A new high-precision strong lensing model of the galaxy cluster MACS J0416.1-2403
Pietro Bergamini, Piero Rosati, Eros Vanzella, Gabriel Bartosch Caminha, Claudio Grillo, Amata Mercurio, Massimo Meneghetti, Giuseppe Angora, Francesco Calura, Mario Nonino, Paolo Tozzi
AAstronomy & Astrophysics manuscript no. paper © ESO 2020November 20, 2020
A new high-precision strong lensing model of the galaxy clusterMACS J0416.1 − Robust characterization of the cluster mass distribution from VLT/MUSE deepobservations
P. Bergamini (cid:63) , (cid:63)(cid:63) , P. Rosati , , E. Vanzella , G. B. Caminha , , C. Grillo , , A. Mercurio , M. Meneghetti ,G. Angora , , F. Calura , M. Nonino , and P. Tozzi INAF – OAS, Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93 /
3, I-40129 Bologna, Italy Dipartimento di Fisica e Scienze della Terra, Università degli Studi di Ferrara, via Saragat 1, I-44122 Ferrara, Italy Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV Groningen, The Netherlands Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen, Denmark INAF – Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Napoli, Italy INAF – Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143, Trieste, Italy INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi, I-50125, Firenze, ItalyReceived February 14, 2020; accepted February 14, 2020
ABSTRACT
We present a new high-precision parametric strong lensing model of the galaxy cluster MACS J0416.1 − z = . . < z < .
2, and 171 clustermember galaxies. Several multiple images are associated to individual clumps in multiply lensed resolved sources. By defining anew metric, which is sensitive to the gradients of the deflection field, we show that we can accurately reproduce the positions ofthese star-forming knots despite their vicinity to the model critical lines. The high signal-to-noise ratio of the MDLF spectra enablesthe measurement of the internal velocity dispersion of 64 cluster galaxies, down to m F W =
22. This allowed us to independentlyestimate the contribution of the subhalo mass component of the lens model from the measured Faber-Jackson scaling relation. Our bestreference model, which represents a significant step forward compared to our previous analyses, was selected from a comparative studyof di ff erent mass parametrizations. The root-mean-square displacement between the observed and model-predicted image positionsis only 0 . (cid:48)(cid:48) , which is 33% smaller than in all previous models. The mass model appears to be particularly well constrained in theMDLF region. We characterize the robustness of the magnification map at varying distances from the model critical lines and the totalprojected mass profile of the cluster. Key words.
Galaxies: clusters: general – Gravitational lensing: strong – cosmology: observations – dark matter – galaxies: kinematicsand dynamics
1. Introduction
In recent years, strong gravitational lensing has become one ofthe most e ff ective techniques to characterize the total mass dis-tribution in the inner regions of galaxy clusters, where multipleimages of background sources are formed and provide crucialconstrains on lens models. The total mass profile of galaxy clus-ters from kiloparsec up to megaparsec scales can be derived bycombining strong lensing with other mass tracers, such as weaklensing (e.g., Umetsu et al. 2014; Hoekstra et al. 2015; Melchioret al. 2015), dynamical methods using inner stellar kinematicsof the central brightest cluster galaxy (BCG, Sartoris et al. 2020)and the phase space of cluster galaxies (Biviano et al. 2013; (cid:63) E-mail: [email protected] (cid:63)(cid:63)
Based on observations collected at the European Southern Observa-tory for Astronomical research in the Southern Hemisphere under ESOprogrammes with ID 0100.A-0763(A), 094.A-0115B, 094.A-0525(A).
Stock et al. 2015), as well as X-ray hydro-static analysis (e.g.,Ettori et al. 2013).By comparing the observed mass density profiles with thosepredicted by N -body and hydrodynamical simulations, one cantest the Λ cold dark matter (CDM) paradigm of structure forma-tion (e.g., Merten et al. 2015). In particular, a robust character-ization of the mass distribution in the cores of galaxy clusters,separating the baryonic and the dark matter (DM) components,can reveal missing physical ingredients in cosmological simula-tions or possibly constrain physical properties of DM (e.g., New-man et al. 2013; Grillo et al. 2015; Natarajan et al. 2017; An-nunziatella et al. 2017; Bonamigo et al. 2018; Meneghetti et al.2020).High-precision strong lensing models can be e ff ectively usedto study the mass distribution of cluster substructures (or sub-halos), particularly when internal kinematics of cluster galax-ies is taken into account (see Bergamini et al. 2019, hereafter Article number, page 1 of 16 a r X i v : . [ a s t r o - ph . GA ] N ov & A proofs: manuscript no. paper
B19). A recent study by Meneghetti et al. (2020) finds an incon-sistency between the number of observed galaxy-galaxy stronglensing (GGSL) systems in massive clusters with those predictedby state-of-the-art cosmological simulations. This study relies onhigh-precision strong lensing models, such as the one presentedhere, to dissect the subhalo population from the total projectedmass distribution of clusters.In addition to total mass distributions, magnification mapsobtained from strong lensing models of galaxy clusters are fun-damental tools to study the astrophysical properties of lensedand highly magnified background sources. Specifically, the lens-ing magnification allows one to resolve details on scales of a fewtens of parsecs at redshift 2-6 (see Vanzella et al. 2017a,b; John-son et al. 2017; Rigby et al. 2017; Cava et al. 2018; Vanzella et al.2019, 2020b). Thus, until the advent of new observational facil-ities, such as the James Webb Space Telescope (JWST) and theExtremely Large Telescope (ELT), strong lensing represents theonly available technique to characterize the high redshift pop-ulation of faint sources, that is the progenitors of present daygalaxies and globular clusters, which are expected to play an im-portant role in the re-ionization process of the Universe (e.g.,Boylan-Kolchin 2018; Ma et al. 2020; He et al. 2020; Vanzellaet al. 2020a).Over the last decade, we have witnessed important progressin cluster lens models thanks to several observational campaignsthat have provided high-quality photometric and spectroscopicdata on a sizable sample of massive galaxy clusters, which aregenerally powerful gravitational lenses. The first of these cam-paigns, the Cluster Lensing And Supernova survey with Hubble(CLASH, Postman et al. 2012) provided panchromatic data on25 massive clusters with the WFC3 and ACS cameras on Hub-ble Space Telescope (HST). The Re-ionization Lensing Clus-ter Survey (RELICS, Coe et al. 2019) extended this HST studyto 41 clusters. Significantly deeper HST observations, in sevenACS / WFC3 bands, were carried as part of the Hubble Fron-tier Field program (HFF, Lotz et al. 2014, 2017) on six clus-ters selected to be powerful lenses. One of the HFF targets,MACS J0416.1 − z = .
396 (Ebeling et al. 2001), with a weak lensing mass of M = (1 . ± . × M (cid:12) (Umetsu et al. 2014).In addition to the HST imaging campaigns, spectroscopicfollow-up programs were carried out on a subsample of clusters.The CLASH-VLT Large Programme (Rosati et al. in prep.), forexample, collected ∼ ∼
900 spectroscopic members, combined with theChandra X-ray and VLA radio observations, revealed a com-plex, mostly bi-modal mass distribution, which is likely the re-sult of a pre-merging phase. The 30 spectroscopically confirmedmultiple images identified in the CLASH-VLT campaign wereused in the lens model of Grillo et al. (2015), thus improvingon previous photometric studies (Zitrin et al. 2013). A free-form lens model was presented by Hoag et al. (2016), using anextended spectroscopic coverage of MACS 0416 based on theGrism Lens-Amplified Survey from Space (GLASS, Treu et al.2015; Schmidt et al. 2014). A significant step forward was thenmade by Caminha et al. (2017) (hereafter C17) by exploitingthe first MUSE observations of MACS 0416. This study pre-sented a high-precision lens model based on a new large sam-ple of 102 spectroscopic multiple images (from 37 backgroundsources) and an extended catalog of cluster galaxies (including144 spectroscopic members). In Bonamigo et al. (2018), we re-fined the C17 lens model by including the mass component as-sociated to the hot-gas, which is traced by deep Chandra X-rayobservations (Ogrean et al. 2015) and dominates the cluster bary-onic content.In this paper, we further improve the lens model ofMACS 0416 by exploiting additional MUSE observations inthe northeast part of the cluster, the MUSE Deep Lens Field(MDLF), which is presented in a accompanying paper byVanzella et al. (2020b). When compared to our previous mod-els (C17 and B19), the combination of the MDLF and the HFFdata lead us to identify ∼
80% more multiple images, ∼ The paper is organized as follows. In Sec. 2, we present theMACS 0416 dataset used to develop our new lens model. Sec. 3describes in detail the mass parametrization of a selected numberof lens models. In Sec. 4, we discuss the results of the optimiza-tion of four lens models and the criteria leading to the selectionof the best reference model. The latter is used to study the ro-bustness of the magnification values of multiple images and tocharacterize the projected total mass profile of MACS 0416. Themain conclusions of our work are drawn in Sec. 5.Throughout this work, we adopt a flat Λ CDM cosmologywith Ω m = . H =
70 km s − Mpc − . Using this cosmol-ogy, a projected distance of 1 (cid:48)(cid:48) corresponds to a physical scaleof 5.34 kpc at the MACS 0416 redshift of z = .
