(2021). Similar to MACSJ1752, the radio relics are found at opposing ends of
an S-shaped X-ray morphology. However, in comparison to MACSJ1752, the
location of the relics are in better agreement with the merger axis, being
directly inline with the elongated X-ray. Jones et al. (2021) also detected a
radio bridge (connection) between the southern radio relic and the southern
BCG. G19 were able to separate the cluster galaxies into two populations of
nearly equal velocity dispersion and suggested an equal mass merger.

WL result: The WL analysis (Figure 31) of ZwCl1856 is presented in Finner et
al. (2021). The mass distribution is dominated by two mass peaks that agree
with their respective BCGs and align along the merger axis that is defined by
the radio relics. A signature of a third, lower significance peak is found to
the north that is located near the northern radio relic. We found the N (S)
subclusters to have masses of $1.6\pm 0.8$ ($1.5\pm 0.7$) $\times 10^{14}\
M\odot$. Intriguingly, ZwCl1856 is a low-mass binary merger that hosts bright
radio relics. The low mass is in agreement with the expected mass from our
XMM-Newton temperature estimate of $3.6^{+0.6}_{-0.5}$ keV (Finner et al.,
2021).

Merger insight: The S-shaped X-ray morphology suggests a merger with a small
impact parameter. The mass peaks in both the north and south are slightly
inside the X-ray brightness peaks, which could indicate that the cluster is
near apocenter and the ICM has been slingshot through the potential. However,
this interpretation, which is based on poor X-ray observations, should be
remedied soon with upcoming deeper Chandra observations. MACSJ1752 and
ZwCl1856 provide two examples of simple cluster mergers that have much
different masses but host bright radio relics.

Figure 32: ZwCl2341.A complex cluster merger with a bullet-shaped X-ray
emission. WL contours increase in steps of $0.5\sigma$.

###  4.26 ZwCl 2341+0000 ($z=0.27$)

The ICM of ZwCl2341 is rich in merging features. van Weeren et al. (2009)
found double radio relics, which were verified in Giovannini et al. (2010) and
Parekh et al. (2022). The polarization properties of the radio relics of
ZwCl2341 were also studied in Stuardi et al. (2022). The radio relics are
roughly aligned with each other and separated by about 2 Mpc. In the
northwest, a bullet is apparent in X-ray emission. In front of the bullet
shape is the northern radio relic with an X-ray shock detected in Ogrean et
al. (2014b). The X-ray emission extends to the northeast and south. A similar
X-ray distribution is seen in A3411. The southern ICM contains an X-ray shock
and the southern radio relic. Zhang et al. (2021) go into detail about many of
the merging features that are found in the Chandra observation. Cluster
galaxies were identified in Boschin et al. (2013) and used to show the SE-NW
elongation of the cluster. Benson et al. (2017) performed a multiwavelength
analysis of ZwCl2341. Their modeling of the cluster galaxies determined that
the cluster is composed of three subclusters. G19 also detect three
subclusters with two in the NW and one in the SE.

Benson et al. (2017) also provide a WL analysis of the galaxy cluster. Their
WL analysis on the available Subaru r-band imaging shows a single peak with a
broad distribution that covers the brightest galaxies in the cluster.

WL result: Our WL analysis (Figure 32) shows a peak in the center that traces
an overdensity of cluster member galaxies. From the central peak there are
extensions to the south, east, and north. The southern extension overlaps the
brightest and densest galaxy region. The northern extension contains the
bullet X-ray emission and more bright cluster galaxies. The eastern extension
is the most peculiar. It seems to not trace many cluster galaxies. The peak of
the eastern extension lies on top of a deep yellow elliptical galaxy. It may
be possible that there is a background cluster in the east of the X-ray
emission that is contaminating the WL signal. The histogram of spectroscopic
redshifts in Figure 4 of Benson et al. (2017) shows a possible background
structure at $z=0.31$ but more spectroscopy would be needed to confirm it. We
fit a three-halo model to the NW, C, and S subclusters and find masses of
$0.8\pm 0.6$, $3.1\pm 1.2$, and $1.3\pm 0.7\times 10^{14}\ M\odot$.

