RELICS: Strong Lensing Analysis of the Galaxy Clusters Abell S295, Abell 697, MACS J0025.4-1222, and MACS J0159.8-0849

Our current understanding of early cosmic history is based on two main pictures: the first provides a view of the early universe through measurements of the cosmic microwave background (CMB; Komatsu et al. 2011; Planck Collaboration et al. 2016a), last scattered about 400,000 yr after the Big Bang, while the second relies on observations of the first galaxies, which formed a few hundred million years later (Rees 1998; Barkana & Loeb 2001). At that era, the universe was filled with neutral hydrogen, which was then gradually reionized by z ∼ 6 (Fan et al. 2006; Dijkstra 2014; Robertson et al. 2015). During this reionization epoch, lasting only a few hundred Myr, the universe went through a particularly rapid evolution (see Zaroubi 2013 and Stark 2016 for reviews).

One way to explore the reionization epoch is based on the statistics of high-redshift galaxies expressed in terms of the galaxy luminosity function (Bromm & Yoshida 2011 and references therein; McLure et al. 2013; Finkelstein et al. 2015; Bouwens et al. 2017b; Livermore et al. 2017), which can then be translated to an ionizing photon budget (Morales & Wyithe 2010; Atek et al. 2015; Robertson et al. 2015; Mason et al. 2018). This is of particular interest, since it is currently uncertain whether galaxies could fully account for reionization. The detection of increasing numbers of high-redshift galaxies is thus crucial. Observing high-redshift galaxies, however, is challenging. Moreover, current observations of galaxies at z ≳ 6 typically correspond to the brighter (L ≳ L*) objects, hence conclusions about the faint end of the luminosity function, representing the more abundant population of galaxies, cannot be directly obtained. Over the past decade, there have been growing efforts to observe samples statistically representative of the underlying, fainter high-redshift galaxy population that may have been responsible for reionization. Significant progress has been achieved with deep and high-resolution observations carried out by the Hubble Space Telescope (HST) under various programs (e.g., Beckwith et al. 2006; Scoville et al. 2007; Grogin et al. 2011; Bradley et al. 2012; Ellis et al. 2013; Bouwens et al. 2015; Finkelstein 2016; Livermore et al. 2017; see Stark 2016 for a review).

A second route to studying galaxies at high redshifts and their contribution to reionization is via spectroscopy. However, spectroscopy of Lyα or other UV metal lines from very distant objects in the reionization era is extremely challenging (Stark et al. 2010; Pentericci et al. 2011; Schenker et al. 2014; Schmidt et al. 2016; Hoag et al. 2017; Laporte et al. 2017). As bright galaxies at high redshifts are scarce, there has been a growing need to discover more high-z candidates that are apparently bright enough to be studied spectroscopically (e.g., Roberts-Borsani et al. 2016).

The discovery of galaxies at high redshift is enhanced by strong gravitational lensing. Acting as natural telescopes, massive galaxy clusters magnify background sources that are intrinsically faint and would otherwise remain undetectable (e.g., Zheng et al. 2012; Coe et al. 2013; Zitrin et al. 2014). Indeed, recent cluster lensing campaigns have been detecting increasing numbers of magnified high-redshift sources (e.g., Bradley et al. 2014; Monna et al. 2014; Jauzac et al. 2015; Huang et al. 2016; Zitrin et al. 2017), enabling the community to probe the fainter end of the luminosity function, up to z ∼ 9–10 (McLure et al. 2013; Oesch et al. 2014; Atek et al. 2015; McLeod et al. 2016; Bouwens et al. 2017b; Ishigaki et al. 2018; Kawamata et al. 2018; Livermore et al. 2017).

Apart from the magnification effect useful for studying lensed galaxies, gravitational lensing also provides a unique way to map the total mass content of clusters, including both the baryonic and dark matter (DM) components (Schneider et al. 1992; Clowe et al. 2006; Bartelmann 2010; Kneib & Natarajan 2011).

Following the success of previous lensing surveys like the Cluster Lensing and Supernovae Survey with Hubble (Postman et al. 2012; Coe et al. 2013; Bradley et al. 2014) and the Hubble Frontier Fields (HFF; Coe et al. 2015; Lotz et al. 2017), the Reionization Lensing Cluster Survey (RELICS; D. Coe et al. in preparation) was designed to study 41 clusters and reveal high-redshift galaxies (z ∼ 6–12), in particular, apparently bright, highly magnified examples that could be followed up spectroscopically from the ground or with the James Webb Space Telescope (JWST; e.g., Salmon et al. 2017).

In order to account for the strong lensing (SL) effect and correctly interpret the results, such as the intrinsic properties of lensed and high-redshift galaxies, it is necessary to derive a detailed lens model, which is our goal here. We present an SL analysis of four clusters observed in the framework of the RELICS¹⁸ program. We briefly introduce the program and observations in Section 2, where we also provide an overview of the clusters analyzed in this work. In Section 3, we describe the adopted strong lens-modeling technique. We present the individual lens modeling for each of the clusters in Section 4. Our findings are presented and discussed in Section 5 and summarized in Section 6.

Throughout this work, we adopt the standard ΛCDM flat cosmological model with a Hubble constant of ${H}_{0}=70\,\mathrm{km}\,{{\rm{s}}}^{-1}{\mathrm{Mpc}}^{-1}$ and Ω_M = 0.3. Magnitudes are quoted in the AB system. Errors correspond to the 68.3% confidence level unless otherwise specified. The errors we quote for the Einstein radius and mass throughout are 10% and 15%, respectively. These were found to also encompass typical differences between different lens-modeling techniques (e.g., Zitrin et al. 2015).

Our SL analysis is based on observations from the RELICS program (PI: D. Coe). The RELICS is a 188-orbit HST Treasury Program (GO-14096; PI: Coe) that has observed 41 galaxy clusters. The goal of the project is to analyze these massive clusters and find magnified, high-redshift galaxies. A Spitzer imaging campaign (PI: Bradac, PI: Soifer) of more than 500 hr accompanies the program and is intended to improve the search for high-redshift galaxies and complement the studies of the observed galaxy properties. A detailed description of the RELICS program and the sample selection will be presented in an upcoming paper (D. Coe et al. 2018, in preparation).

The selection of clusters for the program was mainly based on the Sunyaev–Zel'dovich (SZ) effect. The first half of the sample consists of 21 of the 34 most massive Planck clusters (Planck Collaboration et al. 2016b) for which the SZ masses are similar to or greater than those of the HFF clusters and no HST/IR imaging was available. The remaining clusters were selected based on several criteria, such as mass estimates from X-ray (MCXC; Mantz et al. 2010; Piffaretti et al. 2011) and weak lensing (Applegate et al. 2014; Sereno & Paraficz 2014; Umetsu et al. 2014; von der Linden et al. 2014; Hoekstra et al. 2015), SZ mass estimates from the South Pole Telescope (SPT; Bleem et al. 2015), and Atacama Cosmology Telescope (ACT; Hasselfield & ACT Collaboration 2013) data, as well as following an analysis of clusters from the Sloan Digital Sky Survey (SDSS; Wen et al. 2012; Wong et al. 2013). This combined selection based primarily on mass was aimed at increasing the probability of detecting high-redshift galaxies, since, overall, massive clusters tend to be more efficient lenses.

