arXiv subjects

S. Schuldt

Publications and source records attributed to S. Schuldt.

53 records · Page 3Linked to original sources

arXiv2024

Augmenting the power of time-delay cosmography in lens galaxy clusters by probing their member galaxies I. Type Ia supernovae

We present a simple and promising new method to measure the expansion rate and the geometry of the universe that combines observations related to the time delays between the multiple images of time-varying sources, strongly lensed by galaxy clusters, and Type Ia supernovae, exploding in galaxies belonging to the same lens clusters. By means of two different statistical techniques that adopt realistic errors on the relevant quantities, we quantify the accuracy of the inferred cosmological parameter values. We show that the estimate of the Hubble constant is robust and competitive, and depends only mildly on the chosen cosmological model. Remarkably, the two probes separately produce confidence regions on the cosmological parameter planes that are oriented in complementary ways, thus providing in combination valuable information on the values of the other cosmological parameters. We conclude by illustrating the immediate observational feasibility of the proposed joint method in a well-studied lens galaxy cluster, with a relatively small investment of telescope time for monitoring from a 2 to 3m class ground-based telescope.

astro-ph.CO↗

arXiv2024

Augmenting the power of time-delay cosmography in lens galaxy clusters by probing their member galaxies. II. Cosmic chronometers

We present a novel approach to measuring the expansion rate and the geometry of the Universe, which combine time-delay cosmography in lens galaxy clusters with pure samples of 'cosmic chronometers' (CCs) by probing the member galaxies. The former makes use of the measured time delays between the multiple images of time-varying sources strongly lensed by galaxy clusters, while the latter exploits the most massive and passive cluster member galaxies to measure the differential time evolution of the Universe. We applied two different statistical techniques, adopting realistic errors on the measured quantities, to assess the accuracy and the gain in precision on the values of the cosmological parameters. We demonstrate that the proposed combined method allows for a robust and accurate measurement of the value of the Hubble constant. In addition, this provides valuable information on the other cosmological parameters thanks to the complementarity between the two different probes in breaking parameter degeneracies. Finally, we showcase the immediate observational feasibility of the proposed joint method by taking advantage of the existing high-quality spectro-photometric data for several lens galaxy clusters.

astro-ph.CO↗

arXiv2023

HOLISMOKES -- XI. Evaluation of supervised neural networks for strong-lens searches in ground-based imaging surveys

While supervised neural networks have become state of the art for identifying the rare strong gravitational lenses from large imaging data sets, their selection remains significantly affected by the large number and diversity of nonlens contaminants. This work evaluates and compares systematically the performance of neural networks in order to move towards a rapid selection of galaxy-scale strong lenses with minimal human input in the era of deep, wide-scale surveys. We used multiband images from PDR2 of the HSC Wide survey to build test sets mimicking an actual classification experiment, with 189 strong lenses previously found over the HSC footprint and 70,910 nonlens galaxies in COSMOS. Multiple networks were trained on different sets of realistic strong-lens simulations and nonlens galaxies, with various architectures and data pre-processing. The overall performances strongly depend on the construction of the ground-truth training data and they typically, but not systematically, improve using our baseline residual network architecture. Improvements are found when applying random shifts to the image centroids and square root stretches to the pixel values, adding z band, or using random viewpoints of the original images, but not when adding difference images to subtract emission from the central galaxy. The most significant gain is obtained with committees of networks trained on different data sets, and showing a moderate overlap between populations of false positives. Nearly-perfect invariance to image quality can be achieved by training networks either with large number of bands, or jointly with the PSF and science frames. Overall, we show the possibility to reach a TPR0 as high as 60% for the test sets under consideration, which opens promising perspectives for pure selection of strong lenses without human input using the Rubin Observatory and other forthcoming ground-based surveys.

astro-ph.GA↗

arXiv2023

HOLISMOKES -- IX. Neural network inference of strong-lens parameters and uncertainties from ground-based images

