RELICS: A Very Large (θ_E ∼ 40'') Cluster Lens—RXC J0032.1+1808

Over the past decade, extensive galaxy cluster lensing campaigns have been undertaken with the Hubble Space Telescope (HST; Postman et al. 2012; Schmidt et al. 2014; Treu et al. 2015; Lotz et al. 2017; Coe et al. 2019; Steinhardt et al. 2020). HST's unique combination of sensitivity and resolution allows for the identification of lensed galaxies that are multiply imaged by the targeted galaxy clusters (see, for instance, Franx et al. 1997; Frye & Broadhurst 1998; Broadhurst et al. 2005; Diego et al. 2018; Caminha et al. 2019; Jauzac et al. 2019; Lagattuta et al. 2019). These multiple images, in turn, allow us to construct mass models for the clusters, describing the underlying matter distribution. Most (albeit not all) of the clusters targeted with HST, which are typically estimated to be massive based on X-rays, the Sunyaev–Zeldovich effect (SZ; Sunyaev & Zeldovich 1970), or optical-richness criteria (and lensing signatures such as giant arcs in ground-based data), show multiply imaged background galaxies in numbers that generally increase with the strong-lens scale (or critical area, i.e., the area enclosed by the critical curves of infinite magnification; see, for instance, Vega-Ferrero et al. 2019).

Following the high projected-mass densities in their centers, the strong-lens scale of galaxy clusters is expected to reach θ_E of the order of tens of arcseconds, with θ_E being the effective Einstein radius (i.e., the radius of the area enclosed by the critical curves of the lens, were it a circle). This is indeed the range of typical Einstein radii found in lensing analyses of well-known clusters (Richard et al. 2010b; Oguri et al. 2012; Zitrin et al. 2015; Sharon et al. 2020). Since the Einstein radius size essentially depends on the mass enclosed in the core of the cluster, the distribution of Einstein radii can be used to probe cosmological models as well as structure formation and evolution in its framework (Turner et al. 1984; Narayan & White 1988; Oguri & Blandford 2009). Due to the shape of the cosmic mass function (Tinker et al. 2008), more massive clusters and thus, generally, larger Einstein radii become rarer (see, for instance, Oguri et al. 2012; Zitrin et al. 2012). Other effects are also known to boost the lensing cross section such as a major axis oriented along the line of sight, a high degree of substructure, cluster concentration, or the dynamical state (Meneghetti et al. 2010). Indeed, only a handful of clusters are known to have Einstein radii of θ_E ≳ 35'', for a source at z_s ∼ 2, nominally (see the list in Table 1). The number of clusters with particularly large Einstein radii is therefore very important because it can place useful constraints on structure formation and evolution models (e.g., Oguri & Blandford 2009).

Larger strong lenses (presenting larger Einstein radii) should have on average larger areas of high magnification and thus lens more background sources (Vega-Ferrero et al. 2019), especially if the faint-end slope of the luminosity function is steep enough (Broadhurst et al. 1995; Bradley et al. 2014; Coe et al. 2015), as is currently estimated, for example, for high-redshift galaxies (Bouwens et al. 2015; Finkelstein et al. 2015; Mason et al. 2015). Building a census of this particular class of large cluster lenses is thus also important for efficiently searching for high-redshift galaxies with current and future observations.

The Reionization Lensing Cluster Survey (RELICS; PI: Coe; Coe et al. 2019) is a large Hubble Space Telescope program that has observed 41 galaxy clusters chosen largely based on SZ-mass estimates from the Planck PSZ2 all-sky catalog (Planck Collaboration et al. 2016). One of the chief goals of the RELICS survey is to identify bright high-redshift galaxies that could be followed up from the ground and with the upcoming James Webb Space Telescope. Lens models for the observed clusters are needed to study the dark matter (DM) distribution, as well as the intrinsic properties of newly uncovered high-redshift galaxy candidates lensed by these clusters (Salmon et al. 2018, 2020).

In our systematic analysis of RELICS clusters (Acebron et al. 2018, 2019; Cerny et al. 2018; Cibirka et al. 2018; Paterno-Mahler et al. 2018; Mahler et al. 2019), we have analyzed RXC J0032.1+1808²⁵ (Böhringer et al. 2001; Ebeling et al. 2001), located at R.A. = 00^h32^m110, decl. = +18^d07^m490 at a redshift of z = 0.3956. RXC J0032.1+1808 (RXC0032 hereafter) is the second-richest galaxy cluster in the Sloan Digital Sky Survey DR8 redMaPPer cluster catalog (only after RMJ224319.8-093530.9; see Rykoff et al. 2014), but only the 85th most massive cluster in the PSZ2 catalog (Planck Collaboration et al. 2016), with a mass of ${M}_{500}={7.61}_{-0.63}^{+0.57}\times {10}^{14}{M}_{\odot }$, where M₅₀₀ is defined as the cluster mass within the radius R₅₀₀ inside which the mean mass density is 500 times the critical density δ_c.

A first strong-lensing (SL) model for this cluster, prior to RELICS imaging, was used in Dessauges-Zavadsky et al. (2017) to estimate the properties of a notable multiply imaged system at a redshift of z = 3.6314 (system 1 and 2 here). Here we present our SL analysis of RXC0032 in RELICS data, revealing a very large critical area, similar to only a handful of other clusters known to date (see Table 1). We report here this discovery and detail our SL modeling.

The outline of the paper is as follows: in Section 2 we briefly describe the data and observations used to identify multiple images for the SL analysis, which is presented in Section 3. The results are shown and discussed in Section 4. We compare our results with those obtained from the Lenstool and GLAFIC pipelines²⁶ in Section 5. Finally, we discuss the previously detected lensed star-forming galaxy in Section 6 before summarizing our work in Section 7. Throughout we assume a ΛCDM cosmology with ${{\rm{\Omega }}}_{{\rm{m}}0}=0.3$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}0}=0.7$, H₀ = 70 km s⁻¹ Mpc⁻¹ where 1'' = 5.337 kpc at the redshift of RXC J0032.1+1808.