2. Data
In this section, we briefly describe the MACS 0416 data-set pre-sented in Vanzella et al. (2020b), specifically the new catalogsof multiple images and cluster member galaxies used in our lensmodels. When this paper was close to be submitted, Richard et al. (2020)presented the results from a lens model of MACS 0416 based on thenew MUSE deep observations. A comparison between the two modelswill be possible when their model details will be made public.Article number, page 2 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − Fig. 1.
RGB image (F105W + F125W + F140W + F160W,F606W + F814W, F435W) of MACS 0416 showing the footprintsof the northeast and southwest MUSE pointings in red and yellow,respectively. The new deep MUSE observation in NE region, totaling to17.1h, is shown in green. Circles mark the 213 cluster galaxies includedin our lens models. Red and green circles refer to the 171 spectroscopicmembers. Green circles correspond to the 64 cluster galaxies for whichwe measure a reliable internal velocity dispersion from MUSE spectra.The remaining 42 photometric members are encircled in magenta.
This work is based on the HST multi-band imaging data from theCLASH survey and the HFF program. MACS 0416 has been thetarget of several spectroscopic follow-up campaigns. The red-shift measurements over a wide ( ∼
20 arcmin across) field wereobtained as part of the CLASH-VLT program with VLT / VIMOSand presented in Balestra et al. (2016). Grillo et al. (2015) con-structed a lens model using 30 spectroscopically confirmed mul-tiple images from this initial spectroscopic dataset. The numberof multiple images with a spectroscopic confirmation increasedby more than a factor of three with the advent of MUSE obser-vations, as described by C17. MUSE data-cubes have a field-of-view of 1 arcmin that is spatially sampled with 0 . (cid:48)(cid:48) × . (cid:48)(cid:48) pix-els. The wavelength range extends from 4700 Å to 9350 Å witha dispersion of 1 .
25 Å / pix, and a spectral resolution of ∼ . . (cid:48)(cid:48) seeing) and one to the south-west (SW) of 11h integration (1 . (cid:48)(cid:48) seeing), the latter howeverprovided spectra with a signal-to-noise ratio below expectations(see comment in that paper). In this work, we take advantage ofthe MDLF dataset, which extended the integration time in mostof the NE field to a total of 17.1h, with a final seeing of 0 . (cid:48)(cid:48) (Vanzella et al. 2020b). In Fig. 1, we show the footprints of the programs IDs: NE (2h): GTO 094.A-0115B (P.I. J. Richard), SW(11h): 094.A0525(A) (P.I. F.E. Bauer), NE-deep (15.1h): 0100.A-0763(A) (P.I. E. Vanzella) three MUSE pointings overlaid onto the HST / RGB image of thecluster.
The latest public catalog of spectroscopically confirmed multipleimages in MACS0416, before this work, was released by C17.It included a total of 102 images from 37 background sources.The same catalog was used in the lens models presented byBonamigo et al. (2018) and B19. As described in Vanzella et al.(2020b), the combined analysis of the MDLF observations andthe CLASH / HFF multi-band images has led to the identificationof 182 secure multiple images, at 0 . < z < .
2, which are usedin our new lens model.We assign to each multiple image an ID containing a num-ber and a letter so that images with the same number but dif-ferent letters belong to the same family of multiple images. InMACS 0416, we find eight resolved lensed sources, each con-taining two or more point-like knots (or clumps) at the sameredshift, which are often embedded into a di ff use light emis-sion. Multiple images of the same clump form a family, whilewe call the set of images coming from the same resolved source,a “system” (abbreviated Sys) of multiple images. Thus, the 182multiple images are found associated to 66 independent fami-lies, drawn from 48 di ff erent background sources. Images be-longing to a system are characterized by an identical integer partof their ID number, but by a di ff erent fractional part. Our anal-ysis demonstrates that these multiply imaged clumps are verye ffi cient in constraining the position of the critical lines of thelens models (e.g., see Sys-12 discussed below).Two examples of systems of multiple images are given in thecut-outs of Fig. 2. Sys-12 is made by six image families, withIDs from 12.1 to 12.6, containing two multiple images ( b , c ) ofthe same clumps at z = . a , b , c ).These resolved sources are indicated with rectangles in Fig. 2,with a di ff erent color highlighting each of the eight systems. Redcrosses correspond to multiple images belonging the previouscatalog by C17, while green crosses are the newly discoveredimages. We note that two images of the same background sourceat z = .
923 (identified as 22b and 22c in the C17) have beenexcluded from our final catalog since the elongation of their Ly α emission does not allow a precise determination of their posi-tions. The northern region of the cluster includes 80 out of 82 ofthe new multiple images, where the deepest MUSE pointing wascarried out. The remaining two images, identified as 20.3a and20.3c, are new clumps identified in the source at z = .
222 in theSW field.In Fig. 3 (lower panel), we show in gray the redshift dis-tribution of the new set of 182 multiple images, ranging from z = .
940 to z = . Exploiting the high signal-to-noise ratio of the cluster memberspectra extracted from the MDLF pointing in the NE region ofMACS 0416, we extend the publicly released cluster membercatalog used in the lens model of C17. The latter contained 144spectroscopically confirmed cluster members, which are definedas those galaxies in the HST / WFC3 field-of-view, brighter than
Article number, page 3 of 16 & A proofs: manuscript no. paper ′ BCG - N BCG - S Foreground galaxy
Lensed Systems
Sys - - - -
12 Sys - - - - a a a a a a b b c b c c c b c c b b Sys - b , c ( z = 0.940) ′ NE b b b b b b b c c c c c c b Sys - b , c ( z = 1.893)Sys - a ( z = 1.893) ′ a ′ MACS 0416 ( z = 0.396) Fig. 2.
HST RGB image (F814W, F606W, F435W) of the central region of the galaxy cluster MACS 0416 at z = z = .
112 are encircledin white. Colored rectangles highlight the systems of multiple images produced by eight background sources resolved into multiple clumps (seeSec. 2.2). The dotted cyan circle marks the position of the third predicted image of Sys-12. The bottom panels show three zoom-in images ofsystems 12 and 5, obtained from a median stack of the F814W, F606W, and F435W HST filters. The red and blue circles correspond to the redand green crosses in the main panel. m F W =
24, and with a velocity within ± − in clusterrest frame centered at z = .
396 (this corresponds to the red-shift range [0 . . Article number, page 4 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − m F W log ( M * / M ⊙ ) N z N Multiple imagesMember galaxies
This workSpec - membersMeasured σ galap All (this work)Bergamini 2019North images
Fig. 3.
Top:
Distribution of cluster member galaxies as a function oftheir magnitudes in the HST / F160W filter. The new sample of clus-ter members used in our lens models is plotted in gray, with the spec-troscopic members indicated in blue. Cluster members with a reliablemeasurement of their internal velocity dispersion are in red. The stellarmass on the top axis is based on the empirical relation in Grillo et al.2015.