The 41 clusters were observed in the optical and near-infrared. The images were obtained using three Advanced Camera for Surveys (ACS) filters (F435W, F606W, and F814W) for one orbit each and four Wide Field Camera 3 (WFC3/IR) filters (F105W, F125W, F140W, and F160W) for half an orbit each. In total, each cluster was thus observed for five orbits, with the exception of some cases where HST archival data was available. All subexposures in each filter were combined to create a deep image in that band. The final drizzled images were produced in pixel scales of 003 and 006 after matching the filters to the same pixel frame and correcting the astrometry to the Wide-field Infrared Survey Explorer point-source catalog (Wright et al. 2010). An interval of about one to two months between observations of the same cluster was typically designed to allow the identification of variable sources.

RELICS has delivered reduced HST images, photometry, and photometric redshift catalogs for all of the observed fields (see also Cerny et al. 2018). Source catalogs are produced by running SExtractor (Bertin & Arnouts 1996) in dual-image mode, where a weighted, stacked data frame combined from all near-infrared and one combined from all optical and near-infrared bands are used as reference. Most of the multiple images we identify or consider are detected by SExtractor with the initial chosen parameters, but for some fainter or blended objects, we independently rerun SExtractor after manually varying the parameters until our objects of interest are detected and measured (typically, this entails increasing the number of deblending subthresholds by a factor ×2–×4, lowering the deblending minimum contrast by a factor ×5–× 10, or changing the background mesh size by a factor ×2). Photometric redshift estimates (also referred to here as photo-z) are then derived using the Bayesian photometric redshifts algorithm (BPZ; Benítez 2000; Benítez et al. 2004; Coe et al. 2006) with 11 templates for the spectral energy distribution, including ellipticals, late types, and starbursts (Benitez et al. 2014; Rafelski et al. 2015). RELICS also generates several color-composite images for each cluster, which we use here, constructed from optical and near-infrared bands as indicated in Figures 1–4 (we sometimes refer to the colors of multiple images with respect to these composite images). The reduced data and catalogs are available for the community through the Mikulski Archive for Space Telescopes (MAST).¹⁹

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Out of the four clusters analyzed in this work (Table 1), two are Abell clusters (Abell et al. 1989), one of which belongs to the supplementary Abell catalog. The other two clusters are part of the MAssive Cluster Survey (MACS; Ebeling et al. 2001). Throughout the work, we discuss the four clusters in an increasing R.A. order.

Table 1.  Properties of RELICS Clusters Considered in This Work

Cluster	R.A.	Decl.	Redshift	Planck SZ Mass
	[J2000]	[J2000]		M500 $[{10}^{14}\,{M}_{\odot }]$
MACS J0025.4-1222	00:25:29	−12:22:54	0.586	⋯
MACS J0159.8-0849	01:59:54	−08:51:32	0.405	7.20
Abell S295	02:45:28	−53:02:32	0.300	6.78
Abell 697	08:42:59	+36:21:09	0.282	11.00

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The first cluster we analyze is MACS J0025.4-1222 (the "baby bullet"; hereafter MACS0025). This is a massive, major merger cluster at z = 0.586 with the collision taking place approximately in the plane of the sky (Bradač et al. 2008; Ma et al. 2010). It is also among the most X-ray-luminous clusters at z > 0.5 (Ebeling et al. 2007). A clear separation between the DM component and the intracluster gas can be observed in a similar way to the "Bullet Cluster" (1E 065756; Clowe et al. 2004). While both the galaxy and DM components show a bimodal distribution corresponding to two subclusters with similar masses, the gas shows a single peak located between the two substructures. Bradač et al. (2008) studied the distribution of the different components using strong- and weak-lensing information obtained from multicolor HST images (observed with the ACS F555W and F814W filters and WFPC2 F450W filter) and X-ray data from Chandra. They found that the observed offset between the hot gas and the mass distribution based on the galaxies and lensing maps is in agreement with what is expected for collisionless cold DM (CDM). Approximately 200 objects in the cluster were targeted as part of the DEIMOS/Keck spectroscopic campaign for the MACS survey (Ebeling et al. 2007), providing a precise measurement of the cluster redshift and its velocity dispersion. Additionally, multiple-image candidates selected from the HST data available at the time (GO 9722, 10703, PI: Ebeling; GO 11100, PI: Bradač) were observed with LRIS/Keck (Bradač et al. 2008), and a couple of lens models for this cluster have been published prior to the RELICS data (Bradač et al. 2008; Zitrin et al. 2011).

The second cluster is MACS J0159.8-0849 (MACS0159), a hot and luminous X-ray cluster at z = 0.406 showing a regular X-ray morphology (Maughan et al. 2008). This cluster is part of the sample comprising the 34 brightest MACS clusters (Ebeling et al. 2010). MACS0159 is also one of only two RELICS clusters that are part of the Wong et al. (2013) rank of prominent lenses, based on SDSS luminous red galaxy data. The presence of an extended X-ray source detected as a filamentary structure was reported by Kotov & Vikhlinin (2006), who analyzed the mass-temperature relation of clusters at z ∼ 0.5. An extended and diffuse emission around the brightest cluster galaxy (BCG) was also observed in the radio by Giacintucci et al. (2014) in an analysis searching for radio minihalos in a sample of X-ray-luminous clusters. For this cluster, we made use of the available archival HST data from GO programs 11103 and 12166 (PI: Ebeling).

Abell S295, the third cluster we analyze (AS295; Abell et al. 1989), is located at z = 0.30. This cluster is also known as SPT-CL J0245-5302 and ACT-CL J0245-5302 and is part of the sample comprising the 26 most massive clusters detected by their SZ effect in the SPT survey (Williamson et al. 2011). Edge et al. (1994) reported the discovery of a giant arc, dubbed "the Mexican hat," in the northwest region of the cluster, detected during optical follow-up of the ROSAT All-Sky Survey as part of an ESO Key Program. At that time, the data revealed the presence of three knots in the arc. The cluster AS295 was also studied by Menanteau et al. (2010), who discussed the optical and X-ray properties of the first ACT cluster sample and noted that this cluster is likely a merger. Further works that have included AS295 are those by Ruel et al. (2014) and Bayliss et al. (2016), who presented and analyzed spectroscopic data for a sample of clusters detected in the SPT survey. For this cluster, we used the available archival HST data from GO program 13514 (PI: Pacaud).

The last cluster we analyze here is Abell 697 (A697; Abell et al. 1989; Crawford et al. 1995), a rich and massive cluster located at z = 0.282. This cluster is the 10th-highest SZ mass cluster in the Planck catalog and is part of the ROSAT Brightest Cluster Sample (Ebeling et al. 1998), being a hot and luminous cluster in the X-ray. Kempner & Sarazin (2001) suggested the presence of diffuse cluster-scale radio emission, later classified as a giant radio halo (Venturi et al. 2008). Subsequent radio, optical, and X-ray analyses hint that A697 is the result of a complex multiple-merger history occurring along the line of sight (Girardi et al. 2006; Macario et al. 2010). This finding is supported by the first lens model of this cluster based on data from the Keck Observatory (Metzger & Ma 2000). The authors observed a gravitationally lensed arc to the south of the big cD central structure, which entailed a very elliptical potential in the derived SL model.