Merger insight: ZwCl2341 is another example of a very complex cluster merger.
The formation of the radio relics likely resulted from a collision between the
central and southern subclusters, the central and northern clusters, or both
collisions. The complexity of ZwCl2341 makes it a difficult cluster to
simulate. A better approach may be to search for analogs from cosmological
simulations.

###  4.27 Non-detections

For three of the clusters that are analyzed in G19, we are unable to provide
WL results. Our WL analyses of A3365 ($z=0.093)$ and RXC J1053.7+5452
($z=0.072$) result in no detection. Further analysis of the imaging will be
required to understand the failure. For RXCJ1053, bright stars in the Subaru
imaging are the probable cause of no detection. Bright stars can be removed by
careful subtraction of a 2D model from the stacked image. For example,
observations with the HSC targeting A746 were planned to fix a bright star at
the center of the focal plane, which allowed an easier subtraction of a
symmetric model (HyeongHan et al., 2024b). Future imaging of RXCJ1053 could
use this technique to alleviate the issue of the bright star. At the time of
this analysis, A2443 ($z=0.110$) had no WL quality imaging. However, the
cluster was recently observed with the HSC (PI: H. Cho) and the observations
are starting to be analyzed. The WL result will be presented in Kim et al. in
prep.

##  5 DISCUSSION

###  5.1 WL Mass Estimate Comparison

The $c-M$ relation is typically derived from simulations that have a fixed
volume and in some cases a fixed cosmology (e.g., Duffy et al., 2008; Dutton &
Macciò, 2014), although, some are valid for any cosmology (e.g., Diemer &
Joyce, 2019; Ishiyama et al., 2021). Under these constraints, the $c$ and $M$
of clusters in the simulation are limited and may not represent the full range
of observed clusters. This may be particularly true for the merging cluster
sample where recent, powerful gravitational interactions have taken place that
could have a significant impact on the concentration of the dark matter
distribution. To test whether the $c-M$ relation holds for merging clusters,
it would be ideal to compare a measured concentration from the merging cluster
sample to the expected concentration of the $c-M$ relation. However,
concentrations are poorly constrained for most individual clusters (see Umetsu
et al., 2020, for stacked WL constraints on concentration). An alternative is
to test the effect of a merger on the concentration by comparing the mass
derived from a $c-M$ relation to that of an independent method such as our
two-parameter fitting method (2PNFW).

Figure 33 compares the subcluster mass estimates from the Duffy et al. (2008)
$c-M$ relation to the 2PNFW method. The fuchsia line is a linear fit with the
95% confidence interval filled blue. The best-fit line is steeper than the 1:1
ratio (black-dashed line), which suggests that the $c-M$ relation tends to
provide higher (lower) mass than the 2PNFW method for low- (high-) mass
merging clusters. The $2\sigma$ confidence interval of the linear fit overlaps
with the 1:1 ratio at the low-mass end but is offset at the high-mass end.

The high-mass discrepancy is likely caused by the merging nature of the
sample. If we assume that the mass discrepancy is directly related to the
concentration of the subclusters, then the steeper relation hints that low-
mass (high-mass) subclusters have lower (higher) concentration than expected
from the power-law $c-M$ Duffy et al. (2008) relation. This is not unexpected
since Ludlow et al. (2012); Meneghetti & Rasia (2013); Diemer & Joyce (2019)
have all shown that unrelaxed clusters follow a different $c-M$ relation with
an upturn in concentration at high mass. Therefore, it is not surprising that
our sample of merging clusters departs from the Duffy et al. (2008) $c-M$
relation at the high-mass end. However, this does not explain the low-mass
end. One possibility is that the increased gravitational potential that the
low-mass subcluster feels during pericenter passage increases the velocity
dispersion of galaxies (e.g. Pinkney et al., 1996; Takizawa et al., 2010) and
dark matter, which in turn lowers the dark matter concentration and
subsequently leads to an overestimation of mass from the Duffy et al. (2008)
$c-M$ relation. This would preferentially affect the low-mass subclusters
since, in this sample, they are more likely to merge with a cluster of
comparable or higher mass (see Section 5.3 for mass ratios).

