# A major galaxy cluster merger caught by eROSITA: weak lensing mass
distribution and kinematic description

Rogério Monteiro-Oliveira1
1Universidade Estadual de Santa Cruz, Depto. de Ciências Exatas e
Tecnológicas, Lab. de Astrofísica Teórica e Observacional, 45650-000, Ilhéus,
BA, Brazil
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the weak lensing mass distribution of a triple merging cluster
candidate at $z_{\rm photo}\sim 0.36$ belonging to a supercluster recently
discovered during the eROSITA Performance Verification phase. Our analysis
solved a previous tension in the merger classification by confirming that the
cluster pair eFEDS J093513.3+004746 and eFEDS J093510.7+004910 is undergoing a
major merger with a mass ratio $1.7_{-0.7}^{+0.5}$. According to our two-body
kinematic description, the encounter happened $0.58_{-0.20}^{+0.15}$ Gyr ago,
in a scenario that supports the observed radio relic position at the cluster
outskirts. However, the same analysis showed that the companion cluster, eFEDS
J093501.1+005418, is not gravitationally bound to the interacting system and
therefore it is not part of the supercluster. We also checked the impact of
adopting a scaling relation to determine the halo concentration $c_{200}$. At
the observed merger stage, where the clusters have travelled $\sim$55 per cent
of the path to reach the apoapsis, the choice of the $c_{200}$ (whether from a
scaling relation or a free parameter in the mass model) does not change
significantly either the cluster masses or the kinematic description.

###### keywords:

gravitational lensing: weak – dark matter – galaxies: clusters: general –
galaxies: clusters: individual: eFEDS J093513.3+004746 – galaxies: clusters:
individual: eFEDS J093510.7+004910 – galaxies: clusters: individual: eFEDS
J093501.1+005418

††pubyear: 2015††pagerange: A major galaxy cluster merger caught by eROSITA:
weak lensing mass distribution and kinematic description–A major galaxy
cluster merger caught by eROSITA: weak lensing mass distribution and kinematic
description

##  1 Introduction

Given the large amount of energy involved ($\gtrsim 10^{64}$ ergs; e.g.,
Sarazin, 2004), major galaxy cluster mergers111We adopt the definition of
Martel et al. (2014), and consider as major merger an event in which the mass
ratio between the two most massive (in case of multiple clusters) is less than
two. are often called astrophysical particle colliders (Harvey et al., 2015).
Therefore, they are an excellent laboratory to investigate properties of the
key constituents of clusters, namely the intracluster medium gas (ICM; e.g.
Keshet et al., 2021), the galaxies (e.g enhancement/quenching of star
formation; Kelkar et al., 2020; Hernández-Lang et al., 2021), and the dark
matter (e.g. Fischer et al., 2021), even if they are observed at late stages
of a merger (e.g., Tam et al., 2020). As the ICM carries the most discernible
signatures of the merger process (such as shock waves, cold fronts, and
sloshing; e.g., Ha et al., 2018; Pandge et al., 2019; Ueda et al., 2020, 2021;
Cho et al., 2021; Machado et al., 2022), ongoing high-quality X-ray surveys as
the extended ROentgen Survey with an Imaging Telescope Array (eROSITA; Merloni
et al., 2012; Predehl et al., 2021) will dramatically increase the number of
identified cluster mergers.

Analysing the eROSITA Final Equatorial Depth Survey (eFEDS), Ghirardini et al.
(2021, hereafter G21), reported the discovery of a supercluster at photometric
redshift (photo-$z$) $z_{\rm photo}\sim 0.36$. It consists of a chain of eight
clusters spanning a length of 27 Mpc on the plane of the sky. Despite the fact
that most of them present regular X-ray and optical properties, the authors
identified a triple major merger candidate consisting the galaxy clusters
eFEDS J093513.3+004746 (hereafter eFEDS4746; $z_{\rm photo}=0.367$), eFEDS
J093510.7+004910 (hereafter eFEDS4910; $z_{\rm photo}=0.367$) and eFEDS
J093501.1+005418 (hereafter eFEDS5418; $z_{\rm photo}=0.385$). More details of
these clusters are presented in Table 1. Using LOFAR and uGMRT observations,
G21 found two radio relics at the outskirts of eFEDS4746 and eFEDS4910 as well
as a radio halo coincident with those clusters in projection. Radio relics are
characterized by diffuse radio synchrotron emission, which are known to be
good tracers of the shock wave propagating though the ICM after the
pericentric passage during a merger (Zhang et al., 2020; Knowles et al.,
2021), even though their origin are not yet fully understood (e.g. van Weeren
et al., 2019). On the other hand, due to its regular optical and X-ray
properties, together with the absence of any radio feature, G21 argued that
eFEDS5418 is in a pre-merger state given its short projected distance to its
companions.

In contrast with the radio observations, the major merger scenario between
eFEDS4746 and eFEDS4910 does not find any support either from the estimated
clusters’ richness or mass ratios, $\sim$4:1. It is also worth mentioning that
mass estimates based on the $L_{X}-M$ scaling relations, like those provided
by G21, can be highly biased for clusters involved in a merger (e.g. Torri et
al., 2004). Unfortunately, G21 stated that the shallow data from eROSITA
prevented them to extract more details about the merger dynamics, because the
X-ray counterparts of the radio relics cannot be detected. To solve this
discrepancy in the merger classification, an essential piece of information is
the map of the underlying dark matter distribution from which a comprehensive
description of the current merger status can be obtained.