2.1. Spectroscopic Observations

We obtained Gemini-N GMOS spectroscopic observations for A697 (program ID: GN-2018A-Q-903, PI: Zitrin), primarily targeting the multiply imaged galaxies listed in Figure 4 (see also Table 6). The mask included slits placed on images 1.1, 1.4, 2.3, and 3.1, so that at least one image from each multiple-image system was observed. Observations were carried out in queue mode on the night of 2018 March 18. Raw weather conditions for execution were 50th percentile cloud cover, 20th percentile background brightness, and 70th percentile image quality. From acquisition stars on our mask, we measure an average seeing of 055, and online webcam data²⁰ suggest some possible cloud coverage. A total integration time of 3872 s was achieved using four 968 s long exposures in a nod-and-shuffle mode with a nod of ±075. We adopted the GG455 filter and R400 grating and a central wavelength (CW) of 700 and 710 Å (two CWs were used to fill gaps between the CCDs). The resulting coverage was ∼4700–9500 Å, depending on the position of the slit in the mask, with a resolution of 1.516 Å pixel^–1. No preimaging was required given the available HST images. Data were retrieved through the Gemini Observatory Archive and processed with the Gemini IRAF package using the standard procedure that includes bias, dark, flats, and wavelength calibration. The processed images were combined to a final two-dimensional (2D) spectrum for each slit, which was then also extracted to one dimension. In each slit of the four raw images, we also planted artificial sources to track the reduction progress, given the lack of a continuum emission, to verify that the offsets assigned in various reduction steps matched the expectations.

In addition to multiple images, we also placed slits on high-redshift candidates from Salmon et al. (2017; see Table 2 here), although we note that the relatively short exposure time was dictated by multiple images, typically brighter than high-redshift candidates, which were secondary targets here.

Table 2.  High-z (z ∼ 6) Candidates

Galaxy IDa	R.A.	Decl.	J125b	${z}_{\mathrm{phot}}^{\mathrm{EAZY}}$	${z}_{\mathrm{phot}}^{\mathrm{BPZ}}$	μbestc	μd	MUV,1500e
	(J2000)	(J2000)	[AB]					[AB]
MACS 0025-12-0169	00:25:29.89	−12:22:12.76	25.96 ± 0.24	${6.0}_{-5.6}^{+0.1}$	${5.4}_{-5.0}^{+0.5}$	2.28	${2.31}_{-0.04}^{+0.03}$	$-{20.00}_{-0.32}^{+0.30}$
MACS 0025-12-0450	00:25:32.74	−12:22:40.02	25.80 ± 0.22	${5.9}_{-5.4}^{+0.0}$	${5.3}_{-4.8}^{+0.5}$	2.17	${2.23}_{-0.05}^{+0.03}$	$-{20.41}_{-0.32}^{+0.30}$
MACS 0025-12-0554	00:25:25.01	−12:22:51.38	25.79 ± 0.22	${6.5}_{-4.8}^{+0.3}$	${6.3}_{-0.5}^{+0.3}$	1.38	${1.40}_{-0.01}^{+0.01}$	$-{20.62}_{-0.30}^{+0.30}$
MACS 0025-12-0770	00:25:33.92	−12:23:08.83	26.82 ± 0.36	${5.7}_{-4.7}^{+0.2}$	${0.7}_{-0.2}^{+4.7}$	2.07	${2.13}_{-0.05}^{+0.03}$	$-{19.05}_{-0.33}^{+0.30}$
MACS 0025-12-0851	00:25:35.76	−12:23:14.03	26.90 ± 0.32	${5.9}_{-5.6}^{+0.1}$	${0.8}_{-0.3}^{+5.0}$	1.88	${1.91}_{-0.02}^{+0.01}$	$-{18.94}_{-0.31}^{+0.30}$
MACS 0025-12-1037	00:25:30.79	−12:23:34.51	27.23 ± 0.38	${6.4}_{-5.7}^{+0.5}$	${1.3}_{-0.6}^{+5.3}$	11.58	${12.48}_{-0.79}^{+0.68}$	$-{16.97}_{-0.55}^{+0.36}$
MACS 0025-12-0748	00:25:34.01	−12:23:05.87	27.17 ± 0.34	${6.7}_{-5.9}^{+1.1}$	${6.4}_{-5.5}^{+0.9}$	2.11	${2.12}_{-0.05}^{+0.03}$	$-{18.67}_{-0.32}^{+0.30}$
MACS 0159-08-0085	01:59:48.71	−08:48:58.43	27.01 ± 0.40	${6.0}_{-5.2}^{+0.2}$	${5.4}_{-4.7}^{+0.5}$	2.10	${2.06}_{-0.17}^{+0.20}$	$-{19.32}_{-0.37}^{+0.32}$
MACS 0159-08-0137	01:59:47.69	−08:49:08.00	27.04 ± 0.37	${6.3}_{-5.4}^{+0.4}$	${6.0}_{-5.2}^{+0.5}$	2.15	${2.15}_{-0.18}^{+0.23}$	$-{19.06}_{-0.37}^{+0.32}$
MACS 0159-08-0661	01:59:47.18	−08:50:14.18	26.58 ± 0.25	${5.8}_{-0.7}^{+0.2}$	${5.6}_{-0.7}^{+0.3}$	14.63	${14.85}_{-5.12}^{+3.95}$	$-{17.24}_{-1.22}^{+0.55}$
MACS 0159-08-0621	01:59:50.29	−08:50:08.05	27.96 ± 0.49	${7.1}_{-5.6}^{+1.1}$	${6.6}_{-5.5}^{+1.0}$	5.62	${5.69}_{-0.72}^{+0.79}$	$-{17.30}_{-0.54}^{+0.36}$
Abell 295-0250	02:45:29.87	−53:01:50.40	24.93 ± 0.11	${6.3}_{-0.2}^{+0.3}$	${6.3}_{-0.2}^{+0.3}$	2.60	${2.70}_{-0.34}^{+0.51}$	$-{20.68}_{-0.31}^{+0.32}$
Abell 295-0355	02:45:27.92	−53:02:00.14	27.76 ± 0.50	${6.2}_{-5.6}^{+0.4}$	${5.9}_{-5.4}^{+0.4}$	2.66	${2.83}_{-0.61}^{+0.73}$	$-{18.33}_{-0.47}^{+0.33}$
Abell 295-0796	02:45:27.26	−53:02:49.85	28.23 ± 0.67	${6.1}_{-5.1}^{+0.3}$	${1.0}_{-0.4}^{+5.1}$	6.60	${6.82}_{-1.25}^{+1.41}$	$-{17.14}_{-0.46}^{+0.35}$
Abell 295-1055	02:45:25.15	−53:03:18.86	25.52 ± 0.17	${6.0}_{-0.5}^{+0.4}$	${5.9}_{-0.3}^{+0.3}$	2.43	${2.43}_{-0.17}^{+0.28}$	$-{20.13}_{-0.32}^{+0.31}$
Abell 295-0737	02:45:30.05	−53:02:43.65	27.13 ± 0.29	${7.4}_{-6.4}^{+0.8}$	${6.5}_{-5.5}^{+1.2}$	12.66	${16.02}_{-6.23}^{+6.29}$	$-{16.48}_{-0.49}^{+0.48}$
Abell 295-0568	02:45:36.25	−53:02:25.87	26.02 ± 0.17	${8.1}_{-1.7}^{+0.3}$	${7.7}_{-0.9}^{+0.4}$	1.74	${1.77}_{-0.13}^{+0.22}$	$-{20.20}_{-0.36}^{+0.31}$
Abell 697-0095	08:42:58.55	36:22:46.46	26.12 ± 0.24	${5.7}_{-0.8}^{+0.2}$	${5.5}_{-0.5}^{+0.3}$	1.24	${1.28}_{-0.01}^{+0.01}$	$-{20.62}_{-0.33}^{+0.30}$
Abell 697-0184	08:42:57.33	36:22:19.10	26.41 ± 0.28	${6.1}_{-5.3}^{+0.3}$	${5.7}_{-5.0}^{+0.4}$	1.70	${1.79}_{-0.05}^{+0.05}$	$-{19.96}_{-0.47}^{+0.31}$
Abell 697-0636f	08:43:01.24	36:21:35.75	25.69 ± 0.18	${6.0}_{-5.2}^{+0.6}$	${6.0}_{-5.1}^{+0.5}$	1.49	${1.55}_{-0.02}^{+0.03}$	$-{20.31}_{-0.46}^{+0.31}$
Abell 697-0972g	08:43:00.14	36:20:58.02	26.98 ± 0.31	${6.3}_{-6.0}^{+0.8}$	${0.8}_{-0.4}^{+5.4}$	⋯	⋯	⋯

Notes.