Modeling of strong gravitational lenses is a necessity for further applications in astrophysics and cosmology. Especially with the large number of detections in current and upcoming surveys such as the Rubin Legacy Survey of Space and Time (LSST), it is timely to investigate in automated and fast analysis techniques beyond the traditional and time consuming Markov chain Monte Carlo sampling methods. Building upon our convolutional neural network (CNN) presented in Schuldt et al. (2021b), 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 $e_x$ and $e_y$, Einstein radius $θ_E$) and the external shear ($γ_{ext,1}$, $γ_{ext,2}$) from ground-based imaging data. In contrast to our CNN, this ResNet further predicts a 1$σ$ uncertainty for each parameter. To train our network, we use our improved pipeline from Schuldt et al. (2021b) 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 overall very good recoveries for the SIE parameters, while differences remain in predicting the external shear. From our tests, most likely the low image resolution is the limiting factor for predicting the external shear. Given the run time of milli-seconds per system, our network is perfectly suited to predict the next appearing image and time delays of lensed transients in time. Therefore, we also present the performance of the network on these quantities in comparison to our simulations. Our ResNet is able to predict the SIE and shear parameter values in fractions of a second on a single CPU such that we are able to process efficiently the huge amount of expected galaxy-scale lenses in the near future.

astro-ph.IM↗

arXiv2023

HOLISMOKES -- X. Comparison between neural network and semi-automated traditional modeling of strong lenses

Modeling of strongly gravitationally lensed galaxies is often required in order to use them as astrophysical or cosmological probes. With current and upcoming wide-field imaging surveys, the number of detected lenses is increasing significantly such that automated and fast modeling procedures for ground-based data are urgently needed. This is especially pertinent to short-lived lensed transients in order to plan follow-up observations. Therefore, we present in a companion paper (submitted) a neural network predicting the parameter values with corresponding uncertainties of a Singular Isothermal Ellipsoid (SIE) mass profile with external shear. In this work, we present a newly-developed pipeline glee_auto.py to model consistently any galaxy-scale lensing system. In contrast to previous automated modeling pipelines that require high-resolution images, glee_auto.py is optimized for ground-based images such as those from the Hyper-Suprime-Cam (HSC) or the upcoming Rubin Observatory Legacy Survey of Space and Time. We further present glee_tools.py, a flexible automation code for individual modeling that has no direct decisions and assumptions implemented. Both pipelines, in addition to our modeling network, minimize the user input time drastically and thus are important for future modeling efforts. We apply the network to 31 real galaxy-scale lenses of HSC and compare the results to the traditional models. In the direct comparison, we find a very good match for the Einstein radius especially for systems with $θ_E \gtrsim 2$". The lens mass center and ellipticity show reasonable agreement. The main discrepancies are on the external shear as expected from our tests on mock systems. In general, our study demonstrates that neural networks are a viable and ultra fast approach for measuring the lens-galaxy masses from ground-based data in the upcoming era with $\sim10^5$ lenses expected.

astro-ph.CO↗

arXiv2023

TDCOSMO X. Automated Modeling of 9 Strongly Lensed Quasars and Comparison Between Lens Modeling Software

To use strong gravitational lenses as an astrophysical or cosmological probe, models of their mass distributions are often needed. We present a new, time-efficient automation code for uniform modeling of strongly lensed quasars with GLEE, a lens modeling software, for high-resolution multi-band data. By using the observed positions of the lensed quasars and the spatially extended surface brightness distribution of the lensed quasar host galaxy, we obtain a model of the mass distribution of the lens galaxy. We apply this uniform modeling pipeline to a sample of nine strongly lensed quasars with HST WFC 3 images. The models show in most cases well reconstructed light components and a good alignment between mass and light centroids. We find that the automated modeling code significantly reduces the user input time during the modeling process. The preparation time of required input files is reduced significantly. This automated modeling pipeline can efficiently produce uniform models of extensive lens system samples which can be used for further cosmological analysis. A blind test through a comparison with the results of an independent automated modeling pipeline based on the modeling software Lenstronomy reveals important lessons. Quantities such as Einstein radius, astrometry, mass flattening and position angle are generally robustly determined. Other quantities depend crucially on the quality of the data and the accuracy of the PSF reconstruction. Better data and/or more detailed analysis will be necessary to elevate our automated models to cosmography grade. Nevertheless, our pipeline enables the quick selection of lenses for follow-up monitoring and further modeling, significantly speeding up the construction of cosmography-grade models. This is an important step forward to take advantage of the orders of magnitude increase in the number of lenses expected in the coming decade.