RXC0032 is part of the RELICS cluster sample (Coe et al. 2019). Each cluster field in the RELICS program was observed for two orbits with Wide Field Camera 3 (WFC3/IR) in the F105W, F125W, F140W, and F160W bands, and complemented archival observations with the Advanced Camera for Surveys (ACS) so that each field is observed for three orbits—one in each of the passbands F435W, F606W, and F814W. In addition, 30 hr per band in each of the Spitzer-IRAC channels (PI: M. Bradac, PI: Soifer) were observed. As one orbit of HST archival observations already existed for RXC0032 (program GO 12166, PI: Ebeling), RELICS observed this galaxy cluster for only two orbits with each of the ACS bands, in addition to the two WFC3 orbits (Coe et al. 2019).

Data reduction of the HST images is described in Coe et al. (2019). We used the photometric source catalogs generated with Source Extractor (Bertin & Arnouts 1996) in dual-image mode from the final drizzled 006 pixel⁻¹ images. The photometric redshifts we use (hereafter z_phot), were derived using the Bayesian Photometric Redshift program (BPZ, Benítez 2000; Benítez et al. 2004; Coe et al. 2006) from seven HST band imaging data (from the combined RELICS and the aforementioned archival HST data). The reduced imaging, catalogs, and data products are available for the community through the Mikulski Archive for Space Telescopes (MAST).²⁷

We construct lens models for RXC0032 using two methodologies. Primarily, we use the light-traces-mass (LTM) methodology, outlined in Zitrin et al. (2015) and references therein; see also Broadhurst et al. (2005). With this methodology, we uncover multiple-image sets and publish the first estimate for the size of the lens. We also use a fully parametric formalism for comparison, dubbed hereafter dPIEeNFW. Both methodologies assume two main mass components for the mass distribution: one that accounts for the cluster galaxies, and a second that represents the DM distribution. While member galaxies are also represented differently, the main difference between the two methodologies is that in the LTM methodology the DM distribution is assumed to follow the light distribution, whereas in the parametric model it is independent and follows a combination of symmetric, analytic forms. Both methodologies are implemented on a grid where the resolution can be changed for computational time purposes. Both methodologies are implemented in the same pipeline (Zitrin et al. 2015), and are briefly detailed below.

3.1. LTM

3.2. dPIEeNFW

3.3. Minimization

3.4. SL Analysis of RXC J0032.1+1808

The starting point of our modeling relies on the construction of a cluster member catalog based on the red-sequence method (Gladders & Yee 2000). We use the magnitudes measured from the F606W and F814W filters to draw a color–magnitude diagram. We only consider galaxies down to 24 AB within ±0.3 mag of the sequence (De Lucia & Helmi 2008). To exclude stars from our selection we do not include objects whose magnitudes are brighter than 17 AB or have a stellarity index below <0.95. In addition, we take advantage of the delivered photometric catalog by RELICS to check that all selected cluster members were within z_phot ± 0.1 of the mean redshift of the cluster (as measured by the BPZ software). We finally perform a visual inspection of the selected cluster members. This allows us to discard further interloping galaxies (bright foreground galaxies for instance) or artifacts (such as faint and diffuse objects or double detections), or add missing galaxies that appear to be cluster members based on their colors (not selected initially due to the strict magnitude cuts applied).

For the minimization, we consider (for both models) a positional uncertainty of 14. This value has been found to encompass both the underlying statistical uncertainties, the systematic uncertainties between our LTM and dPIEeNFW methods, as well as possible uncertainties arising from structure along the line of sight (Host 2012). To speed up the runtime of the models, and while RELICS HST images have a resolution of 006 pixel⁻¹, we adopt a resolution of 024 pixel⁻¹ for the minimization.

We present now the multiply imaged systems used in this work, labeled in Table A1 and Figure 1. We first include the multiply imaged system reported by Dessauges-Zavadsky et al. (2017) who identified six two-knot images belonging to this lensed object. We label this as systems 1 and 2 in Table A1 and Figure 1, corresponding to the two different emission knots.

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In addition, based on the full HST ACS + WFC3/IR data set, now including RELICS imaging, we detect additional multiply imaged systems (without a spectroscopic confirmation yet), which we identify based on their morphology and color similarity, as well as predictions from a preliminary LTM model. These images are included as lensing constraints in our final models and we briefly review them here. Systems 3, 4, and c5 straddle the critical curve each forming two bright knots stretching into an arc with their counterimages on the opposite side of the cluster (we note, however, that the identification of the counterimage remains less secure; the LTM, dPIEeNFW, and GLAFIC models implement this image in the SL modeling while the Lenstool model does not). System 6 is lensed into three images appearing as two distinct pink and blue knots (systems 6.1 and 6.2 in our modeling) in the composite HST ACS and WCF3/IR images (Figure 1), making this identification reliable. System 7 is lensed into three images which are identified mainly thanks to their similar, red dropout colors as they do not present any peculiar morphology. This system is a relatively high-redshift dropout object (z_phot ∼ 4.41 based on its first image for which the photometric redshift is the most reliable), hence its redshift is assumed correct and we fix it in our model. Finally, system 9 is lensed into three diffuse red images (in the ACS and WFC3 composite image). We also identify additional, potential systems that were, however, not included in our SL modeling as constraints. System c8 appears to be lensed into two images straddling the critical curve. We do not, however, identify a third counterimage on the other side of the critical curves so we report it only as a candidate system. System c10 lies next to system 9 and is comprised of two faint emission knots. Our SL models predict additional counterimages near a group of cluster member galaxies located at R.A. = 8.0470889 deg; decl. = 18.117521 deg. However, due to the light contamination, we cannot make a reliable identification. Systems c11, c12, and c13 consist each of two arcs straddling the critical curve in the northern region of the cluster. Lacking a WFC3/IR coverage in that northern area of the cluster, and BPZ yielding significantly different photometric redshift estimates, we decided to keep these systems as candidate systems. Spectroscopic confirmation of these systems would help to more accurately constrain the mass distribution in the most northern region, where no other lensing constraints are seen.