Table 1: Summary of the cluster properties found by G21. The original $M_{500}$ were converted to $M_{200}$ assuming the halo mass density profile is well described by an NFW profile with a concentration parameter $c_{200}$ given by the $M_{200}$–$c_{200}$ scaling relation presented by Duffy et al. (2008). Full name | Short name | $z_{\rm photo}$ | Richness | $M_{200}$
---|---|---|---|---
|  |  |  | ($10^{14}$ M⊙)
eFEDS J093501.1+005418 | eFEDS5418 | 0.385 | $143\pm 12$ | $4.2\pm 0.8$
eFEDS J093510.7+004910 | eFEDS4910 | 0.367 | $62\pm 9$ | $2.2\pm 1.1$
eFEDS J093513.3+004746 | eFEDS4746 | 0.367 | $208\pm 15$ |  $8.9\pm 1.7$ [

In this work, we use the public data release of the Hyper Suprime-Cam Subaru
Strategic Program (HSC-SSP; Aihara et al. 2019) to perform the first total
mass reconstruction in the field containing these clusters based on the weak
gravitational lensing technique. Next, we measure the masses of individual
clusters and check if the major merger scenario is feasible. We also address
the kinematics of the merger through an analytical two-body description and
examine whether the positions of the radio relics are consistent with the
proposed merger history. Recently, Chadayammuri et al. (2022, hereafter C22)
pointed that dark matter halo shapes can dramatically change during the merger
event, therefore the use of a scaling relation to determining the halo
concentration, $c_{200}$ (e.g., Duffy et al., 2008), in the mass modeling
could lead to an overestimate of cluster mass. To investigate the possible
impact on the mass determinations and the kinematic description, we test two
scenarios, where in one we include $c_{200}$ as a free parameter in the model,
while in the other a scaling relation is assumed.

The paper is organised as follows. In Section 2 we describe the weak lensing
analysis and the respective results. The proposed scenario for the merger
kinematic is detailed in Section 3. Then, our findings are discussed in
Section 4 and sumarized in Section 5. In this paper, we adopt the standard
$\Lambda$CDM cosmology, with parameters $\Omega_{m}=0.27$,
$\Omega_{\Lambda}=0.73$, $\Omega_{k}=0$, and $h=0.7$.

Figure 1: Composite $gri$ image showing the location of the three galaxy
clusters investigated in this work (labelled in the figure). This is only part
of a larger field for which the data (imaging and photometric catalogue) were
extracted from the HSC-SSP PDR2. The yellow diamonds are placed at the galaxy
clusters’ positions as identified by G21, based on photo-$z$s. The green
contours show the projected density distribution of the red sequence galaxies
selected through a statistical subtraction method in the colour-colour space
(more details given in Section 2.2). The pair eFEDS4746 and eFEDS4910 forms a
bimodal system, possibly in interaction, and, together with eFEDS5418 will
have their kinematic investigated in this work in order to confirm (or not) if
a triple merger is taking place. The cyan dots labeled A–F represent the
positions of mass peaks identified in the weak lensing analysis presented in
this work (Section 2.5). Each of the above-mentioned clusters coincides with a
mass peak (A, B, and D, respectively).

##  2 Projected mass reconstruction

###  2.1 Data

The data underlying this work was retrieved from the wide layer of the second
public data release of the HSC-SSP222Available at https://hsc-
release.mtk.nao.ac.jp/doc/index.php/sample-page/pdr2/. The survey is now
completed, covering $\approx 1100$ deg2 of the sky at $z<1.5$, with five broad
bands $grizy$ reaching a depth of 5$\sigma$ for a point source at $i_{\rm
AB}\approx 26$.

To accomplish our goals of measuring the galaxy shapes with the highest
precision possible, we downloaded the $i$-band imaging from the Data Archive
System (DAS333https://hsc-release.mtk.nao.ac.jp/das_search/pdr2/). We focused
on a region with $0.5\times 0.5$ deg2 in area centred at RA = 09:35:12, DEC =
+00:48:00, roughly matching the position of eFEDS4746. The optical photometric
catalogue ($griz$) and the underlying photo-z’s ($z_{\rm photo}\equiv$ photo-
z_best; Tanaka et al., 2018; Nishizawa et al., 2020) were also retrieved for
all objects in the field.

###  2.2 Red sequence selection

Despite accounting for only $\approx$5 per cent of the total mass, the stellar
content of galaxies works as a good qualitative tracer of the cluster’s
gravitational potential, which is dominated by the dark matter. The projected
density of galaxies allows for the characterization of the cluster’s
morphology (e.g., if the system is uni- or multi-modal) and, ultimately, the
verification of the correspondence between the baryonic and dark matter,
through a comparison of the projected density of galaxies and that of dark
matter, the latter obtained via gravitational lensing (e.g., Monteiro-Oliveira
et al., 2017b; Monteiro-Oliveira et al., 2020; Wittman et al., 2018).

The selection of galaxies belonging to the clusters’ red sequence (Visvanathan
& Sandage, 1977) was done through the application of the statistical
subtraction technique in the $g-r$ versus $i-z$ colour-colour (CC) space. Two
heterogeneous regions were considered in this process: the first one centred
at eFEDS4748 (a circular region with a radius of $2^{\prime}$) in which red
cluster member galaxies are supposed to be numerically dominant, and a
distant, “control” annulus ($10^{\prime}\leq{\rm radius}\leq 14^{\prime}$),
where the counts of the field galaxies are expected to be dominant. As red
sequence galaxies have similar photometric properties at a given redshift
(e.g., Stott et al., 2009), we expect them to form a well-defined locus in the
CC plane, as can be seen in Fig. 2. The locus boundaries become even more
prominent after subtracting the contribution of the outer region, allowing for
a confident CC selection. In the end, we found 765 red sequence members. Their
photo-$z$ distribution, presented in Fig. 3, shows a prominent peak around the
clusters’ location, suggesting the selection was reasonably accurate. For the
sake of comparison, we applied the photo-$z$ cut introduced by Wen & Han
(2013)444According to these authors, cluster members correspond to all
galaxies inside the slice $z_{\rm cluster}-0.04(1+z_{\rm cluster})\leq z\leq
z_{\rm cluster}+0.04(1+z_{\rm cluster})$, finding that 3/4 of our red sequence
members match this criteria.

