VENUS: Strong Lens Mass Models and Einstein Radii of 60 Galaxy Clusters

Introduction

A massive cluster of galaxies can produce more than one hundred multiple images of galaxies behind it by the gravitational lensing effect, which enable various studies that are difficult without gravitational lensing (see [116, 151], for reviews). For example, the magnification and multiple images help identify high-redshift galaxies (e.g., [52, 10] (Willman et al. 2023)) as well as resolve their internal structures (e.g., [202, 95, 5, 147, 28]). Time delays between multiple images, which can be measured for variable sources such as quasars and supernovae, not only allow us to measure the Hubble constant (e.g., [114, 196] (Paleske et al. 2025b)) but also open up the possibility of reconstructing the light curve of a source from single-epoch imaging (e.g., [47, 212]). By utilizing microlensing, one can even detect individual stars in gravitationally lensed arcs to study their stellar population (e.g., [113, 78]). Accurate and precise strong lens mass models are crucial for all these applications.

In addition to studying lensed sources, strong gravitational lensing probes mass distributions of lensing clusters, which provides a useful means of testing the standard structure formation model and exploring the nature of dark matter. For instance, the abundance of arcs in observations have been confronted with theoretical predictions based on the $\Lambda$ cold dark matter ($\Lambda$CDM) paradigm (e.g., Meneghetti et al. [139], for a review). The density profiles of massive clusters measured by strong and weak gravitational lensing have also been directly confronted with the $\Lambda$CDM predictions to find good agreements between them (e.g., [159, 154]). Measurements of the very central density profiles with strong gravitational lensing place tight constraints on the dark matter self-interaction cross-section (e.g., Andrade et al. [8]). The abundance of galaxy-galaxy strong lenses near cluster centers also contains useful cosmological information (e.g., Meneghetti et al. [141]). Furthermore, the statistics of the large Einstein radii serve as another useful test of the $\Lambda$CDM paradigm (e.g., [32, 160]).

A significant advance in studies of cluster strong lensing is achieved by the Hubble Frontier Fields (HFF; Lotz et al. [129]) for which deep imaging of six massive galaxy clusters with the Hubble Space Telescope (HST) was conducted. Cluster Lensing And Supernova survey with Hubble (CLASH; Postman et al. [169]) and Reionization Lensing Cluster Survey (RELICS; Coe et al. [53]) provide HST imaging data of clusters that are shallower than HFF but for larger numbers of clusters. Massive clusters continue to be important targets in the era of James Webb Space Telescope (JWST), see e.g., Zitrin [214]. In addition to the galaxy cluster SMACS J0723.3−7327 being one of five targets of the JWST Early Release Observations (Pontoppidan et al. [168]), there are several related JWST programs including the Canadian NIRISS Unbiased Cluster Survey (CANUCS; Sarrouh et al. [185]), Ultradeep NIRSpec and NIRCam Observations before the Epoch of Reionization (UNCOVER; Bezanson et al. [21]), All the Little Things (ALT; Naidu et al. [148]), GLIMPSE (Atek et al. [11]), GLASS-JWST (Watson et al. [206]), Prime Extragalactic Areas for Reionization and Lensing Science (PEARLS; Windhorst et al. [210]), Cosmic Spring (Welch et al. [207]), Medium-band Astrophysics with the Grism of NIRCam In Frontier Fields (MAGNIF; Fu et al. 2025), and Strong LensIng and Cluster Evolution (SLICE; Cerny et al. [43]).

Vast Exploration for Nascent, Unexplored Sources (VENUS) is a new JWST program in Cycle 4 (GO-6882, PIs: S. Fujimoto and D. Coe) that conducts extensive NIRCam imaging observations to construct a sample of 60 clusters with uniform NIRCam images in 10 filters (S. Fujimoto et al. in preparation). In this paper, we present strong lens mass models of all 60 VENUS clusters. The mass modeling is conducted with the GLAFIC software ([157] (Oguri 2021)), which shows the best performance in a mass modeling comparison project using simulated clusters (Meneghetti et al. [140]). We then compare and discuss their lensing strengths quantified by areas of high-magnification regions as well as the Einstein radii. We compare the observed Einstein radii with theoretical predictions based on the $\Lambda$CDM paradigm. In a companion paper (J. F. V. Allingham et al., in preparation), independent mass models of 57 VENUS clusters using a different software and technique are presented.

This paper is organized as follows. We describe the data and the method used for the analysis in § 2. The result of the mass modeling is given in § 3. We derive lensing strengths of individual clusters in § 4. The comparison of observed Einstein radii with the prediction of the $\Lambda$CDM model is given in § 5. Finally, we conclude in § 6. The strong lens mass modeling is performed assuming the matter density of $\Omega_{\rm m}=0.3$ and the cosmological constant of $\Omega_{\Lambda}=0.7$, although we note that the mass modeling result essentially does not depend on the choice of the cosmological parameters. On the other hand, the $\Lambda$CDM model prediction of the largest Einstein radius in the Universe is made assuming the best-fitting cosmological parameters from the latest cosmological data set (Louis et al. [130]).

Data and Method

VENUS Data

In this paper, we use NIRCam imaging data obtained by the VENUS program for our analysis. The NIRCam observations consist of imaging in ten bands including F090W, F115W, F150W, F200W, F210M, F277W, F300M, F356W, F410M, and F444W, with exposure times of 0.35–0.57 hours per filter. The exposure time is chosen to achieve a $5\sigma$ point source detection limit of $\sim28$ mag across all the NIRCam filters.

The procedure to reduce the data is as follows. We reduce JWST level-2 products from the Mikulski Archive for Space Telescopes with the GRIZLI pipeline (Brammer [29]), similarly to the public DAWN JWST Archive (Valentino et al. [200]). Additional post-processing steps include additional background, $1/f$ noise, and diffraction spike subtraction procedures (e.g., Kokorev et al. [119]). The final drizzled image has a pixel size of 0\farcs03. The JWST images are supplemented by archival Hubble Space Telescope images of the same clusters, which are taken from Gaia-aligned mosaics from the CHArGE archive (Kokorev et al. [118]) and are drizzled in the same manner as the JWST data.

From the reduced images, object detection and photometry in different bands are performed using the SEP software ([19] (Barbary 2016)). Photometric redshifts of detected objects are derived with the EAZY software (Brammer et al. [30]) with the (aperture-corrected) aperture photometry with the aperture size of 0\farcs2.

We note that JWST NIRCam imaging is yet to be conducted for three clusters, RXC J0948.8+1707, Abell 1835, and RXC J1347.5−1145, for which we use existing HST imaging data for our mass modeling. We plan to update our strong lens mass models after JWST images of these clusters are taken.

Identification of Multiple Images

Since the VENUS program selects target clusters based on the availability of strong lens mass models, for each cluster, multiple images that are previously identified are mostly available. Spectroscopic redshifts of these previously identified multiple images are taken from the literature when available. After constructing an initial mass model for each cluster following the strategy described in § 2.3, we search for new multiple images taking advantage of the high-quality VENUS data. Specifically, we select multiple images based on the combination of image morphologies, photometric redshifts, and the consistency of image configurations with the current mass model. After identifying secure multiple images, we add these new multiple images to refine the mass model, and again search for new multiple images. We repeat this procedure iteratively to finalize the multiple image set used for mass modeling. In total, we use $\sim4500$ multiple images for our mass modeling of 60 clusters, $\sim1150$ of which are newly identified in this paper. Our selection of multiple images for the mass modeling analysis is rather conservative such that we adopt only high-confidence systems that are judged from photometric redshifts, morphologies, and the consistency with mass modeling. References for multiples images and spectroscopic redshifts are listed in Appendix A.¹

Mass Modeling Method

We conduct strong lens mass modeling using the glafic software (Oguri [157] (2021)), which takes the so-called parametric approach. In parametric mass modeling, a lens potential of a cluster is described by a superposition of lens potentials of a small number of lens components. Those lens components typically include dark matter halos, member galaxies, and various perturbations. The resulting precision and accuracy of mass modeling critically depend on which lens components are assumed and which parameters of those lens components are fit to the data. Here we describe our strategy to determine a set of lens components and parameters used for mass modeling of each cluster.

We use the elliptical Navarro et al. [152] density profile for halo components. Its radial density profile is given by

where $\rho_{\mathrm{s}}$ and $r_{\mathrm{s}}$ are the characteristic density and the scale radius, respectively. The ellipticity is introduced in the isodensity contours of its surface mass density. We use an approximation proposed in Oguri (2021) for fast calculations of the lensing properties of the elliptical NFW profile. Each halo component is characterized by six parameters, including the mass, the concentration parameter, the centroid position, the ellipticity, and the position angle. The mass and concentration parameter are defined using the virial overdensity (Nakamura & Suto [149]).