^aFollowing the Salmon et al. (2017) notations. ^bApparent magnitude in the F125W band. ^cMagnification from the best-fit model. ^dMean magnification and 1σ errors from 100 random MCMC realizations. ^eAbsolute magnitude at λ = 1500 Å. Errors include the uncertainty in the fit to the UV slope and propagated photometric and magnification errors. ^fz_spec = 5.800, obtained from our GMOS spectroscopic observations for A697. ^gOutside the modeled FOV.

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We did not identify any prominent emission lines in the three multiple-image systems, so we do not obtain new spectroscopic redshifts for the modeling. However, we do detect a prominent line for one of the high-redshift galaxies, namely Abell 697-0636, listed in Table 2, which we interpret as Lyα, yielding a spectroscopic redshift of z_Lyα = 5.800 ± 0.001 (another possible interpretation is [O ii] at a redshift of ∼1.2). The final one- and two-dimensional spectra are shown in Figure 9 and presented in more detail in Section 5.3.

We adopt a light-traces-mass (LTM) approach to model the projected central mass distribution of the clusters. One of the advantages of this approach is its predictive power, guiding the identification of multiple-image systems. The method relies on the simple assumption that the distribution of the observed galaxies traces the overall DM distribution of the cluster. In the following, we give an overview of the method. For a detailed description, we refer to readers to Zitrin et al. (2009, 2015; see also Broadhurst et al. 2005).

The modeling starts with the construction of a catalog of cluster members using the red-sequence method (Gladders & Yee 2000). For each cluster, we use the magnitudes measured from the F606W and F814W filters to draw the color–magnitude diagram and identify the location of the red sequence. We typically consider galaxies to be cluster members when lying within ±0.3 mag of the sequence, selecting galaxies down to 23 AB. We follow this first selection with a visual inspection of the HST color image and manually exclude objects with a doubtful morphology or color. Similarly, we sometimes manually include galaxies that may have been lost in the initial selection. Additionally, the flux of a few galaxies typically has to be reduced manually, especially galaxies that are designated as cluster members but show bright spiral structure and so have, in practice, a significantly lower M/L than the one effectively adopted for cluster ellipticals. We discuss this point in some more detail in Section 5.

The next step is to construct the mass distribution based on the final catalog of selected cluster galaxies. We start by assigning a power-law surface mass–density radial profile to each galaxy:

$$\begin{eqnarray}&&{\rm{\Sigma }}(r)={{Kr}}^{-q}.\end{eqnarray} \tag{ 1 }$$

The amplitude of this profile, K, is linearly scaled with the observed luminosity. The power-law exponent q is the same for all galaxies and is a free parameter of the model. The enclosed mass of a galaxy within an angular distance θ is then given by

$$\begin{eqnarray}&&M(\lt \theta )=\displaystyle \frac{2\pi K}{2-q}{({D}_{l}\theta )}^{2-q},\end{eqnarray} \tag{ 2 }$$

and its deflection field is

$$\begin{eqnarray}&&\alpha (\theta )=\displaystyle \frac{4{GM}(\lt \theta )}{{c}^{2}\theta }\displaystyle \frac{{D}_{\mathrm{ls}}}{{D}_{{\rm{l}}}{D}_{{\rm{s}}}},\end{eqnarray} \tag{ 3 }$$

where D_ls, D_l, and D_s are the angular diameter distances between lens–source, observer–lens, and observer–source, respectively. The last equation can be rephrased and written as

$$\begin{eqnarray}&&\alpha (\theta )={K}_{q}F{\theta }^{1-q},\end{eqnarray} \tag{ 4 }$$

where F is the measured flux, and K_q is a constant that depends on the power-law index q and encompasses all of the previous constants and proportion relations. The deflection angle at a certain position due to all cluster member galaxies, ${{\boldsymbol{\alpha }}}_{\mathrm{gal}}$, is given by a linear sum of all individual galaxy contributions. The sum of all galaxy mass–density distributions therefore defines the galaxy component of the model, which should now be supplemented by a DM distribution.

As the DM distribution should be smoother than the contribution of the galaxies, we apply a 2D Gaussian smoothing to the coadded galaxy component. We define S to be the width of the 2D Gaussian and the second free parameter of the model. The deflection field resulting from the smooth component is defined as ${{\boldsymbol{\alpha }}}_{\mathrm{DM}}$. The overall normalization (essentially K_q) and the relative weight of the galaxies to the DM component (which we denote as K_gal) are two other free parameters of the model.

Since some difference is expected between the distribution of galaxies and DM, a two-parameter external shear is added to the deflection field for further flexibility (also, the external shear effectively introduces ellipticity to the magnification map). The external shear is described by its amplitude and position angle, which are also free parameters of the model. The deflection field from the external shear is marked as ${{\boldsymbol{\alpha }}}_{\mathrm{ex}}$. The basic model thus has a total of six global free parameters.

The total deflection field is then obtained by adding the three components and accounting for their relative contributions:

$$\begin{eqnarray}&&{{\boldsymbol{\alpha }}}_{T}({\boldsymbol{\theta }})={K}_{\mathrm{gal}}{{\boldsymbol{\alpha }}}_{\mathrm{gal}}({\boldsymbol{\theta }})+(1-{K}_{\mathrm{gal}}){{\boldsymbol{\alpha }}}_{\mathrm{DM}}({\boldsymbol{\theta }})+{{\boldsymbol{\alpha }}}_{\mathrm{ex}}({\boldsymbol{\theta }}).\end{eqnarray} \tag{ 5 }$$

Typically, the model can be further improved by allowing the weight of a few central BCGs to vary as free parameters. Similarly, a core radius and ellipticity can be introduced to these BCGs in order to improve the fit. In addition, redshifts of systems lacking spectroscopic measurements can be left as free parameters and optimized in the fitting routine.

We use a Monte Carlo Markov chain (MCMC) code with several thousand steps to obtain the best-fit model, adopting a χ² criteria,

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{n}\displaystyle \frac{{({x}_{i}^{{\rm{m}}}-{x}_{i}^{\mathrm{obs}})}^{2}+{({y}_{i}^{{\rm{m}}}-{y}_{i}^{\mathrm{obs}})}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 6 }$$

where ${x}_{i}^{\mathrm{obs}}$, ${y}_{i}^{\mathrm{obs}}$ are the observed multiple-image positions; ${x}_{i}^{{\rm{m}}}$, ${y}_{i}^{{\rm{m}}}$ are the corresponding coordinates predicted by the model; and σ_i is the positional uncertainty. This minimization quantifies the distance of the multiple-image positions inferred by the model with respect to the observed ones (we assume here a positional uncertainty of 05; e.g., Newman et al. 2013). To assess the uncertainty of the best-fit model, we consider the rms of the reproduced images in the image plane,

$$\begin{eqnarray}&&\mathrm{rms}=\sqrt{\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{n}{({x}_{i}^{{\rm{m}}}-{x}_{i}^{\mathrm{obs}})}^{2}+{({y}_{i}^{{\rm{m}}}-{y}_{i}^{\mathrm{obs}})}^{2}},\end{eqnarray} \tag{ 7 }$$

given the predicted (${x}_{i}^{{\rm{m}}},{y}_{i}^{{\rm{m}}}$) and observed (${x}_{i}^{\mathrm{obs}},{y}_{i}^{\mathrm{obs}}$) positions and the total number N of multiple images.

