arXiv subjects

R. Massey

Publications and source records attributed to R. Massey.

At least 109 records · Page 6Linked to original sources

arXiv2021

TOI-1634 b: an Ultra-Short Period Keystone Planet Sitting Inside the M Dwarf Radius Valley

Studies of close-in planets orbiting M dwarfs have suggested that the M dwarf radius valley may be well-explained by distinct formation timescales between enveloped terrestrials, and rocky planets that form at late times in a gas-depleted environment. This scenario is at odds with the picture that close-in rocky planets form with a primordial gaseous envelope that is subsequently stripped away by some thermally-driven mass loss process. These two physical scenarios make unique predictions of the rocky/enveloped transition's dependence on orbital separation such that studying the compositions of planets within the M dwarf radius valley may be able to establish the dominant physics. Here, we present the discovery of one such keystone planet: the ultra-short period planet TOI-1634 b ($P=0.989$ days, $F=121 F_{\oplus}$, $r_p = 1.790^{+0.080}_{-0.081} R_{\oplus}$) orbiting a nearby M2 dwarf ($K_s=8.7$, $R_s=0.45 R_{\odot}$, $M_s=0.50 M_{\odot}$) and whose size and orbital period sit within the M dwarf radius valley. We confirm the TESS-discovered planet candidate using extensive ground-based follow-up campaigns, including a set of 32 precise radial velocity measurements from HARPS-N. We measure a planetary mass of $4.91^{+0.68}_{-0.70} M_{\oplus}$, which makes TOI-1634 b inconsistent with an Earth-like composition at $5.9σ$ and thus requires either an extended gaseous envelope, a large volatile-rich layer, or a rocky portion that is not dominated by iron and silicates to explain its mass and radius. The discovery that the bulk composition of TOI-1634 b is inconsistent with that of the Earth favors the gas-depleted formation mechanism to explain the emergence of the radius valley around M dwarfs with $M_s\lesssim 0.5 M_{\odot}$.

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arXiv2021

Euclid: Estimation of the impact of correlated readout noise for flux measurements with the Euclid NISP instrument

The Euclid satellite, to be launched by ESA in 2022, will be a major instrument for cosmology for the next decades. \Euclid\ is composed of two instruments: the Visible (VIS) instrument and the Near Infrared Spectromete and Photometer (NISP). In this work we estimate the implications of correlated readout noise in the NISP detectors for the final in-flight flux measurements. Considering the multiple accumulated (MACC) readout mode, for which the UTR (Up The Ramp) exposure frames are averaged in groups, we derive an analytical expression for the noise covariance matrix between groups in the presence of correlated noise. We also characterize the correlated readout noise properties in the NISP engineering grade detectors using long dark integrations. For this purpose, we assume a $(1/f)^{\, α}$-like noise model and fit the model parameters to the data, obtaining typical values of $σ= 19.7^{+1.1}_{-0.8}$ e$^{-} \rm{Hz}^{-0.5}$, $f_{\rm{knee}} = (5.2^{+1.8}_{-1.3}) \times 10^{-3} \, \rm{Hz}$ and $α= 1.24 ^{+0.26}_{-0.21}$. Furthermore, via realistic simulations and using a maximum likelihood flux estimator we derive the bias between the input flux and the recovered one. We find that using our analytical expression for the covariance matrix of the correlated readout noise we diminish this bias by up to a factor of four with respect to the white noise approximation for the covariance matrix. Finally, we conclude that the final bias on the in-flight NISP flux measurements should still be negligible even in the white noise approximation, which is taken as a baseline for the Euclid\on-board processing

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arXiv2021

Euclid preparation: XII. Optimizing the photometric sample of the Euclid survey for galaxy clustering and galaxy-galaxy lensing analyses