First, a halo component is added at the position of the brightest cluster galaxy to optimize model parameters. Additional halo components may be added at around positions of bright cluster member galaxies, only if adding additional halo components significantly improves the fit. All the parameters of each halo component, including its centroid, are allowed to float when a sufficient number of multiple images are found around it. In contrast, if the halo component is located relatively in the outskirts of the strong lensing region, its centroid position may be fixed to the position of a bright cluster member galaxy. In that situation, the concentration parameter $c$ may also be fixed to $c=10$. When included as model parameters, the fitting range of the concentration parameter is always restricted to $1<c<40$. The ellipticity $e$, which is defined by one minus the ratio of the lengths of minor and major axes of the isodensity contour, is always restricted to $0<e<0.8$.

Cluster member galaxies are modeled by pseudo-Jaffe ellipsoids ([111] (Jaffe 1983)). Its radial density profile is given by

where $\rho_0$ is the characteristic density and $r_{\mathrm{core}}$ and $r_{\mathrm{trun}}$ are the core and break radii, respectively. Since we always fix the core radius to $r_{\mathrm{core}}=0$ when modeling member galaxies, each pseudo-Jaffe ellipsoid is characterized by six parameters, including the velocity dispersion, the centroid position, the ellipticity, the position angle, and the break radius.

In order to reduce the number of parameters, we adopt a scaling relation for some of the parameters characterizing the pseudo-Jaffe ellipsoids. We do so by assuming that velocity dispersions and break radii of individual pseudo-Jaffe ellipsoids scale with luminosities $L$ of member galaxies as

and treat $\sigma_*$, $r_{\mathrm{trun},*}$, and $\eta$ as parameters. We usually take the fiducial luminosity $L_*$ as that of the brightest cluster galaxy. Ellipticities and position angles of individual member galaxies are fixed to observed values for their light profiles. These parameters are restricted to the ranges $100\ {\rm km/s}<\sigma_*<400\ {\rm km/s}$, $1''<r_{\mathrm{trun},*}<100''$, and $0.2<\eta<1.5$. The parameter $\eta$ may be fixed to $\eta=1$ if there are not many multiple images. Cluster member galaxies are selected from the imaging data using either the color-magnitude diagram or photometric redshifts.

In some rare cases, a cluster member galaxy may be modeled separately without assuming the scaling relation if that member galaxy is critically important for accurately reproducing some multiple images. Examples of these rare cases include a member galaxy that produces the Einstein cross images of the supernova Refsdal (Kelly et al. [114]) and a member galaxy that produces the three supernova images in Abell 370 (Chen et al. [47]). Furthermore, when it is needed to reproduce central images accurately, the brightest cluster galaxy may be modeled separately assuming the elliptical Hernquist [93] profile, although we note that for most of the clusters the bright cluster galaxies are simply modeled by the pseudo-Jaffe ellipsoids adopting the scaling relation mentioned above.

In addition, we include multipole perturbations to improve the fit. Its lens potential is generally given by

where, for given $n$ and $m$, $\epsilon$ and $\theta_{\epsilon}$ are parameters. The case $m = 2$ and $n = 2$ corresponds to the external shear. Throughout the paper, we fix $n = 2$, and consider $m = 2$ and higher ($m = 3$, $4$, $\ldots$) multipole perturbations. As for halo components, we add higher order terms only when it significantly improves the fit.

We optimize model parameters so as to reproduce positions of multiple images. Source redshifts are fixed to the spectroscopic redshifts when available, otherwise they are also treated as model parameters. When accurate photometric redshifts of strongly lensed systems are available from the VENUS data (see § 2.1), we include the Gaussian priors with a width of typically $\sigma_z = 0.2$. For the cluster MACS J1149.5+2223, we also include time delays and flux ratios of multiple images of the supernova Refsdal (see Liu & Oguri [127]) and include the Hubble constant as a free parameter. We compute $\chi^2$ in the source plane, but taking account of the full magnification tensor as detailed in Appendix 2 of Oguri [157]. This method not only provides an accurate approximation of the image plane $\chi^2$, but also exhibits excellent performance in a mass modeling challenge using simulated clusters (Meneghetti et al. [140]). Positional errors of individual multiple images are set to values typically much larger than measurement errors, considering effects of substructures in mass distributions, and are adjusted so that the reduced $\chi^2$ of the best-fitting model becomes roughly equal to one (see discussions in e.g., Nishida et al. [155]). Specifically, typical values of positional errors are $\sim0.4^{\prime\prime}$–$0.7^{\prime\prime}$, although we sometimes assume much smaller errors for e.g., knots of extended arcs or multiple images with small image separations. Errors on model parameters are derived using the Markov Chain Monte Carlo (MCMC) methods. There are some exceptional cases (e.g., Abell 209) for which the reduced $\chi^2$ is much larger than one because of the small number of observational constraints available for mass modeling, which forces us to assume a very simple mass model with the small number of model parameters.

Mass Modeling Results

Table 1 summarizes the information on the input multiple images as well as $\chi^2$ and the number of the degrees of freedom for the best-fitting model for all the 60 clusters.² Table 1 also presents the root-mean-square (rms) of differences between observed multiple image positions $\theta_i^{\mathrm{obs}}$ and model-predicted multiple image positions $\theta_i^{\mathrm{model}}$ defined by