• 1.  
  LTM. The LTM model is built by considering the weight of the five brightest cluster members (labeled in Figure 1) as free parameters, i.e., allowing their M/L ratio to vary. We also consider as free parameters the ellipticity (varying within a flat, small prior of ±0.05 from the measured value from the light distribution) and position angle (varying within ±5° from the input value we measured) of the bright galaxies located at R.A. = 8.04691, decl. = 18.118922; and R.A. = 8.039185, decl. = 18.115616, (galaxies B and C in Figure 1).We scale our model to the spectroscopic redshift of systems 1 and 2 (see Table A1). The redshift of the remaining systems, except dropout system 7, are left as free parameters to be optimized in the minimization procedure (allowing the relative ${D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$ ratio for each system, corresponding to its best-fit z_phot value, to vary by up to ±0.05; with D_S and D_LS being the angular diameter distances to the source and between the lens and the source). Taking into account the additional freely optimized cluster members and source redshifts, our final model includes a total of 20 free parameters. The resulting critical curves (for a source at z_s = 2 and z_s = 9) for our final best-fit model which has an image reproduction rms = 160, are shown in Figure 1. Finally, the image reproductions by our LTM lens model are shown in Figure B1, showing that the overall shape, orientation, and internal details are satisfactorily recovered.
• 2.  
  dPIEeNFW. In merging clusters such as the one analyzed here, multiple DM halos are usually incorporated in the modeling in order to better explain the mass distribution. Our best-fit model comprises three large-scale DM halos whose centers are fixed to the position of the cluster members labeled A, B, and C in Figure 1. The large-scale DM halos are parameterized with eNFWs, where their ellipticity parameters, concentration, and mass are optimized. Cluster members are modeled with a dPIE profile with a fixed core radius of 0.2 kpc for the reference galaxy, a velocity dispersion that is allowed to vary between 80–120 km s⁻¹, and a cutoff radius varying from 45–65 kpc. They are modeled as spherical with a mag0 = −21.56 (a reference magnitude for the scaling relations; Faber & Jackson 1976; Jullo et al. 2007). Similarly as in the LTM model, we leave the ellipticities and position angles of the two bright galaxies presented above to be optimized.We adopt the same multiple images and cluster member catalogs as for the LTM model, leaving as free parameters the redshifts of background sources that have not been spectroscopically confirmed (except for system 7). Our final model includes a total of 24 free parameters. Our best-fit model has an image reproduction of rms = 143 and for comparison, we also show in Figure 1 the resulting critical curves for a source at z_s = 9.

Figure 1 shows the critical curves from our lens models. Both the LTM and dPIEeNFW models, despite having a very different representation for the different mass components, overall yield similar critical curves. This is perhaps somewhat expected, given that similar sets of multiple images were used as constraints, although notable differences exist as well—especially in regions of high magnification or regions with fewer constraints.

Figure 2 shows the convergence profile for RXC0032. The SL region is dominated by a large number of substructures, accounting for the shallow inner profile. We also note that the resulting mass distribution has a high overall elongation, or ellipticity, computed as $e=({a}^{2}-{b}^{2})/({a}^{2}+{b}^{2})$, of e ∼ 0.74(0.81) ± 0.03 from the LTM(dPIEeNFW) models.

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We compute the value of the effective Einstein radius as ${\theta }_{{\rm{E}}}=\sqrt{A/\pi }$, with A defined as the area enclosed within the critical curves. Our SL analysis reveals a particularly prominent lens with a resulting Einstein radii of ${\theta }_{{\rm{E}}}({z}_{s}=2)\,\sim 38\buildrel{\prime\prime}\over{.} 50(39\buildrel{\prime\prime}\over{.} 60)\pm 0\buildrel{\prime\prime}\over{.} 20$ and θ_E(z_s = 9) ∼ 4810(4810) ± 025 from our LTM(dPIEeNFW) best-fit models, corresponding to an enclosed mass of $M(\lt {\theta }_{{\rm{E}}})=2.13(2.67)\pm 0.2\,\times {10}^{14}{M}_{\odot }$ within the z_s = 2 critical curves. The high degree of substructures aggregated in the center yields this very large Einstein radius, similar to only a few other clusters. While the SZ signal, and gas probes in general, lead to the discovery of clusters of high virial mass (Williamson et al. 2011; Planck Collaboration et al. 2016), a high total mass does not guarantee a large SL region, and in the case of RXC0032, it seems that the high cluster richness, for example, better traces the large Einstein radius. RXC0032 portrays the loose relation—or at least the large scatter in the relation—between gas probes (and the SZ effect in particular) and the central SL area, which depends more closely on various other factors (Giocoli et al. 2016) and especially the amount of matter concentrated or projected in the very center, the cluster triaxial shape, and its orientation along the line of sight (King & Corless 2007; Oguri & Blandford 2009).

We show in Figure 3 the z_s = 9 magnification map from our LTM best-fit model as well as the location of the detected high-redshift galaxy candidates within RXC0032's field of view covered by both ACS and WFC3 (Salmon et al. 2020). We compare in Table 2 the magnification estimates from both methods and provide the intrinsic M_uv at λ = 1500 Å for the high-redshift candidates.