The accuracy of photometric redshifts (photo-zs) particularly affects the results of the analyses of galaxy clustering with photometrically-selected galaxies (GCph) and weak lensing. In the next decade, space missions like Euclid will collect photometric measurements for millions of galaxies. These data should be complemented with upcoming ground-based observations to derive precise and accurate photo-zs. In this paper, we explore how the tomographic redshift binning and depth of ground-based observations will affect the cosmological constraints expected from Euclid. We focus on GCph and extend the study to include galaxy-galaxy lensing (GGL). We add a layer of complexity to the analysis by simulating several realistic photo-z distributions based on the Euclid Consortium Flagship simulation and using a machine learning photo-z algorithm. We use the Fisher matrix formalism and these galaxy samples to study the cosmological constraining power as a function of redshift binning, survey depth, and photo-z accuracy. We find that bins with equal width in redshift provide a higher Figure of Merit (FoM) than equipopulated bins and that increasing the number of redshift bins from 10 to 13 improves the FoM by 35% and 15% for GCph and its combination with GGL, respectively. For GCph, an increase of the survey depth provides a higher FoM. But the addition of faint galaxies beyond the limit of the spectroscopic training data decreases the FoM due to the spurious photo-zs. When combining both probes, the number density of the sample, which is set by the survey depth, is the main factor driving the variations in the FoM. We conclude that there is more information that can be extracted beyond the nominal 10 tomographic redshift bins of Euclid and that we should be cautious when adding faint galaxies into our sample, since they can degrade the cosmological constraints.

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arXiv2021

Euclid: Effect of sample covariance on the number counts of galaxy clusters

Aims. We investigate the contribution of shot-noise and sample variance to the uncertainty of cosmological parameter constraints inferred from cluster number counts in the context of the Euclid survey. Methods. By analysing 1000 Euclid-like light-cones, produced with the PINOCCHIO approximate method, we validate the analytical model of Hu & Kravtsov 2003 for the covariance matrix, which takes into account both sources of statistical error. Then, we use such covariance to define the likelihood function that better extracts cosmological information from cluster number counts at the level of precision that will be reached by the future Euclid photometric catalogs of galaxy clusters. We also study the impact of the cosmology dependence of the covariance matrix on the parameter constraints. Results. The analytical covariance matrix reproduces the variance measured from simulations within the 10 per cent level; such difference has no sizeable effect on the error of cosmological parameter constraints at this level of statistics. Also, we find that the Gaussian likelihood with cosmology-dependent covariance is the only model that provides an unbiased inference of cosmological parameters without underestimating the errors.

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arXiv2021

Euclid: Forecast constraints on the cosmic distance duality relation with complementary external probes

In metric theories of gravity with photon number conservation, the luminosity and angular diameter distances are related via the Etherington relation, also known as the distance-duality relation (DDR). A violation of this relation would rule out the standard cosmological paradigm and point at the presence of new physics. We quantify the ability of Euclid, in combination with contemporary surveys, to improve the current constraints on deviations from the DDR in the redshift range $0<z<1.6$. We start by an analysis of the latest available data, improving previously reported constraints by a factor of 2.5. We then present a detailed analysis of simulated Euclid and external data products, using both standard parametric methods (relying on phenomenological descriptions of possible DDR violations) and a machine learning reconstruction using Genetic Algorithms. We find that for parametric methods Euclid can (in combination with external probes) improve current constraints by approximately a factor of six, while for non-parametric methods Euclid can improve current constraints by a factor of three. Our results highlight the importance of surveys like Euclid in accurately testing the pillars of the current cosmological paradigm and constraining physics beyond the standard cosmological model.

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arXiv2021

Euclid preparation: XI. Mean redshift determination from galaxy redshift probabilities for cosmic shear tomography

The analysis of weak gravitational lensing in wide-field imaging surveys is considered to be a major cosmological probe of dark energy. Our capacity to constrain the dark energy equation of state relies on the accurate knowledge of the galaxy mean redshift $\langle z \rangle$. We investigate the possibility of measuring $\langle z \rangle$ with an accuracy better than $0.002\,(1+z)$, in ten tomographic bins spanning the redshift interval $0.2 99.8\%$. The zPDF approach could also be successful if we debias the zPDF using a spectroscopic training sample. This approach requires deep imaging data, but is weakly sensitive to spectroscopic redshift failures in the training sample. We improve the debiasing method and confirm our finding by applying it to real-world weak-lensing data sets (COSMOS and KiDS+VIKING-450).

