Lensing degeneracies and mass substructure

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 scale factor. Apart from illustrating that the mass sheet degeneracy is not broken by the presence of multiple sources at different redshifts, it will become apparent that the newly constructed mass distribution necessarily alters the existing mass density precisely at the locations of the images in the lens system, and that this change in mass density is linked to the factors with which the sources were rescaled. Combined with the fact that the monopole degeneracy introduces a large amount of uncertainty about the density in between the images, this means that both degeneracies are in fact closely related to substructure in the mass distribution. An example simulated lensing situation based on an elliptical version of a Navarro-Frenk-White profile explicitly shows that such degeneracies are not easily broken by observational constraints, even when multiple sources are present. Instead, the fact that each lens inversion method makes certain assumptions, implicit or explicit, about the smoothness of the mass distribution means that in practice the degeneracies are broken in an artificial manner rather than by observed properties of the lens system.Comment: 10 pages, accepted for publication by MNRA

Non-parametric strong lens inversion of SDSS J1004+4112

In this article we study the well-known strong lensing system SDSS J1004+4112. Not only does it host a large-separation lensed quasar with measured time-delay information, but several other lensed galaxies have been identified as well. A previously developed strong lens inversion procedure that is designed to handle a wide variety of constraints, is applied to this lensing system and compared to results reported in other works. Without the inclusion of a tentative central image of one of the galaxies as a constraint, we find that the model recovered by the other constraints indeed predicts an image at that location. An inversion which includes the central image provides tighter constraints on the shape of the central part of the mass map. The resulting model also predicts a central image of a second galaxy where indeed an object is visible in the available ACS images. We find masses of 2.5x10^13 M_O and 6.1x10^13 M_O within a radius of 60 kpc and 110 kpc respectively, confirming the results from other authors. The resulting mass map is compatible with an elliptical generalization of a projected NFW profile, with r_s = 58_{-13}^{+21} arcsec and c_vir = 3.91 +/- 0.74. The orientation of the elliptical NFW profile follows closely the orientation of the central cluster galaxy and the overall distribution of cluster members.Comment: 11 pages, accepted for publication in MNRA

A genetic algorithm for the non-parametric inversion of strong lensing systems

We present a non-parametric technique to infer the projected-mass distribution of a gravitational lens system with multiple strong-lensed images. The technique involves a dynamic grid in the lens plane on which the mass distribution of the lens is approximated by a sum of basis functions, one per grid cell. We used the projected mass densities of Plummer spheres as basis functions. A genetic algorithm then determines the mass distribution of the lens by forcing images of a single source, projected back onto the source plane, to coincide as well as possible. Averaging several tens of solutions removes the random fluctuations that are introduced by the reproduction process of genomes in the genetic algorithm and highlights those features common to all solutions. Given the positions of the images and the redshifts of the sources and the lens, we show that the mass of a gravitational lens can be retrieved with an accuracy of a few percent and that, if the sources sufficiently cover the caustics, the mass distribution of the gravitational lens can also be reliably retrieved. A major advantage of the algorithm is that it makes full use of the information contained in the radial images, unlike methods that minimise the residuals of the lens equation, and is thus able to accurately reconstruct also the inner parts of the lens.Comment: 11 pages, accepted for publication by MNRA

Non-parametric strong lens inversion of Cl~0024+1654: illustrating the monopole degeneracy

The cluster lens Cl 0024+1654 is undoubtedly one of the most beautiful examples of strong gravitational lensing, providing five large images of a single source with well-resolved substructure. Using the information contained in the positions and the shapes of the images, combined with the null space information, a non-parametric technique is used to infer the strong lensing mass map of the central region of this cluster. This yields a strong lensing mass of 1.60x10^14 M_O within a 0.5' radius around the cluster center. This mass distribution is then used as a case study of the monopole degeneracy, which may be one of the most important degeneracies in gravitational lensing studies and which is extremely hard to break. We illustrate the monopole degeneracy by adding circularly symmetric density distributions with zero total mass to the original mass map of Cl 0024+1654. These redistribute mass in certain areas of the mass map without affecting the observed images in any way. We show that the monopole degeneracy and the mass-sheet degeneracy together lie at the heart of the discrepancies between different gravitational lens reconstructions that can be found in the literature for a given object, and that many images/sources, with an overall high image density in the lens plane, are required to construct an accurate, high-resolution mass map based on strong-lensing data.Comment: 9 pages, accepted for publication by MNRA