Lee et al. (2023) investigated the bias of a WL mass estimate of merging
clusters for both the $c-M$ relation and the 2PNFW method. Their simulations
showed that the dark matter concentration increased after pericenter for both
subclusters involved in the merger, with the lower mass subcluster having a
larger increase than the higher mass subcluster. For the $c-M$ relation, this
lead to an overestimation of the mass of both subclusters because of the low
concentration. This effect persisted from pericenter to approximately 0.75
Gyrs after the collision. At 0.75 Gyrs, the mass bias switched to an
underestimation for the low-mass subcluster while remaining an overestimation
for the high-mass subcluster. On the other hand, the 2PNFW method resulted in
a smaller overall bias shortly after collision (0.25 Gyr), but the mass bias
for the lower-mass subcluster flipped from an overestimation to an
underestimation by 0.5 Gyrs (earlier than the $c-M$ relation). We do not have
a constraint on the time since the collision for the clusters in our sample
and thus cannot make a definitive statement on whether the WL mass bias that
Lee et al. (2023) showed could be the cause of the steeper slope in Figure 33.
However, it is important to note that these biases could arise in these WL
measurements.

Figure 33: Comparison of the mass derived from the Duffy et al. (2008) $c-M$
relation to the mass derived from the two-parameter fit (values from Table 4).
The black-dashed line represents the 1:1 relation. The violet line is the
best-fit linear regression. Uncertainties are considered on both axes. The
$c-M$ relation tends to overestimate/underestimate the mass of low-mass/high-
mass clusters that exhibit radio relics.

###  5.2 WL Mass - Velocity Dispersion Scaling Relation

Merging galaxy clusters are the most energetic events since the big bang
(Sarazin, 2002). This energy is dissipated into the cluster and heats up the
ICM. In addition, the increase in gravitational potential caused by the
introduction of another cluster will in turn boost the velocity dispersion of
galaxies and dark matter for a limited time (e.g. Pinkney et al., 1996;
Takizawa et al., 2010). Therefore, in merging clusters that are observed
shortly after collision it is expected that the mass derived from the velocity
dispersion of galaxies would be biased high.

In Figure 34, we plot our WL derived masses from the 2PNFW method against the
velocity dispersion of galaxies that are reported in G19. The black-dashed
line is the Evrard et al. (2008) scaling relation. The best-fit line sits
below the scaling relation, which is in agreement with the expectation that
velocity dispersion is boosted by the recent merger. It is also interesting
that the best-fit line is steeper than the Evrard et al. (2008) scaling
relation. This may be an indication that low-mass clusters in mergers are more
affected than the high-mass counterparts. The scaling relation based on the
best-fit line is:

$M_{200}=\left(\frac{\sigma_{v}}{1156\pm 381\ \mathrm{km\
s}^{-1}}\right)^{3.8\pm 0.4}\ 10^{15}\ \mathrm{M}_{\odot}$ (18)

An alternative explanation for the steeper slope could be systematics.
Measuring the velocity dispersion relies on the assignment of galaxies into
groups. In the crowded environment of a merging galaxy cluster it is difficult
to perform this assignment. G19 used a GMM technique to perform the
assignment. Naturally, it is more likely that a galaxy belongs to a high-mass
halo than a low-mass halo. In addition, the global velocity dispersion will
favor that of the high-mass halo. Thus, one would expect that the systematic
of assigning galaxies to halos would preferentially affect low-mass halos, to
the point that even some low-mass halos are completely missed.

Figure 34: WL mass - $\sigma_{v}$ scaling relation for subclusters of merging
systems that exhibit radio relics. Masses and system index letters are from
Table 4. The black-dashed line is the Evrard et al. (2008) scaling relation.
WL mass estimates are preferentially lower than the Evrard et al. (2008)
relation, especially at the low-mass end.