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arXiv2020

Euclid: Identification of asteroid streaks in simulated images using StreakDet software

The ESA Euclid space telescope could observe up to 150 000 asteroids as a side product of its primary cosmological mission. Asteroids appear as trailed sources, that is streaks, in the images. Owing to the survey area of 15 000 square degrees and the number of sources, automated methods have to be used to find them. Euclid is equipped with a visible camera, VIS (VISual imager), and a near-infrared camera, NISP (Near-Infrared Spectrometer and Photometer), with three filters. We aim to develop a pipeline to detect fast-moving objects in Euclid images, with both high completeness and high purity. We tested the StreakDet software to find asteroids from simulated Euclid images. We optimized the parameters of StreakDet to maximize completeness, and developed a post-processing algorithm to improve the purity of the sample of detected sources by removing false-positive detections. StreakDet finds 96.9% of the synthetic asteroid streaks with apparent magnitudes brighter than 23rd magnitude and streak lengths longer than 15 pixels ($10\,{\rm arcsec\,h^{-1}}$), but this comes at the cost of finding a high number of false positives. The number of false positives can be radically reduced with multi-streak analysis, which utilizes all four dithers obtained by Euclid. StreakDet is a good tool for identifying asteroids in Euclid images, but there is still room for improvement, in particular, for finding short (less than 13 pixels, corresponding to 8$\,{\rm arcsec\,h^{-1}}$) and/or faint streaks (fainter than the apparent magnitude of 23).

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arXiv2020

Euclid: impact of nonlinear prescriptions on cosmological parameter estimation from weak lensing cosmic shear

Upcoming surveys will map the growth of large-scale structure with unprecented precision, improving our understanding of the dark sector of the Universe. Unfortunately, much of the cosmological information is encoded by the small scales, where the clustering of dark matter and the effects of astrophysical feedback processes are not fully understood. This can bias the estimates of cosmological parameters, which we study here for a joint analysis of mock Euclid cosmic shear and Planck cosmic microwave background data. We use different implementations for the modelling of the signal on small scales and find that they result in significantly different predictions. Moreover, the different nonlinear corrections lead to biased parameter estimates, especially when the analysis is extended into the highly nonlinear regime, with both the Hubble constant, $H_0$, and the clustering amplitude, $σ_8$, affected the most. Improvements in the modelling of nonlinear scales will therefore be needed if we are to resolve the current tension with more and better data. For a given prescription for the nonlinear power spectrum, using different corrections for baryon physics does not significantly impact the precision of Euclid, but neglecting these correction does lead to large biases in the cosmological parameters. In order to extract precise and unbiased constraints on cosmological parameters from Euclid cosmic shear data, it is therefore essential to improve the accuracy of the recipes that account for nonlinear structure formation, as well as the modelling of the impact of astrophysical processes that redistribute the baryons.

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arXiv2020

Euclid preparation: IX. EuclidEmulator2 -- Power spectrum emulation with massive neutrinos and self-consistent dark energy perturbations

We present a new, updated version of the EuclidEmulator (called EuclidEmulator2), a fast and accurate predictor for the nonlinear correction of the matter power spectrum. Percent-level accurate emulation is now supported in the eight-dimensional parameter space of $w_0w_a$CDM$+\sum m_ν$models between redshift $z=0$ and $z=3$ for spatial scales within the range 0.01 $h$/Mpc $\leq k \leq$ 10 $h$/Mpc. In order to achieve this level of accuracy, we have had to improve the quality of the underlying N-body simulations used as training data: (1) we use self-consistent linear evolution of non-dark matter species such as massive neutrinos, photons, dark energy and the metric field, (2) we perform the simulations in the so-called N-body gauge, which allows one to interpret the results in the framework of general relativity, (3) we run over 250 high-resolution simulations with $3000^3$ particles in boxes of 1 (Gpc/$h$)${}^3$ volumes based on paired-and-fixed initial conditions and (4) we provide a resolution correction that can be applied to emulated results as a post-processing step in order to drastically reduce systematic biases on small scales due to residual resolution effects in the simulations. We find that the inclusion of the dynamical dark energy parameter $w_a$ significantly increases the complexity and expense of creating the emulator. The high fidelity of EuclidEmulator2 is tested in various comparisons against N-body simulations as well as alternative fast predictors like Halofit, HMCode and CosmicEmu. A blind test is successfully performed against the Euclid Flagship v2.0 simulation. Nonlinear correction factors emulated with EuclidEmulator2 are accurate at the level of 1% or better for 0.01 $h$/Mpc $\leq k \leq$ 10 $h$/Mpc and $z\leq3$ compared to high-resolution dark matter only simulations. EuclidEmulator2 is publicly available at https://github.com/miknab/EuclidEmulator2 .