Figure 2: Left:  $i-z$ vs $g-r$ colour-colour (CC) diagram of galaxies lying
in two different locations of the field. The inner region, where red sequence
galaxies are expected to dominate in terms of number, was selected as a circle
with radius of $2^{\prime}$ centered at eFEDS4746. We also selected galaxies
in an outer annulus, $10^{\prime}\leq{\rm radius}\leq 14^{\prime}$ from
eFEDS4746, shown in the inset. After a statistical subtraction of the CC
diagrams, we defined the cluster locus (white contour) preferably inhabited by
red sequence member galaxies (Section 2.2). Right:  CC diagram of all galaxies
in the field. The cluster locus is highlighted in green. For the weak lensing
analysis, we are only interested in the galaxies located behind the cluster
($z>z_{\rm cluster}$), and any contribution from both the red sequence and
foreground galaxies ($z<z_{\rm cluster}$) will decrease the lensing signal-to-
noise (S/N). We defined an empirical foreground locus (yellow polygon) and
considered as potential source candidates all galaxies located outside. The
magenta points represent the final source sample selected after some quality
cuts in the shape parameters (Section 2.4). Figure 3: Photometric redshift
distribution of the 765 galaxies identified as red sequence members through
the statistical subtraction in the CC space. The vertical dotted (dashed) line
shows the location of eFEDS4748/eFEDS4910 (eFEDS5418) as found by G21. The
inset panel corresponds to those galaxies matching the selection criteria of
Wen & Han (2013, i.e., $0.31\leq z_{\rm photo}\leq 0.44$) considering the
lowest and highest redshift limits of the three clusters. This sub-sample
comprises 76 per cent of the CC-selected red sequence members.

We built the projected red sequence distribution after smoothing the discrete
field with the function

$D(\vec{\xi})=\sum_{i=1}^{N}K(\vec{\xi_{i}},\sigma_{\xi})\ \mbox{,}$ (1)

which represents a sum over all $N$ galaxies with radial coordinate
$\vec{\xi}$ located inside the smoothing scale $\sigma_{\xi}$, set to
$1^{\prime}$. We adopted the Epanechnikov kernel,

$\displaystyle
K(\vec{\xi_{i}},\sigma_{\xi})=\left\\{\begin{array}[]{ll}\frac{3}{4}\left[1-\left(\frac{\vec{\xi_{i}}}{\sigma_{\xi}}\right)^{2}\right],&\vec{\xi_{i}}\leq\sigma_{\xi}\\\
0,&\vec{\xi_{i}}>\sigma_{\xi}.\end{array}\right.$ (4)

The resulting galaxy surface density map, shown in Fig. 1, is fully consistent
with the cluster positions given by G21. It clearly shows a bimodal
distribution, corresponding to the pair eFEDS4748 and eFEDS4910, and a third
clump reasonable away from the previous overdensity that is related to
eFEDS5418. An interesting feature is the two prickle-like structures emanating
from eFEDS4910, which could be the end of large scale filaments that we
believe to be feeding galaxy clusters (Kuchner et al., 2022).

###  2.3 Shape measurements

The weak lensing effect is described in terms of two quantities, the
convergence

$\kappa=\frac{\Sigma(\rm{\xi})}{\Sigma_{\rm cr}}\ \mbox{,}$ (5)

and the shear,

$\gamma=\gamma_{1}+i\gamma_{2}\ \mbox{.}$ (6)

The former, a scalar, reflects the projected mass density of the gravitational
lens and describes the change in size on background galaxies (note that the
surface brightness is conserved at the same time). It is written in units of
the lensing critical density,

$\Sigma_{\rm cr}=\frac{c^{2}D_{\rm s}}{4\pi GD_{\rm ds}D_{\rm d}}\ \mbox{,}$
(7)

where $D_{\rm s}$, $D_{\rm ds}$ and $D_{\rm d}$ are, respectively, the angular
diameter distances to the source555Another way to refer to background
galaxies, between the lens and the source, and to the lens.

The second quantity, a spin-2 tensor, refers to the image stretching. Similar
to the convergence, the shear is related to the projected gravitational
potential of the lens (e.g., Umetsu, 2020). The total effect caused by the
lens is called the reduced shear, and involves a combination of convergence
and shear,

$g\equiv\frac{\gamma}{1-\kappa}\ \mbox{.}$ (8)

The role of the galaxy cluster is to induce a coherent distortion on the
source galaxies, changing their ellipticity in the sense that, on averaged,

$\langle e\rangle\simeq g\ \mbox{.}$ (9)

By definition, weak gravitational lensing is a statistical phenomenon which
means it can only be measured over a large sample of background galaxies.

As the shape parameters of the galaxies in our region of interest are not made
publicly available at the time this work was been carried out, we measured
them by ourselves. We started by checking the image quality. Despite the
source galaxies being the only objects carrying the gravitational lensing
signal, the identification of non-saturated stars are crucial for the
evaluation imaging quality and the assessment of the point spread function
(PSF).

We built our own photometric catalogue of the $i$-band imaging by running the
software SExtractor (Bertin & Arnouts, 1996) for object identification. Then,
the star/galaxy classification was done on their full width half-maximum
(FWHM) as the following: objects within $0.60^{\prime\prime}\leq{\rm FWHM}\leq
0.71^{\prime\prime}$ were classified as stars (point sources) whereas those
having ${\rm FWHM}>0.73^{\prime\prime}$ were considered as galaxies (extended
objects). Our final galaxy catalogue comprises all objects matched with those
from the HSC-SSP catalogue having the highest confident photo-$z$ estimations
(photo-z_conf_best $>0.13$; Medezinski et al., 2018)

The Bayesian code im2shape (Bridle et al., 1998) was adopted to measure the
ellipticity components $e_{1}$ and $e_{2}$ of stars and galaxies. It works by
modelling the objects as a sum (or a single, in the case of stars) of Gaussian
functions. Given that the observed unsaturated star profiles are the result of
the convolution between the PSF with a delta function, the discrete set of
($e_{1}$, $e_{2}$) can be spatially interpolated across the image to create an
analytical description of the PSF. We accomplished this task resorting to the
Thin Place Regression (Nychka et al., 2014) function in the R environment (R
Core Team, 2014). The interpolation was done three times, each one removing
the 10 per cent worst objects (i.e., those with the largest absolute
residuals). At the end, we arrived at a tight fit as manifested by the very
small residuals ($0.0000\pm 0.0007$ for $e_{1}$ and $e_{2}$) as shown in Fig.
4.

Figure 4: PSF modelling based on bright and unsaturated stars homogeneously
distributed across the field. The 1024 black dots show the raw distribution of
the ellipticity components $e_{1}$ and $e_{2}$, with $\langle
e_{1}\rangle=-0.002\pm 0.013$ and $\langle e_{2}\rangle=-0.006\pm 0.009$.
After removing outliers to create an analytical function to describe the PSF
across the image, we found the residuals represented by the 934 red points as
low as $0\pm 7\times 10^{-4}$ for both $e_{1}$ and $e_{2}$.