With only six global parameters and relying on a simple assumption, the LTM method is a powerful tool to identify multiple images and probe the cluster mass distribution.

We start by constructing a preliminary model (using typical parameter values) based on the red-sequence cluster member selection and photometry. Thanks to the LTM assumption, this initial guess is sufficiently successful to guide the identification of multiple images, which are ultimately identified by eye, based on the morphology, position, and color information obtained from the HST images and compared with the model's prediction. We search for new multiple-image candidates predicted by the model in an iterative way, refining the model in each step. For the final model, we only use the most secure systems, i.e., those that agree with the model's prediction and are conspicuously similar to the prediction and each other in terms of symmetry, color, internal details, photometric redshift, etc.

Most of our multiple-image systems do not have spectroscopic redshifts, and for these cases, our modeling primarily uses the photometric redshift estimates from BPZ averaged over all images of a given system. We individually checked the solutions given by BPZ, and in cases where the photo-zs of multiple images from the same system do not match, we manually assigned a higher weight to brighter and isolated images when determining the adopted system's photo-z. In addition, we typically set for each cluster one multiple-image system with a very good photo-z estimate and leave the redshift of the other systems to be optimized in the minimization around the best-fit value of their photo-z probability distribution function. This will be detailed for each cluster individually in Section 4.

In the following subsections, we present for each cluster the strong lens modeling and some of its immediate results.

The projected mass–density distributions from the best-fit models for a source located at z_s = 2 are shown in Figure 5, and the corresponding mass–density profiles are shown in Figure 6. We define the mass–density distribution of the systems in terms of the convergence, κ, a dimensionless surface density given by

$$\begin{eqnarray}&&\kappa ({\boldsymbol{r}})=\displaystyle \frac{{\rm{\Sigma }}({\boldsymbol{r}})}{{{\rm{\Sigma }}}_{\mathrm{cr}}},\end{eqnarray} \tag{ 8 }$$

where Σ_cr is the critical density for lensing given by

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{cr}}=\displaystyle \frac{{c}^{2}}{4\pi G}\displaystyle \frac{{D}_{s}}{{D}_{{ls}}{D}_{l}},\end{eqnarray} \tag{ 9 }$$

and ${\rm{\Sigma }}({\boldsymbol{r}})$ is the projected surface mass density.

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High-redshift magnification maps, scaled to a source redshift of z_s = 9, are shown in Figure 7. A list of the multiple-image systems used as constraints can be found in Appendix A. We also list therein candidate multiple images (labeled with a c). These are images or systems whose identification was more ambiguous and were not included as SL constraints (for example, images with similar color and/or morphology but a notable discrepancy between the observed and model-predicted location or orientation).

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4.1. MACS J0025.4-1222

Our model for MACS0025 is constructed based on two systems of multiple images. System 1 forms at the northwestern subcluster and is composed of four counterimages showing a two-component morphology, which includes a blue bright spot and a diffuse feature. Three images of this system were identified in Bradač et al. (2008) as system AB. Bradač et al. (2008) also measured a spectroscopic redshift for this system of z = 2.38 using Keck/LRIS data. A second lens model by Zitrin et al. (2011) predicted a fourth image for this system, which they also identified in the data and is used here.

The second system we consider here was labeled system C in Bradač et al. (2008), and we define three multiple images located on the east side of the main BCGs as constraints. This system was also spectroscopically targeted by Bradač et al. (2008) but did not yield a spectroscopic redshift. Bradač et al. (2008) estimated a photometric redshift of ${z}_{\mathrm{phot}}={1.0}_{-0.2}^{+0.5}$ for this system due to the 4000 Å break possibly lying between the F555W and F814W bands. The BPZ estimate from RELICS gives a higher z_phot = 3.8 (corresponding to the value for the counterimages denoted as 2.2 and 2.3 here), a redshift that we fix for this system in our modeling. We note that this estimate uses information from the seven HST band observations available for RELICS, while Bradač et al. (2008) used the information from the F450W/F555W/F814W bands available at the time. This choice of fixing both redshifts in our model was made in order to constrain both mass clumps.

Bradač et al. (2008) also included a third system in their SL model, labeled system D. Here we do not include this as a constraint. Based on the new information from our multiband observations, the northern counterimage in the region of the second BCG (labeled c3.3) does not seem to match the southern arc-shaped images (c3.1-2) in terms of color. In addition, our model does not precisely predict the northern counterimage or the unexpected orientation of the southern arc. So, we designate this system as a candidate system only (labeled as system c3). We note also that Zitrin et al. (2011) included a fourth two-image system that their older model agreed with. We decided to discard this system from our analysis, as it appears to be a single elongated image with no counterimage predicted by our model, and only mention it for completeness as a candidate system (c4).

In our modeling, we include the weight of the northwestern BCG as a free parameter to be optimized, allowing it to vary with respect to the original M/L ratio. We set the ellipticity and position angle for this BCG to the values given by SExtractor. The resulting critical curves, computed for a source at z = 2.38, and the location of the multiple images can be seen in Figure 1.

The multiple-image reproduction is shown in Figure 10. Our model precisely predicts the four counterimages of system 1 with respect to the position and orientation. The prediction for system 2 returns an arc-shaped image with the orientation matching the observed disposition of the system counterimages, although the specific position of the predicted images is less accurate and likely influenced by the local cluster member around which the arc partially revolves. We acknowledge that given the lack of internal details in system 2, its exact configuration remains ambiguous. The final rms of the best-fit model is 057.

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Due to the small number of constraints, i.e., multiple images that could be reliably identified, the model for MACS0025 is, in a sense, a simple model, with a minimum number of free parameters. Future use of the presented lens model should thus acknowledge its limitations.

4.2. MACS J0159.8-0849

We identify four multiple-image systems in MACS0159 with which we construct our model. The first and second systems represent different constraints related to the same source (although it is unclear if the source is one or, e.g., two merging galaxies). The third system shows a distinctive green color in the color-composite RELICS HST image that aided in its identification, and the fourth system corresponds to a small, relatively bright source.

We set the main source to be at z = 1.55, corresponding to the mean z_phot from counterimages 1.1 and 1.2 (the more reliable BPZ estimates, since image 1.3 suffers contamination from the BCG light), and fix the redshift of systems 1 and 2 to this value. The redshifts of systems 3 and 4 are left as free parameters to be constrained by the model.

The mass distribution of the cluster is modeled with both the weight and the core radius of the BCG as free parameters, where the latter can reach values up to 100 kpc. Our best-fit model does not include an ellipticity for the BCG, which seems to be fairly spherical.

Our final model produces well all observed multiple images (Figure 11), resulting in an rms of ≃096. For each of systems 3 and 4, the model also predicts a third, considerably fainter image close to the cluster center that we do not clearly identify in the data, likely due to the BCG's light and the expected faintness of these images. Figure 2 shows the critical curves for our best-fit model, for a source at z = 1.55, and the position of the multiple images.

