arXiv subjects

Liliya L. R. Williams

Publications and source records attributed to Liliya L. R. Williams.

At least 55 records · Page 3Linked to original sources

arXiv2020

Free-form Grale lens inversion of galaxy clusters with up to 1000 multiple images

In the near future, ultra deep observations of galaxy clusters with HST or JWST will uncover $300-1000$ lensed multiple images, increasing the current count per cluster by up to an order of magnitude. This will further refine our view of clusters, leading to a more accurate and precise mapping of the total and dark matter distribution in clusters, and enabling a better understanding of background galaxy population and their luminosity functions. However, to effectively use that many images as input to lens inversion will require a re-evaluation of, and possibly upgrades to the existing methods. In this paper we scrutinize the performance of the free-form lens inversion method Grale in the regime of $150-1000$ input images, using synthetic massive galaxy clusters. Our results show that with an increasing number of input images, Grale produces improved reconstructed mass distributions, with the fraction of the lens plane recovered at better than $10\%$ accuracy increasing from $40-50\%$ for $\sim\!\!150$ images to $65\%$ for $\sim\!1000$ images. The reconstructed time delays imply a more precise measurement of $H_0$, with $\lesssim 1\%$ bias. While the fidelity of the reconstruction improves with the increasing number of multiple images used as model constraints, $\sim 150$ to $\sim 1000$, the lens plane rms deteriorates from $\sim 0.11''$ to $\sim 0.28''$. Since lens plane rms is not necessarily the best indicator of the quality of the mass reconstructions, looking for an alternative indicator is warranted.

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arXiv2020

Extended lens reconstructions with Grale: exploiting time domain, substructural and weak-lensing information

The information about the mass density of galaxy clusters provided by the gravitational lens effect has inspired many inversion techniques. In this article, updates to the previously introduced method in Grale are described, and explored in a number of examples. The first looks into a different way of incorporating time delay information, not requiring the unknown source position. It is found that this avoids a possible bias that leads to "over-focusing" the images, i.e. providing source position estimates that lie in a considerably smaller region than the true positions. The second is inspired by previous reconstructions of the cluster of galaxies MACS J1149.6+2223, where a multiply-imaged background galaxy contained a supernova, SN Refsdal, of which four additional images were produced by the presence of a smaller cluster galaxy. The inversion for the cluster as a whole, was not able to recover sufficient detail interior to this quad. We show how constraints on such different scales, from the entire cluster to a single member galaxy, can now be used, allowing such small scale substructures to be resolved. Finally, the addition of weak lensing information to this method is investigated. While this clearly helps recover the environment around the strong lensing region, the mass sheet degeneracy may make a full strong and weak inversion difficult, depending on the quality of the ellipticity information at hand. We encounter ring-like structure at the boundary of the two regimes, argued to be the result of combining strong and weak lensing constraints, possibly affected by degeneracies.

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arXiv2020

Power law pseudo phase-space density profiles of dark matter halos: fluke of physics?

It has been known for nearly 20 years that the pseudo phase-space density profile of equilibrium simulated dark matter halos, $ρ(r)/σ^3(r)$, is well described by a power law over 3 decades in radius, even though both the density $ρ(r)$, and the velocity dispersion $σ(r)$ deviate significantly from power laws. The origin of this scale-free behavior is not understood. It could be an inherent property of self-gravitating collisionless systems, or it could be a mere coincidence. To address the question we work with equilibrium halos, and more specifically, the second derivative of the Jeans equation, which, under the assumptions of (i) Einasto density profile, (ii) linear velocity anisotropy - density slope relation, and (iii) $ρ/σ^3\propto r^{-α}$, can be transformed from a differential equation to a cubic algebraic equation. Relations (i)-(iii) are all observed in numerical simulations, and are well parametrized by a total of 4 or 6 model parameters. We do not consider dynamical evolution of halos; instead, taking advantage of the fact that the algebraic Jeans equation for equilibrium halos puts relations (i)-(iii) on the same footing, we study the (approximate) solutions of this equation in the 4 and 6 dimensional spaces. We argue that the distribution of best solutions in these parameter spaces is inconsistent with $ρ/σ^3\propto r^{-α}$ being an fundamental property of gravitational evolution, and conclude that the scale-free nature of this quantity is likely to be a fluke.