Non-parametric inversion of gravitational lensing systems with few images using a multi-objective genetic algorithm

Galaxies acting as gravitational lenses are surrounded by, at most, a handful of images. This apparent paucity of information forces one to make the best possible use of what information is available to invert the lens system. In this paper, we explore the use of a genetic algorithm to invert in a non-parametric way strong lensing systems containing only a small number of images. Perhaps the most important conclusion of this paper is that it is possible to infer the mass distribution of such gravitational lens systems using a non-parametric technique. We show that including information about the null space (i.e. the region where no images are found) is prerequisite to avoid the prediction of a large number of spurious images, and to reliably reconstruct the lens mass density. While the total mass of the lens is usually constrained within a few percent, the fidelity of the reconstruction of the lens mass distribution depends on the number and position of the images. The technique employed to include null space information can be extended in a straightforward way to add additional constraints, such as weak lensing data or time delay information.Comment: 9 pages, accepted for publication by MNRA

FLASC: A Flare-Sensitive Clustering Algorithm: Extending HDBSCAN* for Detecting Branches in Clusters

We present FLASC, an algorithm for flare-sensitive clustering. Our algorithm builds upon HDBSCAN* -- which provides high-quality density-based clustering performance -- through a post-processing step that differentiates branches within the detected clusters' manifold, adding a type of pattern that can be discovered. Two variants of the algorithm are presented, which trade computational cost for noise robustness. We show that both variants scale similarly to HDBSCAN* in terms of computational cost and provide stable outputs using synthetic data sets, resulting in an efficient flare-sensitive clustering algorithm. In addition, we demonstrate the algorithm's benefit in data exploration over HDBSCAN* clustering on two real-world data sets.Comment: 20 pages, 11 figures, submitted to ACM TKD

A generalisation of the mass-sheet degeneracy producing ring-like artefacts in the lens mass distribution

The inversion of a gravitational lens system is, as is well known, plagued by the so-called mass-sheet degeneracy: one can always rescale the density distribution of the lens and add a constant-density mass-sheet such that the, also properly rescaled, source plane is projected onto the same observed images. For strong lensing systems, it is often claimed that this degeneracy is broken as soon as two or more sources at different redshifts are available. This is definitely true in the strict sense that it is then impossible to add a constant-density mass-sheet to the rescaled density of the lens without affecting the resulting images. However, often one can easily construct a more general mass distribution -- instead of a constant-density sheet of mass -- which gives rise to the same effect: a uniform scaling of the sources involved without affecting the observed images. We show that this can be achieved by adding one or more circularly symmetric mass distributions, each with its own center of symmetry, to the rescaled mass distribution of the original lens. As it uses circularly symmetric distributions, this procedure can lead to the introduction of ring shaped features in the mass distribution of the lens. In this paper, we show explicitly how degenerate inversions for a given strong lensing system can be constructed. It then becomes clear that many constraints are needed to effectively break this degeneracy.Comment: 6 pages, accepted for publication by MNRA

Multi-scale cluster lens mass mapping I. Strong Lensing modelling

We propose a novel technique to refine the modelling of galaxy clusters mass distribution using gravitational lensing. The idea is to combine the strengths of both "parametric" and "non-parametric" methods to improve the quality of the fit. We develop a multi-scale model that allows sharper contrast in regions of higher density where the number of constraints is generally higher. Our model consists of (i) a multi-scale grid of radial basis functions with physically motivated profiles and (ii) a list of galaxy-scale potentials at the location of the cluster member galaxies. This arrangement of potentials of different sizes allows to reach a high resolution for the model with a minimum number of parameters. We apply our model to the well studied cluster Abell 1689. We estimate the quality of our mass reconstruction with a Bayesian MCMC sampler. For a selected subset of multiple images, we manage to halve the errors between the predicted and observed image positions compared to previous studies. This owes to the flexibility of multi-scale models at intermediate scale between cluster and galaxy scale. The software developed for this paper is part of the public lenstool package which can be found at www.oamp.fr/cosmology/lenstool.Comment: 15 pages, 17 figures, accepted for publication in MNRA

