Precision MARS Mass Reconstruction of A2744: Synergizing the Largest Strong-lensing and Densest Weak-lensing Data Sets from JWST

We present a new high-resolution free-form mass model of A2744 that combines both weak-lensing (WL) and strong-lensing (SL) data sets from JWST. The SL data set comprises 286 multiple images, presenting the most extensive SL constraint to date for a single cluster. The WL data set, employing photo-z selection, yields a source density of ∼350arcmin−2 , marking the densest WL constraint ever. The combined mass reconstruction enables the highest-resolution mass map of A2744 within the ∼1.8 Mpc × 1.8 Mpc reconstruction region to date, revealing an isosceles triangular structure with two legs of ∼1 Mpc and a base of ∼0.6 Mpc. Although our algorithm, which is called MAximum-entropy ReconStruction (MARS), is entirely blind to the cluster galaxy distribution, the resulting mass reconstruction traces the brightest cluster galaxies remarkably well. The five strongest mass peaks coincide with the five most luminous cluster galaxies within ≲2″. We do not detect any unusual mass peaks that are not traced by the cluster galaxies, unlike the findings in previous studies. Our mass model shows the smallest scatter of SL multiple images in both source (∼0.″05) and image (∼0.″1) planes, which is lower than in previous studies by a factor of ∼4. Although MARS represents the mass field with an extremely large number of free parameters (∼300,000), it converges to a solution within a few hours because we use a deep-learning technique. We make our mass and magnification maps publicly available.

Weak lensing can provide measurements for a wide area, sometimes extending beyond the virial radius of galaxy clusters.Its limitation is the intrinsic shape noise/alignment of background galaxies and systematics from instruments.In general, WL requires averaging of background galaxy shapes to reduce the shape noise, which inevitably smooths out substructures.The quality and resolution of WL mass reconstruction depend on the density of the source galaxies and on the reconstruction algorithms (Bartelmann & Schneider 2001;Schneider 2006).
Strong lensing enables us to obtain more precise and higherresolution mass maps.Because SL uses the positions of individual multiply-lensed features, it is immune to intrinsic source shapes and instrumental systematics, providing better performance in signal-to-noise ratio (S/N) and resolution.However, because SL features are observed only in the vicinity of cluster centers, SL mass reconstruction techniques alone cannot provide direct constraints beyond this so-called SL regime.The quality of SL mass reconstruction techniques improves with the number of multiple images and the redshift information available for them.
To attain a cohesive understanding of the cluster mass profile from the central region to the outskirts, one of the most effective approaches is to combine the WL and SL signals (e.g., Bradač et al. 2005;Cacciato et al. 2006;Jee et al. 2007b;Merten et al. 2009;Oguri et al. 2012;Giocoli et al. 2014;Zitrin et al. 2015).However, mass reconstruction with the optimal combination of WL and SL signals is challenging for various reasons.For the mass models based on analytic profiles, the degrees of freedom are too low to properly account for both SL and WL signals simultaneously and sufficiently with high fidelity (underfitting).Free-form methods may have advantages in this regard because of their high degrees of freedom.However, the free parameters usually outnumber the observables, and thus, the solutions are not unique unless they are carefully regularized (overfitting).In addition to these technical challenges, the scarcity of high-quality data sets providing WL and SL has also been an obstacle.Whereas SL data have been predominantly provided by Hubble Space Telescope observations, wide-field imaging data for WL measurements have mostly been obtained through ground-based observations.Thus, the severe disparity in mass resolution between the two regimes makes it difficult to implement an optimal smoothing scheme that preserves significant substructures while suppressing noisy fluctuations.
In this study, we present a new high-resolution free-form mass model of Abell 2744 (A2744 hereafter) at z = 0.308 from the central region to the outskirts by combining unprecedentedly large WL and SL data sets that use the recent high-quality wide-field ( 7. 6 7. 6 ~¢ ´¢ ) JWST images.The SL data set consists of 286 multiple images, whereas the WL source density, which is based on photo-z selection, reaches ∼350 arcmin −2 , which is the largest data set ever used for cluster mass reconstruction.
Recently, under the program Ultra-deep NIRCam and NIRSpec ObserVations before the Epoch of Reionization (UNCOVER; Bezanson et al. 2022;Weaver et al. 2024), deep wide-field (∼45 arcmin 2 ) JWST imaging observations were carried out.Based on the unprecedented depth and resolution of the new A2744 JWST data, several interesting scientific results have been published (e.g., Atek et al. 2023;Castellano et al. 2023;Furtak et al. 2023a;Morishita et al. 2023;Vulcani et al. 2023).However, in the case of the lens model, only lensing models based on parametric SL techniques have been presented.They employed the light-trace-mass (LTM) assumption to model the masses of the cluster galaxies.The center positions of the cluster-scale dark matter halos are allowed to move within the prior intervals centered on the brightest cluster galaxies (BCGs), with 3″ and 5″-30″ for Furtak et al. (2023b) and Bergamini et al. (2023b), respectively.
This study provides the first non-LTM and profile-independent mass reconstruction using not only multiple images (SL), but also distortion (WL) signals, which are densely distributed across the entire A2744 JWST field.We employ the algorithm called MAximum-entropy ReconStruction (MARS; Cha & Jee 2022, 2023).MARS is a free-form mass reconstruction method that uses cross entropy to regularize its solution, which results in a quasi-unique solution even though the number of free parameters greatly exceeds that of the observables.Cha & Jee (2022) tested the fidelity of MARS with synthetic cluster data (Meneghetti et al. 2017) and demonstrated that MARS is one of the best-peforming methods.The test with real HFF cluster data showed that the image-plane scatter of the multiple images is lower than in any of the publicly available mass models (Cha & Jee 2023).In the current study, we extend the previous MARS algorithm of Cha & Jee (2022) to accommodate WL constraints as well.
This paper is structured as follows.In Section 2, we introduce the JWST NIRCam imaging data and reduction steps.In Section 3, we describe the WL analysis method and our algorithm for the mass reconstruction by combining WL and SL.We show our results in Section 4. In Section 5, we discuss our results, and we conclude in Section 6.Unless stated otherwise, we assume a flat ΛCDM cosmology with the dimensionless Hubble constant parameter h = 0.7 and the matter density Ω M = 1 − Ω Λ = 0.3.The plate scale at the cluster redshift (z = 0.308) is 4.536 kpc arcsec 1 -.