The galaxy ellipticities were then extracted after im2shape has performed the
PSF deconvolution. For the sake of quality, we disregarded all galaxies having
measured ellipticity uncertainties greater than 0.2, or showing any evidence
of blending. Unfortunately, as we are not able to know a priori the shapes of
the unlensed galaxies, $e_{1}$ and $e_{2}$ are at best noisy proxies of the
shear field. When required, we adopted $\sigma_{\rm int}=0.3$ as the
dispersion of the intrinsic ellipticity distribution of source galaxies (e.g.
Leauthaud et al., 2007).

###  2.4 Source selection

The cluster and the foreground members have to be carefully removed from the
source sample, in order not to critically dilute the weak lensing signal. To
this end, we tested three selection criteria, one based on a CC cut, and two
others relying on the galaxies photo-$z$, but excluding the cluster locus in
one of them. Using the examples presented in Medezinski et al. (2010) as
guidelines, we defined an empirical region in the CC diagram comprising the
most probable location of foreground candidates plus the cluster locus
(Section 2.2). The CC based selection process, illustrated in Fig. 2, resulted
in a source density of 13.2 galaxies per arcmin-2. For the photo-$z$ based
selection, we considered all galaxies beyond the upper limit suggested by Wen
& Han (2013, i.e., $z_{\rm photo}>0.44$), yielding 16.1 galaxies per arcmin-2.
Excluding galaxies within the cluster locus, the density decreases to 15.6
galaxies per arcmin-2.

We referred to the mass aperture statistic (Schneider, 1996) to create a shear
signal-to-noise map,

${\rm S/N}=\dfrac{\sqrt{2}}{\sigma_{\rm int}^{2}}\
\dfrac{\sum_{i=1}^{N_{\theta_{0}}}e_{+_{i}}(\theta_{i})Q_{\rm
NFW}(\theta_{i},\theta_{0})}{\left[\sum_{i=1}^{N_{\theta_{0}}}Q_{\rm
NFW}^{2}(\theta_{i},\theta_{0})\right]^{1/2}}\ \mbox{,}$ (10)

based on the tangential ellipticity,

$\displaystyle e_{+}=-e_{1}\cos(2\phi)-e_{2}\sin(2\phi)\mbox{,}$ (11)
$\displaystyle\phi=\arctan\left(\frac{y_{i}-y_{\rm bin}}{x_{i}-x_{\rm
bin}}\right)\ \mbox{,}$

averaged over the $N_{\theta_{0}}$ galaxies inside a circular region of
$\theta_{0}=8^{\prime}$ computed in each of the $1.5\times 10^{2}$ spatial
bins centred at the Cartesian coordinates $x_{\rm bin},y_{\rm bin}$. The other
quantities in Eq. 10 are the radial position of the $i$-th galaxy $\theta_{i}$
and the dispersion of the intrinsic ellipticity distribution $\sigma_{\rm
int}$. We adopted a filter (Schirmer, 2004) that roughly matches an NFW shear
profile,

$\noindent Q_{\rm
NFW}(\theta_{i},\theta_{0})=[1+e^{a-b\chi(\theta_{i},\theta_{0})}+e^{-c+d\chi(\theta_{i},\theta_{0})}]^{-1}\times\\\
\dfrac{\tanh[\chi(\theta_{i},\theta_{0})/\chi_{c}]}{\pi\theta_{0}^{2}[\chi(\theta_{i},\theta_{0})/\chi_{c}]}\mbox{,}$
(12)

where $\chi=\theta_{i}/\theta_{0}$. Following Hetterscheidt et al. (2005), we
adopted $a=6$, $b=150$, $c=47$, $d=50$, and $\chi_{c}=0.15$.

The resultant S/N maps based on the three selection methods are presented in
Fig. 5. Overall the three maps appear very similar, and all of them have the
highest S/N in the same region of the clusters eFEDS4746 and eFEDS4910. When
comparing the maps quantitatively within a circular region with $4^{\prime}$
radius centred at eFEDS4746, we found that the CC based map reaches the
highest S/N among the samples. The 97.5 per cent percentile (maximum) of the
S/N is 7.65 (8.10), 6.61 (7.17) and 6.24 (6.92) respectively for the CC based,
the photo-$z$ minus the cluster locus and the photo-$z$ samples. This
conclusion is in line with Medezinski et al. (2018), who found that CC cuts
are more efficient than photo-$z$’s in removing contributions from both
cluster and foreground galaxies and thus maximizing the weak lensing signal.
We will adopt the CC-based selection as our fiducial source sample henceforth.
The 36,589 source galaxies correspond to a critical density $\Sigma_{\rm
cr}=2.8\pm 0.6\times 10^{9}$ M⊙ kpc-2 (Eq. 7). The quoted error on the
critical density reflects the spread of the source redshifts, which is
characterized by a mean photo-$z$ of $\approx 1.1$.

Figure 5: Weak lensing mass aperture statistic S/N. As guidelines, the white
$\times$ signs show the positions of the clusters as found by G21. The black
$+$ signs mark the positions of the most relevant mass peaks according to the
analysis presented in Section 2.5. We tested three different methods for
selecting the source galaxies; the first is solely based on photo-$z$ ($z_{\rm
photo}>0.44$, left), the second being a hybrid one, where we excluded from the
previous sample all galaxies inside the cluster locus (centre), and the third
one, which is only based on the CC cuts (Fig. 2, right). The resulting source
density is indicated in the lower right corner. We showed that the CC-based
selection maximises the S/N in the vicinity of the interacting cluster
candidates eFEDS4748 and eFEDS4910 (see details in the text), and is thus
regarded as our fiducial sample.

###  2.5 Projected mass field and modelling

The Bayesian code LensEnt2 (Marshall et al., 2002) was applied to translate
the source ellipticities into the cluster projected mass distribution. In a
nutshell, this maximum entropy algorithm works by maximizing the evidence of
the reconstructed mass field with respect to the data. Since each individual
galaxy is a noisy proxy of the shear and is correlated with its neighbourhood,
an intrinsic correlation function (ICF) must be applied to smooth the data. We
adopted a Gaussian ICF (e.g. Monteiro-Oliveira et al., 2018, 2020, 2021) with
a FWHM $\sigma_{\rm ICF}=90^{\prime\prime}$. The final convergence map is
presented in Fig. 1. The noise level, $\sigma_{\kappa}=0.04$, was calculated
as follows. For each of 100 iterations, the mass map was computed after every
galaxy orientation was rotated by a random angle in the interval [0,180] to
remove the lensing signal.