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Table 2.  High-z (z ∼ 6) Lensed Candidates

Galaxy IDa	R.A.	Decl.	J125b	${z}_{\mathrm{phot}}^{\mathrm{BPZ}}$ c	${z}_{\mathrm{phot}}^{\mathrm{EZ}}$ c	μLTMd	μdPIEeNFWd	μLenstoold	μGLAFICd	Muv,1500e
	(deg)	(deg)
RXC0032+18-0571	8.052543	18.131884	26.42 ± 0.25	${6.46}_{-0.8}^{+0.7}$	${6.7}_{-1.5}^{+0.7}$	${88.43}_{-22.99}^{+21.89}$	${19.96}_{-4.40}^{+4.55}$	${4.27}_{-1.18}^{+1.25}$	${53.20}_{-61.73}^{+12.49}$	$-{16.10}_{-0.60}^{+0.45}$
RXC0032+18-0052	8.046714	18.147703	26.07 ± 0.26	${5.5}_{-0.7}^{+0.2}$	${5.8}_{-0.8}^{+0.2}$	${4.82}_{-0.23}^{+0.17}$	${5.17}_{-0.24}^{+0.35}$	${1.82}_{-0.17}^{+0.18}$	${2.74}_{-0.37}^{+0.38}$	$-{18.96}_{-0.35}^{+0.30}$
RXC0032+18-0355	8.062891	18.138279	26.64 ± 0.28	${5.8}_{-5.1}^{+0.3}$	${6.1}_{-5.3}^{+0.3}$	${3.75}_{-0.08}^{+0.08}$	${2.81}_{-0.12}^{+0.09}$	${2.08}_{-0.81}^{+0.71}$	${2.30}_{-0.22}^{+0.25}$	$-{18.60}_{-0.90}^{+0.30}$

Notes.

^aGalaxy ID, following Salmon et al. (2020) notations. ^bApparent magnitude in the F125W band. ^cRedshift estimation based on the BPZ and EAZY pipelines along with their 1σ uncertainties. ^dBest-fit magnification estimates (at the respective source redshift) from the LTM, dPIEeNFW, Lenstool, and GLAFIC models. The statistical uncertainty is computed as the standard deviation from 100 MCMC models. The LTM best-fit value is the one used for all relevant computations. ^eAbsolute magnitude, M_uv, at λ = 1500 Å for which the errors have been propagated from the photometric and magnification uncertainties based on our best-fit LTM model. The resulting rest-frame UV luminosities (corrected for lensing magnifications) have a mean of M_uv ∼ −17.90 with a standard deviation of 1.56.

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The derived lensing strength (in the right panel of Figure 3) shows that RXC0032 is a very prominent lens, with a large area of high magnification of about ∼4.4(3.4) arcmin² with μ > 5 to ∼2.4(1.8) arcmin² with μ > 10 from our LTM(dPIEeNFW) best-fit models. RXC0032's lensing strength is significantly larger than that of other RELICS clusters modeled with the LTM pipeline (Acebron et al. 2018, 2019; Cibirka et al. 2018) and similar or higher than those provided by most HFF cluster lenses.

However, the field of RXC0032 seems to be a unique sight line, compared to similarly-strong cluster lenses. Despite its lensing strength and large critical area, RXC0032 reveals only three z ∼ 6 high-redshift galaxy candidates detected in the field (Salmon et al. 2020); these are characterized in Table 2. RXJ0152.7-135 (RXJ0152 hereafter; Acebron et al. 2019) constitutes an interesting counterexample. The strong-lens modeling of both clusters, following the distribution of their member galaxies, reveals two clusters with similar morphologies, i.e., very elongated and showing a high degree of substructure. These effects have been shown to significantly boost the cluster total cross section (Meneghetti et al. 2007). This is clearly evident in the case of RXJ0152, which, despite being a much smaller lens (θ_E(z_s = 2) ∼ 9''; equivalent to a critical area of 0.06 arcmin²), lenses 24 high-redshift galaxy candidates.

The field of RXC0032 provides the lowest yield of high-redshift candidates in comparison to those listed in the right panel of Figure 3. While such a small high-redshift candidate sample can be attributed to cosmic variance (Somerville et al. 2004; Trenti & Stiavelli 2008) especially as RELICS targets the brightest distant objects, the low number of high-redshift candidates could also be explained in part by the fact that the HST/WFC3IR's field of view (123'' × 137'') is fairly small compared to the size of the lens. In Figure 3 (left panel) we show that a significant proportion of high-magnification regions fall outside of the instrument's field of view. Another effect that may play a role in the low number of uncovered high-redshift galaxies is the lower completeness found around the critical curves (Oesch et al. 2015), and conceivably within them where the intracluster light is brighter. Future deeper observations in a wider field of view around RXC0032 should be able to detect more high-redshift galaxies as well as to examine these hypotheses.

The high-z galaxy candidate RXC0032+18-0571 (see Table 2) lies in a very high-magnification area. In the composite (ACS+WFC3) image, this object appears as two distinct light emitting knots, one of which appears to be stretched into an ∼15 arc. This is in agreement with our lens model, which predicts a similar stretching for the arc, further supporting the high-z nature of this object. Our SL models also predict a counterimage on the other side of the cluster. However, based on the RELICS high-z photometric study (Salmon et al. 2020), all apparent counterimage candidates have a lower, best-fit photometric redshift estimate. We show the predicted position of the counterimage in Figure 4 based on our LTM model. Our lens model also indicates that the candidates RXC0032+18-0052 and RXC0032+18-0355 could be two images from the same background source, supporting the high-z nature of these objects as well.

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As part of the RELICS survey, RXC J0032 has also been modeled with the Lenstool (Jullo et al. 2007) and GLAFIC (Oguri 2010; Kawamata et al. 2016) pipelines whose high-end products are publicly available through the MAST archive²⁷. In this section, we compare these two models to our modeling results. We give here a short summary of these two models and refer the reader to the references mentioned above for further details.