JWST NIRCam Images
Our WL+SL analysis uses the publicly available NIRcam mosaic images of A2744 processed by the UNCOVER team, 4 who combined the three JWST programs (1) JWST-DD-ERS-1324 (PI: T. Treu;Treu et al. 2022), (2) JWST-GO-2561 (PIs: I. Labbe and R. Bezanson;Bezanson et al. 2022), and (3) JWST-DD-2756 (PI: W. Chen; Chen et al. 2022).We created color-composite images from the seven filter-imaging data (F115W, F150W, F200W, F277W, F356W, F410M, and F444W) and identified additional multiple images used for SL constraints based on their morphological properties and colors.We use F200W to measure WL because it provides the optimal scale for the point-spread function (PSF) sampling (Finner et al. 2023).In order to obtain the PSF model on the mosaic images, it is necessary to derive PSF models for input frames and stack them with proper rotations and dithers.Thus, we retrieved the input CAL files from the Mikulski Archive for Space Telescopes (MAST) 5 and found the coordinate transformation from the detector reference frame to the mosaic reference frame for each input CAL image.For more details on the mosaic image, we refer to Weaver et al. (2024).

Strong-lensing Data
We combine SL data from various sources in the literature and identify new multiple images.We classify the multiple images into gold and silver class.Gold-class images are those with spectroscopic redshifts.Silver-class images do not possess spectroscopic redshifts, but they either have photometric redshifts or have been identified as multiple images by various studies (e.g., Jauzac et al. 2015;Kawamata et al. 2016;Mahler et al. 2018;Bergamini et al. 2023aBergamini et al. , 2023b;;Furtak et al. 2023b).
Figure 1 shows the multiple-image distributions that we compiled for the current study.In region A, we adopt the catalog of Cha & Jee (2023), who compiled the SL data from Jauzac et al. (2015), Kawamata et al. (2016), Mahler et al. (2018), andBergamini et al. (2023a).All multiple images in Bergamini et al. (2023a) have spectroscopic redshifts.If a multiple-image system without a spectroscopic redshift is agreed to be a valid system by the three papers (Jauzac et al. 2015;Kawamata et al. 2016;Mahler et al. 2018), the system is classified as silver class.In regions B and C, we adopt 79 multiple images from Furtak et al. (2023b) and 9 multiple images that are knots (distinctive features such as star-forming regions in the extended multiple images) of system 68 from Bergamini et al. (2023b).If neither spectroscopic nor photometric redshift information is available, or if there is a considerable disagreement among the photometric redshifts within the same system (greater than , where z phot indicates the mean of the photometric redshifts within the same system), we treat the redshift of the system as a free parameter.We also free the redshift when MARS cannot converge multiple images with its input photometric redshift (see Section 3.2 for details).
In addition to the compiled catalog above, we have identified 16 new multiple-image candidates from six systems in region B (Figure 2), and all of them are classified as silver-class images.When multiple photometric redshifts are available for a system, we adopt the mean value as the system redshift.
A total of 286 images (136 gold and 150 silver images), including 91 knots, are used for the mass reconstruction of A2744.We list the multiple images in Table 1, where we follow the numbering scheme of Furtak et al. (2023b) for the multiple images in regions B and C.

Point-spread Function Modeling
In order to accurately perform a WL analysis, one must model and correct for the PSF.The relatively small field of view of NIRCam provides only a small number of stars per pointing.Thus, in some cases, it is not feasible to produce an empirical PSF model across the detector based on the limited number of observed stars.Jee et al. (2007a) overcame this problem for HST by building a PSF library from the observations of dense stellar fields such as globular clusters and by then using the observed stars in a WL field to find the The color-composite image is created using the F444W filter for red, the F277W filter for green, and the F115W filter for blue.The notation from Bergamini et al. (2023b) is followed for the labeling of the five brightest galaxies.The displayed field of view is 400″ × 400″.matching library.However, the number of NIRCam data for dense stellar fields is currently not sufficient to build an extensive PSF library such as in Jee et al. (2007a).Instead, in the current study, we employ the wavefront sensor data of JWST, which provide maps of the optical path difference (OPD).To use these OPD maps, we use the package WebbPSF (Perrin et al. 2012(Perrin et al. , 2014)).
WebbPSF provides two OPD maps: one map before and the other after the input (observation) date.The OPD map closest to the date of observation cannot always be guaranteed to be the best choice because the JWST optical alignments are monitored and adjusted on a regular basis.For the A2744 data, we verified that both OPD maps give similar results when compared with the PSF in the input frame.Therefore, we decided to choose the OPD map closest to each observation date to maintain consistency.
To verify that the PSFs are properly reconstructed by WebbPSF, we first collect star postage-stamp images for each detector from the A2744 data.Then, the PSF reconstruction for each star is produced using the selected OPD map for the given observation date.These star stamps are taken from calibrated images that are not corrected for distortion with the native pixel scale of the NIRCam short-wavelength channel.The residuals between the stars and the model are computed for the size and ellipticity.These shape parameters are measured using the following quadrupole moments (Jee et al. 2007a): q q q q q q q q q q = --Î I(θ) is the pixel intensity at θ, ¯i j , q is the center of the star, and W(θ) is a circular Gaussian weight function that we used to suppress noise in the outer regions of the PSF.We can then define the complex ellipticity components (e 1 , e 2 ) and size (R), For an ellipse with semimajor and -minor axes a and b and position angle f (measured counterclockwise from the reference axis), e 1 and e 2 correspond to   WebbPSF does not sufficiently take detector effects into account, such as interpixel capacitance or intrapixel sensitivity variations.To remedy this issue, a difference kernel is applied to the PSFs; we used a Gaussian kernel with an empirically determined kernel size.Represented by the black histogram, the residuals after applying the kernel are approximately centered at R = 0, with the median being ( R 3.35 ´-.Given the small magnitudes for both the ellipticity and size residuals (∼10 −4 ), our PSF model produced by WebbPSF and the difference kernel should be sufficient for a WL analysis.
After ensuring that the chosen model can effectively reproduce the detector effects seen in NIRCam data, a distortion-corrected and oversampled PSF model is required for use with the A2744 mosaic image.For each of the eight NIRCam detectors (NRCA1-4 and NRCB1-4) and for each observation date, 100 evenly spaced (distortion corrected and resampled with a pixel scale of 0.02) 31 pixel ×31 pixel PSFs are created using WebbPSF.A principal component analysis (PCA) is then performed to allow for the production of a 31 pixel ×31 pixel PSF at any detector position.Although WebbPSF could be used to produce a PSF at each galaxy centroid position, the high source count and the potential need for multiple stacked and rotated PSFs per galaxy makes PCA a more practical choice given the long execution time required by WebbPSF.A more detailed description of PSF modeling with PCA is provided in Jee et al. (2007a).To model a PSF at each galaxy position in the mosaic image, we identify all contributing input frames, retrieve their PSF models, and stack the results with proper weights and rotations.
The distribution of stars across the A2744 field allows for PSF modeling with PCA using the mosaic image in the same manner as in Finner et al. (2023), which can be compared to the modeling method used in this study.Further analysis and a justification for the choice of our PSF modeling strategy can be found in Appendix A.