Name	$z_{\rm cl}$	R.A.$_{\rm cen}$ [deg]	Decl.$_{\rm cen}$ [deg]	Analysis region [″]	$N_{\rm img}$	$N_{\rm sys}$	$N_{\rm spec}$	$\chi^2/{\rm dof}$	$\Delta_{\rm rms}$ [″]	$\sigma_{\rm JK}$
Abell 2744	0.3080	3.5861023	−30.4002210	[−150, 300],[−150, 300]	187	63	52	238.05/212	0.47	0.99
RXC J0018.5+1626	0.5460	4.6398933	16.4377372	[−120, 120],[−120, 120]	38	15	2	33.50/24	0.58	1.29
MACS J0025.4−1222	0.5860	6.3876685	−12.3879814	[−120, 120],[−120, 120]	19	6	1	6.95/11	0.25	1.44
RXC J0032.1+1808	0.3956	8.0476944	18.1409483	[−120, 120],[−180, 120]	160	55	4	190.33/177	0.64	1.04
MACS J0035.4−2015	0.3520	8.8589284	−20.2623036	[−120, 120],[−120, 120]	64	23	7	69.34/67	0.43	0.97
Abell 68	0.2550	9.2785051	9.1567431	[−120, 120],[−120, 120]	42	15	7	30.11/31	0.49	1.52
Abell 2813	0.2920	10.8527516	−20.6282542	[−120, 120],[−120, 130]	50	16	0	44.25/41	0.51	1.04
El Gordo	0.8730	15.7407763	−49.2720144	[−120, 120],[−120, 140]	164	60	25	153.97/166	0.65	1.23
Abell 209	0.2060	22.9689369	−13.6112565	[−120, 120],[−120, 120]	7	3	1	2.96/1	0.54	0.64
WL J013719−08284	0.5660	24.3539070	−8.4563710	[−120, 120],[−120, 120]	26	9	6	25.42/19	0.47	1.60
CL J0152.7−1357	0.8270	28.1876184	−13.9510105	[−120, 120],[−120, 120]	30	10	1	15.28/26	0.26	2.94
MACS J0159.8−0849	0.4050	29.9555766	−8.8330430	[−120, 120],[−120, 120]	39	17	0	40.08/33	0.58	1.08
RXC J0232.2−4420	0.2840	38.0773351	−44.3466849	[−120, 120],[−120, 120]	45	12	1	50.99/47	0.51	1.38
Abell 370	0.3750	39.9696704	−1.5719347	[−130, 120],[−120, 120]	202	68	57	213.94/219	0.66	0.83
Abell S295	0.3000	41.3534474	−53.0292889	[−130, 120],[−120, 120]	25	9	4	20.49/22	0.42	1.09
Abell 383	0.1870	42.0141412	−3.5291701	[−120, 120],[−120, 120]	30	9	7	35.85/25	0.88	1.47
MACS J0257.1−2325	0.5049	44.2865235	−23.4348168	[−120, 120],[−120, 120]	20	7	1	14.72/15	0.36	1.43
MACS J0257.6−2209	0.3220	44.4211330	−22.1549437	[−120, 120],[−120, 120]	94	27	25	118.82/117	0.68	0.97
MACS J0308.9+2645	0.3560	47.2332750	26.7604961	[−120, 120],[−120, 120]	37	16	0	37.24/31	0.46	1.19
PLCK G171.9−40.7	0.2700	48.2370668	8.3719286	[−120, 120],[−120, 120]	45	14	0	46.38/46	0.50	1.29
MACS J0329.7−0211	0.4500	52.4231973	−2.1962697	[−120, 120],[−120, 120]	37	14	10	33.00/31	0.62	1.04
Abell 3192	0.4250	59.7253342	−29.9253315	[−120, 120],[−120, 120]	52	18	2	52.77/49	0.65	0.98
MACS J0416.1−2403	0.3971	64.0381563	−24.0675044	[−120, 120],[−130, 120]	290	105	94	309.01/340	0.66	0.76
MACS J0417.5−1154	0.4430	64.3945823	−11.9089217	[−120, 120],[−120, 120]	118	40	16	119.97/134	0.58	0.91
MACS J0429.6−0253	0.3990	67.4000869	−2.8851795	[−120, 120],[−120, 120]	21	6	3	13.39/24	0.37	0.89
MACS J0451.9+0006	0.4290	72.9777295	0.1050402	[−120, 120],[−120, 120]	84	30	25	99.91/86	0.53	0.97
Abell 521	0.2480	73.5287383	−10.2235399	[−120, 120],[−120, 120]	38	16	15	32.41/35	0.16	0.62
MS 0451.6−0305	0.5500	73.5451650	−3.0143277	[−120, 120],[−120, 120]	62	22	5	50.68/52	0.44	1.48
MACS J0553.4−3342	0.4120	88.3572793	−33.7077786	[−120, 180],[−120, 120]	147	53	4	165.28/165	0.72	0.88
MACS J0600.1−2008	0.4320	90.0340519	−20.1358576	[−280, 130],[−120, 190]	209	67	20	233.51/233	0.61	1.12
S J0615−5746	0.9720	93.9656197	−57.7801883	[−120, 120],[−120, 120]	60	20	12	57.30/53	0.40	1.41
Bullet cluster	0.2960	104.6587123	−55.9571445	[−120, 280],[−120, 160]	86	30	24	86.39/91	0.65	0.93
MACS J0717.5+3745	0.5450	109.3982478	37.7457290	[−120, 190],[−120, 130]	237	81	7	277.05/253	0.62	1.08
S J0723.3−7327	0.3900	110.8270090	−73.4547390	[−120, 120],[−120, 120]	50	17	6	33.18/40	0.32	1.06
RXC J0911.1+1746	0.5049	137.7979921	17.7747099	[−120, 120],[−120, 120]	11	5	0	1.77/1	0.15	0.93
MACS J0940.9+0744	0.3350	145.2237677	7.7403798	[−120, 120],[−120, 120]	25	8	7	26.29/25	0.30	1.03
RXC J0949.8+1707*	0.3826	147.4658585	17.1195056	[−120, 120],[−120, 120]	20	6	2	5.24/11	0.21	0.58
MACS J1115.8+0129	0.3550	168.9662572	1.4986333	[−120, 120],[−120, 120]	22	7	0	24.01/15	0.80	1.47
PLCK G165.7+67.0	0.3480	171.8196015	42.4773058	[−120, 250],[−230, 150]	118	41	14	61.81/121	0.07	1.18
MACS J1149.5+2223	0.5410	177.3987491	22.3985308	[−120, 120],[−120, 120]	117	38	28	110.58/126	0.45	1.01
PLCK G287.0+32.9	0.3830	177.7089905	−28.0821928	[−140, 120],[−120, 120]	97	31	14	94.27/99	0.68	0.97
MACS J1206.2−0847	0.4390	181.5506031	−8.8009300	[−120, 120],[−120, 120]	124	41	37	128.32/139	0.52	0.78
MACS J1311.0−0310	0.4940	197.7575102	−3.1777062	[−120, 120],[−120, 120]	13	4	1	14.33/8	0.73	2.16
Abell 1689	0.1870	197.8729233	−1.3410994	[−140, 120],[−120, 120]	165	55	17	155.08/183	0.55	0.99
RXC J1347.5−1145*	0.4510	206.8775419	−11.7526347	[−120, 120],[−120, 120]	129	40	38	138.17/157	0.65	0.89
Abell 1835*	0.2530	210.2586212	2.8784684	[−120, 120],[−120, 120]	21	7	2	21.97/8	0.79	3.89
MACS J1423.8+2404	0.5450	215.9494781	24.0784501	[−120, 120],[−120, 120]	27	8	5	36.23/27	0.64	1.42
RXC J1532.9+3021	0.3630	233.2240780	30.3498152	[−120, 120],[−120, 120]	16	6	0	8.71/13	0.28	0.76
PLCK G004.5−19.5	0.5160	289.2711616	−33.5223393	[−120, 120],[−120, 120]	50	17	1	55.36/48	0.51	2.23
MACS J1931.8−2635	0.3520	292.9567621	−26.5775431	[−120, 120],[−120, 120]	40	15	8	39.54/37	0.49	1.03
S J2031.8−4036	0.3420	307.9719397	−40.6252203	[−120, 120],[−120, 120]	66	22	19	78.47/73	0.43	0.97
MACS J2129.4−0741	0.5890	322.3587881	−7.6910536	[−120, 120],[−120, 120]	63	20	12	74.99/67	0.63	1.10
RXC J2129.7+0005	0.2340	322.4164769	0.0892336	[−120, 120],[−120, 120]	38	14	11	21.99/37	0.37	0.86
S J2131.1−4019	0.4420	322.7705195	−40.3252598	[−120, 120],[−120, 120]	27	10	9	13.36/9	0.47	1.64
MS 2137−2353	0.3130	325.0632041	−23.6611906	[−120, 120],[−120, 120]	22	8	4	13.37/20	0.31	0.81
RXC J2211.7−0350	0.3970	332.9413169	−3.8290195	[−120, 120],[−120, 120]	146	43	3	174.99/182	0.61	1.10
MACS J2214.9−1359	0.5020	333.7386868	−14.0035677	[−120, 120],[−120, 120]	56	18	16	54.54/57	0.65	1.02
Abell S1063	0.3480	342.1831870	−44.5308979	[−120, 120],[−120, 120]	173	62	27	205.60/193	0.42	1.20
Abell 2537	0.2966	347.0926033	−2.1921637	[−120, 120],[−120, 120]	57	15	1	74.55/67	0.69	1.06
SPT−CL J2325−4111	0.3580	351.2988390	−41.2035767	[−120, 120],[−120, 120]	56	16	6	60.27/57	0.59	1.13

where $N_{\mathrm{img}}$ is the number of multiple images used for mass modeling of each cluster. We show JWST images and critical curves of best-fitting models in Figure 1.

Figure 2 shows the histogram of $N_{\mathrm{img}}$. It is found that each cluster typically contains $N_{\mathrm{img}} \sim50$ multiple images, and about 25% of clusters contain more than 100 multiple images. The large number of multiple images is crucial for tight constraints on mass distributions of individual galaxy clusters.

Vector (PDF)

Figure 3 shows the histograms of the numbers of strongly lensed systems $N_{\mathrm{sys}}$ and those with spectroscopic redshifts $N_{\mathrm{spec}}$. Since $\sim2$–$4$ multiple images are typically produced for each strongly lensed background galaxy, $N_{\mathrm{sys}}$ is roughly one-third of $N_{\mathrm{img}}$. On the other hand, $N_{\mathrm{spec}}$ varies significantly across clusters (see also Table 1), reflecting the fact that spectroscopic redshifts are obtained by various follow-up observations, typically with the Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope (Caminha et al. [38]; Richard et al. [180]), the HST (e.g., Treu et al. [197]), and the JWST (e.g., Noirot et al. [156]; Fu et al. 2025; Sarrouh et al. [185]). We note that, as part of VENUS, spectroscopy with the JWST micro-shutter assembly (MSA) for 10 out of the 60 clusters is planned, which will increase significantly $N_{\mathrm{spec}}$ for these clusters.

Vector (PDF)

Figure 4 shows the histogram of $\Delta_{\mathrm{rms}}$, which is often used to quantify the precision of a mass model. We find that $\Delta_{\mathrm{rms}}$ of our mass models is typically $\sim0.5^{\prime\prime}$, which is comparable to the rms of other parametric lens models using the similar number of multiple images (see e.g., Cha & Jee [46], for a recent compilation of the rms for HFF mass models). All our mass models have $\Delta_{\mathrm{rms}}$ smaller than $1^{\prime\prime}$.