The Frontier Fields Lens Modeling Comparison Project

Gravitational lensing by clusters of galaxies offers a powerful probe of their structure and mass distribution. Deriving a lens magnification map for a galaxy cluster is a classic inversion problem and many methods have been developed over the past two decades to solve it. Several research groups have developed techniques independently to map the predominantly dark matter distribution in cluster lenses. While these methods have all provided remarkably high precision mass maps, particularly with exquisite imaging data from the Hubble Space Telescope (HST), the reconstructions themselves have never been directly compared. In this paper, we report the results of comparing various independent lens modeling techniques employed by individual research groups in the community. Here we present for the first time a detailed and robust comparison of methodologies for fidelity, accuracy and precision. For this collaborative exercise, the lens modeling community was provided simulated cluster images -- of two clusters Ares and Hera -- that mimic the depth and resolution of the ongoing HST Frontier Fields. The results of the submitted reconstructions with the un-blinded true mass profile of these two clusters are presented here. Parametric, free-form and hybrid techniques have been deployed by the participating groups and we detail the strengths and trade-offs in accuracy and systematics that arise for each methodology. We note in conclusion that lensing reconstruction methods produce reliable mass distributions that enable the use of clusters as extremely valuable astrophysical laboratories and cosmological probes.Comment: 38 pages, 25 figures, submitted to MNRAS, version with full resolution images can be found at http://pico.bo.astro.it/~massimo/papers/FFsims.pd

Different forms of superspreading lead to different outcomes:heterogeneity in infectiousness and contact behavior relevant for the case of SARS-CoV-2

Superspreading events play an important role in the spread of several pathogens, such as SARS-CoV-2. While the basic reproduction number of the original Wuhan SARS-CoV-2 is estimated to be about 3 for Belgium, there is substantial inter-individual variation in the number of secondary cases each infected individual causes—with most infectious individuals generating no or only a few secondary cases, while about 20% of infectious individuals is responsible for 80% of new infections. Multiple factors contribute to the occurrence of superspreading events: heterogeneity in infectiousness, individual variations in susceptibility, differences in contact behavior, and the environment in which transmission takes place. While superspreading has been included in several infectious disease transmission models, research into the effects of different forms of superspreading on the spread of pathogens remains limited. To disentangle the effects of infectiousness-related heterogeneity on the one hand and contact-related heterogeneity on the other, we implemented both forms of superspreading in an individual-based model describing the transmission and spread of SARS-CoV-2 in a synthetic Belgian population. We considered its impact on viral spread as well as on epidemic resurgence after a period of social distancing. We found that the effects of superspreading driven by heterogeneity in infectiousness are different from the effects of superspreading driven by heterogeneity in contact behavior. On the one hand, a higher level of infectiousness-related heterogeneity results in a lower risk of an outbreak persisting following the introduction of one infected individual into the population. Outbreaks that did persist led to fewer total cases and were slower, with a lower peak which occurred at a later point in time, and a lower herd immunity threshold. Finally, the risk of resurgence of an outbreak following a period of lockdown decreased. On the other hand, when contact-related heterogeneity was high, this also led to fewer cases in total during persistent outbreaks, but caused outbreaks to be more explosive in regard to other aspects (such as higher peaks which occurred earlier, and a higher herd immunity threshold). Finally, the risk of resurgence of an outbreak following a period of lockdown increased. We found that these effects were conserved when testing combinations of infectiousness-related and contact-related heterogeneity