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4.3. Abell S295

The modeling of AS295 is based on the identification of six sets of constraints. The first four are part of a known giant arc located in the northwest region of the cluster, close to the BCG. These four subsets can be distinguished by the differences in their color and relative positions, where we identify three multiple images for each set. The arc has a spectroscopic redshift of 0.93 reported in the literature (Edge et al. 1994), and we adopt this value for all four subsets related to the arc.

The two other systems (systems 5 and 6) are found in the southeast concentration of galaxies. These are new identifications, with three multiple images each. System 5 has an elongated image on the east and two counterimages lying close to each other south of the BCG. Between these two images, there is an additional faint galaxy that might locally contribute somewhat to the lensing as well. System 6 is a faint and long arc northwest of the respective BCG. For these systems, there are no spectroscopic redshift measurements available, so we leave their redshifts as free parameters of the model. We note that while we refer to systems 5 and 6 as secure here, since they match the model's prediction, they lack clear internal details, and thus their identification should be taken with slightly more caution.

Our model for AS295 has the weight of the two BCGs as free parameters to be optimized in the minimization. For the northern BCG, we set the ellipticity and position angle to the values given by SExtractor. The southern BCG presents a spherical morphology, so we decided not to apply an ellipticity. The weight of two bright, spiral-looking cluster members (R.A. = 02:45:25.710, decl. = −53:01:43.031 and R.A. = 02:45:37.573, decl. = −53:02:58.595) lying near the two BCGs was reduced by factor of 2 (this slightly improved the reproduction of multiple images; see also the discussion in Section 5.1 for MACS0025).

For systems 1 to 4, our model predicts well the three observed counterimages in each subset. It also predicts a fourth, faint/small counterimage close to the BCG for these systems. System 5 has a reasonable reproduction, and for system 6, we obtain a long arc predicted by the model, in agreement with the position and orientation of our image constraints. The total rms of the predicted images with respect to the observed locations is ≃077. The reproduction of multiple images can be seen in Figure 12.

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The resulting critical curves from our best-fit model (for a source redshift of z = 1), along with the multiple images, are seen in Figure 3.

4.4. Abell 697

We construct the model for A697 using three multiple-image systems. Because there is no spectroscopic redshift available for any of the systems, we fix the main source redshift to the value of the system with the most reliable photometric redshift estimate (corresponding to system 1), leaving the redshift of the remaining systems as free parameters.

The first system corresponds to a blue source showing a distinct morphology, leading to a reliable identification of four counterimages that are well predicted by our model. The second system is also a blue-looking galaxy, and we identify three counterimages south and southeast of the BCG. The third is an arc-shaped object lying south of the cluster center, for which we identify two multiple images. This system was previously reported in the literature by Metzger & Ma (2000), who estimated a lower limit for the redshift of the source of z > 1.3 by combining the arc spectrum and its color.

Our model for A697 allows the weight of the BCG to vary as a free parameter, as well as its ellipticity and position angle. The source redshift for system 1 is set to the mean of the z_phot distributions of images 1.2 and 1.4 (as image 1.1 suffers contamination from the BCG light and, for image 1.3, there was no z_phot solution found). Redshifts for systems 2 and 3 are allowed to vary and are optimized by the minimization procedure. Our best-fit model yields a redshift of $z={2.97}_{-0.38}^{+0.04}$ for system 3, consistent with the lower limit found by Metzger & Ma (2000) within 1σ.

The final model reproduces well systems 1 and 2. For system 1, it also predicts a smaller, fifth counterimage very close to the BCG that we cannot identify, possibly due to the contamination by the BCG light. The model correctly predicts the arc shape and orientation of system 3, and as expected from the lensing symmetry, it also predicts a third, fainter counterimage on the east side of the cluster center that we do not securely identify. The reproduction of multiple images by the best-fit model, which has a final rms of ≃082, is shown in Figure 13. The resulting critical curves for a source at z = 2 and the multiple images used as constraints are shown in Figure 4.

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5.1. Lens Modeling Results

Our best-fit model for MACS0025, as expected from the cluster galaxy distribution, exhibits a bimodal mass distribution (Figures 1 and 5), in agreement with the findings by Bradač et al. (2008) and Zitrin et al. (2011). Such morphology supports the picture that this cluster is undergoing a major merger (Bradač et al. 2008; Ma et al. 2010). The Einstein radius obtained from the modeling for the northwestern subcluster is θ_E(z = 2) = (81 ± 08) and encloses a mass of $M=(1.60\pm 0.24)\times {10}^{13}\,{M}_{\odot }$, while for the southeastern substructure, we found θ_E(z = 2) = (7.2 ± 07) and enclosed mass $M=(1.18\pm 0.18)\times {10}^{13}\,{M}_{\odot }$. Note that Zitrin et al. (2011) reported a substantially larger radius for MACS0025 of θ_E = (30 ± 2)'' for z = 2.38. Our model is similar to theirs around the northwestern subclump but implies a much smaller critical curve around the main (southeastern) clump (as can be seen by comparing Figure 3 from Zitrin et al. 2011 to our Figure 1). Our model was constructed with two major differences compared with the previous model by Zitrin et al. (2011). First, as was implied by the RELICS data, here we assign a significantly higher redshift for system 2, so that the critical curves of the main mass clump are required to be smaller per given redshift. The second difference is the lower weight manually given by us to some of the bright galaxies around the southeastern main clump. Specifically, these are (apparently) cluster galaxies that show bright spiral structure and therefore seem to significantly deviate from the general, early-type cluster member M/L ratio, in the sense that they are about an order of magnitude too bright for their actual mass contribution compared to cluster ellipticals (Maraston 2005; Bahcall & Kulier 2014; Courteau et al. 2014). We therefore manually reduced the mass (i.e., input flux) of these galaxies by a factor of 10.²¹

MACS0159 is a massive X-ray-selected lens and has the largest (single-clump) Einstein radius among the clusters analyzed in this work, corresponding to θ_E(z = 2) = (24.9 ± 25). The mass inside the critical curves corresponds to $M=(1.15\pm 0.17)\,\times {10}^{14}\,{M}_{\odot }$. Our model exhibits a fairly round and regular morphology (Figure 5) that, together with the regular X-ray morphology (Maughan et al. 2008), might indicate that this is a relaxed cluster. The central flattening in the projected mass–density profile seen in Figure 6 (top right) may be, in part, a result of the superposition of other elliptical cluster members onto the BCG (Figure 2).

The model for AS295 shows a bimodal mass distribution following the cluster galaxy concentrations (Figures 3 and 5). The northwestern clump hosting the giant arc has an Einstein radius of θ_E(z = 2) = (12.6 ± 13) and mass of $M=(1.96\pm 0.29)\times {10}^{13}\,{M}_{\odot }$ enclosed by the critical curves. For the southeastern subcluster, our best-fit model gives an Einstein radius of θ_E(z = 2) = (12.7 ± 13) and an enclosed mass of $M=(2.04\pm 0.31)\times {10}^{13}\,{M}_{\odot }$.

Our analysis for A697 implies a rather elliptical mass distribution in projection (Figure 5) and elongated critical curves (Figure 7). The Einstein radius given by the model is θ_E(z = 2) = (11.1 ± 01), and the critical curves encompass a total mass of $M=(1.45\pm 0.23)\times {10}^{13}\,{M}_{\odot }$. The somewhat elliptical nature of the lens found in our analysis agrees with the model from Metzger & Ma (2000), the BCG orientation, and the X-ray morphology reported in Girardi et al. (2006), which reinforces the possible recent-merger scenario.