Vector (PDF)

While the value of $\Delta_{\mathrm{rms}}$ mainly reflects the complexity of the cluster mass distribution that is not fully captured by simple parametric mass modeling, it may also be affected by overfitting of multiple image positions (see e.g., Nishida et al. [155]). In Figure 5, we check the possible dependence of $\Delta_{\mathrm{rms}}$ on $N_{\mathrm{img}}$. We find that $\Delta_{\mathrm{rms}}$ does not strongly depend on $N_{\mathrm{img}}$, although there appears a weak trend of increasing $\Delta_{\mathrm{rms}}$ with increasing $N_{\mathrm{img}}$. This trend may partly be due to overfitting of multiple image positions when $N_{\mathrm{img}}$ is small. An alternative explanation is larger $\Delta_{\mathrm{rms}}$ for more massive clusters with larger Einstein radii. Figure 5 also indicates that PLCK G165.7+67.0 is an outlier in the $\Delta_{\mathrm{rms}}$–$N_{\mathrm{img}}$ relation with the very small rms $\Delta_{\mathrm{rms}} \simeq0.07^{\prime\prime}$ for $N_{\mathrm{img}} = 118$, which is because many knots in lensed giant arcs are included as constraints in order to constrain $H_0$ with the lensed supernova Hope as precisely as possible (Pascale et al. [165]).

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We also check the dependence of $\Delta_{\mathrm{rms}}$ on the source redshift $z_s$, which may become important for studying physical properties of strongly lensed high-redshift galaxies. The result shown in Figure 6 suggests that there is a trend of increasing $\Delta_{\mathrm{rms}}$ with increasing $z_s$, although

Note.— Parameters listed in the table include the cluster redshift $z_{\rm cl}$, the reference coordinates (R.A.$_{\rm cen}$, Decl.$_{\rm cen}$), the size of the region [$x_{\min}, x_{\max}$] [$y_{\min}, y_{\max}$] ($x$ and $y$ are relative coordinates from reference coordinates with the West being $+x$ and North being $+y$) for the analysis of the lensing strength in Section 4, the number of multiple images $N_{\rm img}$, the number of strongly lensed systems $N_{\rm sys}$, the number of strongly lensed systems with spectroscopic redshifts $N_{\rm spec}$, the $\chi^2$ of the best-fitting model and the number of degrees of freedom (dof), the root-mean-square of differences between observed and model-predicted multiple image positions $\Delta_{\rm rms}$, and the standard deviation of the Jackknife distribution $\sigma_{\rm JK}$ with $4\sigma$ clipping. Asterisks in the cluster names indicate that JWST VENUS images are not yet available and therefore are not used for mass modeling of these clusters.

the trend is weak. More quantitatively, $\Delta_{\rm rms} \simeq0.6$ for $z_{\rm s} \simeq1$--$2$, and increases to $\simeq0.7$ for $z_{\rm s} \gtrsim3$ and to $\simeq0.8$ for $z_{\rm s} \gtrsim10$. This trend is presumably because source galaxies at higher redshifts are typically located in the outer parts of multiple image regions where constraints on the mass models are less stringent due to small numbers of multiple images in those regions.

Nishida et al. [155] propose the Jackknife method, for which best-fitting models with one of multiple image systems removed are used to check the predictive power of mass models, as a method of checking the accuracy of mass models (see also Liu & Oguri [127], for its application to the time-delay cosmography). In the Jackknife

method, the degree of “correctness” of a mass model is quantified by the standard deviation of the so-called Jackknife distribution, which refers to a histogram of ratios of differences between observed and model-predicted positions of removed images to assumed positional errors of those images (see Nishida et al. [155], for more details). The derived standard deviation $\sigma_{\rm JK}$ with $4\sigma$ clipping to mitigate effects of e.g., isolated multiple image systems for each cluster is listed in Table 1. Figure 7 shows the dependence of $\sigma_{\rm JK}$ on $N_{\rm img}$. We find that $\sigma_{\rm JK} \sim1$ for most of the clusters, indicating no sign of overfitting. This is particularly true for clusters with large $N_{\rm img}$, which is consistent with an argument in Nishida et al. [155] that a large number of multiple images is crucial for mitigating the overfitting. We find that there are a few clusters with $\sigma_{\mathrm{JK}}>2$ for which our current mass models are likely overfitting multiple image positions. In fact, the cluster with the highest $\sigma_{\mathrm{JK}}$ is Abell 1835 for which the JWST NIRCam imaging is lacking. We expect that new JWST NIRCam images for this cluster will enable us to identify more multiple images that help reduce $\sigma_{\mathrm{JK}}$.

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Quantifying Lensing Strengths

Area of the High Magnification Region

The area of the high magnification region in the image plane is often adopted as an indicator of the lensing strength of each cluster (e.g., Richard et al. [178]; Fox et al. [72]). In this paper, we mainly employ $A_{|\mu|>3}$, which is defined by the image-plane area in which a source at redshift $z_{\mathrm{s}}$ is magnified by a factor of 3 or more, to quantify the lensing strength. The threshold magnification value of 3 is chosen to ensure sufficient magnifications for studying high-redshift galaxies. The analysis is conducted within the region shown in Table 1. For all the clusters, we assume the minimum analysis region size of $240^{\prime\prime}\times240^{\prime\prime}$ centered at the brightest cluster galaxy to make sure that it fully covers the high magnification region associated with the main cluster halo. For some of the clusters, the analysis region is extended further to fully include JWST images of interest. For each cluster, we estimate the mean and the error of $A_{|\mu|>3}$ from the mean and scatter of $A_{|\mu|>3}$ values for 50 mass models that are randomly drawn from the MCMC chain. In addition to $A_{|\mu|>3}$, for the source redshift of $z_{\mathrm{s}}=9$, we also check the area for the higher magnification threshold, $A_{|\mu|>10}$. Table 2 summarizes $A_{|\mu|>3}$ for the source redshift $z_{\mathrm{s}}=1$, 3, and 9, as well as $A_{|\mu|>10}$ for $z_{\mathrm{s}}=9$. For $z_{\mathrm{s}}=9$, we identify PLCK G287.0+32.9, MACS J0600.1−2008, and Bullet cluster as clusters with the largest $A_{|\mu|>3}$ among our sample.