###  5.3 Mass ratio

The energy required to create megaparsec scale merger shocks is expected to be
high and is dependent on cluster mass (e.g., de Gasperin et al., 2014b). To
better understand the types of mergers that are occurring that lead to the
creation of radio relics, we plot the mass ratio of the merging subclusters.

Figure 35 compares the WL mass of the primary and secondary subclusters. These
were selected from the substructures of each merging cluster by their
alignment with the X-ray- and radio-defined merger axes, their agreement with
BCG locations, and their WL $S/N$. The figure shows that many of the mergers
are close to 1:1 mass ratio, independent of the mass scale. Furthermore, most
of the clusters ($\scriptstyle\sim$70%) are in major mergers with mass ratio
$<4$:1. The few outliers tend to be clusters with very massive primaries, such
as PLCKG287 and A2163. Perhaps this is an indication that very massive major
mergers are rare. However, there may be a selection effect because low-mass
primaries ($\lesssim 10^{14}M_{\odot}$) would require very low-mass
secondaries ($<10^{14}M_{\odot}$), which are below the detection threshold of
this Subaru WL dataset.

Figure 35: Mass ratio of merging clusters that exhibit radio relics. Masses
are from the 2PNFW method. Masses and system index letters are from Table 4.
The solid (dashed) black line represents the 1:1 (1:4) mass ratio. Most of the
radio relic systems are in major mergers (ratio $<4$).

###  5.4 Peak Separation

Given the large number of subclusters analyzed in this work, we can compare
the separation of various tracers of the dark matter potential, which may
provide insight into the physics of the collision (e.g., Kim et al., 2017). In
Section 3.7, we identified WL peaks by correlating them with nearby galaxies
and/or X-ray brightness peaks. Therefore, this comparison is done knowing that
the luminous tracer was initially used in identification of the WL peaks.

Figure 36 shows the projected separation of the WL peak from the nearest BCG,
luminosity peak, and number density peak for each subcluster. For each system,
the uncertainty on the location of the mass peak for each subcluster was
estimated by collecting the mass peak location in the 1000 bootstrapped mass
maps. The peaks were then processed with a k-means clustering algorithm with
the number of clusters fixed to the expected value (i.e., 2 for a merging
cluster that has 2 subclusters). The distribution of each of the k-means-
defined subclusters were then used to derive the error ellipse that contained
68% of the peaks.

For the majority of subclusters, the uncertainty on the mass peak has a
$1\sigma$ overlap with the positions of the luminous tracer. We find that the
BCG is closer than the luminosity (number density) peak to the mass peak in
72% (68%) of the subclusters. The median values (vertical lines in Figure 36)
show that the BCG is the best tracer of the mass peak. The median projected
separation of the BCG, the luminosity peak, and the number density peak from
their respective WL peak is $79\pm 14$, $90\pm 15$, and $119\pm 15$ kpc,
respectively. Our median projected separation of the BCG from the mass peak
agrees with Oguri et al. (2010) who found that the separation follows a
Gaussian distribution with $\sigma=90$ kpc. George et al. (2012) compared the
WL mass peaks to BCGs in X-ray detected galaxy groups and found the average
projected separation to be less than 75 kpc. Using the large sample of 10,000
SDSS clusters, Zitrin et al. (2012) studied the offset of the BCG from the DM
peak and found that the average offset is $\scriptstyle\sim$13 kpc. For our
sample, only 1 of the 58 subclusters has a BCG within 13 kpc of the mass peak.
However, the uncertainties on the mass peak are on average approximately 100
kpc. Therefore, these results are not statistically conclusive.

An additional tracer of the DM potential that is beyond the scope of this work
is the intracluster light (ICL). Yoo et al. (2022, 2024) show in simulations
that the ICL is a good tracer of the DM potential. We will explore the
possibility of using the ICL as a tracer of the DM potential in future work.

Figure 36: Projected separation of the BCGs (blue stars), luminosity peaks
(red squares), and number density peaks (green ) from the WL peak. The
horizontal grey bars represent the uncertainty on the mass peak position. The
vertical colored lines show the median separations of each luminous tracer
from the mass peak and their 68% uncertainty.