The convergence map is fully consistent with the mass aperture statistic S/N
(Fig. 5). We identified six significant mass concentrations (i.e., $\geq
4\sigma_{\kappa}$), labelled A–F. The clumps A, B and D can be correlated
respectively to the galaxy clusters eFEDS4910 (RA = 09:35:13, DEC =
+00:49:38), eFEDS4746 (RA = 09:35:15 , DEC = +00:47:26) and eFEDS5418 (RA =
09:35:00, DEC = +00:54:53) as they show a good agreement with the cluster
locations reported by G21 (diamonds in Fig 1 and crosses in Fig. 5) and also
match the distribution of cluster red sequence members (Fig. 1). Given the
lack of any optical counterpart in the field, the peaks C, E, and F probably
consist of a combination of structures seen in projection through the line-of-
sight (e.g., Liu & Haiman, 2016; Wei et al., 2018).

The next step was to measure the individual halos masses. The gravitational
lensing signal induced in each source galaxy corresponds to the sum of the
effect due to $N$ halos, $\kappa=\sum_{i=1}^{N}\kappa_{i}$,
$\gamma_{j}=\sum_{i=1}^{N}\gamma_{i}$, with $j={1,2}$. We assumed that the
halos density profile can be described by the NFW profile (Navarro et al.,
1996). In this case, the model encompasses four basic parameters, namely the
lens centre ($x_{c},y_{c}$), halo mass $M_{200}$666The total mass enclosed
within a sphere whose density is 200 times the critical density of the
Universe, $\rho_{c}=\frac{3H^{2}(z)}{8\pi G}$. and concentration $c_{200}$.

When modelling the halo masses, we were also interested in checking if the way
halos are selected affects the final mass measurements. For example, is it
possible that a model that only considers halos with a secure optical
counterpart would produce discrepant mass measurements than a model that
accounts for all high S/N halos? To address this question, three models were
computed. In model #1, we measured all halos masses (assuming $z_{\rm
photo}\approx 0.36$ for all), whereas in model #2, only the three known
clusters had their masses computed. In both cases, we set the halo
concentration using the $M_{200}-c_{200}$ scaling relation proposed by Duffy
et al. (2008). Aiming to check the potential impact of the merger age on the
measured concentration, in model #3, we included $c_{200}$ as a free parameter
when measuring the halo masses. The full parameter vectors $\Theta$ are
presented in Table 2. For all models, the halo centres were kept fixed at the
position found by our algorithm.

Table 2: Models description. $N_{\rm par}$ corresponds to the number of parameters in each model. $M$ and $c$ refer respectively to $M_{200}$ and $c_{200}$. Model | Halos | $\Theta$ | $N_{\rm par}$
---|---|---|---
#1 | 6 (A–F) |  $M^{\rm A}$, $M^{\rm B}$, $M^{\rm C}$, $M^{\rm D}$, $M^{\rm E}$, $M^{\rm F}$ | 6
#2 | 3 (A, B, D) |  $M^{\rm A}$, $M^{\rm B}$, $M^{\rm D}$ | 3
#3 | 3 (A, B, D) |  $M^{A}$, $M^{B}$, $M^{D}$, $c^{\rm A}$, $c^{\rm B}$, $c^{\rm D}$ | 6

The $\chi^{2}$ statistic for each model is

$\chi^{2}=\sum_{j=1}^{N_{{\rm
sources}}}\sum_{i=1}^{2}\frac{[g_{i}(M_{200},x_{c},y_{c})-e_{i,j}]^{2}}{\sigma_{\rm
int}^{2}+\sigma_{{\rm obs}_{i,j}}^{2}},$ (13)

where $g_{i}$ is the theoretical reduced shear (Eq. 8), $e_{i,j}$ is the
measured ellipticity of source galaxies (Sec. 2.3), and $\sigma_{{\rm
obs}_{i,j}}$ is the error on shape measurement given by im2shape. The
likelihood is

$\mathcal{L}\propto\exp\left(-\frac{\chi^{2}}{2}\right)\,.$ (14)

Finally, we write the posterior of our problem as

$\noindent{\rm Pr}(\Theta|{\rm data})\propto\mathcal{L}({\rm
data}|\Theta)\times\mathcal{P}(\Theta)\,.$ (15)

For the models #1 and #2, we applied a flat prior $\mathcal{P}(\Theta)$ for
the masses, $0<M_{200}\leq 10^{16}$ M⊙, to avoid non-physical values and to
accelerate the convergence. In model #3 we also added a prior on the
concentration, $0<c_{200}\leq 15$.

The posterior in Equation 15 was sampled by the MCMC algorithm with a
Metropolis sampler MCMCMETROP1R (Martin et al., 2011). Four chains of $10^{5}$
elements plus $10^{4}$ as ’burn-in’ were generated for each model, all of
which were considered convergent at the end (Plummer et al., 2006). The
marginalized posteriors are shown in Table 3.