Ellipticity Measurement and Source Selection
We use a forward-modeling approach to measure the ellipticity of each galaxy before it is convolved by the JWST PSF.The PSF model predicted at the location of each source was convolved with an elliptical Gaussian and fit to the galaxy using the MPFIT optimizer (Markwardt 2009).We fixed the background and centroid of a source to the values output by SExtractor (Bertin & Arnouts 1996).The free parameters are the position angle (f), semimajor and -minor axes (a and b), and the normalization.Ideally, the average of the galaxy ellipticity ( ) ( ) e e e e , c o s2 , s i n2 should be an unbiased estimator of the reduced shear g.However, because of a number of factors, in practice, the estimator is biased.Two outstanding contributors are noise bias and model bias.The noise bias occurs due to the nonlinear relation between pixel noise and parameter noise, whereas the model bias is caused by the fact that the galaxy model (in this case, the elliptical Gaussian) is different from the actual galaxy profile.In addition, the blending effect is also a significant source of shear bias.Instead of characterizing these biases individually, we performed WL image simulations matching the JWST quality and derived multiplicative factors of 1.11 and 1.07 for g 1 and g 2 , respectively (Jee et al. 2013;Finner et al. 2023).
To select the source galaxies, we used the photometric redshift catalog provided by the UNCOVER team. 6As a conservative measure, we selected sources with a photometric redshift greater than 0.4 as background objects.Additionally, we imposed shape quality criteria based on the fitting status and the recovered shapes.We discarded sources whose MPFIT STATUS parameter was different from unity because this typically indicates unstable fitting.Moreover, the minimum ellipticity measurement error was set to δe = 0.4.When the object size is reported to be too small, the source is typically either point source-like or unrealistically compact.We avoided these cases by imposing that the semiminor axis is greater than 0.4 pixels.The mean photometric redshift of the sources is 2.5, and the source density is 350 arcmin 2 ~-, which is the highest of all existing WL studies (the typical source density in HSTbased WL is 100 arcmin 2 ~-).We note that we did not explicitly mask out the SL areas when selecting WL sources.In general, because the galaxy shapes become curved in the SL regime, their ellipticities may underrepresent the local reduced shear.We find that approximately 60 objects are located within 0 2 of the multiple images.Because they comprise only ∼0.6% of the WL sources, we do not think that the bias caused by these sources is significant.Furthermore, given the much stronger constraining power from the SL multiple images in the SL regime, the bias, if any, should be negligible.

Lensing Theory
In this section, we provide a brief review of lensing theory, covering the range from the outskirts (WL) to the central regions (SL) of galaxy clusters.For more details, we refer to review papers (e.g., Bartelmann & Schneider 2001;Kochanek 2006;Kneib & Natarajan 2011;Hoekstra et al. 2013).In the WL regime, the characteristic scale of the variation in the distortion of the background galaxy image becomes much smaller than the galaxy size, and thus the change in the galaxy shape is approximated by the following matrix A: where κ indicates the convergence and g 1(2) denotes the first (second) component of the reduced shear The reduced shear g is computed as g = γ/(1 − κ), where γ is the shear.
The convergence κ is given by where Σ (Σ c ) is the (critical) surface mass density.Σ c can be computed as follows: where c is the speed of light, D s(d) denotes the angular diameter distance to the source (lens), and D ds represents the angular diameter distance between the lens and the source.The shear γ is related to the convergence κ through where the kernel D at the position (x 1 , x 2 ) is defined as In the SL regime, the absolute value of the reduced shear can exceed unity (|g| > 1).In this case, we replace g with 1/g * in Equation (5), where g * represents the complex conjugate of g.The relation between the observed image position θ and the source position β follows the lens equation, where α is called the deflection angle.The deflection angle α can be computed through the convolution of the convergence κ or the differentiation of the deflection potential Ψ.The MARS algorithm uses the convolution to obtain the deflection angle α as follows: Because the mass outside the reconstruction field affects both deflections (Equation ( 11)) and shears (Equation ( 8)) within the reconstruction field, we make the field size of the model 40% larger (∼2.5 Mpc × 2.5 Mpc) than the actual reconstruction field (∼1.8 Mpc × 1.8 Mpc).

MARS WL+SL Mass Reconstruction Algorithm
We employ the MARS algorithm (Cha & Jee 2022, 2023) to reconstruct the mass distribution of A2744.In our previous studies, the application of MARS was limited to SL mass modeling.In the current study, we revised MARS so that it can now also use WL signals.The new MARS minimizes the following function: where SL 2 c and WL 2 c represent the χ 2 terms for SL and WL observables, respectively, and R is the regularization term.The weight parameters m, w, and r determine the relative importance of the SL, WL, and regularization terms, respectively.
The reduction of SL 2 c decreases the scatter of the multiple images in the source plane (i.e., the positions of the delensed multiple images).SL 2 c is defined as where is the total number of systems, and J is the number of multiple images from each system.As is in Cha & Jee (2023), we treat each "knot" (distinctive feature, e.g., a star-forming region) within a multiple image as an individual image.We refer to Cha & Jee (2023) for details.
We define WL 2 c as follows: where g i,j (ò i,j ) indicates the jth component of the expected reduced shear (observed ellipticity) evaluated at the position and redshift of the ith WL source.σ m,i is the measurement error for the ith WL source, and σ s (g) is the shape noise per ellipticity component for the expected reduced shear g.In general, the shape noise decreases as g increases (i.e., every source galaxy becomes stretched in a nearly identical way, regardless of its intrinsic shape when g approaches unity).We use the following function to model the effect: where σ s (0) = 0.25 is the intrinsic shape noise in the region where there is no shape distortion by gravitational lensing.
By adopting the maximum cross entropy, MARS regularizes the mass reconstruction to prevent overfitting and to achieve the smoothest possible solutions, unless the substructures are strongly required by the data.The regularization term R is given by where κ and p are the convergence and prior, respectively.The prior is updated for each epoch of minimization by smoothing the convergence map obtained from the previous iteration using a Gaussian kernel.For more details, we refer to Cha & Jee (2022, 2023).The kernel sizes are σ = 0.6, 1.2, and 2.4 pixels for the 140 × 140, 280 × 280, and 560 × 560 mass grids, respectively.We double the kernel size when only WL data are used.
The WL+SL mass reconstruction run is carried out in the following steps: 1. Perform an SL-only (i.e., set w = 0 in Equation ( 12)) lowresolution mass reconstruction with a 140 × 140 mass grid, which includes 20-cell-thick stripes at the boundaries outside the reconstruction field (the actual mass reconstruction field has a resolution of 100 × 100).We start the minimization with a flat κ = 0.1 convergence field.The minimization ends when the multiple-image positions converge in the source plane.2. Add the WL 2 c term to evolve the SL-only solution from step 1 to the WL+SL solution.3. Increase the resolution of the grid from step 2 to 280 × 280 (by a factor of two) and restart the WL+SL mass reconstruction.4. Repeat step 3 by further increasing the resolution to 560 × 560 (now the marginal stripe is 80 cells thick, and the resolution within the reconstruction field is 400 × 400).We end the reconstruction process when the WL 2 c per WL component reaches unity, while the multiple-image scatter in the source plane is consistent with noise.
We treat the redshift of a multiple-image system as a free parameter if neither its spectroscopic nor its photometric redshift are known.When MARS cannot converge a multipleimage system with its known photometric redshift, we also free its redshift.These freed redshifts are constrained along with the mass reconstruction, and we refer to the values as model redshifts.As in Cha & Jee (2023), we set a flat prior for a model redshift with z model = [z cluster + 0.1, 15], where z cluster = 0.308 is the redshift of A2744.
When the peripheral grid cells are included, the final resolution of the resulting mass map is 560 × 560, requiring ∼300,000 free parameters.To minimize our target function f (Equation ( 12)) with this large number of free parameters, we use the Adam (adaptive moment) optimizer (Kingma & Ba 2014).The Adam optimizer is commonly used in deep learning to optimize complex models with an extremely large number of free parameters.Thanks to its efficiency, MARS converges to a solution within just a few hours.Although our main result is the result that is constrained by both WL and SL data, it is instructive to examine how the solution changes with respect to the full WL+SL reconstruction when only one of the two data sets is used.Therefore, we repeat the above mass reconstruction run with one data set at a time [for the WL-only (SL-only) reconstruction, we set m = 0 (w = 0) in Equation ( 12)] and include the comparisons in our discussion.