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arXiv2020

The BUFFALO HST Survey

The Beyond Ultra-deep Frontier Fields and Legacy Observations (BUFFALO) is a 101 orbit + 101 parallel Cycle 25 Hubble Space Telescope Treasury program taking data from 2018-2020. BUFFALO will expand existing coverage of the Hubble Frontier Fields (HFF) in WFC3/IR F105W, F125W, and F160W and ACS/WFC F606W and F814W around each of the six HFF clusters and flanking fields. This additional area has not been observed by HST but is already covered by deep multi-wavelength datasets, including Spitzer and Chandra. As with the original HFF program, BUFFALO is designed to take advantage of gravitational lensing from massive clusters to simultaneously find high-redshift galaxies which would otherwise lie below HST detection limits and model foreground clusters to study properties of dark matter and galaxy assembly. The expanded area will provide a first opportunity to study both cosmic variance at high redshift and galaxy assembly in the outskirts of the large HFF clusters. Five additional orbits are reserved for transient followup. BUFFALO data including mosaics, value-added catalogs and cluster mass distribution models will be released via MAST on a regular basis, as the observations and analysis are completed for the six individual clusters.

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arXiv2019

Image Flux Ratios of Gravitationally Lensed HS 0810+2554 with High Resolution Infrared Imaging

We report near simultaneous imaging using LMIRCam on the LBTI of the quadruply imaged lensed quasar HS 0810+2554 at wavelengths of 2.16, 3.7 and $4.78~μ$m with a Full Width Half Max (FWHM) spatial resolution of $0^{\prime\prime}\!\!.13$, $0^{\prime\prime}\!\!.12$ and $0^{\prime\prime}\!\!.15$ respectively, comparable to HST optical imaging. In the $\rm{z} = 1.5$ rest frame of the quasar, the observed wavelengths correspond to 0.86, 1.48, and $1.91~μ$m respectively. The two brightest images in the quad, A and B, are clearly resolved from each other with a separation of $0.187^{\prime\prime}$. The flux ratio of these two images (A/B) trends from 1.79 to 1.23 from 2.16 to $4.78~μ$m. The trend in flux ratio is consistent with the $2.16~μ$m flux originating from a small sized accretion disk in the quasar that experiences only microlensing. The excess flux above the contribution from the accretion disk at the two longer wavelengths originates from a larger sized region that experiences no microlensing. A simple model employing multiplicative factors for image B due to stellar microlensing $(m)$ and sub-structure millilensing $(M)$ is presented. The result is tightly constrained to the product $m\times M=1.79$. Given the observational errors, the 60\% probability contour for this product stretches from $m= 2.6$, $M = 0.69$ to $m= 1.79$, $M = 1.0$, where the later is consistent with microlensing only.

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arXiv2019

Model-independent and model-based local lensing properties of B0128+437 from resolved quasar images

The galaxy-scale gravitational lens B0128+437 generates a quadrupole-image configuration of a background quasar that shows milli-arcsecond-scale subcomponents in the multiple images observed with VLBI. As this multiple-image configuration including the subcomponents has eluded a parametric lens-model characterisation so far, we determine local lens properties at the positions of the multiple images with our model-independent approach. Using PixeLens, we also succeed in setting up a global free-form mass density reconstruction including all subcomponents as constraints. We compare the model-independent local lens properties with those obtained by PixeLens and those obtained by the parametric modelling algorithm Lensmodel. A comparison of all three approaches and a model-free analysis based on the relative polar angles of the multiple images corroborate the hypothesis that elliptically symmetric models are too simplistic to characterise the asymmetric mass density distribution of this lenticular or late-type galaxy. In addition, the model-independent approach efficiently determines local lens properties on the scale of the quasar subcomponents, which are computationally intensive to obtain by free-form model-based approaches. As only 40% of the small-scale subcomponent local lens properties overlap within the 1-$σ$ confidence bounds, mass density gradients on milli-arcsecond scales cannot be excluded. Hence, aiming at a global reconstruction of the deflecting mass density distribution, increasingly detailed observations require flexible free-form models that allow for density fluctuations on milli-arcsecond scale to replace parametric ones, especially for asymmetric lenses or lenses with localised inhomogeneities like B0128.