Bottom:
Redshift distribution of the 182 multiple images used toconstrain the lens models (gray). The previous sample of multiple im-ages from C17 is shown in blue. The red histogram refers to only themultiple images from the deep NE field (above the white dashed line inFig. 8). photometric bands, with m F W <
24, as described in Grilloet al. 2015.The revised catalog of cluster members, to be used for thenew lens models, includes 19 additional spectroscopic membersbased on MDLF observations. Four of these galaxies are fainterthan m F W =
24, they are however included in the sample dueto their secure membership. We also add to this catalog sevenextra spectroscopic members based on new HFF / F160W pho-tometry, which is needed for the lens model. Moreover, we addto the final sample a bright galaxy ( m F W = z = . − above the upperbound of the velocity range chosen for the cluster member se-lection. Due to its high F160W luminosity, we expect this galaxyto have a non-negligible impact on the cluster mass model. Weremove a faint cluster member (ID Gal -739 in C17) since it isbelow the magnitude limit based on revised photometry.Summarizing, our final cluster member catalog counts a totalof 213 galaxies, 171 have a secure spectroscopic redshift, while42 members are still photometric members. We note that a re-cent method based on a convolution neural network technique,which was developed to identify cluster members using multi- band HST image cutouts (Angora et al. 2020), confirms 40 outof the 42 photometric members included in this work.Following the study by B19, we measure the line-of-sightstellar velocity dispersion, σ galap , of a large number of clustergalaxies by taking advantage of the increased mean signal-to-noise ratio ( (cid:104) S / N (cid:105) ) of galaxy spectra in the MDLF (a factor ∼ . (cid:48)(cid:48) radius, which yieldsthe best compromise between high (cid:104) S / N (cid:105) and low contamina-tion from nearby bright sources. Velocity dispersions are mea-sured by cross-correlating the observed galaxy spectra with a setof stellar templates, using the 02 / (cid:104) S / N (cid:105) >
10 and σ galap >
60 km s − . Wealso apply small corrections to σ galap and its uncertainties, d σ galap ,based on spectral simulations (see appendix A in B19). The finalsample of cluster members with internal kinematics includes 64galaxies. We note that the MDLF pointing allows us to measurereliable velocity dispersions for 15 additional galaxies down to m F W ∼
22, that is approximately a magnitude fainter than thelimit adopted in B19. The sample of cluster members with mea-sured internal kinematics is indicated in Fig. 1 (green circles). InFig. 4, we show the measured σ galap values as a function of mag-nitude in the F160W filter.
3. Strong lensing models
To model the mass distribution of MACS 0416 we use the publicsoftware
LensTool (Kneib et al. 1996; Jullo et al. 2007; Jullo &Kneib 2009) for strong lensing modeling. Following the methodadopted in our previous studies, we decompose the cluster totalmass into several mass components, based on simple parametricmodels. A set of free-parameters ( ξξξ ) is determined by constrain-ing the mass models with the positions of the 182 multiple im-ages (Sec. 2.2), x obs , thereby maximizing the following posteriorprobability distribution function (see Caminha et al. 2016): P (cid:16) ξξξ | x obs (cid:17) ∝ P (cid:16) x obs | ξξξ (cid:17) × P ( ξξξ ) ,,, (1)where P ( ξξξ ) correspond to prior probability distributions for themodel free-parameters, while the likelihood P (cid:16) ξξξ | x obs (cid:17) is givenby: L ≡ P (cid:16) x obs | ξξξ (cid:17) ∝ exp (cid:32) − χ ( ξξξ ) (cid:33) . (2)The lens model χ quantifies the displacement on the lens planebetween the observed and model-predicted positions of the mul-tiple images ( x pred ), given the set of model parameters ξξξ . To com- Article number, page 5 of 16 & A proofs: manuscript no. paper m F W σ g a l a p [ k m s − ] ⟨ S / N ⟩ BCG - SBCG - N Measured σ galap Fitted relationLens model
Gal - Fig. 4.
Measured internal stellar velocity dispersions of 64 cluster mem-ber galaxies as a function of their magnitudes in the HST / F160W fil-ter (filled circles). Their colors encode the mean signal-to-noise ratioof galaxy spectra ( (cid:104) S / N (cid:105) ). The magenta triangle refers to the brightgalaxy member in Sys-14 ( Gal -8971). The green solid line is the best-fit σ galap - m F W relation obtained as described in Sec. 3.3, while the lightgreen area corresponds to measured mean scatter around the best-fit( ∆ σ ap ). The red band corresponds to the 68% confidence level of the σ - m F W relation obtained from the optimization of our reference lensmodel ( LM-4HALOS ). The magenta square indicates the velocity disper-sion of
Gal -8971 and its 1- σ error, as predicted by the lens model (seeSec. 3). pute the χ value an isotropic uncertainty, ∆ x i , j , on the observedpositions of the images is assumed (Jullo et al. 2007): χ ( ξξξ ) : = N fam (cid:88) j = N jim (cid:88) i = (cid:13)(cid:13)(cid:13)(cid:13) x obsi , j − x predi , j ( ξξξ ) (cid:13)(cid:13)(cid:13)(cid:13) ∆ x i , j . (3)The index i runs over the multiple images belonging to the same j -th family (e.g., 1a, 1b, 1c for system 1), while the index j runsover all families (in our case N f am = N jim is the numberof observed multiple images coming from the same j -th family(e.g., N j = im = N totim = (cid:80) N fam j = N jim observed multiple images, by defining N par as the totalnumber of model free-parameters, we can write the number ofdegree-of-freedom (DoF) of the lens model as:DoF = × N totim − × N f am − N par = N con − N par . (4)The term 2 × N f am stems from the fact that the unknown posi-tions of the N f am background sources are additional free parame-ters of the model. Thus, N con is the e ff ective number of availableconstraints.To sample the lens model posterior distribution in Eq.2, LensTool exploits a Bayesian Markov chain Monte Carlo(MCMC) technique. After the removal of a large burn-in phase( ∼ × samples, obtained using ten walkers), the final MCMCchains of our lens models contain at least 10 samples of free pa-rameters’ values. The uncertainties on the model free-parameters are determined by re-sampling the posterior distribution once theerror on the positions of the observed multiple images ( ∆ x i , j ) isre-scaled so that the reduced χ is close to one (the initial value ∆ x i , j is set to 0.5 (cid:48)(cid:48) (1.0 (cid:48)(cid:48) ) for HST (MUSE) detected images).To quantify the goodness of our lens models, we use threemain indicators. The first, as customary, is the root-mean-squareseparation between the observed and model-predicted positionsof multiple images, ∆ rms (see e.g., Caminha et al. 2016): ∆ rms = (cid:118)(cid:117)(cid:116) N totim N totim (cid:88) i = (cid:107) ∆ i (cid:107) , with ∆ i = x obsi − x predi , (5)where ∆ i is the displacement between the i -th observed andpredicted image. The second and the third indicators are theBayesian information criterion (BIC, Schwarz 1978), and theAkaike information criterion (AIC, Akaike 1974) defined as:BIC ≡ − L max ) + N par ln( N con ) , AIC ≡ − L max ) + N par . (6) L max is the maximum value of the likelihood in Eq. 2.These information criteria ensure that the introduction of ex-tra free parameters in a lens model is justified by a correspondingincrease of the model likelihood, thus avoiding over-fitting. As ageneral rule of thumb, the best cluster lens model has the lowest ∆ rms , BIC and AIC values. The overall total mass distribution of MACS 0416 (or equallyits total gravitational potential, φ tot ) is divided into the followingsum of parametric mass profiles: φ tot = N h (cid:88) i = φ haloi + N g (cid:88) j = φ galj + φ f oreg + φ κ,γ . (7)The first term takes into account the N h cluster-scale smooth ha-los of the cluster potential ( φ haloi ), while the second one is as-sociated to the N g cluster member galaxies (or subhalos), eachwith gravitational potentials φ galj . The third term, φ f oreg , is thecontribution from a prominent foreground galaxy residing in theSW region of the cluster. The last term, φ κ,γ , refers to a possibleconstant convergence, or shear, associated to extra mass unac-counted for in the cluster field.In our study, we have explored a large number of lens modelswith di ff erent mass parametrizations and number of subcompo-nents. In the following two subsections, we describe the best fourlens models selected on the basis of the aforementioned criteria,which are referred to as LM-4HALOS , LM-BCGs , LM-HLBCGs , and
LM-SHEAR . In Sec. 4, we explain why the
LM-4HALOS model isfinally adopted as our reference model.