Name	$A_{|\mu|>3}$ [arcmin$^2$], $z_s=1$	$A_{|\mu|>3}$ [arcmin$^2$], $z_s=3$	$A_{|\mu|>3}$ [arcmin$^2$], $z_s=9$	$A_{|\mu|>10}$ [arcmin$^2$], $z_s=9$	$\theta_{\mathrm{Ein}}$ [arcsec], $z_s=1$	$\theta_{\mathrm{Ein}}$ [arcsec], $z_s=3$	$\theta_{\mathrm{Ein}}$ [arcsec], $z_s=9$
Abell 2744	1.921 ± 0.077	4.206 ± 0.211	4.989 ± 0.271	1.071 ± 0.043	16.56 ± 0.10	23.58 ± 0.08	26.53 ± 0.14
RXC J0018.5+1626	0.634 ± 0.042	3.020 ± 0.335	4.006 ± 0.461	1.220 ± 0.191	7.67 ± 0.89	24.85 ± 0.40	31.85 ± 0.35
MACS J0025.4−1222	0.231 ± 0.022	1.734 ± 0.403	2.491 ± 0.539	0.521 ± 0.124	3.71 ± 0.74	11.91 ± 0.38	15.08 ± 0.45
RXC J0032.1+1808	2.809 ± 0.097	5.149 ± 0.245	5.972 ± 0.301	1.771 ± 0.112	23.15 ± 0.57	43.46 ± 0.26	48.31 ± 0.22
MACS J0035.4−2015	0.862 ± 0.029	1.861 ± 0.091	2.269 ± 0.127	0.565 ± 0.025	12.93 ± 0.24	21.77 ± 0.18	24.93 ± 0.21
Abell 68	2.315 ± 0.242	3.958 ± 0.508	4.621 ± 0.609	1.200 ± 0.119	8.11 ± 0.40	22.18 ± 0.89	26.32 ± 0.34
Abell 2813	2.116 ± 0.247	4.016 ± 0.658	4.772 ± 0.824	1.119 ± 0.132	19.20 ± 0.94	27.34 ± 0.69	30.62 ± 0.61
El Gordo	0.006 ± 0.001	3.216 ± 0.116	4.645 ± 0.229	1.366 ± 0.086	0.65 ± 0.07	30.78 ± 0.56	44.80 ± 0.53
Abell 209	0.370 ± 0.134	0.565 ± 0.241	0.639 ± 0.286	0.139 ± 0.043	7.21 ± 0.65	9.42 ± 0.69	10.32 ± 0.78
WL J013719−08284	0.453 ± 0.030	1.520 ± 0.403	1.945 ± 0.597	0.531 ± 0.194	11.01 ± 2.05	26.52 ± 0.34	31.56 ± 0.49
CL J0152.7−1357	0.006 ± 0.002	1.026 ± 0.134	1.757 ± 0.219	0.426 ± 0.059	0.35 ± 0.05	13.00 ± 0.31	19.07 ± 0.44
MACS J0159.8−0849	1.194 ± 0.087	2.629 ± 0.280	3.226 ± 0.356	0.961 ± 0.141	14.86 ± 0.66	27.64 ± 0.21	32.40 ± 0.46
RXC J0232.2−4420	1.618 ± 0.118	2.704 ± 0.284	3.133 ± 0.344	0.915 ± 0.098	19.03 ± 0.48	27.66 ± 0.14	30.19 ± 0.22
Abell 370	2.330 ± 0.032	4.325 ± 0.096	5.052 ± 0.131	1.822 ± 0.037	21.52 ± 0.10	37.77 ± 0.13	43.94 ± 0.21
Abell S295	2.071 ± 0.344	4.110 ± 0.836	4.853 ± 1.022	1.103 ± 0.181	7.67 ± 0.32	17.38 ± 1.46	21.10 ± 1.19
Abell 383	1.011 ± 0.169	1.412 ± 0.268	1.558 ± 0.306	0.466 ± 0.077	13.42 ± 0.31	17.43 ± 0.21	18.84 ± 0.24
MACS J0257.1−2325	0.315 ± 0.016	1.310 ± 0.129	1.757 ± 0.192	0.434 ± 0.042	6.17 ± 0.25	17.01 ± 0.25	20.79 ± 0.30
MACS J0257.6−2209	1.436 ± 0.050	2.697 ± 0.132	3.198 ± 0.161	0.906 ± 0.043	17.19 ± 0.16	27.51 ± 0.14	30.60 ± 0.18
MACS J0308.9+2645	2.016 ± 0.092	3.555 ± 0.245	4.055 ± 0.305	1.080 ± 0.100	26.48 ± 0.38	40.09 ± 0.14	44.11 ± 0.28
PLCK G171.9−40.7	2.606 ± 0.247	4.374 ± 0.584	5.075 ± 0.709	1.284 ± 0.139	23.51 ± 0.46	34.11 ± 0.22	37.44 ± 0.26
MACS J0329.7−0211	0.959 ± 0.038	2.332 ± 0.097	2.883 ± 0.127	0.850 ± 0.046	11.31 ± 0.32	26.78 ± 0.32	31.22 ± 0.42
Abell 3192	1.332 ± 0.083	2.931 ± 0.270	3.563 ± 0.364	1.127 ± 0.106	18.63 ± 0.43	30.81 ± 0.45	35.16 ± 0.55
MACS J0416.1−2403	1.063 ± 0.015	1.967 ± 0.040	2.271 ± 0.050	0.629 ± 0.016	15.98 ± 0.15	28.36 ± 0.17	32.30 ± 0.14
MACS J0417.5−1154	1.411 ± 0.025	3.824 ± 0.148	4.701 ± 0.200	1.147 ± 0.037	14.07 ± 0.11	34.09 ± 0.16	39.51 ± 0.16
MACS J0429.6−0253	0.393 ± 0.030	0.794 ± 0.100	0.942 ± 0.127	0.271 ± 0.037	10.91 ± 0.45	17.98 ± 0.16	20.32 ± 0.22
MACS J0451.9+0006	0.927 ± 0.018	2.044 ± 0.069	2.467 ± 0.093	0.673 ± 0.022	11.59 ± 0.33	28.13 ± 0.17	32.01 ± 0.16
Abell 521	0.382 ± 0.144	0.944 ± 0.462	1.210 ± 0.587	0.160 ± 0.042	5.20 ± 0.14	8.25 ± 0.25	9.53 ± 0.55
MS 0451.6−0305	0.663 ± 0.036	3.697 ± 0.522	5.116 ± 0.700	1.391 ± 0.192	6.55 ± 0.73	24.92 ± 0.20	31.87 ± 0.37
MACS J0553.4−3342	2.038 ± 0.086	4.509 ± 0.278	5.531 ± 0.354	1.452 ± 0.093	14.99 ± 0.55	37.00 ± 0.14	42.64 ± 0.15
MACS J0600.1−2008	4.165 ± 0.065	8.429 ± 0.192	9.781 ± 0.228	3.308 ± 0.088	13.03 ± 0.28	53.31 ± 0.38	62.62 ± 0.22
S J0615−5746	0.000 ± 0.000	1.385 ± 0.046	2.280 ± 0.113	0.678 ± 0.029	0.23 ± 0.00	23.49 ± 0.12	32.96 ± 0.19
Bullet cluster	2.913 ± 0.260	6.922 ± 0.755	8.363 ± 0.945	1.411 ± 0.132	12.70 ± 0.69	24.05 ± 0.29	26.75 ± 0.24
MACS J0717.5+3745	2.725 ± 0.025	6.328 ± 0.108	7.306 ± 0.169	2.373 ± 0.067	8.90 ± 0.63	57.22 ± 0.19	65.37 ± 0.17
S J0723.3−7327	1.039 ± 0.097	2.780 ± 0.431	3.610 ± 0.569	0.909 ± 0.093	9.05 ± 0.18	20.59 ± 0.25	25.36 ± 0.38
RXC J0911.1+1746	0.171 ± 0.018	0.744 ± 0.149	1.033 ± 0.245	0.250 ± 0.054	5.58 ± 0.96	13.77 ± 0.84	17.28 ± 0.68
MACS J0940.9+0744	0.147 ± 0.034	0.244 ± 0.086	0.281 ± 0.107	0.071 ± 0.028	7.49 ± 0.59	10.69 ± 0.08	11.53 ± 0.12
RXC J0949.8+1707*	0.940 ± 0.088	1.879 ± 0.321	2.224 ± 0.427	0.630 ± 0.118	15.54 ± 1.22	25.80 ± 0.57	29.72 ± 0.18
MACS J1115.8+0129	0.929 ± 0.103	1.864 ± 0.366	2.265 ± 0.459	0.629 ± 0.113	14.21 ± 0.75	22.23 ± 0.51	25.24 ± 0.58
PLCK G165.7+67.0	0.519 ± 0.026	1.214 ± 0.112	1.538 ± 0.160	0.383 ± 0.026	4.78 ± 0.33	14.84 ± 0.63	17.49 ± 0.75
MACS J1149.5+2223	0.747 ± 0.178	3.542 ± 0.729	4.647 ± 0.939	1.410 ± 0.354	6.81 ± 0.24	24.55 ± 1.19	34.98 ± 4.16
PLCK G287.0+32.9	4.042 ± 0.090	7.937 ± 0.335	9.337 ± 0.365	3.211 ± 0.126	22.32 ± 0.52	49.26 ± 0.34	57.75 ± 0.28
MACS J1206.2−0847	1.184 ± 0.014	2.851 ± 0.051	3.515 ± 0.076	1.038 ± 0.019	15.12 ± 0.12	30.24 ± 0.10	35.65 ± 0.14
MACS J1311.0−0310	0.323 ± 0.029	0.971 ± 0.129	1.225 ± 0.176	0.370 ± 0.069	7.72 ± 0.66	17.58 ± 0.41	21.52 ± 0.49
Abell 1689	4.725 ± 0.175	5.922 ± 0.274	6.336 ± 0.301	1.815 ± 0.081	43.22 ± 0.19	51.19 ± 0.17	53.91 ± 0.20
RXC J1347.5−1145*	1.350 ± 0.022	2.956 ± 0.104	3.498 ± 0.138	0.961 ± 0.051	20.59 ± 0.37	36.90 ± 0.09	42.03 ± 0.12
Abell 1835*	2.680 ± 0.394	4.163 ± 0.703	4.714 ± 0.835	1.345 ± 0.224	26.19 ± 0.63	35.91 ± 0.49	39.38 ± 0.61
MACS J1423.8+2404	0.376 ± 0.015	1.526 ± 0.111	2.017 ± 0.171	0.599 ± 0.044	7.17 ± 0.25	21.35 ± 0.21	26.68 ± 0.25
RXC J1532.9+3021	0.556 ± 0.068	1.378 ± 0.269	1.735 ± 0.375	0.429 ± 0.070	7.82 ± 0.65	15.55 ± 0.22	18.36 ± 0.20
PLCK G004.5−19.5	1.414 ± 0.033	3.657 ± 0.118	4.392 ± 0.161	1.429 ± 0.067	17.29 ± 0.45	39.29 ± 0.21	46.37 ± 0.20
MACS J1931.8−2635	0.951 ± 0.026	1.857 ± 0.073	2.208 ± 0.096	0.598 ± 0.024	14.74 ± 0.22	23.92 ± 0.13	26.89 ± 0.14
S J2031.8−4036	1.347 ± 0.213	2.312 ± 0.381	2.650 ± 0.459	0.741 ± 0.137	14.31 ± 1.94	29.28 ± 2.24	32.31 ± 2.53
MACS J2129.4−0741	0.420 ± 0.016	2.281 ± 0.161	3.170 ± 0.282	0.916 ± 0.057	7.56 ± 0.22	24.38 ± 0.22	31.44 ± 0.40
RXC J2129.7+0005	0.697 ± 0.074	1.191 ± 0.154	1.395 ± 0.189	0.297 ± 0.027	9.77 ± 0.14	13.76 ± 0.17	15.21 ± 0.24
S J2131.1−4019	0.632 ± 0.020	1.514 ± 0.091	1.816 ± 0.121	0.575 ± 0.057	10.34 ± 0.52	22.83 ± 0.31	26.57 ± 0.22
MS 2137.3−2353	0.438 ± 0.031	0.670 ± 0.061	0.745 ± 0.072	0.191 ± 0.018	14.09 ± 1.12	18.97 ± 0.15	20.54 ± 0.21
RXC J2211.7−0350	2.409 ± 0.046	4.277 ± 0.123	4.884 ± 0.142	1.453 ± 0.043	30.08 ± 0.49	46.47 ± 0.25	51.97 ± 0.26
MACS J2214.9−1359	0.532 ± 0.028	1.643 ± 0.134	2.155 ± 0.192	0.604 ± 0.049	9.58 ± 0.25	21.58 ± 0.29	25.86 ± 0.36
Abell S1063	1.975 ± 0.039	3.770 ± 0.084	4.430 ± 0.108	1.269 ± 0.030	20.00 ± 0.10	33.29 ± 0.38	38.43 ± 0.32
Abell 2537	1.854 ± 0.122	3.061 ± 0.236	3.474 ± 0.287	1.041 ± 0.096	19.68 ± 0.57	29.47 ± 0.28	32.97 ± 0.24
SPT−CL J2325−4111	2.762 ± 0.176	6.166 ± 0.538	7.566 ± 0.709	2.130 ± 0.158	19.55 ± 0.63	36.45 ± 0.25	42.42 ± 0.41