It should be noted that often in comparison studies, the same constraints are used throughout with the goal of comparing the different methodologies explicitly (e.g., Zitrin et al. 2015; Meneghetti et al. 2017). In contrast, here our goal is mainly to probe the credibility of our results, and especially the large Einstein radius estimation. We therefore incorporate the Lenstool and GLAFIC models as well, since these were constructed completely independently by other groups within the RELICS collaboration, including independently identified multiple-image sets (presented in Tables A1 and A2). The differences between the results of these different methods also provide the reader with a more quantitative assessment of the magnitude of underlying systematic uncertainties in the presented analysis. It should also be mentioned that while the image recovery rms is often used in assessing the reliability of strong-lens models, a comparison of the rms values of models that use different constraints is of little use (e.g., Johnson & Sharon 2016).

We first review additional multiple images that were identified independently by the Lenstool (L) and GLAFIC (G) teams and were incorporated in their modeling. System L14 is composed of two multiple images straddling the critical curves in the very northern part of the cluster. However, due to a large rms value for image L14.2, we would consider this a candidate identification. The two multiple images of system L15 appear as orange in the composite image and L15.1 is stretched into an arc and may consist of two merging counterimages. System L16 is lensed into two diffuse arcs, the second being a radial arc. Systems L17 and L18, each composed of three multiple images, appear as brown and pink emission knots in the central region of the cluster, respectively. System L19 comprises three multiple images, each presenting a similar morphology with two emission knots. System L20 lies next to system c8, and is composed of two diffused knots. System G21 is stretched into a blue arc in the composite image with two bright emission knots at each end, and is the only system identified in the southeastern region of this cluster. Finally, system L22 is the most southern multiply imaged system identified and lies next to a small group of cluster members which may introduce a more local galaxy–galaxy lensing effect. To avoid confusion, all multiple images presented throughout this work are shown in Figure 1 and summarized in Table A1. Table A1 also indicates which images were used as constraints in each model, displayed in Figure B2.

We review the Lenstool and GLAFIC models here.

• 1.  
  Lenstool. This model is built with four large-scale halos parameterized with a dPIE density profile. Their central coordinates, as well as their ellipticity, position angle, core radius, and velocity dispersion, are left to be freely optimized. The best-fit positions for the large-scale DM halos are found to be: R.A., decl. = (8.048919, 18.1298051); (8.045599, 18.120679); (8.049396, 18.146159); (8.040309, 18.115001), and are shown in Figure 4. The small-scale halos associated with galaxy members, identified via the red-sequence method, are modeled with a dPIE profile with a fixed core radius of 0.15 kpc, while both the velocity dispersion and the cutoff radius of a fiducial reference galaxy are freely optimized following the adopted scaling relations (Faber & Jackson 1976; Jullo et al. 2007). Aside from systems 1 and 2 that have a spectroscopic redshift measurement, the redshifts of all other multiple images used in the modeling are freely optimized. All multiple images are included in the models with a positional uncertainty of 03 and the optimization is performed in the image plane. The best-fit model results in an image reproduction of rms = 059.
• 2.  
  GLAFIC. This model includes four eNFW large-scale halos which have been fixed to the following coordinates (displayed in Figure 4): R.A, decl. = (8.049478, 18.143654); (8.047251, 18.116557); (8.039177, 18.115615); (8.040402, 18.123657), while the mass, ellipticity, position angle, and concentration parameters are left as free parameters. The SL model also includes cluster members identified with the red-sequence method that are modeled as pseudo-Jaffe ellipsoids (Keeton 2001) and following the scaling relations. The redshifts of all multiple images used in the GLAFIC SL model, bar systems 1 and 2, are optimized assuming a Gaussian prior (with δ_z = 0.5) around their photo-z estimates. A positional uncertainty of 06 is assumed for all multiple images. The GLAFIC best-fit model has an image reproduction of rms = 049.

As can be seen (see also Figures 1 and 4), the parametric models (i.e., dPIEeNFW, Lenstool, and GLAFIC) have each followed a different modeling prescription: the dPIEeNFW has incorporated three large-scale halos; the Lenstool and GLAFIC both have four halos, although their positions slightly vary. The LTM, in contrast, is only semiparametric and adopts a substantially different methodology. In addition, unlike the other parametric models, the dPIEeNFW is constructed on a fixed grid.

We compare here the main outputs of all four SL models, i.e., the convergence profile, magnification estimates, the resulting critical curves, Einstein radius, and the redshift estimates for the multiple images.

We show in Figure 2 a comparison of the convergence profiles. All convergence profiles are in good agreement within the 1σ error bars. However, the innermost region of the cluster is less well constrained, possibly due to uncertainties related to the chosen modeling techniques. Another issue is that the statistical uncertainties in the LTM and dPIEeNFW models seem to be smaller than those in the Lenstool and GLAFIC models. We are in the process of examining the origin for this discrepancy, which might be, for instance, related to the finite and lower resolution of the LTM and dPIEeNFW models (which also tends to boost the official χ² quoted for them). In addition, our fixing of system 7, the highest-redshift system used here, to its best photometric redshift value, can also contribute to lowering the errors compared to the two other models.

The resulting critical curves are found to be in good agreement between all models as shown in Figure 4. More significant differences are found in regions with no SL constraints such as the most northern or southeastern regions of the cluster. In the case of the LTM lens model, the differences in the modeling of the southeastern region of the cluster may explain the higher rms value reported for the images 1.5 and 2.5.

We find that all modeling tools consistently reveal a large Einstein radius (see Table 3), with a mean and standard deviation estimates of θ_E(z_s = 2.0) = 39.44 ± 2.0 and θ_E(z_s = 9.0) = 48.15 ± 1.47.