Projected Mass Distribution
We present our mass reconstruction results in Figure 4.The overall mass structure of A2744 within the current mass reconstruction field revealed by the WL+SL data sets is isosceles triangular and characterized by the three main subclusters: the northern (G3), northwestern (G1+G2), and southern (BCG-N + BCG-S) mass substructures.The legs of the isosceles triangle (the distances between BCG-S and G2 and between BCG-S and G3) are ∼1 Mpc, whereas the base (the distance between G3 and G2) is ∼0.6 Mpc.The northwestern (southern) substructure is further resolved into two smaller peaks: G1 and G2 (BCG-N and BCG-S).It is remarkable that the five strongest mass peaks are precisely aligned with the five most luminous cluster galaxies (2″), although MARS is entirely blind to the cluster galaxy distribution and never uses the LTM assumption.
A comparison of the WL+SL result with the WL-only (upper left) and the SL-only (upper right) ones delivers a few important takeaway messages.First, the centroids of the five strongest mass peaks are well-constrained by either data set.Although it is not surprising to see these alignments with the SL-only result, it is unprecedented that the WL data alone can constrain the mass centroids at this high-significance level (S/N  9σ).We believe that this is enabled by the unprecedentedly high WL source density (∼350 arcmin −2 ).Second, the SL-only mass reconstruction provides strong constraints only within the SL regimes, which are confined to the r  0.3 Mpc region in the southern subcluster, the r  0.2 Mpc region in the northwestern subcluster, and the r  0.1 Mpc region in the northern subcluster.Outside the SL regime, the mass density in the SL-only result is determined by the initial prior (i.e., a flat convergence with κ = 0.1).We need the WL data set to place meaningful constraints outside the SL regimes.Third, the WL data set does not detect any significant mass peaks other than the aforementioned five mass peaks (BCG-N, BCG-S, G1, G2, and G3).Jauzac et al. (2016) presented WL+SL analysis and reported identifications of eight significant substructures in A2744.Of these, four (N, NW, S3, and core in their notation) coincide with our mass peaks.The other four are not present in our mass reconstruction.Of these four, one substructure (Wbis in their notation) is located near the field boundary of the JWST footprint, and thus, our result cannot be used to rule out its presence.
We compare our mass map to the cluster member galaxy distributions in Figure 5.We select cluster member candidates whose photometric redshifts are within the 0.28 < z phot < 0.32 range.We only select the objects that are brighter than 24 mag.We applied 3σ clipping and performed a linear fit in the color-magnitude diagram.The final member selection is made by identifying galaxies within 1σ of the best-fit relation.The F277W-F444W color is used because the combination clearly highlights the red-sequence galaxies.Because we already demonstrated that the five strongest mass peaks precisely coincide with the five brightest galaxies, good degrees of masslight agreements are somewhat expected in this comparison.However, we note that the number density peaks when smoothed do not always fall exactly on the mass peaks.Although the cluster member catalog is incomplete, the offsets are probably primarily due to asymmetric galaxy distributions around the deepest potential wells.We suspect that the ongoing mergers may contribute to the asymmetry.We also observe that there are some luminosity/number density clumps that lack distinct mass counterparts.They might be groups with low mass-to-light ratios or concentrations of galaxies that are not gravitationally bound and are only projected along the line-ofsight direction.

Cumulative Projected Mass
We present the cumulative mass profiles of the five mass peaks in Figure 6.We also display the total mass profile from the field center (R.A. = 3.568514, decl.= −30.386321),roughly corresponding to the geometric center of the isosceles triangle defined by the three mass peaks G3, G2, and BCG-S.The WL+SL and SL mass profiles are measured directly from the κ map, while the WL results are derived by simultaneously fitting five NFW profiles to the WL data with and without the mass-concentration (M-c) relation of Duffy et al. (2008).We cannot directly use the convergence map obtained from the WL-only result to estimate the mass because the κ value in the SL regime is significantly underestimated, which is the combined effect of the mass-sheet degeneracy and regularization.We refer to Appendix B for more details of the NFW fit.
Overall, the best-fit NFW profiles from WL yield lower masses near the mass-peak centers and higher masses at large radii (r  300 kpc) than the main WL+SL results obtained directly from the convergence map.This implies that the densities at the mass peaks are significantly higher than the best-fit NFW predictions derived from our WL data.We provide two-dimensional comparisons of the issue in Appendix B. The SL-only results are similar to the WL+SL ones in the SL regime (r  100 kpc), but they are systematically lower at larger radii, where the lack of constraints causes the density default to the initial prior.
The WL+SL result shows that the total projected mass within r = 200 kpc from BCG-North is ∼1.73 × 10 14 M e , which is consistent with the values in Bergamini et al. (2023b) and Furtak et al. (2023b).However, for the other mass peaks, our mass model provides lower values than the parametric models.The projected mass within 200 kpc from G1, G2, and G3, are ∼1.14 × 10 14 M e , ∼1.15 × 10 14 M e , and ∼8.77 × 10 13 M e , respectively.The projected mass of G3 differs most from the value of Bergamini et al. (2023b).This is perhaps because there is only one multiple-image system around G3, and thus, our free-form model cannot produce a sharp peak there.Except for G3, the mass profiles from Furtak et al. (2023b) are similar to the profiles derived from our WLonly best-fit NFW model, which are systematically lower than those from our main (WL+SL) model.Our WL+SL model  estimates that the total projected mass within the r = 1 Mpc aperture from the field center is ∼1.19 × 10 15 M e .