Einstein Radius

Another quantity that is commonly used to quantify the lensing strength of each cluster is the Einstein radius (e.g., Zitrin et al. [216]; Fox et al. [72]). The definition of the Einstein radius is, however, not unique when the mass distribution is not circularly symmetric on the sky. Throughout the paper, we employ the Einstein radius $\theta_{\mathrm{Ein}}$ defined by the area enclosed by the tangential critical curve. Specifically, within the analysis region shown in Table 1, we search for the largest contiguous region defined by $1-\kappa-\gamma<0$, where $\kappa$ and $\gamma$ are convergence and shear. Given the area $A_{1-\kappa-\gamma<0}$ of the largest contiguous region, the Einstein radius is defined by

In the same manner as for $A_{|\mu|>3}$, we derive $\theta_{\mathrm{Ein}}$ and its error from 50 mass models that are randomly drawn from the MCMC chain. Table 2 summarizes $\theta_{\mathrm{Ein}}$ values for the source redshift $z_{\mathrm{s}}=1$, 3, and 9. For $z_{\mathrm{s}}=9$, we identify MACS J0717.5+3745, MACS J0600.1−2008, and PLCK G287.0+32.9 as clusters with the largest $\theta_{\mathrm{Ein}}$ among the VENUS clusters (see also Zitrin et al. [218, 225]; Furtak et al. [83]).

We show the histograms of the Einstein radii for the source redshifts of $z_{\mathrm{s}}=1$ and 9 in Figure 8. The Einstein radii are larger for the higher source redshift, because of the increase of the deflection angle as $\propto D_{\mathrm{ls}}/D_{\mathrm{os}}$, where $D_{\mathrm{ls}}$ and $D_{\mathrm{os}}$ are angular diameter distances from the lens and the source and from the observer to the source, respectively. We find that the VENUS clusters typically have the Einstein radii $\theta_{\mathrm{Ein}}\sim30^{\prime\prime}$ for $z_{\mathrm{s}}=9$, which confirms that they have large Einstein radii on average. This

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is not surprising because VENUS selects target clusters based on their high lensing strengths.

Correlation of the High-Magnification Areas and the Einstein Radii

Since both $A_{|\mu|>3}$ and $\theta_{\mathrm{Ein}}$ quantify lensing strengths, we naturally expect a correlation of these two quantities. To explicitly check for such a correlation, in the left panel of Figure 9 we show the distribution of $A_{|\mu|>3}$ and $\theta_{\mathrm{Ein}}$ for the source redshift of $z_{\mathrm{s}}=9$. We find that $A_{|\mu|>3}$ and $\theta_{\mathrm{Ein}}$ are indeed positively correlated with each other with roughly $A_{|\mu|>3} \propto\theta_{\mathrm{Ein}}^2$. Some clusters such as Bullet cluster and Abell 2744 are outliers of the correlation in the sense that they have relatively large $A_{|\mu|>3}$ compared with values expected from their $\theta_{\mathrm{Ein}}$. This is simply because of the existence of large sub-clumps in these clusters; our definitions of $A_{|\mu|>3}$ and $\theta_{\mathrm{Ein}}$ are in such a way that these sub-clumps significantly increase $A_{|\mu|>3}$ while they do not significantly affect $\theta_{\mathrm{Ein}}$ that is solely determined by the size of the critical curve of the main clump in our definition. In addition, the large external shear from other clumps can enhance the area of the high-magnification region in each clump for such complex systems. We caution that some clusters required two pointings to cover the critical curves (see the analysis region column in Table 1), which indicates that $A_{|\mu|>3}$ for some of the clusters cannot be well covered by just one JWST NIRCam pointing.

In the right panel of Figure 9, we also show a plot similar to the left panel of Figure 9, but for the area of the high-magnification region with the higher magnification threshold, $A_{|\mu|>10}$. This Figure suggests that the area is better correlated with the Einstein radius $\theta_{\mathrm{Ein}}$ such that $A_{|\mu|>10} \propto\theta_{\mathrm{Ein}}^2$ for this higher magnification threshold of $|\mu| > 10$. This Figure also confirms our previous argument that, among the 60 VENUS clusters, MACS J0717.5+3745, MACS J0600.1−2008, and PLCK G287.0+32.9, which also have the largest Einstein radii for $z_{\mathrm{s}}=9$, are the best natural telescopes to study high-redshift Universe in terms of their large high-magnification areas. On the other hand, clusters with large $A_{|\mu|>3}$, such as Bullet, probe a complementary parameter space and therefore are also important for studying high-redshift Universe.

The Largest Einstein Radius in the Universe

The distribution of Einstein radii at the high tail serves as one of many observational tests of the standard $\Lambda$CDM model (e.g., [204, 217, 4] (Broadburt & Barkana 2008)). Among others, Oguri & Blandford [160] present detailed predictions of the largest Einstein radius in the Universe based on the halo model approach.

Here we compare observed $\theta_{\mathrm{Ein}}$ for the VENUS clusters with the $\Lambda$CDM prediction of Oguri & Blandford [160], which adopt a triaxial halo model of Jing & Suto [104] to derive the distribution of the Einstein radii, with some minor updates. The use of a triaxial halo is crucial given the strong dependence of the lensing strength on the viewing angle of a cluster. The halo mass function is updated to a fitting formula presented in Despali et al. [57]. Assuming the latest best-fit cosmological parameters of Louis et al. [130] with $\Omega_{\mathrm{m}}=0.3116$, $\Omega_{\mathrm{b}}=0.04920$, $\Omega_{\Lambda}=0.6884$, $h=0.6762$, $n_{\mathrm{s}}=0.9709$, and $\sigma_8=0.8149$, and an effective all-sky area of 40,000 deg$^2$, we generate a mock halo sample, and for each cluster we assign the concentration parameter and axis ratios following the probability distribution function given in Jing & Suto [104]. For each cluster, the viewing angle is also randomly assigned. We then compute the tangential critical curve of each cluster to derive the Einstein radius using the same definition as given in Equation (7). The Einstein radii are computed for source redshifts from $z_{\mathrm{s}}=1$ to $9.5$ with an interval of $\Delta z_{\mathrm{s}}=0.5$, and values of the Einstein radii are recorded only when they are sufficiently large, e.g., $\theta_{\mathrm{Ein}} \geq20''$ for $z_{\mathrm{s}}=1$ and $\theta_{\mathrm{Ein}} \geq40''$ for $z_{\mathrm{s}}=9.5$. We repeat this procedure 800 times to generate 800 realizations of an all-sky sample of clusters with large Einstein radii.