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arXiv2019

Free-form Grale reconstruction of Abell 2744: robustness of uncertainties against changes in lensing data

Abell 2744, a massive Hubble Frontier Fields merging galaxy cluster with many multiple images in the core has been the subject of many lens inversions using different methods. While most existing studies compare various inversion methods, we focus on a comparison of reconstructions that use different input lensing data. Since the quantity and quality of lensing data is constantly improving, it makes sense to ask if the estimated uncertainties are robust against changes in the data. We address this question using free-form Grale, which takes only image information as input, and nothing pertaining to cluster galaxies. We reconstruct Abell 2744 using two sets of strong lensing data from the Hubble Frontier Fields community. Our first and second reconstructions use 55 and 91 images, respectively, and only 10 of the 91 images have the same positions and redshifts as in the first reconstruction. Comparison of the two mass maps shows that Grale uncertainties are robust against these changes, as well as small modifications in the inversion routine. Additionally, applying the methods used in Sebesta et al. (2016) for MACS J0416, we conclude that, in a statistical sense, light follows mass in Abell 2744, with brighter galaxies clustering stronger with the recovered mass than the fainter ones. We also show that the faintest galaxies are anti-correlated with mass, which is likely the result of light contamination from bright galaxies, and lensing magnification bias acting on galaxies background to the cluster.

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arXiv2019

Searching for Highly Magnified Stars at Cosmological Distances: Discovery of a Redshift 0.94 Blue Supergiant in Archival Images of the Galaxy Cluster MACS J0416.1-2403

Individual highly magnified stars have been recently discovered at lookback times of more than half the age of the Universe, in lensed galaxies that straddle the critical curves of massive galaxy clusters. Having confirmed their detectability, it is now important to carry out systematic searches for them in order to establish their frequency, and in turn learn about the statistical properties of high-redshift stars and of the granularity of matter in the foreground deflector. Here we report the discovery of a highly magnified star at redshift $z = 0.94$ in a strongly lensed arc behind a Hubble Frontier Field galaxy cluster, MACS J0416.1-2403, discovered as part of a systematic archival search. The bright transient (dubbed "Warhol") was discovered in Hubble Space Telescope data taken on 2014 September 15 and 16. This single image faded over a period of two weeks, and observations taken on 2014 September 1 show that the duration of the microlensing event was at most four weeks in total. The light curve may also exhibit slow changes over a period of years consistent with the level of microlensing expected from stars responsible for the intracluster light (ICL) of the cluster. Optical and infrared observations taken near peak brightness can be fit by a stellar spectrum with moderate host-galaxy extinction. A blue supergiant matches the measured spectral energy distribution near peak, implying a temporary magnification of at least several thousand. While the spectrum of an O-type star would also fit the transient's spectral energy distribution, extremely luminous O-type stars are much less common than blue supergiants. The short timescale of the event and the estimated effective temperature indicate that the lensed source is an extremely magnified star.

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arXiv2018

The role of multiple images and model priors in measuring $H_0$ from supernova Refsdal in galaxy cluster MACS J1149.5+2223

Multiple image gravitational lensing systems with measured time delays provide a promising one-step method for determining $H_0$. MACS J1149, which lenses SN Refsdal into a quad S1-S4, and two other widely separated images, SX and SY, is a perfect candidate. If time delays are pinned down, the remaining uncertainty arises from the mass distribution in the lens. In MACS J1149, the mass in the relevant lens plane region can be constrained by (i) many multiple images, (ii) the mass of the galaxy splitting S1-S4 (which, we show, is correlated with $H_0$), (iii) magnification of SX (also correlated with $H_0$), and (iv) prior assumptions on the mass distribution. Our goal is not to estimate $H_0$, but to understand its error budget, i.e., estimate uncertainties associated with each of these constraints. Using multiple image positions alone, yields very large uncertainty, despite the fact that the position of SX is recovered to within $\!\le\!0.036$" (rms $\!\le\!0.36$") by GRALE lens inversion. Fixing the mass of the galaxy that splits S1-S4 reduces $1σ$ uncertainties to $\sim 23\%$, while fixing the magnification of SX yields $1σ$ uncertainties of $32\%$. We conclude that smaller uncertainties, of order few percent, are a consequence of imposing prior assumptions on the shapes of the galaxy and cluster mass distributions, which may or may not apply in a highly non-equilibrium environment of a merging cluster. We propose that if a measurement of $H_0$ is to be considered reliable, it must be supported by a wide range of lens inversion methods.