astro-ph.CO↗

arXiv2022

Constraining the multi-scale dark-matter distribution in CASSOWARY 31 with strong gravitational lensing and stellar dynamics

We study the inner structure of the group-scale lens CASSOWARY 31 (CSWA 31) by adopting both strong lensing and dynamical modeling. CSWA 31 is a peculiar lens system. The brightest group galaxy (BGG) is an ultra-massive elliptical galaxy at z = 0.683 with a weighted mean velocity dispersion of $σ= 432 \pm 31$ km s$^{-1}$. It is surrounded by group members and several lensed arcs probing up to ~150 kpc in projection. Our results significantly improve previous analyses of CSWA 31 thanks to the new HST imaging and MUSE integral-field spectroscopy. From the secure identification of five sets of multiple images and measurements of the spatially-resolved stellar kinematics of the BGG, we conduct a detailed analysis of the multi-scale mass distribution using various modeling approaches, both in the single and multiple lens-plane scenarios. Our best-fit mass models reproduce the positions of multiple images and provide robust reconstructions for two background galaxies at z = 1.4869 and z = 2.763. The relative contributions from the BGG and group-scale halo are remarkably consistent in our three reference models, demonstrating the self-consistency between strong lensing analyses based on image position and extended image modeling. We find that the ultra-massive BGG dominates the projected total mass profiles within 20 kpc, while the group-scale halo dominates at larger radii. The total projected mass enclosed within $R_{eff}$ = 27.2 kpc is $1.10_{-0.04}^{+0.02} \times 10^{13}$ M$_\odot$. We find that CSWA 31 is a peculiar fossil group, strongly dark-matter dominated towards the central region, and with a projected total mass profile similar to higher-mass cluster-scale halos. The total mass-density slope within the effective radius is shallower than isothermal, consistent with previous analyses of early-type galaxies in overdense environments.

astro-ph.GA↗

arXiv2022

Strong lensing in UNIONS: Toward a pipeline from discovery to modeling

We present a search for galaxy-scale strong gravitational lenses in the initial 2 500 square degrees of the Canada-France Imaging Survey (CFIS). We designed a convolutional neural network (CNN) committee that we applied to a selection of 2 344 002 exquisite-seeing $r$-band images of color-selected luminous red galaxies (LRGs). Our classification uses a realistic training set where the lensing galaxies and the lensed sources are both taken from real data, namely the CFIS $r$-band images themselves and the Hubble Space Telescope (HST). A total of 9 460 candidates obtain a score above 0.5 with the CNN committee. After a visual inspection of the candidates, we find a total of 133 lens candidates, of which 104 are completely new. The set of false positives mainly contains ring, spiral, and merger galaxies, and to a lesser extent galaxies with nearby companions. We classify 32 of the lens candidates as secure lenses and 101 as maybe lenses. For the 32 highest quality lenses, we also fit a singular isothermal ellipsoid mass profile with external shear along with an elliptical Sersic profile for the lens and source light. This automated modeling step provides distributions of properties for both sources and lenses that have Einstein radii in the range $0.5\arcsec<θ_E<2.5\arcsec$. Finally, we introduce a new lens and/or source single-band deblending algorithm based on auto-encoder representation of our candidates. This is the first time an end-to-end lens-finding and modeling pipeline is assembled together, in view of future lens searches in a single band, as will be possible with Euclid.

astro-ph.CO↗

arXiv2022

TDCOSMO. IX. Systematic comparison between lens modelling software programs: time delay prediction for WGD 2038-4008

The importance of alternative methods for measuring the Hubble constant, such as time-delay cosmography, is highlighted by the recent Hubble tension. It is paramount to thoroughly investigate and rule out systematic biases in all measurement methods before we can accept new physics as the source of this tension. In this study, we perform a check for systematic biases in the lens modelling procedure of time-delay cosmography by comparing independent and blind time-delay predictions of the system WGD 2038-4008 from two teams using two different software programs: GLEE and LENSTRONOMY. The predicted time delays from the two teams incorporate the stellar kinematics of the deflector and the external convergence from line-of-sight structures. The un-blinded time-delay predictions from the two teams agree within $1.2σ$, implying that once the time delay is measured the inferred Hubble constant will also be mutually consistent. However, there is a $\sim$4$σ$ discrepancy between the power-law model slope and external shear, which is a significant discrepancy at the level of lens models before the stellar kinematics and the external convergence are incorporated. We identify the difference in the reconstructed point spread function (PSF) to be the source of this discrepancy. When the same reconstructed PSF was used by both teams, we achieved excellent agreement, within $\sim$0.6$σ$, indicating that potential systematics stemming from source reconstruction algorithms and investigator choices are well under control. We recommend that future studies supersample the PSF as needed and marginalize over multiple algorithms or realizations for the PSF reconstruction to mitigate the systematics associated with the PSF. A future study will measure the time delays of the system WGD 2038-4008 and infer the Hubble constant based on our mass models.