Table 3: Results of the modelling. The median was taken as the representative value for each posterior, and the quoted error corresponds to the 68 per cent c.l. interval. Masses $M_{200}$ are presented in units of $10^{14}$ M⊙. The concentrations shown in parenthesis are those predicted by Duffy et al. (2008). Based on AIC and BIC statistics, model #1 was considered as the preferred one. Model | $M_{200}^{\rm A}$ | $M_{200}^{\rm B}$ | $M_{200}^{\rm C}$ | $M_{200}^{\rm D}$ | $M_{200}^{\rm E}$ | $M_{200}^{\rm F}$ | $c_{200}^{\rm A}$ | $c_{200}^{\rm B}$ | $c_{200}^{\rm D}$ |  $\Delta$AIC |  $\Delta$BIC |
---|---|---|---|---|---|---|---|---|---|---|---|---
| eFEDS4910 | eFEDS4746 |  | eFEDS5418 |  |  |  |  |  |  |  |
$\\#1$ | $2.63_{-1.18}^{+0.96}$ | $2.73_{-1.48}^{+1.08}$ | $4.23_{-1.81}^{+1.53}$ | $2.43_{-1.30}^{+1.07}$ | $1.27_{-0.94}^{+0.61}$ | $3.08_{-1.53}^{+1.20}$ | $(3.4_{-0.1}^{+0.1})$ | $(3.4_{-0.2}^{+0.1})$ | $(3.4_{-0.2}^{+0.1})$ | 0 | 2 |
$\\#2$ | $2.54_{-1.23}^{+0.96}$ | $3.46_{-1.52}^{+1.28}$ | – | $3.42_{-1.55}^{+1.19}$ | – | – | $(3.4_{-0.1}^{+0.1})$ | $(3.3_{-0.1}^{+0.1})$ | $(3.3_{-0.1}^{+0.1})$ | 20 | 0 |
$\\#3$ | $2.80_{-1.70}^{+1.21}$ | $3.27_{-1.95}^{+1.44}$ | – | $2.32_{-1.14}^{+0.87}$ | – | – | $2.9_{-1.7}^{+1.2}$ | $2.6_{-1.4}^{+1.0}$ | $7.1_{-3.9}^{+3.0}$ | 23 | 26 |

It is remarkable that, regardless of the model, the masses of all eFEDS
clusters are comparable within the error bars. To select the best model, we
resorted to the Akaike Information Criterion (AIC777${\rm
AIC}=2k-2\ln{\mathcal{\hat{L}}}$, where $k$ is the number of parameters in the
considered model and $\mathcal{\hat{L}}$ is the corresponding maximum log-
likelihood ) and the Bayesian Information Criterion (BIC888${\rm
BIC}=k\ln{n}-2\ln{\mathcal{\hat{L}}}$, where $n$ is the number of data
points.). Both metrics are based on the idea that simplest models are
preferred over the most complex ones, penalising therefore those with a large
number of parameters. A small difference between the two is that BIC also
takes into account the number of data points, thus avoiding overfitting. Among
a finite number of models, those with the lowest AIC/BIC is considered the
best to describe the data.

The model #1 is the preferred according to the AIC ($\Delta{\rm AIC}>10$; Kass
& Raftery, 1995) whereas the BIC statistic suggests that it is comparable to
model #2 ($\Delta{\rm BIC}<2$). These results make us confident in choosing
the model #1 as the fiducial one henceforth. The full posterior plot,
presented in Fig. 7, shows that, in general, the parameters are not
(anti-)correlated among them, except when the nearest neighbour(s) is
considered (A – B, B – E, D – F). Nevertheless, even in these cases, the anti-
correlation is very weak. Another remarkable feature in the model is the
considerably high mass associated with an isolated halo, labelled “C”,
surpassing even the identified cluster masses. Similarly, for clumps “E” and
“F”, we did not find any clear optical counterparts after computing the
photo-$z$ distribution of galaxies inside a circular region with a radius of
$1.5^{\prime}$ ($\sim$ 0.5 Mpc), implying that they probably do not correspond
to a single halo. Wei et al. (2018) showed that a collection of halos more
massive than $10^{13}$ M⊙ ($z\approx 1$) can be detected at line-of-sight with
a ${\rm S/N}\geq 3$. We end this discussion with an important caveat: the
masses in Table 3 only correspond to the “true” value if the halo redshift is
the same as that considered in the modeling, $z=0.36$.

Regarding the interacting binary system candidate, eFEDS4746 is the more
massive one in 52% of the MCMC samples, as we can see in Fig. 6. Regardless of
which cluster is considered the most massive, the mass ratio (i.e. the ratio
between the corresponding posteriors, $\mathcal{R}\equiv{\rm M}_{200,\rm
cluster}/{\rm M}_{200,\rm subcluster}$) is $\mathcal{R}=1.7_{-0.7}^{+0.5}$. In
63 per cent of the samples, we found $\mathcal{R}<2$, suggesting that the
system can be classified as a major merger, while in another 28 per cent,
$2\leq\mathcal{R}<4$, suggesting that a semi-major merger class is more
suitable to describe the possible interaction (Martel et al., 2014). If we
consider a toy model in which eFEDS4746 and eFEDS4910 will merge within a few
Gyr (e.g. Machado et al., 2015), the final cluster would have a mass of
$M_{200}=5.53_{-1.37}^{+1.24}\times 10^{14}$ M⊙ (Fig. 6), considering the
progenitors masses are conserved.

Figure 6: Probability density function (PDF) of the sum (dashed line) and the
difference (M${}_{200}^{4746}$ – M${}_{200}^{4910}$, continuous line) of the
posteriors of masses of individual clusters. In 52 (48) per cent of the
$4\times 10^{5}$ MCMC samples, eFEDS4748 (eFEDS4910) emerges as the most
massive cluster of the pair. The total mass is $5.53_{-1.37}^{+1.24}\times
10^{14}$ M⊙. Figure 7: Posteriors of the parameters of model #1 (six NFW-
halos) as mapped by the MCMC sampler. Dark (light) blue corresponds to 68 (95)
per cent c.l. Along the diagonal line the marginalised posteriors of each halo
(A – F) are shown. We considered the median as the representative value for
each distribution. The number inside the plots shows the Pearson’s correlation
coefficient for each pair.

The halo concentrations are fully consistent in both models #1 and #2. When
setting $c_{200}$ as a free parameter in model #3, we found comparable values
within the large error bars, despite the face values being systematically
smaller than those obtained by the scaling relation. The exception is
eFEDS5418, with a considerably higher concentration.