Magnification
Figure 7 displays the magnification maps from the SL-only and SL + WL models.Similar to the mass map, the magnification map is characterized by the three main criticalcurve loops.The overall structures of the critical curves broadly agree with those reported in the literature (e.g., Bergamini et al. 2023b;Furtak et al. 2023b).However, the limited resolution and lack of images around compact halos cause MARS to exhibit some lack of detail near the cluster member galaxies.
The shapes of the critical curves in the SL-only model are similar to those in the WL+SL model.These similarities are expected because the critical curves are primarily constrained by the SL data set.However, the WL+SL result provides a significantly higher and more detailed magnification in the outskirts.This is because the lack of constraints causes the SLonly model to predict much lower and simpler mass densities (defaulting to the initial prior) outside the SL regime.
In Figure 8 we also compare our magnification map with the results from Furtak et al. (2023b) 7 and Bergamini et al.  (2023b),8 who kindly made their results publicly available.Although the overall morphology is similar, the details are significantly different among three magnification maps.The magnification contours from Furtak et al. (2023b) and Bergamini et al. (2023b) extend farther out, predicting higher-magnification values in the outskirts.In particular, Bergamini et al. (2023b) suggest broad higher-magnification distributions around G3.This difference might arise because our mass profile, which is constrained by the WL data, decreases faster than the parametric descriptions used in Furtak et al. (2023b) and Bergamini et al. (2023b).

Robustness Test of the Mass Model
A robust mass model is expected to accurately reproduce the observed SL and WL features without overfitting.Here, we assess the quality of our mass model using the following four metrics: the lens-plane scatter, lens-plane image reconstructions, the per-galaxy shear predictability, and tangential shear profiles.It is important to note that a satisfactory performance assessed by these metrics is only a necessary condition, not a sufficient condition, for a robust mass model.

Lens-plane Scatter of Multiple Images
One of the most common metrics used to evaluate SL mass models is the difference between the observed and predicted locations of multiple images on the image plane.We compute the rms of the position differences between the observed and predicted multiple images from our mass model using where M represents the total number of multiple images, and θ truth,m and θ model,m are the locations of the observed and predicted multiple images for the mth image, respectively.
In Figure 9 we plot the distributions of lens-plane scatter.The rms value for the SL-only (SL + WL) mass model is Δ rms = 0 06 (0 11).The SL-only mass model in this study yields an rms value that is slightly higher than our previous result (Cha & Jee 2023, Δ rms = 0 05), where the mass reconstruction was limited to the main cluster region (within the single ACS pointing).The scatter increases approximately by a factor of two when the WL data are included (from 0 05 to 0 11).This increase is not surprising because the inclusion of the WL 2 c term in Equation ( 12) effectively lowers the weight on the SL 2 c term under the same regularization.Nevertheless, we emphasize that this rms value (0 11) is still a factor of four lower than those in other JWST-based SL-only studies; for instance, the scatter is Δ rms = 0 51 and 0 43 for Furtak et al. (2023b) and Bergamini et al. (2023b), respectively.Because the JWST image informs us of the location of the multiple images with an accuracy of a few pixels, our relatively small scatter (0 112) should not be attributed to overfitting.

Lens-plane Image Reconstruction
Although the lens-plane scatter metric (Section 5.1.1)provides a useful statistic to assess the quality of the lens model, it does not inform us about the robustness of the lens model on small scales in the neighborhood of the multipleimage positions.In particular, overfit models from free-form approaches fail to recover lens-plane galaxy morphologies reliably because of high-frequency noise.
Figure 10 displays the reconstructed multiple images in the lens plane from our WL+SL mass model.We choose the four systems that exhibit highly distorted multiple images because they are more sensitive to details in the mass distribution.In general, it is easier to reconstruct the morphology of the system with more resolved features because the mass reconstruction used them.Sources 1 (a) and 68 (c) are the systems where we identified four knots in the mass reconstruction, while sources 4 (b) and 74 (d) are those for which we only used their centroids.Although the images with multiple knots provide better reconstructions, the case with a single constraint also yields good results.We performed these tests with other multiple-image systems and verified that their reconstruction qualities are similar.Therefore, we conclude that our lens model is stable on small scales in the vicinity of the multiple-image positions even after we require the mass model to account for all WL features.

Shear Predictability per Galaxy
Having verified that our combined mass model reproduces the SL features in terms of the multiple-image positions and morphologies in Sections 5.1.1 and 5.1.2,we discuss here how much the predicted shear at the galaxy positions is consistent with the observations.Because each galaxy's ellipticity measurement contains its intrinsic shape and measurement noise as well as the shear, it is important to include them in our judgment of the goodness of the fit.Because Equation (15) is already designed to accommodate this need, we decide to use it and adopt the distribution of the normalized squared residual per-galaxy ellipticity component (hereafter, we refer to it as per-galaxy WL 2 c ) as our metric.and ∼32% higher, respectively, than our WL+SL or WL-only result.This illustrates that our final WL+SL model robustly reproduces the WL features as well as the SL features.Moreover, this serves as an important lesson that a complete mass model within the current JWST A2744 field requires WL constraints.

