds long. In a given field, the 5-{\sigma} depth of the combined exposures typically reaches the 27th magnitude, while that of single visits is 24.5 mag - similar to the expected depth of the upcoming Vera-Rubin LSST. The fluxes of the different lensed images of the targets were reliably de-blended, providing not only light curves with photometric precision down to the photon noise limit, but also high-resolution models of the targets whose features and astrometry were systematically confirmed in Hubble Space Telescope imaging. This was made possible thanks to a new photometric pipeline, lightcurver, and the forward modelling method STARRED. Finally, the time delays between pairs of curves and their uncertainties were estimated, taking into account the degeneracy due to microlensing, and for the first time the full covariance matrices of the delay pairs are provided. Of note, this survey, with 13 square degrees, has applications beyond that of time delays, such as the study of the structure function of the multiple high-redshift quasars present in the footprint at a new high in terms of both depth and frequency. The reduced images will be available through the European Southern Observatory Science Portal.

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GPU-Accelerated Gravitational Lensing & Dynamical (GLaD) Modeling for Cosmology and Galaxies

Time-delay distance measurements from strongly lensed quasars provide a robust, independent method for determining the Hubble constant ($H_0$). This approach cross-checks $H_0$ estimates from the distance ladder in the late universe and the cosmic microwave background in the early universe. However, the mass-sheet degeneracy in lensing models introduces systematic uncertainty, limiting precision. Dynamical modeling complements strong lensing by constraining the mass distribution with independent observational data. We develop a methodology and software framework for joint modeling of stellar kinematics and lensing data. Using simulated data for the lensed quasar RXJ1131$-$1131, we demonstrate that high-quality kinematic data can achieve $\sim$4% precision on $H_0$. Through extensive modeling, we examine the impact of the presence of a supermassive black hole in the lens galaxy and potential systematic biases in kinematic data on $H_0$ measurements. Our results show that imposing priors on black hole mass and orbital anisotropy, or excluding central kinematic bins, mitigates biases in $H_0$ estimates. By testing on mock kinematic data with systematic biases, we highlight the need for sub-percent control of kinematic systematics, which is achievable with current technology. Additionally, we leverage GPU parallelization to accelerate Bayesian inference, reducing a previously month-long process by an order of magnitude. This pipeline offers significant potential for advancing cosmological and galaxy evolution studies with large datasets.

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TDCOSMO. XVI. Measurement of the Hubble Constant from the Lensed Quasar WGD$\,$2038$-$4008

Time-delay cosmography is a powerful technique to constrain cosmological parameters, particularly the Hubble constant ($H_{0}$). The TDCOSMO collaboration is performing an ongoing analysis of lensed quasars to constrain cosmology using this method. In this work, we obtain constraints from the lensed quasar WGD 2038-4008 using new time-delay measurements and previous mass models by TDCOSMO. This is the first TDCOSMO lens to incorporate multiple lens modeling codes and the full time-delay covariance matrix into the cosmological inference. The models are fixed before the time delay is measured, and the analysis is performed blinded with respect to the cosmological parameters to prevent unconscious experimenter bias. We obtain $D_{\Delta t} = 1.68^{+0.40}_{-0.38}$ Gpc using two families of mass models, a power-law describing the total mass distribution, and a composite model of baryons and dark matter, although the composite model is disfavored due to kinematics constraints. In a flat $\Lambda$CDM cosmology, we constrain the Hubble constant to be $H_{0} = 65^{+23}_{-14}\, \rm km\ s^{-1}\,Mpc^{-1}$. The dominant source of uncertainty comes from the time delays, due to the low variability of the quasar. Future long-term monitoring, especially in the era of the Vera C. Rubin Observatory's Legacy Survey of Space and Time, could catch stronger quasar variability and further reduce the uncertainties. This system will be incorporated into an upcoming hierarchical analysis of the entire TDCOSMO sample, and improved time delays and spatially-resolved stellar kinematics could strengthen the constraints from this system in the future.

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Nine lensed quasars and quasar pairs discovered through spatially-extended variability in Pan-STARRS

We present the proof-of-concept of a method to find strongly lensed quasars using their spatially-extended photometric variability through difference imaging in cadenced imaging survey data. We apply the method to Pan-STARRS, starting with an initial selection of 14 107 Gaia multiplets with quasar-like infrared colours from WISE. We identify 229 candidates showing notable spatially-extended variability during the Pan-STARRS survey period. These include 20 known lenses, alongside an additional 12 promising candidates for which we obtain long-slit spectroscopy follow-up. This process results in the confirmation of four doubly lensed quasars, four unclassified quasar pairs and one projected quasar pair. Only three are pairs of stars or quasar+star projections, the false positive rate is thereby 25%. The lenses have separations between 0.81" and 1.24" and source redshifts between z = 1.47 and z = 2.46. Three of the unclassified quasar pairs are promising dual quasars candidates with separations ranging from 6.6 to 9.3 kpc. We expect that this technique will be a particularly efficient way to select lensed variables in the upcoming Rubin-LSST, which will be crucial given the expected limitations for spectroscopic follow-up