##  3 Kinematic analysis

Tailored hydrodynamical simulations have been proven to provide a realistic
description of the merger kinematics (e.g., Molnar et al., 2020; Doubrawa et
al., 2020; Moura et al., 2021). However, when it is not possible to utilize
such a tool, we have to resort to a simplified yet still reliable analytical
description of the two-body interaction, such as the Monte Carlo Merger
Analysis Code (MCMAC; Dawson, 2013). This is a Bayesian model designed to
calculate the possible merger scenarios from the probability density function
(PDF) with just a few input parameters, namely the cluster masses ($M_{200}$,
Table 3), their spatial separation projected at an angle $\alpha$ from the
plane of the sky ($R_{\rm p}$, Fig 8) and their mean redshift ($\bar{z}$,
Table 1). The great improvements offered by this code over other tools are the
treatment of the clusters as spatially extended objects instead of point
masses (Beers et al., 1982) and the availability of the posteriors of the
quantities of interest (e.g., time since/to the pericentric passage, maximum
separation). MCMAC assumes mass conservation (i.e., $M=M_{1}+M_{2}={\rm
constant}$) during the zero impact parameter merger and no angular momentum.
The maximum relative velocity $V_{\rm r}$ is set to be the free-fall velocity
according to the cluster masses. It has two versions, one designed for post
mergers (MCMAC-post) and another for pre-interaction pairs (MCMAC-pre). In the
latter case, the condition for bound pairs,

$V_{\rm r}^{2}R_{\rm p}\leq 2GM\sin^{2}\alpha\cos\alpha,$ (16)

is not necessarily satisfied during the MCMC re-samplings. The probability of
a system being bound (unbound) can be defined as the ratio between the number
of MCMC-pre samples in which Eq. 16 is true (false) and the total number of
samples, 10,000 in this work (for both MCMAC-pre/post). We adopted the cluster
redshifts estimated in G21. To compute the uncertainties, we assumed a squared
sum of a typical error on subcluster redshift ($\sigma_{z}\approx 0.007$;
e.g., Monteiro-Oliveira et al., 2017a, 2018) with an uncertainty of 5 per cent
corresponding to the photo-$z$ precision of the HSC-SSP data (Tanaka et al.,
2018), which will have a much larger weight on the final estimation. We also
considered a realistic prior on the velocity of the merger component on the
plane of sky ($v_{\rm plane}<1500$ km s-1; Monteiro-Oliveira et al., 2022).

Despite the fact that MCMAC is designed to describe two-body interactions, in
Monteiro-Oliveira et al. (2022) we suggested that the algorithm can be applied
in more complex systems, by reducing them to one pair at a time. Although not
dealing with all bodies in the system simultaneously, this approach provides a
realist chronological order for the collision events. In this sense, we use
the MCMAC-pre mode to describe the potential mergers, as illustrated in Fig.
8.

Figure 8: A sketch showing the likely configuration of the galaxy clusters
studied in this work. The dots correspond to their respective halo centre
position according to our weak lensing analysis. The percentages refer to the
probability of each pair being found gravitationally bound according to the
MCMAC code. The same quantity, but computed with the masses delivered by the
model #3 (i.e. $c_{200}$ is given by the modelling), is shown in parenthesis.
When the drawn line is continuous (dashed) the pair is considered bound
(unbound). The number below refers to the projected separation between the
mass peaks (in kpc).

We found that only the pair eFEDS4746/4910 is gravitationally bound, a
scenario recovered in 81 per cent of the MCMC-pre samples. However, when
either of these clusters forms a pair with eFEDS5418, a bound system is only
reported by less than 41 per cent of the samples. Even when we increased the
mass of the cluster to the value of the combined mass of eFEDS4746 and
eFEDS4910 in our toy model, the correspondingly increased percentage (48%) is
not enough to change the kinematic classification. Regarding any possible
impact on the mass reconstruction described with a free $c_{200}$ parameter
(model #3 in Table 3), all of the results presented in this paragraph remain
consistent.

Our proposed post-merger kinematic description is presented in Table 4. MCMAC-
post does not distinguish between two possible scenarios: (1) a collision that
has just happened $TSC_{0}$ Gyr ago and the members are outgoing, or (2) a
system that already reached the apoapsis and the clusters are incoming for a
new encounter $TSC_{1}$ Gyr after the last pericentric passage. This
degeneracy can be broken, however, if any other age proxy is also observed.
The presence of radio relics in the outskirts of eFEDS4746/4910 is itself a
piece of strong evidence that the clusters have had their encounter not too
long ago (e.g., A3376; Machado & Lima Neto, 2013; Monteiro-Oliveira et al.,
2017b). Therefore, based on this observable, we disregarded the incoming
scenario in which the collision happened $\sim$3.13 Gyr ago. Thus, our
kinematic description suggests that the collision between eFEDS4746 and
eFEDS4910 took place $0.58_{-0.20}^{+0.15}$ Gyr ago with a 3-dimensional
velocity of $2148_{-402}^{+393}$ km s-1 in a direction orientated
$42_{-23}^{+24}$ degrees from the plane of the sky. The system is been
observed after running $\sim$55 per cent of the path to the apoapsis, where
the clusters will be $1.8_{-1.1}^{+0.9}$ Mpc apart. Again, these results
remain almost unchanged when considering the alternative model #3 with the
halo concentration as a free parameter.

Table 4: Kinematic description of the post-merger scenario for the cluster pair eFEDS4748–eFEDS4910 according to MCMAC-post. The first five rows show the input quantities (from top to down): the cluster masses $M$, the cluster redshifts $z$, and their projected separation $d_{\rm proj}$. The following 7 rows are: the parameter estimations corresponding to the angle between the merger axis and the plane of the sky, $\alpha$, the current 3D relative velocity, $v_{\rm 3D,obs}$, the current 3D separation between the clusters, $d_{\rm 3D,obs}$, the 3D velocity at collision time, $v_{\rm 3D,col}$, the 3D maximum separation at the clusters’ apoapsis, $d_{\rm 3D,max}$, the time since collision for the outgoing scenario, $TSC_{0}$, and the time since collision for the incoming scenario, $TSC_{1}$. Quantity | Unit | Median | 68 per cent c.l
---|---|---|---
$M_{200}^{\rm 4746}$ |  $10^{14}$ M⊙ | 2.96 | 1.55 – 4.32
$M_{200}^{\rm 4910}$ |  $10^{14}$ M⊙ | 2.74 | 1.58 – 3.85
$z^{\rm 4746}$ | – | 0.367 | 0.351 – 0.382
$z^{\rm 4910}$ | – | 0.367 | 0.352 – 0.382
$d_{\rm proj}$ | Mpc | 0.71 | 0.66 – 0.75
$\alpha$ | degrees | 42 | 19 – 66
$v_{\rm 3D,obs}$ | km s-1 | 1194 | 724 – 1667
$d_{\rm 3D,obs}$ | Mpc | 0.96 | 0.65 – 1.18
$v_{\rm 3D,col}$ | km s-1 | 2148 | 1747 – 2541
$d_{\rm 3D,max}$ | Mpc | 1.8 | 0.7 – 2.7
$TSC_{0}$ | Gyr | 0.58 | 0.38 – 0.72
$TSC_{1}$ | Gyr | 3.13 | 0.87 – 5.81