Most of the total mass of galaxy clusters is in the form of smoothhalos that extend over a scale of hundreds to thousands of kilo-parsec. These cluster-scale halos are dominated by the DM com-ponent, with a non-negligible fraction of hot-gas and stars re-sponsible for the intra-cluster light (ICL) emission.The cluster-scale component of our
LensTool models isparametrized as a sum of elliptical dual pseudo-isothermal mass
Article number, page 6 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − Measured parameters of the scaling relations N ( σ galap ) m refF W σ refap [km s − ] α ∆ σ ap [km s − ] β cut ( γ = .
64 17.02 295 . + . − . . + . − . . + . − . . + . − . Table 1.
Main parameters of the σ galap - m F W and r galcut - m F W scaling relations obtained from the measured stellar velocity dispersions of N ( σ galap ) =
64 cluster member galaxies. For each parameter, we quote the median value and the 16-th, 84-th percentiles of its marginalized posterior distribution(see Fig. 5). The normalization σ refap is computed at the reference magnitude m refF W = .
02 of the northern BCG. The slope β cut of the r galcut - m F W relation is inferred from Eq. 10 (see text). distributions (dPIE, Limousin et al. 2005; Elíasdóttir et al. 2007;Bergamini et al. 2019). The set of free-parameters includes: twoparameters for their sky position, two for the ellipticity e = a − b a + b (where a and b are the semi-major and semi-minor axis of thedPIE profile) and the position angle ( θ ) measured counterclock-wise from the west direction, additional three parameters are thecentral velocity dispersion σ , the core radius r core , and the trun-cation radius r cut .All the lens models we have developed contain four cluster-scale dPIEs that are used to parameterize the hot-gas masscontent, obtained by fitting the Chandra deep X-ray surfacebrightness distribution (see Bonamigo et al. 2018). These arefixed profiles and therefore do not contribute to the number offree-parameters. In the LM-BCGs , LM-HLBCGs , and
LM-SHEAR lens models, the DM and ICL content of the cluster halo isparametrized using three dPIEs of infinite truncation radius. Twoelliptical dPIEs have center positions close to the cluster BCGs,while a third circular dPIE is left free to move in the NE re-gion around a minor galaxy over-density (as in B19 and C17).In the
LM-4HALOS model, we add a fourth cluster-scale ellipticaldPIE whose position is left free to vary in the southern region ofthe cluster. This extra halo significantly reduces the model ∆ rms ,particularly around the BCG-S. The LM-SHEAR model was alsostudied to test the impact of possible undetected massive struc-tures in the cluster outskirts (i.e., φ κ,γ (cid:44) x and y shear components ( γ x and γ y ), and the value of theconvergence ( κ ) are additional free-parameters. Here we describe how we model the total mass (DM plusbaryons) content of cluster member galaxies. Each subhalo isparametrized as a circular dPIE profile, with negligible coreradius, whose position is centered on the peak of the stellarlight emission. As customary, to reduce the number of free-parameters, we adopt the following scaling relations for the cen-tral velocity dispersions ( σ gal ) and truncation radii ( r galcut ), as afunction of galaxy luminosity (Jorgensen et al. 1996; Natarajan& Kneib 1997): σ galLT , i = σ re fLT (cid:32) L i L re f (cid:33) α , (8) r galcut , i = r re fcut (cid:32) L i L re f (cid:33) β cut . (9)In Eq. 8, we introduce the LensTool fiducial velocity dispersion σ LT that is related to the central velocity dispersion of the dPIEby σ = (cid:113) σ LT . The two normalizations σ re fLT and r re fcut are com-puted at the reference luminosity L re f . For the L i luminosities, α = 0.3 +0.03−0.03 σ refap = 295.2 +17.3−16.6 Δ σ ap [km s −1 ] σ refap [km s −1 ] Δ σ ap = 33.1 +3.4−2.9 α Δ σ a p [ k m s − ] σ r e f a p [ k m s − ] Fig. 5.
Marginalized posterior distributions for the fitting parametersof the σ galap - m F W scaling relation (see Fig. 4). The normalization σ refap is computed at the magnitude of the BCG-N, α is the value of slope ofthe scaling relation, ∆ σ ap quantifies the mean scatter of the σ galap valuesaround the best-fit scaling relation. The median values and [16-th, 84-th] percentiles of the marginalized posterior distributions are quoted inthe titles. we use F160W Kron magnitudes of cluster galaxies, as a goodproxy of their total mass (Grillo et al. 2015). The BCG-N mag-nitude, mag re fF W = .
02, is used for the values of L re f in Eqs. 8and 9. A third scaling relation with slope β core = . LensTool to scale the dPIE core radius. However,since we assume a negligible value for the reference core radius( r re fcore = (cid:48)(cid:48) × − = . , wefit the slope α and the normalization σ re fap of the σ galap - m F W relation, that is the Faber-Jackson relation in Eq. 8 (Faber &Jackson 1976), using the 64 stellar velocity dispersions σ galap (see Sec. 2.3). As in B19 (appendix B), we consider 100walkers performing 5000 steps each, with the following pri-ors (cid:104) σ re fmin , σ re fmax (cid:105) = [100 , α min , α max ] = [0 . , . We exploit the python implementation of the A ffi ne-InvariantMCMC Ensemble sampler (Goodman & Weare 2010; Foreman-Mackey et al. 2013, https: // emcee.readthedocs.io / en / latest / ).Article number, page 7 of 16 & A proofs: manuscript no. paper (cid:104) ( ∆ σ ap ) min , ( ∆ σ ap ) min (cid:105) = [0 , ∆ σ ap ) of the relation,which is a free parameter in the fit (see posterior distributionsin Fig. 5). Since the Faber-Jackson relation is one of the projec-tions of the Fundamental Plane, a few galaxies may lie above thisrelation, as more compact galaxies tend to have higher velocitydispersions for a given luminosity.Following B19, by assuming a fixed scaling between thetotal mass of the cluster galaxies and their luminosity, that is M tot , i / L i ∝ L γ i , the slope of the scaling relation in Eq.9 can bedetermined by: β cut = γ − α + , (10)where γ = . α , σ re fap , ∆ σ ap , and the inferred value of β cut .In our lens models, we fix the two slopes α and β cut to thefitted (or inferred) values, while we use a Gaussian prior for thenormalization σ re fLT centered on the measured σ re fap , with standarddeviation equal to ∆ σ ap . To obtain this prior on σ re fLT , we depro-ject the measured σ re fap , as detailed in B19. Conversely, a largeuniform prior between 1 (cid:48)(cid:48) and 50 (cid:48)(cid:48) is adopted for the normaliza-tion r re fcut in Eq. 9.In the four lens models described here, subhalos associatedto 212 out of 213 cluster galaxies are drawn from the scal-ing relations, whereas the galaxy Gal -8971 (RA = = − LM-BCGs model, also the BCGsare optimized outside the scaling relations as circular core-lessdPIE profiles. Other details on the di ff erent mass componentsincluded in the four lens models are reported in Table 2.In the top panel of Table 3, we report the set of input param-eters of the LM-4HALOS model, including the range of the flatpriors adopted for those which are left to vary. The four dPIEprofiles describing the cluster-scale mass distribution introduce22 free-parameters in the model (see Sec. 3.2). We also indicatethe fixed parameters of the four dPIEs used to model the hot-gas mass distribution. The subhalo mass components includeseight additional free-parameters in this model. Two parametersdescribe the foreground galaxy at z = .
112 (see Fig. 6) andtwo are associated to the normalizations of the scaling relations.The Gaussian prior adopted for σ re fLT , which is derived from themeasured stellar kinematics of the cluster members (see above),is indicated in square brackets. In conclusion, the LM-4HALOS lens model includes a total of N par =
30 free parameters, with N con =
232 constrains corresponding to 202 DoF (see Eq. 4).