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arXiv2018

Testing the uniqueness of gravitational lens mass models

The positions of images produced by the gravitational lensing of background sources provide unique insight in to galaxy-lens mass distribution. However, even quad images of extended sources are not able to fully characterize the central regions of the host galaxy. Most previous work has focused either on the radial density profile of the lenses or localized substructure clumps. Here, we concentrate on the azimuthal mass asymmetries near the image circle. The motivation for considering such mass inhomogeneities is that the transition between the central stellar dominated region and the outer dark matter dominated region, though well represented by a power law density profile, is unlikely to be featureless, and encodes information about the dynamical state and assembly history of galaxies. It also happens to roughly coincide with the Einstein radius. We ask if galaxies that have mass asymmetries beyond ellipticity can be modeled with simpler lenses, i.e., can complex mass distributions masquerade as simple elliptical+shear lenses? Our preliminary study indicates that for galaxies with elliptical stellar and dark matter distributions, but with no mass asymmetry, and an extended source filling the diamond caustic, an elliptical+shear lens model can reproduce the images well, thereby hiding the potential complexity of the actual mass distribution. For galaxies with non-zero mass asymmetry, the answer depends on the size and brightness distribution of the source, and its location within the diamond caustic. In roughly half of the cases we considered the mass asymmetries can easily evade detection.

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arXiv2018

Evidence for the line of sight structure in the largest Hubble Frontier Field cluster, MACSJ0717.5+3745

MACS J0717 is the most massive and extended of the Hubble Frontier Field clusters. It is one of the more difficult clusters to model, and we argue that this is in part due to the line of sight structure (LoS) at redshifts beyond 2. We show that the Grale mass reconstruction based on sources at 3<z_s<4.1 has at least 10^{13}M_sun more mass than that based on nearby sources, z_s<2.6, and attribute the excess mass to a putative LoS, which is at least 75" from the cluster center. Furthermore, the lens-model fitted z_s's of the recent Kawamata et al. reconstruction are biased systematically low compared to photometric z_s's, and the bias is a function of images' distance from the cluster center. We argue that these mimic the effect of LoS. We conclude that even in the presence of 100-200 images, lens-model adjusted source redshifts can conceal the presence of LoS, demonstrating the existence of degeneracies between z_s and (sub)structure. Also, a very good fit to image positions is not a sufficient condition for having a high fidelity mass map: Kawamata et al. obtain an rms of 0.52" for 173 images of 60 sources; our Grale reconstruction of the exact same data yields a somewhat different map, but similarly low rms, 0.62". In contrast, a Grale model that uses reasonable, but fixed z_s gives a worse rms of 1.28" for 44 sources with 126 images. Unaccounted for LoS can bias the mass map, affecting the magnification and luminosity function estimates of high redshift sources.

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arXiv2018

The impact of $Λ$CDM substructure and baryon-dark matter transition on the image positions of quad galaxy lenses

The positions of multiple images in galaxy lenses are related to the galaxy mass distribution. Smooth elliptical mass profiles were previously shown to be inadequate in reproducing the quad population. In this paper, we explore the deviations from such smooth elliptical mass distributions. Unlike most other work, we use a model-free approach based on the relative polar image angles of quads, and their position in 3D space with respect to the Fundamental Surface of Quads. The FSQ is defined by quads produced by elliptical lenses. We have generated thousands of quads from synthetic populations of lenses with substructure consistent with $Λ$CDM simulations, and found that such perturbations are not sufficient to match the observed distribution of quads relative to the FSQ. The result is unchanged even when subhalo masses are increased by a factor of ten, and the most optimistic lensing selection bias is applied. We then produce quads from galaxies created using two components, representing baryons and dark matter. The transition from the mass being dominated by baryons in inner radii to being dominated by dark matter in outer radii can carry with it asymmetries, which would affect relative image angles. We run preliminary experiments using lenses with two elliptical mass components with nonidentical axis ratios and position angles, perturbations from ellipticity in the form of nonzero Fourier coefficients $a_4$ and $a_6$, and artificially offset ellipse centers as a proxy for asymmetry at image radii. We show that combination of these effects is a promising way of accounting for quad population properties. We conclude that the quad population provides a unique and sensitive tool for constraining detailed mass distribution in the centers of galaxies.