All models do also estimate RXC0032 to have a prominent lensing strength and are in very good agreement regarding the total area with high magnification (see Figure 3, right panel). However, as expected, large discrepancies between reconstructions appear around the lens critical lines, or the highest magnification regions (Meneghetti et al. 2017), as is demonstrated by the magnification estimates of the high-z candidates in Table 2. These values should thus be used with caution.

Finally, we compare in Figure 5 the model-predicted redshifts among the four models presented in this work, which are also reported in Tables A1 and A2. We find that the LTM and dPIEeNFW models are in good agreement with each other, however they predict systematically higher redshifts than the Lenstool and GLAFIC models. The redshifts obtained with the Lenstool pipeline are slightly overestimated with respect to those derived with GLAFIC but remain in fairly good agreement. Part of the reason for the systematic underestimate of the Lenstool and GLAFIC models compared to the LTM and dPIEeNFW models is that the highest-redshift multiple-image system, system 7, was fixed to its dropout redshift value for the LTM and dPIEeNFW models while in the Lenstool and GLAFIC models it was not, resulting in systematically lower redshifts. It should also be noted that the statistical uncertainties do not encompass the differences between models, which are more representative of the underlying uncertainties.

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Note that all the SL models presented in this work have been built with only one multiply imaged system that has a spectroscopic redshift confirmation (with its two emission knots used as constraints in the modeling). It will be interesting to revise the differences between the models when more secure redshifts are measured.

The lensed star-forming galaxy at z = 3.6314 (corresponding to our systems 1 and 2) was studied by Dessauges-Zavadsky et al. (2017). They used the Lenstool pipeline to constrain the mass distribution of the RXC0032 lensing model based on the z = 3.6314 system (considering the two emission knots as in the other models presented in this work), as well as two other, triply imaged systems which were spectroscopically confirmed (although the redshift was not published). They derived the magnification for the lensed images of systems 1 and 2, as shown in Table 4, which are necessary to study the intrinsic properties of the source.

Here, we compare in Table 4 the magnification estimated for each image of the system derived by the LTM, dPIEeNFW, Lenstool, and GLAFIC algorithms and the model presented in Dessauges-Zavadsky et al. (2017). The magnification values between the strong-lens models differ significantly (the dispersion between models being larger than the statistical uncertainties), which makes a reliable estimation of the intrinsic properties of the background source challenging. It will be interesting, when more spectroscopic data becomes available, to analyze if the lens models converge toward similar magnification values.

The RELICS survey was designed to efficiently discover and characterize bright high-redshift galaxies that are magnified by massive cluster lenses, as well as to identify which galaxy clusters are the most efficient lenses for future follow-up campaigns (Coe et al. 2019). Based mostly on RELICS observations, we present here a full SL analysis of the merging galaxy cluster RXC J0032.1+1808. More recently, efforts have focused on SL systematic uncertainties arising from different modeling techniques (Johnson & Sharon 2016; Meneghetti et al. 2017; Remolina González et al. 2018). In this work, we have adopted two different methodologies, the LTM technique and a fully parametric model, dPIEeNFW. We have also compared our results with the models obtained with the Lenstool and GLAFIC pipelines, that were independently constructed, so that our conclusions can be made more robust. As we show throughout, the results for most quantities of interest seem to agree fairly well between the different models. In that sense, differences between the LTM, dPIEeNFW, Lenstool and GLAFIC resulting models are then more representative of the true underlying uncertainties than the magnitude of the statistical uncertainties from the respective minimization procedures.

The derived mass distribution and Einstein radius of RXC0032 reveals a very prominent lens, with a large effective Einstein radius of θ_E ∼ 40'' at z_s = 2.0, as supported by all models probed here. Since mergers enhance the lensing cross section, merging clusters such as RXC0032 are of particular interest for the statistical study of the strongest gravitational lenses and are particularly useful when comparing Einstein radius distributions from observations to those from theoretical expectations (Redlich et al. 2012, 2014). While RELICS has only uncovered three high-redshift galaxy candidates in this field (Salmon et al. 2020), we find that RXC0032 is a promising lens to carry out wider and deeper imaging follow-up in order to expand the present coverage to all high-magnification regions.

All the lens models presented in this work and their corresponding deflection fields, magnification maps for different redshifts, as well as 100 random models to calculate errors, are made publicly available through the MAST archive²⁷.

We kindly thank the anonymous referee for the insightful and 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. This work was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI grant Nos. JP15H05892 and JP18K03693. K.U. acknowledges support from the Ministry of Science and Technology of Taiwan under the grant MOST 106-2628-M-001-003-MY3. This work is partially supported by the Australian Research Council Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO-3D). S.T. acknowledges support from the ERC Consolidator Grant funding scheme (project ConTExt, grant No. 648179). The Cosmic Dawn Center is funded by the Danish National Research Foundation.

Facilities: HST - Hubble Space Telescope satellite, Spitzer - , MAST. -

We present in Tables A1 and A2 the list of multiple images used in this work, as well as candidate identifications. All systems have been identified independently by different lensing teams.