4. Results
The main results from the optimizations of the four lens mod-els are summarized in Table 2. In addition to the three indicatorsof the goodness of the lens models defined above ( ∆ rms , BIC,and AIC), we introduce a fourth figure of merit, χ kin , whichquantifies the agreement between the lens predicted and mea-sured stellar velocity dispersions of member galaxies. If we call σ S Rap ( m F W , i ) the aperture-averaged projected velocity disper- Fig. 6.
RGB cut-out (F814W, F606W, F435W) centered on the fore-ground galaxy at z = .
112 located ∼ (cid:48)(cid:48) at SW of the BCG-S(RA = = − LM-4HALOS reference lensmodel. The sizes of the ellipses refer to 1- σ errors along the x and y di-rections. The white line is the critical line computed for a source at z = . sion of the i -th galaxy (with luminosity m F W , i ), inferred fromthe best-fit LensTool scaling relation, χ kin is defined as: χ kin = N galm (cid:88) i = σ S Rap ( m galF W , i ) − σ galap , i d σ galap , i , (11)where N galm is the number of galaxies with a measured velocitydispersion σ galap . As general rule, the lower is the χ kin value, thebetter is the agreement between the LensTool best-fit σ - mag scaling relation and the measured Faber-Jackson relation.Based on these four figure of merits, the LM-4HALOS lensmodel emerges as the best model, which reproduces the posi-tions of the multiple images with the lowest ∆ rms , and best matchthe internal kinematics of the cluster member galaxies (lowest χ kin ). This model also possesses the lowest values of the BICand AIC criteria. In the upcoming sections, we therefore charac-terize in detail this reference model by discussing its ability topredict robust positions and magnifications of multiple images,specifically those close to critical lines and around selected clus-ter galaxies. LM-4HALOS lens model: predicted positions andmagnifications of the multiple images
In Fig. 7, we show the distribution of the di ff erences between theobserved and model-predicted positions of the multiple imagesin the x and y directions and corresponding values of ∆ rms . Itis worth noting that the resulting ∆ rms = . (cid:48)(cid:48) is 34% smaller Article number, page 8 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − Properties of selected lens modelsModel ID N par DoF ∆ rms [ (cid:48)(cid:48) ] BIC AIC χ kin Description
LM-4HALOS
30 202 0.40 346 243 4941 Four cluster-scale halos, 212 cluster members including BCGs
LM-BCGs
28 204 0.45 408 311 6653 Three cluster-scale halos, 210 cluster members excluding BCGs
LM-HLBCGs
28 204 0.46 412 316 7217 Three cluster-scale halos, 212 cluster members including BCGs
LM-SHEAR
27 205 0.48 413 320 5042 Three cluster-scale halos, one shear term, one convergence term, 212 clustermembers including BCGs
Table 2.
Description of the four selected best lens models. N par and DoF are the number of model free-parameters and degrees-of-freedom. ∆ rms is the root-mean-square displacement between the positions of observed and model-predicted multiple images (see Eq. 5). The BIC (BayesianInformation Criterion) and AIC (Akaike Information Criterion) values are computed using Eq. 6. The χ kin value, defined in Eq. 11, quantifies theagreement between the model predicted and measured cluster member velocity dispersions. In the last column, we summarize the di ff erences ofthe mass parametrization in the four lens models. The reference model selected on the basis of the best figure of merits (from column four to seven)is indicated in bold. North imagesSouth images North Δ rms = 0.37′ ′ South Δ rms = 0.47′ ′ Total Δ rms = 0.40′ ′ Δ x [arcsec] Δ y [ a r c s ec ] Fig. 7.
2D and 1D distributions of the displacements ∆ i , along x and y directions, between the observed and model-predicted positions of the182 multiple images used to constrain the LM-4HALOS lens model, andcorresponding ∆ rms values (see Eq. 5). The 125 images in the north fieldand the 57 in the south are shown separately as red and green dots,respectively (see the dividing line between the NE and SW regions inFig. 8). than our previous model (B19) despite the significantly largernumber of multiple images (182 vs 102). The ∆ rms value in theMDLF field (0.37 (cid:48)(cid:48) ) is appreciably smaller than the one in thesouthern field (0.47 (cid:48)(cid:48) ). We interpret this as the result of a betterconstrained mass model in the NE field due to the large numberof multiple images (see Vanzella et al. 2020b). The mass distri-bution in the SE region appears more complex, as discussed byBalestra et al. (2016) who found MACS 0416 in a pre-mergingphase based on a dynamical and structural analysis of the cluster.Interestingly, we find that the inclusion of the fourth cluster-scalehalo in the LM-4HALOS lens model is needed to significantly re-duce ∆ rms in the SE field, thus confirming the complex struc-ture of the southern cluster field. The bright foreground galaxy(Fig. 6), which we simply model as a circular dPIE at the clusterredshift, adds further uncertainty to the projected mass distribu- tion in this region. In Fig. 8, we show the spatial distribution ofthe absolute displacements, ∆ i , between the observed and model-predicted positions of multiple images. This also shows that themultiple images in the NE field are, on average, better repro-duced independently from their redshifts.The reference lens model predicts 226 multiple images as-sociated to 66 independent sources. We note that there are 44additional images predicted “a posteriori” by the model, whichare still waiting for a secure identification (see for example Sys-12 at the end of this subsection).To study the statistical uncertainty ( ∆ µ ) on the absolute mag-nification, | µ | , we show in Fig. 9 the relative error on the abso-lute magnification in di ff erent | µ | regimes. This is particularlyrelevant for the study of high- z strongly magnified sources (seeVanzella et al. 2020b). The magnification uncertainties are com-puted as follows. Firstly, we generate 500 realizations of the LM-4HALOS lens model by randomly extracting 500 parametersamples from the
LensTool
MCMC chains. For each realiza-tion, we compute the absolute magnifications at the predictedposition of the multiple images and derive their posterior distri-butions. To confidently assign the correct magnification to eachimage, we verify that each model realization predicts the ex-pected parity for that image. We define | µ | as the median value ofthe magnification distributions, and ∆ µ as half of the di ff erencebetween the 84-th and 16-th percentiles.One can see in Fig. 9 that the relative magnification uncer-tainty progressively increases at larger | µ | . This is due to the factthat the most magnified images are located closer to the criticallines of the lens model where magnification gradients becomesparticularly large. In these regions we also expect that systematicerrors due to model parametrization become increasingly impor-tant. This analysis is deferred to a future paper.To better understand the origin of magnification uncertain-ties, we investigate how ∆ µ/ | µ | varies as a function of thedistance between the multiple images and their closest clus-ter galaxy, and the magnification itself. The result is shown inFig. 10. On average, the images that form close to cluster mem-bers have larger relative errors. This anti-correlation is explainedby the fact that galaxy masses act as small gravitational lensesembedded into the cluster potential and introduce numerous sec-ondary critical lines (with sizes of few arcseconds) into the lensmodel. Images closer to critical lines tend to have larger ∆ µ/ | µ | values. The few highly magnified images with ∆ µ/ | µ | ∼
20% atdistances of 6-8 (cid:48)(cid:48) from cluster galaxies are necessarily locatedaround the main critical lines of the cluster computed for theirredshifts.
Article number, page 9 of 16 & A proofs: manuscript no. paper
Input parameter values and intervals of the
LM-4HALOS lens model x [arcsec] y [arcsec] e θ [ ◦ ] σ LT [km s − ] r core [arcsec] r cut [arcsec] C l u s t er - s c a l e h a l o s st Cluster Halo [ − . , .
0] [ − . , .
0] [0 . , .
9] [100 . , .
0] [350 . , .
0] [0 . , .
0] 2000.0 nd Cluster Halo [15 . , .
0] [ − . , − .
0] [0 . , .
9] [90 . , .
0] [350 . , .
0] [0 . , .
0] 2000.0 rd Cluster Halo [ − . , − .
0] [0 . , .
0] 0 . . . , .
0] [0 . , .
0] 2000.0 th Cluster Halo [ − . , .
0] [ − . , − .
0] [0 . , .
9] [0 . , .