###  5.5 The Golden Sample

Dissociation of the ICM from the cluster potential by ram pressure during a
cluster merger is a powerful feature that can be used to understand the merger
scenario and constrain the physics of mergers (for an example see Zhang et
al., 2019). However, the variety of gas dissociation in merging clusters is
diverse. Clusters such as A115 have broad V-shaped X-ray emission tails that
are as wide as they are long. In contrast, the tails behind the subclusters of
MACSJ1752 extend approximately 0.5 Mpc but are only $\scriptstyle\sim$100 kpc
at their greatest width. The relation of X-ray brightness peaks and WL peaks
also vary. In some dissociative mergers, like A115, the X-ray brightness peaks
are retained in the WL peak. Conversely, in A2034, the X-ray brightness peaks
trail behind the WL peaks. As demonstrated in simulations by ZuHone et al.
(2018), the differences may be caused by impact parameter, pre-merger
concentration of the ICM, total mass, concentration of the DM, or projection.

Achieving a sound understanding of cluster mergers relies on carefully
recreating the collision on a computer. Cases like A2744 and A746 have
multiple subclusters that are merging along many axes, which makes them
difficult to reproduce. For constraining the properties of merging clusters,
it is best to start with the simplest cases. G19 defined a “gold sample” of
cluster mergers that have simple geometry. Having completed WL analyses of the
systems, we provide an updated gold sample. These systems have distinct merger
features and mostly bimodal mass distributions. Our updated gold sample is
A1240, A2034, MACSJ1752, RXCJ1314, ZwCl0008, ZwCl1447 and ZwCl1856. CIZAJ2242
is also a strong candidate but, at this point, the existing optical/IR
observations of the cluster cannot support a WL analysis of the precision that
is needed to fully understand the complexity of the mass distribution. Aside
from A2034, these systems all contain double radio relics, which provide
critical constraints on the merger timescale and viewing angle. There are
additional candidate systems that may be valuable, such as A1612, but the
current multiwavelength data do not support a robust analysis.

##  6 CONCLUSIONS

We present WL analyses of 29 merging galaxy clusters that exhibit megaparsec-
sized radio relics. The WL analyses are performed on Subaru Suprime-Cam and
Hyper-Suprime-Cam observations. Of the 29 systems, WL analyses are
successfully achieved for 26 clusters and 3 clusters are unsuccessful.

Our WL analysis technique uses a PCA to create PSF models for each galaxy and
a Gaussian model to fit the galaxy shape. Source galaxies are selected from
CMDs utilizing the spectroscopically confirmed red-sequence galaxies as a
guide. The source galaxies are used to generated high-resolution mass maps
with the FIATMAP code. Comparing the mass maps with the vast multiwavelength
supplementary data, significant (WL $S/N>3$) subclusters are identified. Mass
estimates for each subcluster of a system are determined by simultaneously
fitting multiple NFW halos to the source galaxies. The mass estimates are
performed by both fitting a $c-M$ relation (Duffy et al., 2008) and by MCMC
sampling the $c$ and $M$ parameter space. With subclusters identified and
masses estimated, an in-depth discussion of each merging system is presented.
From the analysis of each system and the analysis of the radio relic sample as
a whole, the following conclusions are made.

  * •

We combine our WL results with the multiwavelength observations from
literature to develop a better understanding of each merging system. We show
that WL analysis is critical to detecting and quantifying the substructures in
merging clusters and in some cases discover subclusters that were previously
undetected. Utilizing the new information provided through WL analysis, we
update the merging scenarios for each system and give new insight into the
past collisions. In addition, we highlight peculiar systems that warrant
further investigation, such as the dissociative nature of Abell 2061 and its
large radio relic distance.