As a self-consistency check, we note that the Einstein radius and mass agree with the expectation given (for a circular lens) by

$$\begin{eqnarray}&&{\theta }_{E}={\left(\displaystyle \frac{4{GM}(\lt {\theta }_{E})}{{c}^{2}}\displaystyle \frac{{D}_{{ls}}}{{D}_{l}{D}_{s}}\right)}^{1/2}.\end{eqnarray} \tag{ 10 }$$

5.2. Implications for High-redshift Studies

5.3. Redshift and Other Uncertainties

Observing 41 massive clusters with HST and Spitzer, the treasury RELICS program was primarily designed to find magnified high-z candidates, some of which are expected to be apparently bright enough for future spectroscopic follow-up from the ground and with JWST (e.g., Salmon et al. 2017; Acebron et al. 2018). The SL models for the clusters are crucial for studying the magnified sources and, for example, for deriving their intrinsic brightness (Table 2) or star formation properties or for constructing the high-redshift luminosity function.

In this work, we have presented SL models for four RELICS clusters. Our analysis, based on the LTM approach, whose main advantage is its predictive power for finding multiple images, supplies the first published SL models for AS295 and MACS0159. For A697 and MACS0025, we present improved models thanks to the new RELICS data. In our modeling, we used as constraints both some previously known (in MACS0025, A697, and AS295) and, importantly, several new identifications (in MACS0159, A697, and AS295) of multiply imaged galaxies.

Out of the four clusters we analyzed, AS295 has the largest cumulative area of high magnification, possibly as large as that of the largest HFF clusters (Figure 8). This is mainly thanks to its two merging subclumps, which create a large region of high magnification between them (Figure 7; much of the region is outside each subclump's critical curves). Our analysis also indicates that MACS0025 is a highly magnifying lens, with a cumulative area of high magnification comparable to the typical HFF cluster (Figure 8). Also in the case of MACS0025, there is only a small critical area around each subclump, and most of the high-magnification area forms between the two subclumps. In that sense, it should be noted that a large area of high magnification alone may not be sufficient for significantly enhancing the detection of high-redshift galaxies, and a sizable critical area—as was the case for the HFF clusters—is also important. MACS0159 is the largest main-clump lens in the sample we analyze here, with an Einstein radius of θ_E = 249 ± 25 (for a source at z_s = 2), and also has a significant HFF-like area with high magnification. It is thus likely the most efficient lens among the four clusters we analyze here. Despite signatures of recent galaxy interaction with the BCG or suggested merger scenarios along the line of sight (Girardi et al. 2006; Macario et al. 2010), A697 seems fairly regular in terms of SL and constitutes the smallest lens in our sample, as well as the one with the smallest area of high magnification.

We note that the above results, in particular the magnification power of the lenses, should be referred to as initial models, which will be improved as additional model constraints become available.

To address the uncertainties in our SL models arising from the lack of spectroscopic information, we performed a suite of tests where we changed the initial configurations with respect to the redshift of the multiple images. We then examined the effect on the resulting density profiles and lens power (cumulative area of high magnification).

In one set of trials, we constructed new lens models with a range of input, main source redshifts that deviate from their fiducial photo-z value. In a second set of trials, we allowed systems with spectroscopic redshifts to vary, to check whether the redshifts are recovered and probe the effect on the resulting models. Other tests were similarly designed to probe the effect of different redshift combinations in our models. For most of the cases, the different density profiles agree well within the 3σ confidence level. Regarding the magnification values, our tests suggest that the redshift uncertainties typically propagate into differences of up to 50% in the cumulative area of high magnifications (for μ > 5 and 10). We also found that the spectroscopic redshifts in the couple of cases where available were recovered to within 10%.

We obtained GEMINI-N/GMOS spectroscopic observations to target multiply imaged galaxies and high-z candidates in the field of A697. While we did not identify any prominent feature in the multiple-image spectra, we detect a single emission line in the spectrum of the high-z candidate Abell 697-0636. Following its dropout nature and photometric redshift, we interpret this as a Lyα line, yielding a redshift of z_Lyα = 5.800 ± 0.001. The line is detected with ∼15σ confidence and has an observed Gaussian width of (117 ± 15) km s⁻¹, and we measured a rest-frame equivalent width of (52 ± 22) Å.

The lens models we presented here, including mass–density, shear, and magnification maps, are made available through the MAST archive.

We would like to thank the reviewer of this work for useful comments. This work is based on observations taken by the RELICS Treasury Program (GO-14096) with the NASA/ESA HST. Program GO-14096 is supported by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work was performed in part under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. RCL acknowledges support from an Australian Research Council Discovery Early Career Researcher Award (DE180101240). Based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e Inovação (Brazil). Data were retrieved through the Gemini Observatory Archive and processed using the Gemini IRAF package. We thank Michael Lundquist for very useful help with the Gemini data reduction.

In this section we provide details on the multiple images and candidates for each of the clusters analysed in this work.

Table 3.  Multiple Images and Candidates for MACS J0025.4-1222

Arc ID	R.A.	Decl.	zspec	zphota	zmodelb	Individual rmsc
	(J2000)	(J2000)			[95% C.I.]	(arcsec)
1.1	00:25:27.684	−12:22:11.19	2.38d	${2.57}_{-0.19}^{+0.16}$	⋯	0.53
1.2	00:25:27.162	−12:22:21.69	''	⋯	⋯	0.62
1.3	00:25:27.183	−12:22:33.89	''	⋯	⋯	0.67
1.4	00:25:27.524	−12:22:23.50	''	${2.36}_{-0.23}^{+0.21}$	⋯	0.55
2.1	00:25:34.103	−12:23:11.56	⋯	${0.38}_{-0.04}^{+0.03}$	3.80e	0.39
2.2	00:25:34.096	−12:23:14.37	⋯	${3.82}_{-0.13}^{+0.13}$	''	0.35
2.3	00:25:34.068	−12:23:15.38	⋯	${3.78}_{-0.18}^{+0.15}$	''	1.03
c3.1	00:25:31.900	−12:23:06.28	⋯	⋯	∼1.9	⋯
c3.2	00:25:31.900	−12:23:05.01	⋯	⋯	''	⋯
c3.3	00:25:32.160	−12:22:58.13	⋯	${1.34}_{-1.12}^{+1.58}$	''	⋯
c4.1	00:25:29.274	−12:22:42.57	⋯	${2.72}_{-0.15}^{+0.28}$	f	⋯
c4.2	00:25:29.292	−12:22:42.12	⋯	⋯	''	⋯
c5.1	00:25:28.350	−12:22:23.07	⋯	${2.43}_{-0.24}^{+0.10}$	∼2.2	⋯
c5.2	00:25:28.334	−12:22:22.26	⋯	⋯	''	⋯
c5.3	00:25:27.812	−12:22:39.42	⋯	⋯	''	⋯

Notes.

^aPhotometric redshift from BPZ, taken from the RELICS catalog. Uncertainties correspond to 2σ. ^bRedshift prediction from the best-fit model. For candidate systems, this value corresponds to the best prediction given by the model. ^cBetween the observed and predicted location of the multiple images. ^dSpectroscopic redshift reported in Bradač et al. (2008). ^eMean z_phot between images 2.2 and 2.3, fixed in the model. ^fPoorly constrained.