0] [100 . , .
0] [0 . , .
0] 2000.0 st Gas Halo − − − nd Gas Halo − − rd Gas Halo − − − th Gas Halo − − Subh a l o s Gal - .
35 2 .
62 [0 . , .
6] [ − . , .
0] [60 . , .
0] 0 . . , . Foreground gal. . − .
55 0 . . . , .
0] 0 . . , . Scaling relations N gal = m refF W = . α = . σ refLT = (248 ± β cut = . r refcut = [1 . , . γ = . Optimized parameters of the
LM-4HALOS lens model x [arcsec] y [arcsec] e θ [ ◦ ] σ LT [km s − ] r core [arcsec] r cut [arcsec] C l u s t er - s c a l e h a l o s st Cluster Halo − . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . nd Cluster Halo . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . rd Cluster Halo − . + . − . . + . − . . + . − . . + . − . th Cluster Halo . + . − . − . + . − . . + . − . . + . − . . + . − . . + . − . Subh a l o s Gal - . + . − . − . + . − . . + . − . . + . − . Foreground. gal. − .
55 0.0 0.0 138 . + . − . . + . − . Scaling relations N gal = m refF W = . α = . σ refLT = . + . − . β cut = . r refcut = . + . − . γ = . Table 3.
Top:
Input parameters of the
LM-4HALOS reference lens model. Singles numbers refer to fixed parameters. For the free parameters, wequote within square brackets the boundaries of the input flat priors. For the σ refLT parameter, a Gaussian prior is adopted with mean and standarddeviation indicated in round brackets. N gal is the number of cluster member galaxies included in the scaling relations (see Eqs. 8 and 9). Bottom:
Optimized output values of the free parameters of the reference lens model. We quote the median value and the [16-th, 84-th] percentiles from themarginalized posterior distribution.
We now define a new metric, which is sensitive to the gra-dients of the deflection field. We focus on a number of specificlensed systems to test the ability of the lens model to predictthe positions of multiply imaged clumps, associated to the sameresolved source, which are close to the critical lines.We take Sys-5 (see bottom-right panels of Fig. 2) as an ex-ample. This system is composed by six clumps (5.1, 5.2, 5.3,5.4, 5.5, and 5.6) belonging to the same extended source at z = . . a , 5 . b , and 5 . c .Using the observed and model-predicted positions of the mul-tiple images, we determine the “distance vectors” connectingeach pair of lensed clumps belonging to the same extended im-age: for example, d . a , . aobs ( pre ) , d . a , . aobs ( pre ) , d . a , . aobs ( pre ) , etc. (similar com- binations are obtained for images b and c ). The modulus ofthese vectors corresponds to the distance between these knots.Moreover, we measure the angular di ff erence in the orientationof the corresponding observed and predicted distance vectors,so that (cid:12)(cid:12)(cid:12) ∆ φ i , j (cid:12)(cid:12)(cid:12) = arccos (cid:104) ˆ d i , jpre · ˆ d i , jobs (cid:105) , where we use “ · ” to indi-cate the scalar product, while i and j are di ff erent knots (e.g., (cid:12)(cid:12)(cid:12) ∆ φ . a , . a (cid:12)(cid:12)(cid:12) = arccos (cid:104) ˆ d . a , . apre · ˆ d . a , . aobs (cid:105) ).On the right side of Fig. 11, we plot, on the radial axis, theabsolute value of the di ff erence between the predicted and ob-served distance vectors ( | d i , jobs − d i , jpre | ) for the whole set of mul-tiple images belonging to eight di ff erent systems. The values of ∆ φ i , j are plotted along the angular coordinate. In this represen-tation, a perfectly reconstructed pair of images should lie on the Article number, page 10 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − ′ st C 2 nd C4 th C3 rd C Redshift st G 2 nd G3 rd G4 th G ′ ′ ′ ′ Fig. 8.
RGB image of MACS 0416 (as in Fig. 1), with overlaid contours(in white) of the total projected mass distribution obtained from the ref-erence lens model. The contours are expressed in units of 10 M (cid:12) kpc − .The magenta crosses mark the positions of the four dPIE profiles used todescribe the hot-gas component of the cluster. The green crosses markthe centers of the four cluster-scale halos. The circles show the positionsof the 182 observed multiple images, with colors encoding their red-shift. Their sizes scale proportionally to the displacements between theobserved and model-predicted positions of the multiple images ( (cid:107) ∆ i (cid:107) ).The white dashed line separates two sets of multiple images in the north-ern and southern fields. origin of the reference frame. On the left side of the diagram, weshow the angular distribution of the ∆ φ i , j values. In 90% of thecases, the di ff erence in the distances between the observed andpredicted knots is lower than 0 . (cid:48)(cid:48) . Similarly, the ∆ φ i , j valuesare lower than 5 . ◦ for 90% of image pairs. This result demon-strates that our reference lens model not only accurately predictsthe observed position of the multiple images ( ∆ rms = . (cid:48)(cid:48) ),but it is also able to accurately reproduce the extension and theorientation of the inner structure of extended multiple images.To conclude this session characterizing the robustness of ourreference model, we briefly describe the properties of two in-teresting systems of multiple images, namely Sys-12 and Sys-14. The extended background source associated to Sys-12, at z = .
94, is a spiral galaxy which is lensed into three imagesby MACS 0416. The top-left and middle-left panels of Fig. 12show how two of these images ( b and c ) are resolved into sev-eral star-forming clumps which are close to merging onto thecritical line. Vanzella et al. (2020b) characterized these sys-tems as star-forming complexes with extremely low luminos-ity (M UV ∼ −
11) and small sizes ( (cid:46)
30 pc), thanks to theirstrong magnification (see below). We securely identify six point-like knots (12 . . . . .
6) on each side of the critical line, whichare included in our lens model. The clumps with the largest
N Δ μ / | μ | | μ | < 155 ≤ | μ | < 73 ≤ | μ | < 51 ≤ | μ | < 3 | μ | ≥ 15 15 ≤ | μ | < 50 Fig. 9.
Distributions of the relative statistical error on the absolute mag-nification ( ∆ µ/ | µ | ) for the 226 multiple images that are predicted by thereference lens model. Histograms refer to six di ff erent intervals of | µ | .Vertical black dashed lines mark the median values of each distribution(the red dash line corresponds to bin with | µ | ≥ Δ μ / | μ | d gal [arcsec] | μ | z Fig. 10.
Relative statistical uncertainty on the absolute magnification ∆ µ/ | µ | for the 226 predicted multiple images (as in Fig. 9) as a func-tion of their distances from the closest cluster galaxy. The colors of thecircles encode the | µ | values, while their sizes scale according to theirredshifts. separations, namely 12.4 and 12.5, are 10.9 kpc apart on thesource plane, while the closest central knots, 12.2 and 12.3, areonly 1.7 kpc apart. The mirrored images 12.4b and 12.4c arelocated 0 . (cid:48)(cid:48) from the critical line and have in fact the high-est magnifications among the entire sample of multiple images, µ . b = + − and µ . c = + − . We emphasize how our ref-erence lens model reproduces the 12 positions of Sys-12 with Article number, page 11 of 16 & A proofs: manuscript no. paper
Sys - - - - -
14 Sys -
15 Sys -
16 Sys - Fig. 11. Di ff erence between the observed and model-predicted separa-tions and orientations of each pair of clumps (or knots) identified withineight systems of multiple images (see Fig. 2). Each dot corresponds to adi ff erent pair of clumps, colored according to the parent system. We de-fine d ijobs ( d ijpre ) as the measured (predicted) distance between the i -th and j -th clump belonging to the same resolved lensed image (e.g., i = . c , j = . c in the top left panel of Fig. 12). We plot the absolute di ff er-ence | d ijobs − d ijpre | along the radial axis on the right side, while the angle( ∆ φ ij ) between d ijobs and d ijpre vectors (measured counterclockwise) isplotted across the angular coordinate. On the left side of the diagram,we draw a binned distribution of all the data-points. The 90-th, 95-th,and 99-th percentiles of the distributions along the radial and angulardirections are shown as red semi-circles and sectors respectively. very high precision ( ∆ rms = . (cid:48)(cid:48) ), as well as the relative dis-tances and orientations of all knots belonging to a given lensedimage ( b or c ), as one can appreciate from Fig. 11 (cyan dotscluster near the origin of the diagram).The third image of Sys-12 is blindly predicted by the lensmodel ∼ (cid:48)(cid:48) SE of image b / c (see Fig. 2), around a clustermember which contributes to create an Einstein ring configura-tion (see top-right and middle-right panels of Fig. 12). This thirdimage is not included in the catalog of multiple images as thecorresponding lensed knots cannot be readily identified in HSTimages.The MUSE spectra of the multiple images in Sys-12 are char-acterized by a prominent [OII] λ . This analysis yields the velocity mapsshown in the lower panels of Fig. 12. By ray-tracing the six star-forming knots in image b and c on the source plane at z = . ∼ −
100 km s − for 12.4, up to ∼
100 km s − for12.5, across ∼
11 kpc. We defer to a future publication a detailedstudy of this and other galaxy-scale systems of multiple images. We use the MUSE Python data analysis framework (MPDAF) forthis analysis (https: // mpdaf.readthedocs.io / en / latest). Fig. 12.