astro-ph.CO↗

arXiv2022

STRIDES: Automated uniform models for 30 quadruply imaged quasars

Gravitational time delays provide a powerful one step measurement of $H_0$, independent of all other probes. One key ingredient in time delay cosmography are high accuracy lens models. Those are currently expensive to obtain, both, in terms of computing and investigator time (10$^{5-6}$ CPU hours and $\sim$ 0.5-1 year, respectively). Major improvements in modeling speed are therefore necessary to exploit the large number of lenses that are forecast to be discovered over the current decade. In order to bypass this roadblock, building on the work by Shajib et al. (2019), we develop an automated modeling pipeline and apply it to a sample of 30 quadruply imaged quasars and one lensed compact galaxy, observed by the Hubble Space Telescope in multiple bands. Our automated pipeline can derive models for 30/31 lenses with few hours of human time and <100 CPU hours of computing time for a typical system. For each lens, we provide measurements of key parameters and predictions of magnification as well as time delays for the multiple images. We characterize the cosmography-readiness of our models using the stability of differences in Fermat potential (proportional to time delay) w.r.t. modeling choices. We find that for 10/30 lenses our models are cosmography or nearly cosmography grade (<3% and 3-5% variations). For 6/30 lenses the models are close to cosmography grade (5-10%). These results are based on informative priors and will need to be confirmed by further analysis. However, they are also likely to improve by extending the pipeline modeling sequence and options. In conclusion, we show that uniform cosmography grade modeling of large strong lens samples is within reach.

astro-ph.CO↗

arXiv2022

Close, bright and boxy: the superluminous SN 2018hti

SN 2018hti was a very nearby (z=0.0614) superluminous supernova with an exceedingly bright absolute magnitude of -21.7 mag in r-band at maximum. The densely sampled pre-maximum light curves of SN 2018hti show a slow luminosity evolution and constrain the rise time to ~50 rest-frame days. We fitted synthetic light curves to the photometry to infer the physical parameters of the explosion of SN 2018hti for both the magnetar and the CSM-interaction scenarios. We conclude that one of two mechanisms could be powering the luminosity of SN 2018hti; interaction with ~10 Msun of circumstellar material or a magnetar with a magnetic field of B_p~1.3e13 G and initial period of P_spin~1.8 ms. From the nebular spectrum modelling we infer that SN 2018hti likely results from the explosion of a ~40 Msun progenitor star.

astro-ph.HE↗

arXiv2021

HOLISMOKES. VI. New galaxy-scale strong lens candidates from the HSC-SSP imaging survey

We have carried out a systematic search for galaxy-scale strong lenses in multiband imaging from the Hyper Suprime-Cam (HSC) survey. Our automated pipeline, based on realistic strong-lens simulations, deep neural network classification, and visual inspection, is aimed at efficiently selecting systems with wide image separations (Einstein radii ~1.0-3.0"), intermediate redshift lenses (z ~ 0.4-0.7), and bright arcs for galaxy evolution and cosmology. We classified gri images of all 62.5 million galaxies in HSC Wide with i-band Kron radius >0.8" to avoid strict pre-selections and to prepare for the upcoming era of deep, wide-scale imaging surveys with Euclid and Rubin Observatory. We obtained 206 newly-discovered candidates classified as definite or probable lenses with either spatially-resolved multiple images or extended, distorted arcs. In addition, we found 88 high-quality candidates that were assigned lower confidence in previous HSC searches, and we recovered 173 known systems in the literature. These results demonstrate that, aided by limited human input, deep learning pipelines with false positive rates as low as ~0.01% can be very powerful tools for identifying the rare strong lenses from large catalogs, and can also largely extend the samples found by traditional algorithms. We provide a ranked list of candidates for future spectroscopic confirmation.