We compare the observed Einstein radii for the VENUS clusters with the $\Lambda$CDM prediction in Figure 10. The largest Einstein radius in the VENUS cluster sample is that of Abell 1689 for $z_{\mathrm{s}}\lesssim2$ and that of MACS J0717.5+3745 for $z_{\mathrm{s}}\gtrsim2$. We find that the largest Einstein radius in the VENUS cluster sample is well within the 95% confidence limit of the $\Lambda$CDM prediction of the largest Einstein radius in the Universe, and therefore is compatible with the $\Lambda$CDM model. It should be noted that the VENUS cluster sample is not complete in the coverage of clusters with high lensing strengths, and there may be additional clusters that have even higher Einstein radii, especially for high $z_{\mathrm{s}}$.

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Conclusion

We have presented a uniform mass modeling of 60 galaxy clusters observed with the JWST mostly as part of the VENUS program, using the GRAFIX software. The JWST/NIRCam images are available for 57 out of the 60 galaxy clusters. We have identified many new multiple images, taking advantage of the VENUS JWST/NIRCam images. Specifically, we have used $\sim4500$ multiple images for our mass modeling of the 60 clusters, $\sim1150$ of which are newly identified in this paper. As a result, individual VENUS clusters typically contain $\sim50$ multiple images. The large number of multiple images enables us to tightly constrain the mass distribution of each cluster. The root-mean-square of differences between observed and model-predicted multiple image positions is typically \sim0\farcs5, which is consistent with previous studies. We have conducted the Jackknife test to show that there is no sign of overfitting for most of the clusters. The median Einstein radius of the VENUS cluster sample is $\sim30^{\prime\prime}$ and the distribution of the Einstein radii extends out to $\sim60^{\prime\prime}$ for the source redshift of $z_s=9$, which confirms that lensing strengths of VENUS clusters are among the highest known to date. For $z_s=9$, we have identified MACS J0717.5+3745, MACS J0600.1−2008, and PLCK G287.0+32.9 as clusters with the largest Einstein radii ($\theta_{\rm Ein}\sim60^{\prime\prime}$) as well as the largest areas ($A_{|\mu|>10}\sim3\ {\rm arcmin}^{2}$) of the high-magnification regions. We have compared the observed Einstein radii with the $\Lambda$CDM model prediction to conclude that they are compatible with each other.

The strong lens mass models presented in this paper, together with independent mass models presented in a companion paper (J. F. V. Allingham et al., in preparation), will be crucial for studying the high-redshift Universe with the high-quality VENUS data. These regions of highest magnification will enable studies of the faintest objects at high redshift, helping to push the redshift frontier, constrain the faint end of the UV luminosity function, and identify candidate star clusters and metal-poor galaxies for spectroscopic follow-up.

Acknowledgments

This work was conducted and this paper was written without the use of AI tools of any kind. We thank Gabriel Brammer and Hollis Akins for their contributions to data processing and managements. This work is based on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #6882. This work was supported by JSPS KAKENHI Grant Numbers JP25H00662 (MO), JP22K21349 (MO, YF), JP23K13149 (YF), 26K00736 (YF), JP22H04939 (KK), JP23K20035 (KK), JP24H00004 (KK). SF acknowledges support from the Dunlap Institute, funded through an endowment established by the David Dunlap family and the University of Toronto. We acknowledge the support of the Canadian Space Agency (CSA) [25JWGO4A18]. JMD acknowledges the support of projects PID2022-138896NB-C51 (MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, Investigación. MB acknowledges support from the ERC Grant FIRSTLIGHT # 101053208, Slovenian national research agency ARIS through grants N1-0238 and P1-0188, and ESA PRODEX Experiment Arrangements No. 4000146646 and 4000149972. CJC acknowledges support from the European Research Council (ERC) Advanced Investigator Grant EPOCHS (788113). MGO acknowledges financial support from the State Agency for Research of the Spanish MCIU through Center of Excellence Severo Ochoa award to the Instituto de Astrofísica de Andalucía CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033, and from the grant PID2022-136598NB-C32 “Estallidos8” funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. MGO also acknowledges the support by the project ref. AST22_00001_Subp_11 funded from the EU - NextGenerationEU, PPCC Junta de Andalucía.

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RXC J0018.5+1626

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MACS J0025.4−1222

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RXC J0032.1+1808

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MACS J0035.4−2015

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Abell 68

Identifications of multiple images from Smith et al. [191], Cerny et al. [43], Smith et al. [190], Richard et al. [176], this paper. Spectroscopic redshifts of multiple images from Smith et al. [190], Richard et al. [176].

Abell 2813

Identifications of multiple images from this paper.

El Gordo

Identifications of multiple images from Zitrin et al. [219], Caminha et al. [39], Caputi et al. [40], Diego et al. [65], Diego et al. [66], Frye et al. [75], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [39], Caputi et al. [40], Frye et al. [75].

Abell 209

Identifications of multiple images from Zitrin et al. [224]. Spectroscopic redshifts of multiple images from Caminha et al. [38].

WL J013719−08284

Identifications of multiple images from Welch et al. [208], Vanzella et al. [203], Welch et al., in prep., this paper. Spectroscopic redshifts of multiple images from Pascale et al. [163], Welch et al., in prep.

CL J0152.7−1357

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MACS J0159.8−0849

Identifications of multiple images from Cibirka et al. (2018), this paper.

RXC J0232.2−4420

Identifications of multiple images from Cerny et al. [43], this paper. Spectroscopic redshifts of multiple images from Mainali [136].

Abell 370

Identifications of multiple images from Richard et al. [175], Diego et al. [64], Fu et al. (2025), Fudamoto et al., in prep., Richard et al. [178], Kawamata et al. [109], Lagattuta et al. [120], Chen et al. [47], Lagattuta et al. [121], this paper. Spectroscopic redshifts of multiple images from Soucail et al. [194], Richard et al. [175], Diego et al. [64], Lagattuta et al. [120], Gledhill et al. [86], Ghosh et al. [85], Lagattuta et al. [121], Sarrouh et al. [185], Richard et al. [180], Fu et al. (2025).

Abell S295

Identifications of multiple images from Cibirka et al. (2018), this paper. Spectroscopic redshifts of multiple images from Edge et al. [71].

Abell 383

Identifications of multiple images from Cerny et al. [41], Zitrin et al. [220], Smith et al. [189], Richard et al. [174]. Spectroscopic redshifts of multiple images from Cerny et al. [41], Smith et al. [189], Newman et al. [153], Richard et al. [174], Zitrin et al. [224].

MACS J0257.1−2325

Identifications of multiple images from Zitrin et al. [216], this paper, Nakane et al. [150]. Spectroscopic redshifts of multiple images from Nakane et al. [150].

MACS J0257.6−2209

Identifications of multiple images from Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Richard et al. [180].

MACS J0308.9+2645

Identifications of multiple images from Acebron et al. [2], this paper, Abdurro’uf et al. [1].

PLCK G171.9−40.7

Identifications of multiple images from Acebron et al. [2], this paper.

MACS J0329.7−0211

Identifications of multiple images from Caminha et al. [38], Zitrin et al. [224], Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [38], Zitrin et al. [224], Richard et al. [180].

Abell 3192

Identifications of multiple images from Hsu et al. [96], Hamilton-Morris et al. [92], Okabe et al. [161], Cerny et al. [43], this paper. Spectroscopic redshifts of multiple images from Hsu et al. [96].

MACS J0416.1−2403

Identifications of multiple images from Zitrin et al. [222], Jauzac et al. [100], Kawamata et al. [110], Kawamata et al. [109], Diego et al. [60], Caminha et al. (2017a), Richard et al. [180], Bergamini et al. [18], Diego et al. [67], this paper. Spectroscopic redshifts of multiple images from Grillo et al. [89], Christensen et al. [48], Hoag et al. [94], Ritcaršić et al. (2025), Rodney et al. [183], Bergamini et al. [18], Caminha et al. (2017a), Richard et al. [180], Sarrouh et al. [185].

MACS J0417.5−1154

Identifications of multiple images from Mahler et al. [133], this paper, Tsujita et al. [199]. Spectroscopic redshifts of multiple images from Mahler et al. [133], Sarrouh et al. [185], Jauzac et al. [103], Tsujita et al. [199].