Table A1.  Multiple Images and Candidate Identifications for RXC J0032.1+1808

Arc IDa	R.A.	Decl.	zphot [zmin–zmax]b	zspec	${z}_{\mathrm{model}}^{\mathrm{LTM}}$[68% C.I.]c	${z}_{\mathrm{model}}^{\mathrm{dPIEeNFW}}$[68% C.I.]c	LTM/dPIEeNFWd	Lenstoold	GLAFICd	Rmse
	(deg)	(deg)								('')
1.1	8.031533	18.114345	0.24 [0.17–0.64]	3.6314f	-	-	✓	✓	✓	1.32
1.2	8.032142	18.113763	0.37 [0.27–0.43]	"	-	-	✓	✓	✓	0.40
1.3	8.032671	18.112831	0.31 [0.23–0.47]	"	-	-	✓	✓	✓	1.58
1.4	8.034162	18.111284	3.99 [0.25–4.36]	"	-	-	✓	✓	✓	0.65
1.5	8.040679	18.106825	0.44 [0.24–0.69]	"	-	-	✓	✓	✓	2.15
1.6	8.031840	18.113526	-	"	-	-	✓		✓	0.34
2.1	8.031454	18.114526	3.54 [3.42–3.65]	3.6314f	-	-				1.36
2.2	8.032260	18.113637	4.01 [3.90–4.11]	"	-	-	✓	✓	✓	0.70
2.3	8.032492	18.113161	3.61 [3.55–3.74]	"	-	-	✓	✓	✓	1.15
2.4	8.034382	18.111196	3.61 [3.51–3.77]	"	-	-	✓	✓	✓	0.44
2.5	8.040661	18.106926	0.30 [0.11–3.61]	"	-	-	✓	✓	✓	2.06
2.6	8.031893	18.113509	-	"	-	-	✓		✓	0.76
3.1	8.050767	18.131576	1.05 [1.01–1.06]	-	2.10 [2.04–2.20]	2.14 [2.14–2.23]	✓	✓	✓	0.49
3.2	8.049925	18.131756	1.05 [1.01–1.76]	-			✓	✓	✓	2.50
3.3	8.035982	18.134820	1.63 [1.00–1.79]	-			✓		✓	0.83
4.1	8.051190	18.131482	-	-	2.23 [2.00–2.21]	2.15 [2.08–2.16]	✓		✓	1.04
4.2	8.049535	18.131823	-	-			✓		✓	1.86
4.3	8.036149	18.134948	0.97 [0.70–1.96]	-			✓		✓	1.62
c5.1	8.051015	18.131285	1.70 [1.58–1.78]	-	∼2.3	∼2.0			✓	-
c5.2	8.049483	18.131599	1.41 [1.39–1.81]	-					✓	-
c5.3	8.035952	18.134634	1.59 [1.13–1.92]	-					✓	-
6.11	8.042302	18.142446	1.71 [1.71–1.78]	-	1.61 [1.56–1.70]	1.73 [1.73–1.84]	✓	✓	✓	0.24
6.12	8.046450	18.140492	1.10 [1.02–1.14]	-			✓	✓	✓	0.62
6.13	8.059917	18.139378	1.11 [1.02–1.71]	-			✓	✓	✓	1.65
6.21	8.042491	18.142400	1.03 [0.93–1.09]	-	1.64 [1.53–1.60]	1.89 [1.78–1.92]	✓		✓	0.26
6.22	8.046380	18.140578	-	-			✓		✓	0.69
6.23	8.060023	18.139325	-	-			✓		✓	1.70
7.1	8.036775	18.128377	4.41 [0.42–4.66]	-	4.4g	4.4g	✓	✓	✓	1.25
7.2	8.043467	18.126995	0.30 [0.16–3.96]	-			✓	✓	✓	2.13
7.3	8.060019	18.123190		-			✓	✓	✓	3.15
c8.1	8.051712	18.122845	2.82 [2.74–3.11]	-	∼2.7	∼2.7			✓	-
c8.2	8.049750	18.123512	0.05 [0.02–3.07]	-					✓	-
9.1	8.047754	18.114628	3.25 [3.00–3.53]	-	2.83 [2.84–3.06]	2.38 [2.03–2.39]	✓	✓	✓	3.70
9.2	8.053467	18.117267	-	-			✓	✓	✓	2.13
9.3	8.032281	18.122876	2.14 [1.85–2.28]	-			✓	✓	✓	2.39
c10.1	8.053825	18.117562	2.32 [2.17–2.53]	-	∼2.7	∼2.2				-
c10.2	8.047263	18.114624	-	-						-
c10.3	8.032309	18.123215	-	-						-
c11.1	8.049167	18.153004	2.76 [0.07–3.08]	-	∼3	∼3		✓	✓	-
c11.2	8.047482	18.152860	2.71 [0.36–3.10]	-				✓	✓	-
c12.1	8.049383	18.152779	-	-	∼3	∼3				-
c12.2	8.048008	18.152721	1.36 [0.59–2.66]	-						-
c13.1	8.048654	18.152527	-	-	∼3	∼3				-
c13.2	8.047547	18.152442	1.54 [0.16–2.98]	-						-
L14.1	8.046523	18.155275	0.55 [0.46–4.54]	-	-	-		✓		-
L14.2	8.051645	18.155275	0.92 [0.75–1.42]	-	-	-		✓		-
L15.1	8.043844	18.148375	3.11 [0.26–3.32]	-	-	-		✓		-
L15.2	8.061134	18.144614	3.19[1.87–3.27]	-	-	-		✓		-
L16.1	8.054572	18.140649	-	-	-	-		✓		-
L16.2	8.051372	18.140131	-	-	-	-		✓		-
L17.1	8.040382	18.131975	4.23 [0.34–4.49]	-	-	-		✓		-
L17.2	8.042291	18.131664	0.77 [0.19–0.90]	-	-	-		✓		-
L/G18.1	8.039637	18.131971	2.64 [2.35–2.92]	-	-	-		✓	✓	-
L/G18.2	8.043413	18.131387	2.27 [2.22–2.55]	-	-	-		✓	✓	-
L18.3	8.059395	18.127467	2.31 [2.04–2.61]	-	-	-		✓		-
L19.1	8.051801	18.126669	3.94 [3.46–4.30]	-	-	-		✓		-
L19.2	8.052461	18.12658	4.37 [4.24–4.48]	-	-	-		✓		-
L19.3	8.028309	18.131135	3.87 [3.06–4.50]	-	-	-		✓		-
L20.1	8.050102	18.123303	2.43 [0.15–2.90]	-	-	-		✓		-
L20.2	8.051282	18.122912	0.08 [0.05–3.08]	-	-	-		✓		-
G21.1	8.048514	18.107541	1.59 [0.45–2.57]	-	-	-			✓	-
G21.2	8.047208	18.107382	1.58 [0.46–2.55]	-	-	-			✓	-
L22.1	8.035258	18.106743	-	-	-	-		✓		-
L22.2	8.033758	18.107896	-	-	-		✓		-

Notes.