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Table 4.  Multiple Images and Candidates for MACS J0159.8-0849

Arc ID	R.A.	Decl.	zspec	zphota	zmodelb	Individual rmsc
	(J2000)	(J2000)			[95% C.I.]	(arcsec)
1.1	01:59:50.749	−8:50:21.66	⋯	${1.68}_{-0.27}^{+0.09}$	1.55d	1.57
1.2	01:59:48.306	−8:49:57.57	⋯	${1.41}_{-0.01}^{+0.30}$	''	1.87
1.3	01:59:49.267	−8:49:58.49	⋯	${0.49}_{-0.07}^{+0.05}$	''	0.50
2.1	01:59:50.875	−8:50:18.72	⋯	${0.05}_{-0.03}^{+2.75}$	1.55d	1.29
2.2	01:59:48.329	−8:49:59.34	⋯	${0.84}_{-0.08}^{+0.29}$	''	0.65
2.3	01:59:49.196	−8:49:57.89	⋯	⋯	''	0.18
3.1	01:59:49.846	−8:49:41.11	⋯	${3.59}_{-0.07}^{+0.19}$	${2.63}_{-0.01}^{+0.73}$	0.32
3.2	01:59:49.882	−8:50:37.75	⋯	${4.05}_{-0.24}^{+0.14}$	''	0.66
4.1	01:59:49.000	−8:50:13.63	⋯	${2.20}_{-0.07}^{+0.30}$	${1.46}_{-0.11}^{+0.03}$	0.26
4.2	01:59:48.784	−8:49:26.71	⋯	${2.54}_{-0.33}^{+0.14}$	''	0.80
c5.1	01:59:50.911	−8:49:59.05	⋯	${0.91}_{-0.77}^{+1.84}$	∼1.4	⋯
c5.2	01:59:50.882	−8:49:55.27	⋯	${0.91}_{-0.78}^{+2.0}$	''	⋯
c6.1	01:59:49.543	−8:50:08.64	⋯	⋯	∼2.3	⋯
c6.2	01:59:49.127	−8:49:13.75	⋯	${2.36}_{-0.17}^{+0.16}$	''	⋯

Notes.

^aPhotometric redshift from BPZ, taken from the RELICS catalog. Uncertainties correspond to 2σ. ^bRedshift prediction from the best-fit model. For candidate systems, this value corresponds to the best prediction given by the model. ^cBetween the observed and predicted location of the multiple images. ^dMean z_phot between images 1.1 and 1.2 set to be the main source redshift, fixed in the model.

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Table 5.  Multiple Images and Candidates for Abell S295

Arc ID	R.A.	Decl.	zspec	zphota	zmodelb	Individual rmsc
	(J2000)	(J2000)			[95% C.I.]	(arcsec)
1.1	02:45:24.123	−53:01:51.05	0.93d	${0.84}_{-0.14}^{+0.03}$	⋯	0.74
1.2	02:45:24.455	−53:01:42.52	''	⋯	⋯	0.82
1.3	02:45:25.128	−53:01:37.57	''	⋯	⋯	0.34
2.1	02:45:24.102	−53:01:50.01	0.93d	⋯	⋯	1.10
2.2	02:45:24.361	−53:01:42.84	''	⋯	⋯	0.91
2.3	02:45:25.128	−53:01:36.68	''	⋯	⋯	0.19
3.1	02:45:24.213	−53:01:51.42	0.93d	⋯	⋯	0.72
3.2	02:45:24.495	−53:01:43.22	''	⋯	⋯	1.01
3.3	02:45:25.234	−53:01:37.68	''	⋯	⋯	0.43
4.1	02:45:24.093	−53:01:47.34	0.93d	⋯	⋯	1.50
4.2	02:45:24.181	−53:01:44.38	''	⋯	⋯	0.72
4.3	02:45:25.137	−53:01:35.26	''	${1.08}_{-0.40}^{+1.51}$	⋯	0.47
5.1e	02:45:34.874	−53:03:05.50	⋯	${0.84}_{-0.02}^{+0.02}$ f	${1.52}_{-0.07}^{+0.13}$	0.63
5.2	02:45:35.142	−53:03:04.81	⋯	⋯	''	0.83
5.3	02:45:37.287	−53:02:47.51	⋯	⋯	''	1.01
6.1e	02:45:34.685	−53:02:39.42	⋯	${0.33}_{-0.23}^{+3.07}$	${2.17}_{-0.49}^{+0.01}$	0.83
6.2	02:45:34.466	−53:02:40.80	⋯	${3.25}_{-2.91}^{+0.25}$	''	0.91
6.3	02:45:33.468	−53:02:48.00	⋯	${2.74}_{-2.70}^{+0.38}$	''	0.57
c7.1	02:45:34.812	−53:02:41.94	⋯		∼2.4	⋯
c7.2	02:45:33.035	−53:02:54.42	⋯	${0.91}_{-0.38}^{+0.18}$	⋯	⋯

Notes.

^aPhotometric redshift from BPZ, taken from the RELICS catalog. Uncertainties correspond to 2σ. ^bRedshift prediction from the best-fit model. For candidate systems, this value corresponds to the best prediction given by the model. ^cBetween the observed and predicted location of the multiple images. ^dSpectroscopic redshift reported in Edge et al. (1994) and Williams et al. (1999). ^eWhile we refer to systems 5 and 6 as secure, we acknowledge that, due to a lack of internal details, their identification should be taken with somewhat more caution. ^fThis estimate is given by a poor SED fit.

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Table 6.  Multiple Images for Abell 697

Arc ID	R.A.	Decl.	zspec	zphota	zmodelb	Individual rmsc
	(J2000)	(J2000)			[95% C.I.]	(arcsec)
1.1	08:42:57.098	+36:22:03.38	⋯	${1.54}_{-0.25}^{+0.30}$	2.0d	0.73
1.2	08:42:56.664	+36:21:56.43	⋯	${1.91}_{-0.75}^{+0.59}$	''	0.25
1.3	08:42:57.509	+36:21:54.87	⋯	⋯	''	0.54
1.4	08:42:58.841	+36:22:06.24	⋯	${2.18}_{-0.33}^{+0.56}$	''	1.35
2.1	08:42:56.983	+36:21:46.67	⋯	${1.90}_{-0.57}^{+0.58}$	${1.96}_{-0.28}^{+0.06}$	0.38
2.2	08:42:57.828	+36:21:48.42	⋯	⋯	''	0.44
2.3	08:42:58.789	+36:21:55.99	⋯	${2.40}_{-0.66}^{+0.19}$	''	1.33
3.1	08:42:57.139	+36:21:47.34	⋯	${2.78}_{-0.46}^{+0.37}$	${2.97}_{-0.38}^{+0.04}$	0.85
3.2	08:42:57.380	+36:21:47.52	⋯	${2.94}_{-0.39}^{+0.42}$	''	1.15
p3.3e	08:42:58.905	+36:21:59.237	⋯	⋯	''	⋯

Notes.

^aPhotometric redshift from BPZ (using a new run of SExtractor with parameters edited manually). Uncertainties correspond to 2σ. ^bRedshift prediction from the best-fit model. ^cBetween the observed and predicted location of the multiple images. ^dMean z_phot between images 1.2 and 1.4 set to be the main source redshift, fixed in the model. ^eCounterimage predicted by the best-fit model but not identified in the observed image.

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Reproduction of the multiple image systems based on our best-fit model, for each of the clusters analysed in this work.

In this section we present the results from our different tests intended to evaluate the impact of the use of photometric redshifts in our analysis.

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