astro-ph.GA↗

arXiv2021

Photometric Redshift Estimation with a Convolutional Neural Network: NetZ

The redshifts of galaxies are a key attribute that is needed for nearly all extragalactic studies. Since spectroscopic redshifts require additional telescope and human resources, millions of galaxies are known without spectroscopic redshifts. Therefore, it is crucial to have methods for estimating the redshift of a galaxy based on its photometric properties, the so-called photo-$z$. We developed NetZ, a new method using a Convolutional Neural Network (CNN) to predict the photo-$z$ based on galaxy images, in contrast to previous methods which often used only the integrated photometry of galaxies without their images. We use data from the Hyper Suprime-Cam Subaru Strategic Program (HSC SSP) in five different filters as training data. The network over the whole redshift range between 0 and 4 performs well overall and especially in the high-$z$ range better than other methods on the same data. We obtain an accuracy $|z_\text{pred}-z_\text{ref}|$ of $σ= 0.12$ (68% confidence interval) with a CNN working for all galaxy types averaged over all galaxies in the redshift range of 0 to $\sim$4. By limiting to smaller redshift ranges or to Luminous Red Galaxies (LRGs), we find a further notable improvement. We publish more than 34 million new photo-$z$ values predicted with NetZ here. This shows that the new method is very simple and fast to apply, and, importantly, covers a wide redshift range limited only by the available training data. It is broadly applicable and beneficial to imaging surveys, particularly upcoming surveys like the Rubin Observatory Legacy Survey of Space and Time which will provide images of billions of galaxies with similar image quality as HSC.

astro-ph.GA↗

arXiv2021

HOLISMOKES -- II. Identifying galaxy-scale strong gravitational lenses in Pan-STARRS using convolutional neural networks

We present a systematic search for wide-separation (Einstein radius >1.5"), galaxy-scale strong lenses in the 30 000 sq.deg of the Pan-STARRS 3pi survey on the Northern sky. With long time delays of a few days to weeks, such systems are particularly well suited for catching strongly lensed supernovae with spatially-resolved multiple images and open new perspectives on early-phase supernova spectroscopy and cosmography. We produce a set of realistic simulations by painting lensed COSMOS sources on Pan-STARRS image cutouts of lens luminous red galaxies with known redshift and velocity dispersion from SDSS. First of all, we compute the photometry of mock lenses in gri bands and apply a simple catalog-level neural network to identify a sample of 1050207 galaxies with similar colors and magnitudes as the mocks. Secondly, we train a convolutional neural network (CNN) on Pan-STARRS gri image cutouts to classify this sample and obtain sets of 105760 and 12382 lens candidates with scores pCNN>0.5 and >0.9, respectively. Extensive tests show that CNN performances rely heavily on the design of lens simulations and choice of negative examples for training, but little on the network architecture. Finally, we visually inspect all galaxies with pCNN>0.9 to assemble a final set of 330 high-quality newly-discovered lens candidates while recovering 23 published systems. For a subset, SDSS spectroscopy on the lens central regions proves our method correctly identifies lens LRGs at z~0.1-0.7. Five spectra also show robust signatures of high-redshift background sources and Pan-STARRS imaging confirms one of them as a quadruply-imaged red source at z_s = 1.185 strongly lensed by a foreground LRG at z_d = 0.3155. In the future, we expect that the efficient and automated two-step classification method presented in this paper will be applicable to the deeper gri stacks from the LSST with minor adjustments.