MACS J0429.6−0253

Identifications of multiple images from Caminha et al. [38], Zitrin et al. [224], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [38].

MACS J0451.9+0006

Identifications of multiple images from MUSE Atlas, Richard et al. [173], this paper. Spectroscopic redshifts of multiple images from MUSE Atlas, Richard et al. [173].

Abell 521

Identifications of multiple images from Richard et al. [177], Messa et al. [143], this paper. Spectroscopic redshifts of multiple images from Richard et al. [177].

MS 0451.6−0305

Identifications of multiple images from Borys et al. [23], Jauzac et al. [99], MacKenzie et al. [131], this paper. Spectroscopic redshifts of multiple images from Borys et al. [23], Jauzac et al. [99], Knudsen et al. [117].

MACS J0553.4−3342

Identifications of multiple images from Ebeling et al. [70], Cerny et al. [43], this paper. Spectroscopic redshifts of multiple images from Ebeling et al. [70], Cerny et al. [43].

MACS J0600.1−2008

Identifications of multiple images from Furtak et al. [83], Fujimoto et al. [79], this paper. Spectroscopic redshifts of multiple images from Furtak et al. [83], Fujimoto et al. [79], Laporte et al. [123].

S J0615−5746

Identifications of multiple images from Paterno-Mahler et al. [166], Adamo et al. [5], Messa et al. [144], this paper. Spectroscopic redshifts of multiple images from Paterno-Mahler et al. [166], Messa et al. [144].

Bullet cluster

Identifications of multiple images from Mehlert et al. [137], B r a d a c et al. (2009), Paraficz et al. [162], B r a d a c et al. (2006), Richard et al. [180], Cha et al. [44], this paper, Rihtaršičič et al. (2026). Spectroscopic redshifts of multiple images from Mehlert et al. [137], Rihtaršičič et al. (2026), Richard et al. [180], Gonzalez et al. [88], Cha et al. [44], Paraficz et al. [162].

MACS J0717.5+3745

Identifications of multiple images from Zitrin et al. [218], Limousin et al. [126], Vanzella et al. [201], Diego et al. [62], Kawamata et al. [110], this paper. Spectroscopic redshifts of multiple images from Limousin et al. [126], Schmidt et al. [186], Treu et al. [197], Vanzella et al. [201].

S J0723.3−7327

Identifications of multiple images from Golubchik et al. [87], Mahler et al. [134], Pascale et al. [164], this paper. Spectroscopic redshifts of multiple images from Golubchik et al. [87], Mahler et al. [134], Noirot et al. [156].

RXC J0911.1+1746

Identifications of multiple images from this paper, Strait et al. [195].

MACS J0940.9+0744

Identifications of multiple images from Claeyssens et al. [50], Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Claeyssens et al. [50], Richard et al. [180].

RXC J0949.8+1707

Identifications of multiple images from Allingham et al. [6]. Spectroscopic redshifts of multiple images from Allingham et al. [6].

MACS J1115.8+0129

Identifications of multiple images from Zitrin et al. [224], Caminha et al. [38], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [38].

PLCK G165.7+67.0

Identifications of multiple images from Frye et al. [76], Cañameras et al. [33], Cañameras et al. [34], Frye et al. [74]. Spectroscopic redshifts of multiple images from Polletta et al. [167], Cañameras et al. [33], Frye et al. [76].

MACS J1149.5+2223

Identifications of multiple images from Zitrin & Broadhurst [215], Kelly et al. [112], Kelly et al. [114], Smith et al. [192], Rau et al. [171], Zheng et al. [213], Jauzac et al. [102], Treu et al. [198], Kawamata et al. [110], Richard et al. [178], Schuldt et al. [187], this paper. Spectroscopic redshifts of multiple images from Smith et al. [192], Kelly et al. [112], Jauzac et al. [102], Grillo et al. [90], Treu et al. [198], Sarrouh et al. [185], Schuldt et al. [187].

PLCK G287.0+32.9

Identifications of multiple images from Zitrin et al. [225], D’Addona et al. [56], this paper, Betts et al. [20]. Spectroscopic redshifts of multiple images from D’Addona et al. [56].

MACS J1206.2−0847

Identifications of multiple images from Caminha et al. (2017b), Ebeling et al. [69], Zitrin et al. [221], Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Caminha et al. (2017b), Ebeling et al. [69], Zitrin et al. [221], Richard et al. [180].

MACS J1311.0−0310

Identifications of multiple images from Zitrin et al. [224], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [38].

Abell 1689

Identifications of multiple images from Broadhurst et al. [31], Limousin et al. [124], Coe et al. [51], Bina et al. [22], Diego et al. [63], this paper. Spectroscopic redshifts of multiple images from Broadhurst et al. [31], Frye et al. [73], Jullo et al. [106], Frye et al. (2002), Bina et al. [22], Limousin et al. [124].

RXC J1347.5−1145

Identifications of multiple images from Halkola et al. [91], Caminha et al. [38], Bradač et al. [25], Zitrin et al. [224], Richard et al. [180]. Spectroscopic redshifts of multiple images from Ravindranath & Ho [172], Caminha et al. [38], Cohen & Kneib [54], Richard et al. [180].

Abell 1835

Identifications of multiple images from Richard et al. [177], Morandi et al. [146]. Spectroscopic redshifts of multiple images from Richard et al. [177].

MACS J1423.8+2404

Identifications of multiple images from Limousin et al. [125], Zitrin et al. [224], this paper. Spectroscopic redshifts of multiple images from Limousin et al. [125], Sarrouh et al. [185].

RXC J1532.9+3021

Identifications of multiple images from Zitrin et al. [224], this paper.

PLCK G004.5−19.5

Identifications of multiple images from Sifón et al. [188], this paper, Yanagisawa et al. [211]. Spectroscopic redshifts of multiple images from Sifón et al. [188].

MACS J1931.8−2635

Identifications of multiple images from Caminha et al. [38], Zitrin et al. [224], this paper, Allingham et al. [7], Coulter et al. [55]. Spectroscopic redshifts of multiple images from Caminha et al. [38], Zitrin et al. [224], Coulter et al. [55].

S J2031.8−4036

Identifications of multiple images from Richard et al. [179], Richard et al. [180], Cerny et al. [43], this paper. Spectroscopic redshifts of multiple images from Christensen et al. [48], Richard et al. [179], Richard et al. [180].

MACS J2129.4−0741

Identifications of multiple images from Caminha et al. [38], Zitrin et al. [224], Monna et al. [145], Jauzac et al. [99], Cerny et al. [43], this paper. Spectroscopic redshifts of multiple images from Treu et al. [197], Caminha et al. [38], Christensen et al. [48], Monna et al. [145], Huang et al. [97], Jauzac et al. [99].

RXC J2129.7+0005

Identifications of multiple images from Caminha et al. [38], Zitrin et al. [224], Richard et al. [177], Williams et al. [209], Dhanasingham et al. [59], this paper. Spectroscopic redshifts of multiple images from Caminha et al. [38], Belli et al. [15], Williams et al. [209].

S J2131.1-4019

Identifications of multiple images from Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Richard et al. [180].

MS 2137-2353

Identifications of multiple images from Mellier et al. [138], Zitrin et al. [224], Cerny et al. [41], this paper. Spectroscopic redshifts of multiple images from Sand et al. [184], Zitrin et al. [224], Cerny et al. [41].

RXC J2211.7-0350

Identifications of multiple images from Cerny et al. [42], Cerny et al. [43], Zhang et al. [212]. Spectroscopic redshifts of multiple images from Cerny et al. [42], Zhang et al. [212].

MACS J2214.9-1359

Identifications of multiple images from Zitrin et al. [216], Richard et al. [180], this paper. Spectroscopic redshifts of multiple images from Richard et al. [180].

Abell S1063

Identifications of multiple images from Richard et al. [178], Johnson et al. [105], Caminha et al. [35], Diego et al. [61], Kawamata et al. [109], Karman et al. [108], Beauchesne et al. [14], Zitrin et al. [224], Diego et al. [68]. Spectroscopic redshifts of multiple images from Karman et al. [108], Karman et al. [107], Caminha et al. [35], Balestra et al. [12], Treu et al. [197], Beauchesne et al. [14].

Abell 2537

Identifications of multiple images from Cerny et al. [42], this paper. Spectroscopic redshifts of multiple images from Cerny et al. [42].

SPT-CL J2325-4111

Identifications of multiple images from Mahler et al. [135], this paper. Spectroscopic redshifts of multiple images from Mahler et al. [135].