Detailed analysis of the lensed system 12 at z = . . . . . .
6) areidentified within the lensed images a and b , straddling the critical lineat z = .
940 (blue line). Green crosses mark the positions of the ob-served multiple images. Red ellipses show the positions predicted bythe
LM-4HALOS reference lens model, with sizes corresponding to 1- σ errors along the x and y directions. The top and middle right panels showthe third lensed image predicted by the lens model (blue spiral structurearound the cluster galaxy Gal -4677). The blue line is the associated crit-ical line; the red ellipses correspond to the predicted multiple images.In the top panels, RGB cut-outs combine F814W, 606W, F435W fil-ters, while in the middle panels we show median stack images with thesame filters. The bottom panels show the corresponding velocity mapsobtained by tracing the shift of the [OII] emission doublet in the MUSEdatacube (around ∼ z = . Sys-14 (see Fig. 13) was studied in detail by Vanzella et al.(2017b). In this system, two close compact sources (14.1 and14.2) at z = .
221 are lensed into six multiple images each (nineout of twelve multiple images are used as constraints into thelens models). Eight images, 14 . a , b , c , d and 14 . a , b , c , d , arelocated around a pair of cluster galaxies. Since the mass dis-tribution of the brighter galaxy ( Gal -8971) strongly a ff ects thegeometry of the GGSL, its mass halo is parameterized as an el-liptical dPIE profile outside the subhalo scaling relations in alllens models (see Sec. 3.1 and Table 3). We note that the threeimages 14 . d , 14 . e , and 14 . f , are not used in the lens mod-els, due to the lack of a secure counterpart in the HST images. Article number, page 12 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − Fig. 13.
Inspection of system 14 at z = . . .
2) is imaged six times by the cluster mass distribu-tion, around two cluster galaxies (
Gal -8971 and
Gal -8980, with F160Wmagnitudes 19.6 and 21.5, respectively). Green crosses, red ellipses, andblue lines have the same meaning as Fig. 12. The three gray-scale cut-outs on the left are median stack images of F814W, F606W, and F435WHST filters. Right panels show absolute magnification ( | µ | ) maps com-puted at z = . e ) and a sixth ( f )image (middle and bottom panels, see also Fig. 2). Instead, we tentatively identify the 14 . c image and use it in thelens model with the same position as 14 . c and a positional er-ror which includes both images. Compared to our previous lensmodel by B19, the new model predicts an additional counter-image of the 14.1 and 14.2 pair, which however still remains notsecurely identified (mid panel of Fig. 13). For all images associ-ated to Sys-14, we find ∆ rms = . (cid:48)(cid:48) . We refer to Vanzella et al.(2020b) for a discussion on the magnification and physical sizesof the multiple images of Sys-14. As described in Sec. 3.1, the mass distribution of MACS 0416obtained with the
LM-4HALOS lens model includes a cluster-scalecomponent and a subhalo population traced by the cluster mem-bers. The former includes a fixed hot-gas component and thedominant DM halo which is modeled with four (elliptical) dPIEprofiles. The latter includes 212 circular dPIEs with vanishingcore radii and a separate elliptical subhalo to model the brightmember in Sys-14 (Fig. 13). In Table 3, we quote all the inputparameters and the output optimized values of each parameterobtained with of our reference model. In Fig. 8, we show the iso-density contours of the total projected mass distribution overlaidon the HST image of MACS 0416, and we also indicate the cen-ters of the cluster-scale halos. R [kpc] M ( < R )[ M ⊙ ] R [arcsec] LM - - f M Fig. 14.
Top:
Cumulative projected total mass profile of MACS 0416as a function of the distance ( R ) from the northern BCG. The red curverefers to the median and the 68% confidence level obtained from thereference lens model. The previous result by B19 is plotted in blue. Thetotal mass of the subhalos, associated to cluster galaxies, is shown ingreen. Black vertical segments mark the positions of the 182 multipleimages. Bottom:
Ratio between the new mass profile and the one fromB19 (blue) and fractional contribution of the subhalo component to thetotal cumulative mass (in green).
As in C17 and B19, the new mass model does not imply anysignificant o ff set between the position of the cluster-scale halosurrounding the northern BCG and the peak of the light distri-bution (the projected distance of the halo mass peak from theBCG-N is formally 0 (cid:48)(cid:48) . + . − . ). As discussed above, the mass dis-tribution around the southern clump of MACS 0416 is not suf-ficiently well constrained to investigate possible o ff sets with theluminous mass. The optimized position of the third halo in theNE region, whose presence is needed to reduce significantly the ∆ rms value, is very close to a local over-density of the projectedgalaxy distribution (Fig. 8), in keeping with C17 and B19.The cumulative projected total mass profile of MACS 0416as a function of the projected distance, R , from the BCG-N isshown in Fig. 14 (red thin band). The small 1- σ uncertainties arecomputed by randomly extracting parameters from the MCMCchains of 500 realizations of the LM-4HALOS model. The new cu-mulative mass profile is found in very good agreement with thatobtained with our previous B19 model, which had a somewhatdi ff erent parametrization and used 80 fewer multiple images (thedi ff erence is less than 3% over the radial range where the multi-ple images are located).Fig. 14 also shows the cumulative projected mass profile as-sociated to the subhalos, and their relative contribution to thetotal mass. At small projected distances from the BCG-N, thesubhalos contribute for more than 40% to the total mass, whilethis contribution does not exceed 15% at R >
100 kpc.As discussed in Sec. 3.3, the normalization of the σ - m F W scaling relation for the subhalos in all the lens models is obtainedadopting a Gaussian prior derived from the observed σ ap - m F W Faber-Jackson relation.