astro-ph.GA↗

arXiv2020

HOLISMOKES -- IV. Efficient mass modeling of strong lenses through deep learning

Modelling the mass distributions of strong gravitational lenses is often necessary to use them as astrophysical and cosmological probes. With the high number of lens systems ($>10^5$) expected from upcoming surveys, it is timely to explore efficient modeling approaches beyond traditional MCMC techniques that are time consuming. We train a CNN on images of galaxy-scale lenses to predict the parameters of the SIE mass model ($x,y,e_x,e_y$, and $θ_E$). To train the network, we simulate images based on real observations from the HSC Survey for the lens galaxies and from the HUDF as lensed galaxies. We tested different network architectures, the effect of different data sets, and using different input distributions of $θ_E$. We find that the CNN performs well and obtain with the network trained with a uniform distribution of $θ_E$ $>0.5"$ the following median values with $1σ$ scatter: $Δx=(0.00^{+0.30}_{-0.30})"$, $Δy=(0.00^{+0.30}_{-0.29})" $, $Δθ_E=(0.07^{+0.29}_{-0.12})"$, $Δe_x = -0.01^{+0.08}_{-0.09}$ and $Δe_y = 0.00^{+0.08}_{-0.09}$. The bias in $θ_E$ is driven by systems with small $θ_E$. Therefore, when we further predict the multiple lensed image positions and time delays based on the network output, we apply the network to the sample limited to $θ_E>0.8"$. In this case, the offset between the predicted and input lensed image positions is $(0.00_{-0.29}^{+0.29})"$ and $(0.00_{-0.31}^{+0.32})"$ for $x$ and $y$, respectively. For the fractional difference between the predicted and true time delay, we obtain $0.04_{-0.05}^{+0.27}$. Our CNN is able to predict the SIE parameters in fractions of a second on a single CPU and with the output we can predict the image positions and time delays in an automated way, such that we are able to process efficiently the huge amount of expected lens detections in the near future.

astro-ph.GA↗

arXiv2020

HOLISMOKES -- I. Highly Optimised Lensing Investigations of Supernovae, Microlensing Objects, and Kinematics of Ellipticals and Spirals

We present the HOLISMOKES programme on strong gravitational lensing of supernovae as a probe of supernova (SN) physics and cosmology. We investigate the effects of microlensing on early-phase SN Ia spectra using four different SN explosion models, and find that within 10 rest-frame days after SN explosion, distortions of SN Ia spectra due to microlensing are typically negligible ($<$1% distortion within the 1$σ$ spread, and $\lesssim$10% distortion within the 2$σ$ spread). This shows great prospects of using lensed SNe Ia to obtain intrinsic early-phase SN spectra for deciphering SN Ia progenitors. As a demonstration of the usefulness of lensed SNe Ia for cosmology, we simulate a sample of mock lensed SN Ia systems that are expected to have accurate and precise time-delay measurements in the era of the Rubin Observatory Legacy Survey of Space and Time (LSST). Adopting realistic yet conservative uncertainties on their time-delay distances and lens angular diameter distances (of 6.6% and 5%, respectively), we find that a sample of 20 lensed SNe Ia would allow a constraint on the Hubble constant ($H_0$) with 1.3% uncertainty in the flat $Λ$CDM cosmology. We find a similar constraint on $H_0$ in an open $Λ$CDM cosmology, while the constraint degrades to $3\%$ in a flat $w$CDM cosmology. We anticipate lensed SNe to be an independent and powerful probe of SN physics and cosmology in the upcoming LSST era.

astro-ph.CO↗

arXiv2019

The inner dark matter distribution of the Cosmic Horseshoe (J1148+1930) with gravitational lensing and dynamics

We present a detailed analysis of the inner mass structure of the Cosmic Horseshoe (J1148+1930) strong gravitational lens system observed with the Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3). In addition to the spectacular Einstein ring, this systems shows a radial arc. We obtained the redshift of the radial arc counter image $z_\text{s,r} = 1.961 \pm 0.001$ from Gemini observations. To disentangle the dark and luminous matter, we consider three different profiles for the dark matter distribution: a power-law profile, the NFW, and a generalized version of the NFW profile. For the luminous matter distribution, we base it on the observed light distribution that is fitted with three components: a point mass for the central light component resembling an active galactic nucleus, and the remaining two extended light components scaled by a constant M/L. To constrain the model further, we include published velocity dispersion measurements of the lens galaxy and perform a self-consistent lensing and axisymmetric Jeans dynamical modeling. Our model fits well to the observations including the radial arc, independent of the dark matter profile. Depending on the dark matter profile, we get a dark matter fraction between 60 % and 70 %. With our composite mass model we find that the radial arc helps to constrain the inner dark matter distribution of the Cosmic Hoseshoe independently of the dark matter profile.

astro-ph.GA↗