Reduced Tangential Shear Profile
The reduced tangential shear is a measure of how strongly the shapes of background galaxies are tangentially aligned with respect to a reference point.We adopt the center of the mosaic image here as our reference because it maximizes the radius where the measurement is derived from a complete circle.The reduced tangential shear g t is evaluated via the following equation: where g 1(2) denotes the first (second) component of the reduced shear, and f represents the position angle of the object measured counterclockwise from the reference axis.The amplitude of the reduced tangential shear is given by and is sensitive to the overall shape of the radial mass profile.Therefore, it is possible that a mass model that performs well in the above per-galaxy WL 2 c test performs poorly in this tangential shear test, and vice versa.
Figure 12 displays the comparison of the observed tangential shear with the model prediction.We recall that the radial behavior of the current tangential shear is different from the behavior of the typical cases in the literature for two reasons.First, the reference point is at the center of the mass reconstruction field, which is near the geometric center of the three substructures of A2744.Second, the mass distribution of A2744 is by and large trimodal.The relatively low projected mass density near the field center causes the amplitude of the tangential shears remain low until the radius reaches r ∼ 600 kpc, which is approximately the mean distance from the field center to the three substructures.Some noticeable deviations from the observation are present in the SL-only and WL-only mass models.The SL-only mass model predicts that the reduced tangential shear is initially more negative at r  400 kpc, reaches levels similar to the observation at 500 kpc  r  700 kpc, and becomes significantly lower at larger radii (r  800 kpc).This pattern is attributed to the absence of the multiple-image constraints at r  400 kpc or r  800 kpc, where our SL-only mass reconstruction tends to default to the initial prior.The WL-only mass model predicts the tangential shear well up to r  500 kpc.However, beyond this, its prediction is systematically lower than the observation.This is due to the fact that our maximumentropy regularization makes it difficult for the WL-only mass reconstruction to reach high κ values in the SL regime because of the sharper mass peaks at lower entropy.The WL+SL mass model provides the best predictions, which match the observed values in the entire range.Figure 14.WL+SL mass reconstruction of A2744.The blue intensity region represents the mass distributions from the WL+SL mass map.The red intensity region corresponds to the X-ray surface brightness (OBSID: 7915, 8477 and 8557; PI: J. Kempner).The green region displays the radio continuum from radio observations (Venturi et al. 2013;Paul et al. 2019; GMRT data at 325MHz).The color-composite image is created using the Subaru/Suprime-Cam observations, with the z band for red, the R band for green, and the B band for blue (Finner & Jee 2022).The areas enclosed with solid white lines indicate the four known radio relics (Pearce et al. 2017;Rajpurohit et al. 2021).The solid orange lines show the expected merger axes from the mass bridges.The field of view is 13 46 × 13 46.
The current tangential shear test illustrates that it is important to incorporate both SL and WL data sets when a high-fidelity wide-field mass reconstruction is desired in a massive cluster.As observed, the WL-only mass reconstruction is biased because the WL data alone cannot adequately inform us about the mass distribution in the extremely high-density regions.On the other hand, the SL-only reconstruction is only robust in the SL regime, where multiple images are densely distributed.Although this issue can be somewhat mitigated by assuming that the mass profile follows some analytic descriptions, in post-merger clusters, such as A2744, the assumption may diminish our ability to learn how the mass profiles are affected by the subhalo collisions.

Comparison with Previous Studies and Merging Scenarios
Due to its rich and puzzling substructures, A2744 was introduced with the nickname "Pandora's cluster" (Merten et al. 2011).One of the notable features highlighted in Merten et al. 2011 was the "ghost" clump, which lacks any apparent correlation with the cluster galaxies.In addition, the authors reported large offsets between the BCGs and the mass peaks, which were supported by Medezinski et al. (2016).However, other studies (e.g., Jauzac et al. 2016;Bergamini et al. 2023b;Furtak et al. 2023b) based on SL-only or WL+SL data sets found neither significant offsets between BCGs and mass peaks nor the ghost clump.A caveat in the latter studies is that the mass models are reconstructed based on the LTM assumption, and thus the possibility of mass peaks with considerable departure from the galaxies is not extensively explored.The current study is the first free-form mass reconstruction of A2744 based on WL+SL with no LTM assumption.Our JWST result supports neither the existence of the ghost clump nor the mass-galaxy offsets, although our analysis is completely blind to the locations of the cluster galaxies in A2744.
Intracluster stars and globular clusters have been suggested as visible tracers of dark matter (e.g., Jee 2010; Alonso Asensio et al. 2020;Montes & Trujillo 2022;Yoo et al. 2022;Diego et al. 2023) if their formation occurs at high redshift (Ko & Jee 2018;Joo & Jee 2023;Werner et al. 2023).Recently, Harris & Reina-Campos (2023) identified more than 10,000 intracluster globular clusters in A2744 with the same JWST imaging data that we used in the current study.The spatial distribution of the intracluster globular clusters in A2744 closely follows the lensing-based mass map presented in the current study and other studies (Bergamini et al. 2023b;Furtak et al. 2023b).
The scrutiny of our mass map hints at the existence of two mass bridges: one bridge between NW and S, and the other between N and NW.These density enhancements are mainly constrained by the WL data set, which provides an unprecedentedly high source density (∼350 arcmin −2 ).To estimate the significance of these mass bridges, we compute the uncertainty of the WL-only mass map (Figure 13).We reconstruct 1000 mass maps from bootstraps of the original WL catalog and measure the standard deviation (Figure 13(b)), which we adopt as the uncertainty map.The S/N map (Figure 13(c)) is obtained by dividing the WL-only mass map by this uncertainty map.The significance of the contours outlining these mass bridges is ∼6.0σ according to the S/N map.Because A2744 is one of the most massive clusters known to date, the mass density within the current JWST field should be significantly positive, and thus, the true background level can be estimated only from studies with much larger fields.Nevertheless, as a conservative measure, we also evaluated the significance of the mass bridges with respect to the background level estimated within the reconstruction field.When adopting the outermost contour level of the mass map (Figure 13(a)) as the baseline, we find that the significance decreases to ∼4.0σ, which implies that the mass-bridge features are still high in this conservative measure.The mass bridges may be interpreted as arising from the mergers because numerical simulations show that mass bridges develop between the two clusters after the core passage.
Interestingly, the orientations and locations of the two largest radio relics in A2744 are consistent with the hypothesized merger axes inferred by the two mass bridges (Figure 14).The brighter relic (R1) is located ∼1 Mpc away from the northern clump and is on the hypothesized merger axis connecting NW and N. Furthermore, the orientation of the relic is perpendicular to the merger axis.The fainter relic (R2), ∼0.5 Mpc southeast of the southern (main) clump, is also on the axis connecting NW and S, and again, its orientation is orthogonal to the axis.
However, despite the above intriguing possibility, a complete reconstruction of the merging scenario of A2744 is still challenging.First of all, A2744 consists of at least five massive substructures, which provide too many degrees of freedom for a plausible merging scenario reconstruction.Furthermore, the X-ray morphology of A2744 is complex and contains many substructures that are significantly dissociated from the galaxy distributions.Although similar degrees of gas-galaxy offsets have been observed in other binary mergers (e.g., Clowe et al. 2006;Paraficz et al. 2016), the complexity of A2744 cannot be explained by binary encounters like this.