Article number, page 13 of 16 & A proofs: manuscript no. paper
The resulting scaling relation obtained from the optimizationof the
LM-4HALOS lens model is shown in Fig. 4 (see red band).The very good agreement between the inferred scaling relationand the observed one (data points and green band) is not com-mon to the other models we analyzed. Indeed, the normalization σ re fLT of these secondary models tend to lie significantly below theobserved value despite the adopted prior, as it can be appreciatedfrom their higher χ kin values (see Table 2 and Eq. 11).In the e ff ort to accurately reproduce the positions of themultiple images associated to the galaxy scale system 14 (seeFig. 13), we have parameterized the mass of Gal -8971 as an ex-tra elliptical dPIE outside the scaling relations (see Sec. 3.3). Itsmeasured velocity dispersion, 169 . ± . − (see magentatriangle in Fig. 4), is in tension with the aperture-projected ve-locity dispersion obtained from the posterior distributions of the LM-4HALOS model (133 . + . − . km s − , see magenta square). De-spite the high-quality data of this study, the complex geometryof the mass distribution of this system, with a close galaxy com-panion embedded in the cluster halo, makes it di ffi cult to reliablyinfer both the values of the velocity dispersion and truncation ra-dius of Gal − R E is given by (Elíasdóttir et al. 2007 and B19): M ( R E ) = πσ G (cid:18) (cid:113) r core + R E − r core − (cid:113) r cut + R E + r cut (cid:19) . (12)For a fixed value of the core radius, a higher values of r cut pro-duces the same mass M ( R E ) for a smaller central velocity disper-sion σ . Our best fit model yields an r cut value for this specificgalaxy exceeding R E , with a large uncertainty.In Fig. 15, we show the posterior distribution of the referencecentral velocity ( σ re f ) and the reference truncation radius ( r re fcut )of the cluster member scaling relations, which clearly displaysthe same degeneracy discussed above. The constraints on thesetwo parameters obtained for the new reference model are com-pared with the results of B19, which also used a kinematic priorin the lens models. We note that the normalization of the scal-ing relations ( σ re f ) in the new model is consistent with B19 at1- σ level, whereas we find a di ff erence of ∼
30 kpc in the nor-malization of the truncation radius. It is worth noting that such adi ff erence could not be detected without using galaxy kinemat-ics to constrain the scaling relations in the lens model, because inthis case the σ - r cut degeneracy is significantly larger (see B19).Since the B19 model assumed a larger value of the refer-ence core radius of the subhalo dPIE profiles ( r re fcore = . (cid:48)(cid:48) compared to 1 (cid:48)(cid:48) × − in LM-4HALOS ), we re-optimize the lensmodel changing the core radius parameter. The result, illustratedin Fig. 15 (dashed confidence contours), shows that σ scales asexpected with the larger core radius, based on Eq. 12, however r cut does not change significantly. Therefore, the larger trunca-tion radii of the subhalos obtained with the new model are notdue to inherent degeneracies of the dPIE parametrization with r re fcore .Considering the dependence of the total mass of the subhaloson their values of velocity dispersion and cut radius (Eq. 12), thecluster members in the LM-4HALOS model are characterized bylarger masses (by about a factor of two) compared to those in themodel of B19. Based on the previous model, Meneghetti et al.(2020) found that cluster members in MACS 0416 have a signif-icantly larger probability to produce GGSL compared to galaxy-scale subhalos in Λ CDM numerical simulations. We have ver-ified that the GGSL probability obtained from the
LM-4HALOS r refcut [kpc] σ ref [km s −1 ] σ r e f [ k m s − ] Bergamini 2019LM - M ⊙ M ⊙ M ⊙ - ′ r refcore r refcore M ( R = 1.0′ ′ ) r core = 0.05′ ′ - ′ Fig. 15.
Marginalized posterior distributions of the normalizations σ ref and r refcut of the cluster member scaling relations (see Eq. 8 and 9). Nor-malizations are computed at the magnitude of the BCG-N ( mag ref F160W = . LM-4HALOS reference lens model;results from the previous model by B19 model are in blue. Colored con-tours encompass the 1, 2, 3 σ confidence levels; the vertical solid anddashed lines correspond to the 50-th, 16-th and 84-th percentiles of themarginalized distributions. The 1 and 2 σ black dashed contours referto the LM-4HALOS model with r refcore = . (cid:48)(cid:48) , instead of r refcore = (cid:48)(cid:48) × − of the reference model. The green and magenta lines are σ - r cut curveswith constant projected mass, within an aperture of R = (cid:48)(cid:48) ( = .
34 kpcat z = . r refcore = (cid:48)(cid:48) × − ,magenta curves refer r refcore = . (cid:48)(cid:48) (as in B19). model is ∼
30% higher than that measured from the B19model. Thus, our new results confirm the tension reported byMeneghetti et al. (2020) between the observations of the innerstructure of cluster galaxies and the theoretical expectations inthe framework of the Λ CDM model.The constraints on the mass distribution of the subhalo pop-ulation with lens models can be further improved by takinginto account the measured velocity dispersion for each mem-ber galaxy, thus including the intrinsic scatter of the σ - L scalingrelation (Bergamini et al. 2020), which is neglected in the cur-rent lens models. We leave to a future work a further analysis ofthe subhalo component, which fully exploits strong lensing andgalaxy kinematics constraints.
5. Conclusions
We have presented a new high-precision strong lens model forthe galaxy cluster MACS J0416.1 − z = . ff erentbackground sources. In this new sample, we have added several Article number, page 14 of 16. Bergamini et al.: A new high-precision strong lensing model of the galaxy cluster MACS J0416.1 − multiply lensed clumps, belonging to resolved extended sources,which are particularly e ffi cient in constraining the position ofthe critical lines in their vicinity. Moreover, we have extendedthe sample of cluster members to 213 galaxies, 171 of whichare spectroscopically confirmed (27 more than in our previouscatalog).By exploiting new galaxy spectra with high signal-to-noiseratio extracted from the MDLF, we have measured the inner stel-lar kinematics of 64 cluster members (15 more than in the pre-vious analysis by B19), down to m F W (cid:39)
22. We have usedthe newly measured velocity dispersions to constrain the scal-ing relations of the subhalo population in the lens models as inB19. The cluster-scale mass distribution is modeled with a num-ber of DM-dominated halos in addition to the hot-gas componenttraced by the Chandra X-ray data.Among the four lens models in our study, we have selected areference model, which best reproduces the positions of the ob-served multiple images (smallest ∆ rms ) and the observed σ - mag scaling relation for cluster members (see Table 2).We can summarize the results obtained from our referenceMACS 0416 lens model as follows:1. Despite the large number of observed multiple images, thenew reference lens model yields ∆ rms = . (cid:48)(cid:48) , thus re-ducing the value obtained by B19 by a third. The model isparticularly accurate in the MDLF region of MACS 0416( ∆ NErms = . (cid:48)(cid:48) , 0.1 (cid:48)(cid:48) smaller than in the southern field), wherethe newly identified images are located.2. We have studied the robustness of the magnification mapsderived from our reference lens model. We have computedthe uncertainties on the absolute magnification ( | µ | ) of mul-tiple images on their predicted positions by sampling the | µ | distributions from the MCMC chains of the lens model.We have studied how the relative error on the magnification, ∆ µ/ | µ | , varies with the magnification and with the distance ofthe multiple images from secondary critical lines associatedto cluster galaxies. We find that the relative error remainswithin ∼
10% at | µ | (cid:46)
10, at distances beyond 2 (cid:48)(cid:48) from thecenter of member galaxies, while it increases up to ∼ | µ | (cid:38) . (cid:48)(cid:48) and 5 . ◦ , respec-tively, for 90% of image pairs. This analysis lends support tothe interpretation presented by Vanzella et al. (2020b) of thenature of highly magnified clumps in resolved lensed galax-ies in the MLDF. They range from young massive star clus-ters (e.g., in Sys-14, see also Vanzella et al. 2017b), to barelyresolved knots, with sizes and luminosities similar to whatobserved in local star clusters (e.g., Sys-12).4. Using the newly constrained mass distribution of the sub-halo component, we confirm the results of Meneghetti et al.(2020), who find that the probability of producing GGSLevents in several clusters, including MACS 0416, is approx-imately a factor of ten higher than the theoretical predictionsbased on Λ CDM numerical simulations.The new lens model of MACS 0416 presented here, togetherwith the new catalog of cluster members and the updated catalog of multiple images presented in Vanzella et al. (2020b), are madepublicly available . Acknowledgements.
This project is partially funded by PRIN-MIUR2017WSCC32. PB acknowledges financial support from ASI through the agree-ment ASI-INAF n. 2018-29-HH.0. MM acknowledges support from the ItalianSpace Agency (ASI) through contract “Euclid - Phase D” and from the grantPRIN-MIUR 2015 “Cosmology and Fundamental Physics: illuminating theDark Universe with Euclid”. FC acknowledges support from grant PRIN-MIUR20173ML3WW_001. CG acknowledges support by VILLUM FONDEN YoungInvestigator Programme through grant no. 10123. We acknowledge fundingfrom the INAF “main-stream” grants 1.05.01.86.20 and 1.05.01.86.31. GBCacknowledge funding from the European Research Council through the GrantID 681627-BUILDUP, the Max Planck Society for support through the MaxPlanck Research Group for S. H. Suyu and the academic support from theGerman Centre for Cosmological Lensing..
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