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arXiv2020

Euclid preparation: VIII. The Complete Calibration of the Colour-Redshift Relation survey: VLT/KMOS observations and data release

The Complete Calibration of the Colour-Redshift Relation survey (C3R2) is a spectroscopic effort involving ESO and Keck facilities designed to empirically calibrate the galaxy colour-redshift relation - P(z|C) to the Euclid depth (i_AB=24.5) and is intimately linked to upcoming Stage IV dark energy missions based on weak lensing cosmology. The aim is to build a spectroscopic calibration sample that is as representative as possible of the galaxies of the Euclid weak lensing sample. In order to minimise the number of spectroscopic observations to fill the gaps in current knowledge of the P(z|C), self-organising map (SOM) representations of the galaxy colour space have been constructed. Here we present the first results of an ESO@ VLT Large Programme approved in the context of C3R2, which makes use of the two VLT optical and near-infrared multi-object spectrographs, FORS2 and KMOS. This paper focuses on high-quality spectroscopic redshifts of high-z galaxies observed with the KMOS spectrograph in the H- and K-bands. A total of 424 highly-reliable z are measured in the 1.3<=z<=2.5 range, with total success rates of 60.7% in the H-band and 32.8% in the K-band. The newly determined z fill 55% of high and 35% of lower priority empty SOM grid cells. We measured Halpha fluxes in a 1."2 radius aperture from the spectra of the spectroscopically confirmed galaxies and converted them into star formation rates. In addition, we performed an SED fitting analysis on the same sample in order to derive stellar masses, E(B-V), total magnitudes, and SFRs. We combine the results obtained from the spectra with those derived via SED fitting, and we show that the spectroscopic failures come from either weakly star-forming galaxies (at z<1.7, i.e. in the H-band) or low S/N spectra (in the K-band) of z>2 galaxies.

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arXiv2020

Euclid: Reconstruction of Weak Lensing mass maps for non-Gaussianity studies

Weak lensing, which is the deflection of light by matter along the line of sight, has proven to be an efficient method for constraining models of structure formation and reveal the nature of dark energy. So far, most weak-lensing studies have focused on the shear field that can be measured directly from the ellipticity of background galaxies. However, within the context of forthcoming full-sky weak-lensing surveys such as Euclid, convergence maps (mass maps) offer an important advantage over shear fields in terms of cosmological exploitation. While it carry the same information, the lensing signal is more compressed in the convergence maps than in the shear field. This simplifies otherwise computationally expensive analyses, for instance, non-Gaussianity studies. However, the inversion of the non-local shear field requires accurate control of systematic effects caused by holes in the data field, field borders, shape noise, and the fact that the shear is not a direct observable (reduced shear). We present the two mass-inversion methods that are included in the official Euclid data-processing pipeline: the standard Kaiser & Squires method (KS), and a new mass-inversion method (KS+) that aims to reduce the information loss during the mass inversion. This new method is based on the KS method and includes corrections for mass-mapping systematic effects. The results of the KS+ method are compared to the original implementation of the KS method in its simplest form, using the Euclid Flagship mock galaxy catalogue. In particular, we estimate the quality of the reconstruction by comparing the two-point correlation functions and third- and fourth- order moments obtained from shear and convergence maps, and we analyse each systematic effect independently and simultaneously...

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arXiv2020

Euclid: The importance of galaxy clustering and weak lensing cross-correlations within the photometric Euclid survey

The data from the Euclid mission will enable the measurement of the photometric redshifts, angular positions, and weak lensing shapes for over a billion galaxies. This large dataset will allow for cosmological analyses using the angular clustering of galaxies and cosmic shear. The cross-correlation (XC) between these probes can tighten constraints and it is therefore important to quantify their impact for Euclid. In this study we carefully quantify the impact of XC not only on the final parameter constraints for different cosmological models, but also on the nuisance parameters. In particular, we aim at understanding the amount of additional information that XC can provide for parameters encoding systematic effects, such as galaxy bias or intrinsic alignments (IA). We follow the formalism presented in Euclid Collaboration: Blanchard et al. (2019) and make use of the codes validated therein. We show that XC improves the dark energy Figure of Merit (FoM) by a factor $\sim 5$, whilst it also reduces the uncertainties on galaxy bias by $\sim 17\%$ and the uncertainties on IA by a factor $\sim 4$. We observe that the role of XC on the final parameter constraints is qualitatively the same irrespective of the galaxy bias model used. We also show that XC can help in distinguishing between different IA models, and that if IA terms are neglected then this can lead to significant biases on the cosmological parameters. We find that the XC terms are necessary to extract the full information content from the data in future analyses. They help in better constraining the cosmological model, and lead to a better understanding of the systematic effects that contaminate these probes. Furthermore, we find that XC helps in constraining the mean of the photometric-redshift distributions, but it requires a more precise knowledge of this mean in order not to degrade the final FoM. [Abridged]