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arXiv2018

Dynamics of merging: Post-merger mixing and relaxation of an Illustris galaxy

During the merger of two galaxies, the resulting system undergoes violent relaxation and seeks stable equilibrium. However, the details of this evolution are not fully understood. Using Illustris simulation, we probe two physically related processes, mixing and relaxation. Though the two are driven by the same dynamics---global time-varying potential for the energy, and torques caused by asymmetries for angular momentum---we measure them differently. We define mixing as the redistribution of energy and angular momentum between particles of the two merging galaxies. We assess the degree of mixing as the difference between the shapes of their N(E)s, and their N(L^2)s. We find that the difference is decreasing with time, indicating mixing. To measure relaxation, we compare N(E) of the newly merged system to N(E) of a theoretical prediction for relaxed collisionless systems, DARKexp, and witness the system becoming more relaxed, in the sense that N(E) approaches DARKexp N(E). Because the dynamics driving mixing and relaxation are the same, the timescale is similar for both. We measure two sequential timescales: a rapid, 1 Gyr phase after the initial merger, during which the difference in N(E) of the two merging halos decreases by ~80%, followed by a slow phase, when the difference decreases by ~50% over ~8.5 Gyrs. This is a direct measurement of the relaxation timescale. Our work also draws attention to the fact that when a galaxy has reached Jeans equilibrium it may not yet have reached a fully relaxed state given by DARKexp, in that it retains information about its past history. This manifests itself most strongly in stars being centrally concentrated. We argue that it is particularly difficult for stars, and other tightly bound particles, to mix because they have less time to be influenced by the fluctuating potential, even across multiple merger events.

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arXiv2016

Lens Models Under the Microscope: Comparison of Hubble Frontier Field Cluster Magnification Maps

Using the power of gravitational lensing magnification by massive galaxy clusters, the Hubble Frontier Fields provide deep views of six patches of the high redshift Universe. The combination of deep Hubble imaging and exceptional lensing strength has revealed the greatest numbers of multiply-imaged galaxies available to constrain models of cluster mass distributions. However, even with $\mathcal{O}(100)$ images per cluster, the uncertainties associated with the reconstructions are not negligible. The goal of this paper is to show the diversity of model magnification predictions. We examine 7 and 9 mass models of Abell 2744 and MACS J0416, respectively, submitted to the Mikulski Archive for Space Telescopes for public distribution in September 2015. The dispersion between model predictions increases from 30% at common low magnifications ($μ\sim2$) to 70% at rare high magnifications ($μ\sim40$). MACS J0416 exhibits smaller dispersions than Abell 2744 for $2<μ<10$. We show that magnification maps based on different lens inversion techniques typically differ from each other by more than their quoted statistical errors. This suggests that some models underestimate the true uncertainties, which are primarily due to various lensing degeneracies. Though the exact mass sheet degeneracy is broken, its generalized counterpart is not broken at least in Abell 2744. Other, local degeneracies are also present in both clusters. Our comparison of models is complementary to the comparison of reconstructions of known synthetic mass distributions. By focusing on observed clusters, we can identify those that are best constrained, and therefore provide the clearest view of the distant Universe.

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arXiv2016

Testing DARKexp against energy and density distributions of Millennium-II halos

We test the DARKexp model for relaxed, self-gravitating, collisionless systems against equilibrium dark matter halos from the Millennium-II simulation. While limited tests of DARKexp against simulations and observations have been carried out elsewhere, this is the first time the testing is done with a large sample of simulated halos spanning a factor of ~ 50 in mass, and using independent fits to density and energy distributions. We show that DARKexp, a one shape parameter family, provides very good fits to the shapes of density profiles, ρ(r), and differential energy distributions, N(E), of individual simulated halos. The best fit shape parameter $ϕ_{0}$ obtained from the two types of fits are correlated, though with scatter. Our most important conclusions come from ρ(r) and N(E) that have been averaged over many halos. These show that the bulk of the deviations between DARKexp and individual Millennium-II halos come from halo-to-halo fluctuations, likely driven by substructure, and other density perturbations. The average ρ(r) and N(E) are quite smooth and follow DARKexp very closely. The only deviation that remains after averaging is small, and located at most bound energies for N(E) and smallest radii for ρ(r). Since the deviation is confined to 3-4 smoothing lengths, and is larger for low mass halos, it is likely due to numerical resolution effects.