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Accelerating galaxy dynamical modeling using a neural network for joint lensing and kinematics analyses

Strong gravitational lensing is a powerful tool to provide constraints on galaxy mass distributions and cosmological parameters, such as the Hubble constant, $H_0$. Nevertheless, inference of such parameters from images of lensing systems is not trivial as parameter degeneracies can limit the precision in the measured lens mass and cosmological results. External information on the mass of the lens, in the form of kinematic measurements, is needed to ensure a precise and unbiased inference. Traditionally, such kinematic information has been included in the inference after the image modeling, using spherical Jeans approximations to match the measured velocity dispersion integrated within an aperture. However, as spatially resolved kinematic measurements become available via IFU data, more sophisticated dynamical modeling is necessary. Such kinematic modeling is expensive, and constitutes a computational bottleneck which we aim to overcome with our Stellar Kinematics Neural Network (SKiNN). SKiNN emulates axisymmetric modeling using a neural network, quickly synthesizing from a given mass model a kinematic map which can be compared to the observations to evaluate a likelihood. With a joint lensing plus kinematic framework, this likelihood constrains the mass model at the same time as the imaging data. We show that SKiNN's emulation of a kinematic map is accurate to considerably better precision than can be measured (better than $1\%$ in almost all cases). Using SKiNN speeds up the likelihood evaluation by a factor of $\sim 200$. This speedup makes dynamical modeling economical, and enables lens modelers to make effective use of modern data quality in the JWST era.

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Strong gravitational lensing by AGNs as a probe of the quasar-host relations in the distant Universe

The tight correlations found between masses of supermassive black holes (SMBHs) and the luminosities, total stellar masses, and velocity dispersions of their host galaxies are often interpreted as a sign of their co-evolution. Studying these correlations across redshift provides a powerful insight into the evolutionary path followed by the quasar and its host galaxy. While the mass of the black hole is accessible from single-epoch spectra, measuring the mass of its host galaxy is challenging as the active nucleus largely overshines its host. Here, we present a technique to probe quasar-host relations beyond the local universe with strong gravitational lensing, hence overcoming the use of stellar population models or velocity dispersion measurements, both prone to degeneracies. We study in detail one of the three known cases of strong lensing by a quasar to accurately measure the mass of its host and to infer a total lensing mass of $\log_{10}(M_{\rm Tot, h}/M_{\odot}) = 10.27^{+0.06}_{-0.07}~$ within the Einstein radius of 1.2 kpc. The lensing measurement is more precise than any other alternative technique and compatible with the local $M_{BH}$-$M_{\star, h}$ scaling relation. The sample of such quasar-galaxy or quasar-quasar lensing systems should reach a few hundreds with Euclid and Rubin-LSST, thus enabling the application of such a method with statistically significant sample sizes.

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Using wavelets to capture deviations from smoothness in galaxy-scale strong lenses

Modeling the mass distribution of galaxy-scale strong gravitational lenses is a task of increasing difficulty. The high-resolution and depth of imaging data now available render simple analytical forms ineffective at capturing lens structures spanning a large range in spatial scale, mass scale, and morphology. In this work, we address the problem with a novel multiscale method based on wavelets. We tested our method on simulated Hubble Space Telescope (HST) imaging data of strong lenses containing the following different types of mass substructures making them deviate from smooth models: (1) a localized small dark matter subhalo, (2) a Gaussian random field (GRF) that mimics a nonlocalized population of subhalos along the line of sight, and (3) galaxy-scale multipoles that break elliptical symmetry. We show that wavelets are able to recover all of these structures accurately. This is made technically possible by using gradient-informed optimization based on automatic differentiation over thousands of parameters, which also allow us to sample the posterior distributions of all model parameters simultaneously. By construction, our method merges the two main modeling paradigms - analytical and pixelated - with machine-learning optimization techniques into a single modular framework. It is also well-suited for the fast modeling of large samples of lenses. All methods presented here are publicly available in our new Herculens package.

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Modeling lens potentials with continuous neural fields in galaxy-scale strong lenses

Strong gravitational lensing is a unique observational tool for studying the dark and luminous mass distribution both within and between galaxies. Given the presence of substructures, current strong lensing observations demand more complex mass models than smooth analytical profiles, such as power-law ellipsoids. In this work, we introduce a continuous neural field to predict the lensing potential at any position throughout the image plane, allowing for a nearly model-independent description of the lensing mass. We apply our method on simulated Hubble Space Telescope imaging data containing different types of perturbations to a smooth mass distribution: a localized dark subhalo, a population of subhalos, and an external shear perturbation. Assuming knowledge of the source surface brightness, we use the continuous neural field to model either the perturbations alone or the full lensing potential. In both cases, the resulting model is able to fit the imaging data, and we are able to accurately recover the properties of both the smooth potential and of the perturbations. Unlike many other deep learning methods, ours explicitly retains lensing physics (i.e., the lens equation) and introduces high flexibility in the model only where required, namely, in the lens potential. Moreover, the neural network does not require pre-training on large sets of labelled data and predicts the potential from the single observed lensing image. Our model is implemented in the fully differentiable lens modeling code Herculens.

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