Conclusion
Leveraging 286 multiple images and the WL source density of 350 arcmin 2 ~-, we have presented a new WL+SL mass model of A2744 using the MARS free-form algorithm.For the WL analysis, we carefully modeled the PSF and measured the ellipticities of the background sources, whose selection is based on photometric redshifts.For the SL constraints, we compiled multiple images from the literature and also identified new multiple-image candidates.
Our WL+SL mass reconstruction provides the highestresolution mass map of A2744 within the ∼1.8 Mpc ×1.8 Mpc field to date, revealing a giant isosceles triangular structure characterized by two legs of ∼1 Mpc and a base of ∼0.6 Mpc.While our algorithm MARS remains entirely unaware of the distribution of cluster galaxies, the resulting mass reconstruction traces the brightest cluster galaxies remarkably well, with the five strongest mass peaks coinciding with the five most luminous cluster galaxies.The most remarkable features of our mass reconstruction include the two mass bridges: one bridge connects N and NW, and the other connects NW and S.These 4σ features are interpreted as arising from the ongoing mergers because the merger axes defined by the features are consistent with the positions and morphologies of the two brightest radio relics in A2744.
We support the robustness of our mass model with various tests involving lens-plane position scatter, a lens-plane morphology reconstruction, per-galaxy WL 2 c statistics, and tangential shear profiles.We demonstrate that the WL+SL mass model performs well in all these tests, while the performance is not satisfactory when only one of the two (WL and SL) data sets is used for the model construction.The comparison of the current result with those in previous studies shows some important differences in mass profiles and magnifications in the SL and WL regimes.We attribute them to both the usage of the WL data outside the SL regime and the free-form MARS algorithm.
The present study demonstrates that with the advent of the JWST era, the cluster mass reconstruction combining WL and SL has now emerged as a critical and also practical channel to robustly measure the mass distributions of massive clusters from their cores to the outskirts.This will enhance our understanding of cluster physics, dark matter properties, and reionization-epoch galaxies.

Appendix B Substructure Properties
The cumulative mass profiles that we present in Section 4.2 cannot be used to isolate the mass properties of individual substructures because the measured convergence is the result of the superposition of multiple halo profiles.Here, we determine the substructure properties by simultaneously fitting five NFW profiles.A2744 is a complex system with a number of merging features, including sophisticated X-ray morphologies and radio relics.Hence, this NFW fitting is not expected to produce results devoid of bias, as mergers are likely to cause substantial deviations from NFW descriptions in cluster mass profiles (e.g., Lee et al. 2023).Nevertheless, simultaneous multi-halo fitting can significantly diminish the impact of neighboring substructures.
We perform the simultaneous NFW fitting in two approaches.The first method is to fit the five NFW profiles to our WL shape catalogs.In the second approach, we fit the five NFW profiles to the convergence map obtained from the WL+SL mass reconstruction.In Tables 2 and 3, we display the results with and without the M-c relation of Duffy et al. (2008), respectively.We also present the posterior distributions of all free parameters in Figures 18, 19, 20, and 21.When we assume the M-c relation of Duffy et al. (2008), both the WL and WL +SL results show similar concentration values.However, in the results obtained without the M-c relation, the WL+SL model gives considerably higher concentration values, which are attributed to the availability of the SL constraints near the mass peaks.The merging cluster simulations of Lee et al. (2023) demonstrate that the concentrations tend to increase because mass infalls occur in post-collision systems.However, because WL does not sample the signals near the halo centers densely, the concentrations are biased low, which in turn leads to an overestimation of the cluster mass in WL analysis.The extent of this overestimation depends on the state of the merger, with the factor potentially reaching as high as 2-3.The systematic differences in both concentration and mass shown in Table 3 are consistent with the predictions of Lee et al. (2023).
In order to visualize the two-dimensional difference between the NFW fitting results and the main WL+SL mass map, we present the residual (subtraction of the WL+SL mass map from the NFW fitting result) mass maps in Figure 22.Although the details differ between the two results with and without the M-c relations, similar trends are present in the residual maps.The shear-based models produced with the best-fit NFW parameters (first and third rows) yield lower mass densities around the five mass peaks, while they predict higher values in the outskirts.This illustrates that if we attempt to estimate the total mass of A2744 using the extrapolation of the shear-based NFW fitting results, the procedure will result in a severe overestimation.The residuals created with the convergence-based models (second and fourth rows) are somewhat more complex and show larger azimuthal variations.

Notes.
a The total mass is computed by summing the virial mass from five fitted halos.
b For the WL-only result, we use the reduced shear to fit the NFW profiles.In the case of the WL+SL, we use the two-dimensional κ distributions to fit the NFW profiles (see Appendix B for more details).

Notes.
a The total mass is computed by summing the virial mass from five fitted halos.b For the WL-only result, we use the reduced shear to fit the NFW profiles.In the case of the WL+SL, we use the two-dimensional κ distributions to fit the NFW profiles (see Appendix B for more details).

Figure 1 .
Figure 1.Multiple-image distributions in the A2744 field.The orange (cyan) circles indicate the locations of the gold-class (silver-class) multiple images, including the knots of extended multiple images.The color-composite image is created using the F444W filter for red, the F277W filter for green, and the F115W filter for blue.The notation from Bergamini et al. (2023b) is followed for the labeling of the five brightest galaxies.The displayed field of view is 400″ × 400″.

Figure 2 .
Figure 2. Newly identified multiple-image candidates from this study.We find six candidate SL systems around G2 (a) and G1 (b); see Figure 1 for their locations.The cyan circles and numbers indicate the locations and IDs of the multiple images, respectively.The color-composite images are created using F200W (red), F150W (green), and F115W (blue).
− b)/(a + b) referred to as the ellipticity e.The left panel of Figure 3 shows the ellipticities of the observed stars and the residuals (observed star ellipticity-model PSF ellipticity).The detector-induced ellipticity in the observed stars is corrected by the model because the residual points are centered around e 1 , e 2 = 0.The right panel shows the residual size R computed by subtracting the observed model size from the star size.The initial result is given as the blue distribution, with the median being ( The PSFs produced by the WebbPSF package are systematically smaller than the observed stars, and this difference likely arises because

Figure 3 .
Figure 3. WebbPSF model corrections for ellipticity and size.We present the result for F200W, where we measure WL signals.Left: Original complex ellipticity components of stars measured in the A2744 NIRCam images (blue), and residuals (black) computed by subtracting the WebbPSF model values from the observed values.The median and standard error for the residuals are provided in the top right corner of the plot.Right: Residual size R between the observed stars and the PSFs (blue), and the residual computed after applying a difference kernel (black).In the top left corner of the plot, the median residuals before R á ñ and after the kernel R s á ñ, including the associated standard errors, are provided.

Figure 4 .
Figure 4. Mass contours of A2744 overlaid on the color-composite images.The yellow contours indicate the convergence κ.The upper left (upper right) panel displays the mass contours obtained from the WL-only (SL-only) mass map, while the lower panel presents the mass contours derived from the WL+SL mass map.In the WL-only mass map, the contours correspond to κ = [0.15,0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8].For the SL-only and WL+SL mass maps, the contours indicate κ = [0.15,0.2, 0.3, 0.4, 0.6, 0.9, 1.2, 1.8, 2.4].To mitigate pixel-scale artifacts, we apply Gaussian smoothing with a kernel of σ = 2″ (σ = 1″) to the WL-only (WL+SL) mass contours.In the lower right panel, unsmoothed mass distributions are displayed as a color map.The color-composite images are the same as shown in Figure 1.