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arXiv2016

Testing light-traces-mass in Hubble Frontier Fields Cluster MACS-J0416.1-2403

We reconstruct the projected mass distribution of a massive merging Hubble Frontier Fields cluster MACSJ0416 using the genetic algorithm based free-form technique called Grale. The reconstructions are constrained by 149 lensed images identified by Jauzac et al. using HFF data. No information about cluster galaxies or light is used, which makes our reconstruction unique in this regard. Using visual inspection of the maps, as well as galaxy-mass correlation functions we conclude that overall light does follow mass. Furthermore, the fact that brighter galaxies are more strongly clustered with mass is an important confirmation of the standard biasing scenario in galaxy clusters. On the smallest scales, approximately less than a few arcseconds, the resolution afforded by 149 images is still not sufficient to confirm or rule out galaxy-mass offsets of the kind observed in ACO 3827. We also compare the mass maps of MACSJ0416 obtained by three different groups: Grale, and two parametric Lenstool reconstructions from the CATS and Sharon/Johnson teams. Overall, the three agree well; one interesting discrepancy between Grale and Lenstool galaxy-mass correlation functions occurs on scales of tens of kpc and may suggest that cluster galaxies are more biased tracers of mass than parametric methods generally assume.

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arXiv2016

Ubiquity of density slope oscillations in the central regions of galaxy and cluster-sized systems

One usually thinks of a radial density profile as having a monotonically changing logarithmic slope, such as in NFW or Einasto profiles. However, in two different classes of commonly used systems, this is often not the case. These classes exhibit non-monotonic changes in their density profile slopes which we call oscillations for short. We analyze these two unrelated classes separately. Class 1 consists of systems that have density oscillations and that are defined through their distribution function $f(E)$, or differential energy distribution $N(E)$, such as isothermal spheres, King profiles, or DARKexp, a theoretically derived model for relaxed collisionless systems. Systems defined through $f(E)$ or $N(E)$ generally have density slope oscillations. Class 1 system oscillations can be found at small, intermediate, or large radii but we focus on a limited set of Class 1 systems that have oscillations in the central regions, usually at $\log(r/r_{-2})\lesssim -2$, where $r_{-2}$ is the largest radius where $d\log(ρ)/d\log(r)=-2$. We show that the shape of their $N(E)$ can roughly predict the amplitude of oscillations. Class 2 systems which are a product of dynamical evolution, consist of observed and simulated galaxies and clusters, and pure dark matter halos. Oscillations in the density profile slope seem pervasive in the central regions of Class 2 systems. We argue that in these systems, slope oscillations are an indication that a system is not fully relaxed. We show that these oscillations can be reproduced by small modifications to $N(E)$ of DARKexp. These affect a small fraction of systems' mass and are confined to $\log(r/r_{-2})\lesssim 0$. The size of these modifications serves as a potential diagnostic for quantifying how far a system is from being relaxed.

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arXiv2016

Quantifying substructures in {\it Hubble Frontier Field} clusters: comparison with $ΛCDM$ simulations

The Hubble Frontier Fields (HFF) are six clusters of galaxies, all showing indications of recent mergers, which have recently been observed for lensed images. As such they are the natural laboratories to study the merging history of galaxy clusters. In this work, we explore the 2D power spectrum of the mass distribution $P_{\rm M}(k)$ as a measure of substructure. We compare $P_{\rm M}(k)$ of these clusters (obtained using strong gravitational lensing) to that of $Λ$CDM simulated clusters of similar mass. To compute lensing $P_{\rm M}(k)$, we produced free-form lensing mass reconstructions of HFF clusters, without any light traces mass (LTM) assumption. The inferred power at small scales tends to be larger if (i)~the cluster is at lower redshift, and/or (ii)~there are deeper observations and hence more lensed images. In contrast, lens reconstructions assuming LTM show higher power at small scales even with fewer lensed images; it appears the small scale power in the LTM reconstructions is dominated by light information, rather than the lensing data. The average lensing derived $P_{\rm M}(k)$ shows lower power at small scales as compared to that of simulated clusters at redshift zero, both dark-matter only and hydrodynamical. The possible reasons are: (i)~the available strong lensing data are limited in their effective spatial resolution on the mass distribution, (ii)~HFF clusters have yet to build the small scale power they would have at $z\sim 0$, or (iii)~simulations are somehow overestimating the small scale power.

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