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arXiv2020

Euclid: The reduced shear approximation and magnification bias for Stage IV cosmic shear experiments

Stage IV weak lensing experiments will offer more than an order of magnitude leap in precision. We must therefore ensure that our analyses remain accurate in this new era. Accordingly, previously ignored systematic effects must be addressed. In this work, we evaluate the impact of the reduced shear approximation and magnification bias, on the information obtained from the angular power spectrum. To first-order, the statistics of reduced shear, a combination of shear and convergence, are taken to be equal to those of shear. However, this approximation can induce a bias in the cosmological parameters that can no longer be neglected. A separate bias arises from the statistics of shear being altered by the preferential selection of galaxies and the dilution of their surface densities, in high-magnification regions. The corrections for these systematic effects take similar forms, allowing them to be treated together. We calculated the impact of neglecting these effects on the cosmological parameters that would be determined from Euclid, using cosmic shear tomography. To do so, we employed the Fisher matrix formalism, and included the impact of the super-sample covariance. We also demonstrate how the reduced shear correction can be calculated using a lognormal field forward modelling approach. These effects cause significant biases in Omega_m, sigma_8, n_s, Omega_DE, w_0, and w_a of -0.53 sigma, 0.43 sigma, -0.34 sigma, 1.36 sigma, -0.68 sigma, and 1.21 sigma, respectively. We then show that these lensing biases interact with another systematic: the intrinsic alignment of galaxies. Accordingly, we develop the formalism for an intrinsic alignment-enhanced lensing bias correction. Applying this to Euclid, we find that the additional terms introduced by this correction are sub-dominant.

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arXiv2020

Euclid: The selection of quiescent and star-forming galaxies using observed colours

The Euclid mission will observe well over a billion galaxies out to $z\sim6$ and beyond. This will offer an unrivalled opportunity to investigate several key questions for understanding galaxy formation and evolution. The first step for many of these studies will be the selection of a sample of quiescent and star-forming galaxies, as is often done in the literature by using well known colour techniques such as the `UVJ' diagram. However, given the limited number of filters available for the Euclid telescope, the recovery of such rest-frame colours will be challenging. We therefore investigate the use of observed Euclid colours, on their own and together with ground-based u-band observations, for selecting quiescent and star-forming galaxies. The most efficient colour combination, among the ones tested in this work, consists of the (u-VIS) and (VIS-J) colours. We find that this combination allows users to select a sample of quiescent galaxies complete to above $\sim70\%$ and with less than 15$\%$ contamination at redshifts in the range $0.75 65\%$ completeness level and contamination below 20$\%$ at $1<z<2$ for finding quiescent galaxies. In comparison, the sample of quiescent galaxies selected with the traditional UVJ technique is only $\sim20\%$ complete at $z<3$, when recovering the rest-frame colours using mock Euclid observations. This shows that our new methodology is the most suitable one when only Euclid bands, along with u-band imaging, are available.

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arXiv2019

Euclid preparation: V. Predicted yield of redshift 7<z<9 quasars from the wide survey

We provide predictions of the yield of $7 8$ may be selected from Euclid $OYJH$ photometry alone, but selection over the redshift interval $7 7.5$ should be found in the DR1 data release, expected in 2024. It will be possible to determine the bright-end slope of the QLF, $7<z<8$, $M_{1450}<-25$, using 8m class telescopes to confirm candidates, but follow-up with JWST or E-ELT will be required to measure the faint-end slope. Contamination of the candidate lists is predicted to be modest even at $J_{AB}\sim23$. The precision with which $k$ can be determined over $7<z<8$ depends on the value of $k$, but assuming $k=-0.72$ it can be measured to a 1 sigma uncertainty of 0.07.