Figure 5 .
Figure 5. Mass contours overlaid on the cluster galaxy number and luminosity density maps.The left (right) panel shows the color map indicating the smoothed (Gaussian kernel with σ ∼ 10″) cluster galaxy number density (luminosity).The solid black lines represent the mass contours derived from our combined mass map, which are the same as shown in Figure 4.The dashed white line presents the footprint of the JWST observations.

Figure 6 .
Figure 6.Cumulative projected mass profiles of A2744.We present the radial cumulative mass profiles of five halos (BCG-North, BCG-South, G1, G2, and G3).Additionally, we include the cumulative profile from the center of the field of view.The dashed green (cyan) lines indicate the cumulative mass profiles from the WL NFW profile-fitting result without (with) the M-c relation.The dashed blue lines represent the SL-only mass profiles derived from the current study.The dashed purple and brown lines represent the mass profiles from Furtak et al. (2023b) and Bergamini et al. (2023b), respectively.The solid red line and shaded regions display the profiles and the 1σ uncertainties of our combined (WL+SL) mass model, respectively.

Figure 7 .
Figure 7. Magnification maps of the reconstructed lens models at the reference redshift z s = 10.The left (right) panel shows the magnification map from the SL-only (SL + WL) mass model.Unlike Figure 4, we do not apply the smoothing to the magnification map from the combined mass model.

Figure 8 .
Figure 8. Magnification comparison with the literature.The black contours show the magnifications from our combined mass map.The red and blue contours represent the magnifications from Furtak et al. (2023b) and Bergamini et al. (2023b), respectively.For each magnification, the inner (outer) contour indicates |μ| = 4 (|μ| = 2).The reference redshift is z s = 10.

Figure 9 .
Figure9.Lens-plane scatter between the observed and predicted locations of multiple images.Δ x(y) represents the deviation from the observation along the x-axis (y-axis).The red (blue) dots indicate the lens-plane scatter distributions derived from the WL+SL (SL-only) mass map.Δ rms is the rms value of the total lens-plane scatter (see Equation (18)).

Figure 10 .
Figure 10.Lens-plane image reconstructions.The left panels show the observed multiple images, with red circles indicating the images selected for reconstruction.The dashed yellow circles represent the locations of multiple images.The right panels present the reconstructed images in the lens plane, with dimensions matching those of the left panels.The color-composite images are the same as shown in Figure 2.

Figure 11
Figure 11 displays the distributions of the per-galaxy WL 2 c measured for the WL-only, SL-only, and WL+SL mass reconstructions.Both WL-only and WL+SL models provide a mean per-galaxy WL 2 c close to unity, whereas the value is ∼50% higher for the SL-only model.Because the SL-only models from Furtak et al. (2023b) and Bergamini et al. (2023b), who used JWST observations, are publicly available, we retrieved the models and computed their per-galaxy WL 2 c with our WL data.We found that the mean per-galaxy WL 2 c from Furtak et al. (2023b) andBergamini et al. (2023b) are ∼40% and ∼32% higher, respectively, than our WL+SL or WL-only result.This illustrates that our final WL+SL model robustly reproduces the WL features as well as the SL features.Moreover, this serves as an important lesson that a complete mass model within the current JWST A2744 field requires WL constraints.

Figure 11 .
Figure 11.Distributions of the per-galaxy WL 2 c .The green (blue) histograms indicate the distributions of the per-galaxy WL 2 c obtained from the WL-only (SL-only) mass map.The red histograms represent the distributions of the pergalaxy WL 2 c from the WL+SL mass map.The vertical lines correspond to the mean per-galaxy WL 2 c values from each mass map, indicated by the numbers in the lower left corner (matching the color of each histogram).

Figure 12 .
Figure 12.Comparison of the radial reduced tangential shear.We compute the radial tangential shear from the center of the field of view.The dots indicate the radial reduced tangential shear.The error bars and shaded regions show the standard uncertainties.The black, red, blue, and green samples are obtained from the observations, the combined model, the SL-only model, and the WLonly model, respectively.

Figure 13 .
Figure 13.Mass, uncertainty, and S/N maps of the WL-only mass model.The dashed black lines indicate the footprint of the JWST observations.All three maps are smoothed with a Gaussian kernel of σ = 2″.(a) WL-only mass map.The contour labels show the convergence κ.(b) Uncertainty of the WL-only mass model derived from 1000 bootstrap realizations.(c) S/N map of the WL-only mass model.The contour levels are 3σ, 6σ, 9σ, 12σ, and 15σ.The red arrows indicate the mass bridges.

Figure 15 .Figure 16 .
Figure 15.Whisker plots showing the ellipticity direction and magnitude at the mosaic star positions for the two different PSF models.The upper (lower) three plots show from left to right the star ellipticities, the WebbPSF model (empirically obtained PSF model) ellipticities, and the residual ellipticities from subtracting the model ellipticities from the star ellipticities.

Figure 17 .
Figure17.Cumulative projected mass profile of A2744 from the center of the field of view (same as in Figure6).The solid black (red) line and shaded regions indicate the cumulative mass profile and the 1σ uncertainties of the combined mass model using WebbPSF (empirically obtained PSF).

Figure 18 .
Figure 18.WL-only NFW fitted results with the M-c relation of Duffy et al. (2008).The units of the mass and virial radius are 10 14 M e and kpc, respectively.

Figure 19 .
Figure 19.WL+SL NFW fitted results with the M-c relation of Duffy et al. (2008).The units of the mass and the virial radius are the same as in Figure 18.

Figure 20 .
Figure 20.WL-only NFW fitted results without the M-c relation.The units of the mass and the scale radius are 10 14 M e and kpc, respectively.

Figure 21 .
Figure 21.WL+SL NFW fitted results without the M-c relation.The units of the mass and the scale radius are the same as in Figure 20.

Figure 22 .
Figure 22.Residual mass between the NFW fitting and the main WL+SL mass maps.(a) Two-dimensional mass models created from the best-fit NFW parameters.(b) Subtraction of the main WL+SL mass map from the models shown in panel (a).The first and second (third and fourth) rows display the results obtained with (without) the M-c relation of Duffy et al. (2008).The dashed black lines indicate the footprints of the JWST observations.The solid black lines indicate regions where the residual is zero.See the text for discussions.

Table 2
NFW Profile-fitting Results with the M-c Relation

Table 3
Comparison of the WL NFW Fitting Results with the Main Mass Model