  * •

We compare WL mass maps to the X-ray emission from the ICM. In cases such as
MACSJ1752 and ZWCL1856, we find that the X-ray emission, WL mass maps, and
radio relics paint a clear picture of the past merger. In others, the
complexity of the system and the positions of the radio relics are less
straightforward. Gas dissociation is clearly visible in many of the clusters.
In general, the overall distribution of the mass maps is elongated in the same
manner as the X-ray emission.

  * •

We compare the mass distributions to the galaxy light and galaxy number
density distributions. A strong correlation between mass peaks and galaxies
are obvious. In most cases, the most massive subcluster is co-spatial with the
global luminosity and number density peaks of the system. This finding is not
surprising if more massive halos are expected to have more galaxies. However,
in clusters such as A1240 and A1612, the more massive subcluster is not
associated with the global luminosity and number density peaks. The galaxy
distributions in general show the same elongation as the X-ray and WL signals
and in some cases stretch further along the axis of elongation.

  * •

We investigate the influence of method on the mass estimation by comparing the
use of a $c-M$ relation to a mass estimate that samples both $c$ and $M$. We
find that the two methods depart slightly from a 1:1 relation with the fixed
$c-M$ mass estimate technique providing a smaller (larger) mass estimate for
high- (low-) mass subclusters. We suggest the primary reason for this
discrepancy is that the $c-M$ relation is derived from simulations that
include both relaxed and unrelaxed clusters.

  * •

The WL-derived masses are compared to the velocity dispersion measurements, a
proxy for mass, of each subcluster. We find that the scatter of the velocity
dispersion with respect to the WL mass is large. Lower mass subclusters tend
to have more inflated velocity dispersions than high mass systems when
compared to the scaling relation from Evrard et al. (2008). We suggest that
this could be caused by the increased gravitational effect that slow-mass
subclusters feel during pericenter passage, which inflates the velocity
dispersion and leads to the large scatter in comparison to WL mass.

  * •

For each system, primary and secondary subclusters that are responsible for
the generation of the radio relics are distinguished by their mass and
alignment to the radio relics. Comparing the mass of the primary to secondary
subclusters, we find that most merging clusters that exhibit radio relics are
major mergers with a mass ratio less than 4.

  * •

We analyze the separation of the peak of the dark matter distribution from
other luminous peaks. On average, the BCG most closely traces the dark matter
peak followed by the luminosity and then the number density peak. The average
offset of the BCG from its mass peak is found to be $75\pm 14$ kpc.

  * •

An updated sample of 8 ideal merging clusters were defined (a Golden sample).
The ideal cases have clear X-ray features, such as bullet morphology that
highlight the merger path, and have verified merger-induced radio relics.
Additional important selection criteria come from the WL signal: robust
detection of the WL signal, dominant bimodal distribution, and dissociation of
the gas from the dark matter density peaks. This sample of merging clusters
represent the ideal cases that can be reproduced through simulations.
Furthermore, the large separation of the mass peaks from the gas peaks makes
this sample idea for studying the nature of dark matter.

This work was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (RS-2024-00340949). RJvW
acknowledges support from the ERC Starting Grant ClusterWeb 804208. This
research is based on data collected at the Subaru Telescope, which is operated
by the National Astronomical Observatory of Japan. We are honored and grateful
for the opportunity of observing the Universe from Maunakea, which has the
cultural, historical, and natural significance in Hawaii. MGCLS data products
were provided by the South African Radio Astronomy Observatory and the MGCLS
team and were derived from observations with the MeerKAT radio telescope. The
MeerKAT telescope is operated by the South African Radio Astronomy
Observatory, which is a facility of the National Research Foundation, an
agency of the Department of Science and Innovation. Based on observations
obtained with XMM-Newton, an ESA science mission with instruments and
contributions directly funded by ESA Member States and NASA. This research has
made use of data obtained from the Chandra Data Archive and the Chandra Source
Catalog, and software provided by the Chandra X-ray Center (CXC) in the
application packages CIAO and Sherpa. This paper makes use of LSST Science
Pipelines software developed by the Vera C. Rubin Observatory. We thank the
Rubin Observatory for making their code available as free software at
https://pipelines.lsst.io.

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