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arXiv2019

Deep and rapid observations of strong-lensing galaxy clusters within the sky localisation of GW170814

We present observations of two strong-lensing galaxy clusters located within the $90$ per cent credible sky localization maps released following LIGO-Virgo's discovery of the binary black hole (BH-BH) gravitational wave (GW) source GW170814. Our objectives were (1) to search for candidate electromagnetic (EM) counterparts to GW170814 under the hypothesis that it was strongly-lensed, and thus more distant and less massive than inferred by LIGO-Virgo, and (2) to demonstrate the feasibility of rapid target of opportunity observations to search for faint lensed transient point sources in crowded cluster cores located within GW sky localizations. Commencing $20$ hours after discovery, and continuing over $12$ nights, we observed Abell 3084 ($z=0.22$) and SMACSJ0304.3$-$4401 ($z=0.46$) with GMOS on the Gemini-South telescope, and Abell 3084 with MUSE on ESO's Very Large Telescope. We detect no candidate EM counterparts in these data. Calibration of our photometric analysis methods using simulations yield $5σ$ detection limits for transients in difference images of the cores of these clusters of $i=25$. This is the most sensitive photometric search to date for counterparts to GW sources, and rules out the possibility that GW170814 was lensed by these clusters with a kilonova-like EM counterpart. Based on the detector frame masses of the compact objects, and assuming that at least one Neutron Star (NS) is required in the merging system to produce a kilonova-like counterpart, implies that GW170814 was neither a NS-NS nor NS-BH merger at $z>8$ lensed by either of these clusters. Also, in the first ever emission line search for counterparts to GW sources, we detected no lines down to a $5σ$ detection limit of $5\times10^{-17}{\rm erg\,s^{-1}\,cm^{-2}}$.

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arXiv2018

Growing a `Cosmic Beast': Observations and Simulations of MACS J0717.5+3745

We present a gravitational lensing and X-ray analysis of a massive galaxy cluster and its surroundings. The core of MACS\,J0717.5+3745 ($M(R<1\,{\rm Mpc})\sim$\,$2$$\times$$10^{15}\,\msun$, $z$=$0.54$) is already known to contain four merging components. We show that this is surrounded by at least seven additional substructures with masses ranging from $3.8-6.5\times10^{13}\,\msun$, at projected radii $1.6$ to $4.9$\,Mpc. We compare MACS\,J0717 to mock lensing and X-ray observations of similarly rich clusters in cosmological simulations. The low gas fraction of substructures predicted by simulations turns out to match our observed values of $1$--$4\%$. Comparing our data to three similar simulated halos, we infer a typical growth rate and substructure infall velocity. That suggests MACS\,J0717 could evolve into a system similar to, but more massive than, Abell\,2744 by $z=0.31$, and into a $\sim$\,$10^{16}\,\msun$ supercluster by $z=0$. The radial distribution of infalling substructure suggests that merger events are strongly episodic; however we find that the smooth accretion of surrounding material remains the main source of mass growth even for such massive clusters.

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arXiv2018

The shape of galaxy dark matter halos in massive galaxy clusters: Insights from strong gravitational lensing

We assess how much unused strong lensing information is available in the deep \emph{Hubble Space Telescope} imaging and VLT/MUSE spectroscopy of the \emph{Frontier Field} clusters. As a pilot study, we analyse galaxy cluster MACS\,J0416.1-2403 ($z$$=$$0.397$, $M(R<200\,{\rm kpc})$$=$$1.6$$\times$$10^{14}\msun$), which has 141 multiple images with spectroscopic redshifts. We find that many additional parameters in a cluster mass model can be constrained, and that adding even small amounts of extra freedom to a model can dramatically improve its figures of merit. We use this information to constrain the distribution of dark matter around cluster member galaxies, simultaneously with the cluster's large-scale mass distribution. We find tentative evidence that some galaxies' dark matter has surprisingly similar ellipticity to their stars (unlike in the field, where it is more spherical), but that its orientation is often misaligned. When non-coincident dark matter and baryonic halos are allowed, the model improves by 35\%. This technique may provide a new way to investigate the processes and timescales on which dark matter is stripped from galaxies as they fall into a massive cluster. Our preliminary conclusions will be made more robust by analysing the remaining five \emph{Frontier Field} clusters.

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