^aIdentification of all multiple images uncovered. Those identified with a "c" refer to less reliable therefore candidate identifications that were not included in our SL models. Systems identified with an "L" or "G" refer to those independently identified by the Lenstool and GLAFIC modeling teams, respectively. ^bPhotometric redshift with upper and lower limits, based on the BPZ estimates from the RELICS catalog with the 95% confidence range; a "-" sign indicates an image for which its z_phot could not be measured due to light contamination or poor signal-to-noise ratio. ^cRedshift prediction based on our LTM and analytical dPIEeNFW best-fit models, respectively. ^dThe symbol ✓ indicates whether a multiple image is used in the SL modeling by the LTM/dPIEeNFW, Lenstool and GLAFIC pipelines. ^eRms between the observed and model-predicted multiple images from our LTM best-fit model. ^fMeasured by Dessauges-Zavadsky et al. (2017), considered as a fixed redshift in our SL models. ^gWe fix the redshift of system 7 which is a "dropout" object.

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Table A2.  Redshift Estimates for the Multiple Images in the RXC J0032.1+1808 Cluster Field from the Lenstool and GLAFIC Models

Arc IDa	${z}_{\mathrm{model}}^{\mathrm{Lenstool}}$[68% C.I.]b	${z}_{\mathrm{model}}^{\mathrm{GLAFIC}}$[68% C.I.]b	RmsLenstoolc	RmsGLAFICc
			('')	('')
1.1	3.6314d	3.6314d	0.55	0.57
1.2	"	"	0.32	0.05
1.3	"	"	0.41	0.33
1.4	"	"	0.49	0.54
1.5	"	"	0.75	0.96
1.6	"	"	⋯	0.12
2.1	3.6314d	3.6314d	0.56	0.47
2.2	"	"	0.38	0.03
2.3	"	"	0.05	0.43
2.4	"	"	0.58	0.52
2.5	"	"	0.62	1.07
2.6	"	"	⋯	0.19
3.1	1.01 [1.00–1.15]	1.57 [1.47–1.69]	0.11	0.28
3.2	"	"	0.17	1.14
3.3	⋯	"	⋯	0.30
4.1	⋯	1.72 [1.57–1.85]	⋯	0.38
4.2	⋯	"	⋯	0.36
4.3	⋯	⋯		⋯
c5.1	⋯	1.68 [1.55–1.88]	⋯	0.33
c5.2	⋯	"	⋯	0.31
c5.3	⋯	⋯	⋯	⋯
6.11	2.11 [2.05–2.19]	⋯	0.60	⋯
6.12	"	⋯	0.94	⋯
6.13	"	⋯	0.09	⋯
6.21	⋯	1.34 [1.29–1.47]	⋯	0.58
6.22	⋯	"	⋯	0.38
6.23	⋯	"	⋯	0.19
7.1	3.00 [3.00–3.01]	2.82 [2.54–3.07]	0.90	0.91
7.2	"	"	0.31	0.26
7.3	"	"	0.93	0.86
c8.1	⋯	2.83 [2.63–3.12]	0.50	0.15
c8.2	⋯	"	0.40	0.15
9.1	2.22 [2.16–2.26]	1.99 [1.91–2.09]	0.21	0.50
9.2	"	"	0.60	0.58
9.3	"	"	0.80	0.13
c11.1	1.28 [1.22–1.68]	2.78 [2.28–3.22]	0.22	0.13
c11.2	⋯	"	0.25	0.04
L14.1	1.95 [1.74–3.52]	⋯	0.31	⋯
L14.2	"	⋯	0.08	⋯
L15.1	3.01 [3.01–3.05]	⋯	0.53	⋯
L15.2	"	⋯	1.22	⋯
L16.1	2.43 [2.24–2.97]	⋯	0.62	⋯
L16.2	"	⋯	0.73	⋯
L17.1	2.54 [2.40–2.66]	⋯	0.54	⋯
L17.2	"	⋯	0.47	⋯
L/G18.1	2.05 [2.00–2.08]	1.65 [1.51–1.81]	0.60	0.37
L/G18.2	"	"	0.98	0.30
L18.3	"	⋯	0.63	⋯
L19.1	5.00 [4.97–5.00]	⋯	0.64	⋯
L19.2	"	⋯	0.48	⋯
L19.3	"	⋯	1.19	⋯
L20.1	2.70 [2.58–2.85]	⋯	0.48	⋯
L20.2	"	⋯	0.47	⋯
G21.1	⋯	1.74 [1.58–1.97]	⋯	0.03
G21.2	⋯	"	⋯	0.02
L22.1	1.43 [1.33–1.55]	⋯	0.08	⋯
L22.2	"	⋯	0.06	⋯

Notes.

^aIdentification of the multiple images following Figure 1 and Table A1. ^bRedshift prediction based on Lenstool and GLAFIC best-fit models with the 1σ statistical uncertainty, respectively. ^cIndividual rms between the observed and model-predicted multiple images from the Lenstool and GLAFIC models. ^dConsidered as a fixed redshift in the Lenstool and GLAFIC models.

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We present here the reproduction of the multiple images identified in RXC J0032.1+1808's field with our LTM pipeline (see Figure B1), as well as HST cutouts for the additional images independently identified by the Lenstool and GLAFIC lensing teams (see Figure B2).

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