##  4 Discussion

Our weak lensing mass map shows an excellent agreement with the cluster
positions detected by eROSITA and Subaru (Fig. 9). Assuming that the density
profile of each cluster can be satisfactorily described by an NFW profile, we
found that all the clusters have similar masses (Table 3). For the last two in
the Table, these estimates show a good agreement with G21, considering the
error bars. The most outstanding feature arising from our weak lensing
analysis is the absence of a dominant cluster, contrary to the findings of
G21. Their mass estimate for eFEDS4746 based on the $L_{X}-M$ scaling relation
is $\sim 3$ times larger than our value. Even though their estimates agree
with those from the Planck catalogue, we stress that the weak lensing
technique does not make any assumption a priori about the cluster dynamical
state (e.g., Soja et al., 2018; Umetsu, 2020).

Figure 9: Summary of the analysis presented in this work. The $gri$ image
shows a zoomed-in view of the three clusters. The yellow diamonds are cluster
coordinates according to G21, and the red X is the position of the single
X-ray peak. Our weak lensing analysis has identified six significant mass
peaks (i.e., $\geq 4\sigma_{\kappa}$, A–F in decreasing order of
significance), whose locations are shown by the cyan points (halos C and F are
outside of the current field-of-view). Three of them correspond to the known
clusters, eFEDS4910 (A), eFEDS4746 (B), and eFEDS5418 (D). At the outskirts of
eFEDS4746/4910, there are two radio relics (RSE and RN; magenta squares), a
hint of a recent merger activity. The cyan contours represent the convergence
map in units of $\sigma_{\kappa}$, starting from $4\sigma_{\kappa}$.

The ICM distribution is asymmetric in the interacting pair, showing a single
peak related to eFEDS4746 (’X’ in Fig. 9), located $\sim 23^{\prime\prime}$
away from the corresponding mass peak. As this separation is less than the
expected error on the mass peak centroid caused by the effects of shape noise
and smoothing of the mass map (Dietrich et al., 2012), we can not make any
assumption about any possible spatial coincidence (or not) between the mass
and gas peaks. On the other hand, the lack of an X-ray peak leads us to
conclude that eFEDS4910 has temporarily lost its gas counterpart, a feature
only seen in the class of dissociative mergers (Dawson, 2013). Even so, such a
highly asymmetric gas post-merger configuration is supported by the observed
mass ratio, $1.7_{-0.7}^{+0.5}$, as the ICM dynamics is mostly governed by the
initial conditions of the gas distribution (e.g. concentration; Machado et
al., 2015; Moura et al., 2021).

Since there are no spectroscopic redshift measurements for these galaxy
clusters, we have to resort to their photometric redshifts estimated by G21 in
order to address their kinematics. This was accomplished with an analytical
two-body analysis applied to each pair to infer the possible pre- and post-
merger scenarios. We confirmed that the pair eFEDS4746/4910 is highly likely
bounded. On the other hand, the cluster eFEDS5418 is not gravitationally bound
to either of eFEDS4746/4910 (when considered individually). This conclusion
remains the same even in the case when we consider a fictional cluster formed
from the merger between eFEDS4746/4910 (having the combined masses) placed
midway their current positions, although the probability of the unbound
scenario decreases to 52 per cent. Therefore, our analysis does not support
the conjecture proposed by G21 that eFEDS5418 is a pre-merger and part of the
supercluster. Regarding the last feature, this is not surprising since the
selection criterion adopted by G21 is based solely on the local over-density
instead of a more physically motivated one (e.g., occurrence of a future
gravitational collapse; Chon et al., 2015).

Our proposed kinematic description posits that eFEDS4746 and eFEDS4910 are
undergoing a major merger orientated with an axis of $42_{-23}^{+24}$ degrees
from the plane of the sky. Despite the error bars, such a large angle can
explain the absence of any brightness and temperature jumps at the positions
of both radio relics, as these features can be detected even at angles as
large as 30 degrees (C22). The pericentric passage happened
$0.58_{-0.20}^{+0.15}$ Gyr ago. We can also provide an alternative estimation
of the merger age, by computing the time required for the shock to travel from
the cluster centre (in this case the midpoint with RA = 09:35:14, DEC =
+00:49:39) to the radio relic positions. Taking the relic RNE, whose projected
location ($\sim 1.2$ Mpc) matches the expectation for a cluster merger ($\sim
1$ Mpc; e.g., Ha et al., 2018) and considering a shock velocity of $v\sim
3000$ km s-1 (Machado et al., 2015), we arrive at $\sim 0.5$ Gyr, in agreement
with the kinematic estimation. Considering the relic RN instead ($\sim 0.5$
Mpc), we find a considerable lower age, $\sim 0.2$ Gyr. Despite both being
consistent with a merger caught not so far from the pericentric passage, the
differences in distance to the cluster centre of the radio relics could be a
hint that they were not generated during the same event.

The halo concentration $c_{200}$ can significantly change during the several
phases of the cluster merger, mainly in the moments close to the pericentric
passage (C22). Therefore, it could have an important role in the cluster mass
measurement (C22) and consequently on the whole merger kinematic description,
depending on whether it is determined from a scaling relation (obtained from
relaxed halos; Duffy et al., 2008) or included as a free parameter in the
modeling. After testing the two scenarios, we found comparable masses and
therefore, similar merger histories. A possible explanation for such an
agreement is, at the current merger phase (i.e., at the current time since the
pericentric passage), any changes in $c_{200}$ have completely vanished.
Unfortunately, a direct comparison with the temporal evolution of $c_{200}$
presented in C22 (see their Fig. 4) is not possible, as the authors presented
it in terms of an arbitrary time, starting from the beginning of the
simulation.

##  5 Summary

Based on the second public data release of the Hyper Suprime-Cam Subaru
Strategic Program (HSC-SSP PDR2), we present the first weak gravitational
lensing reconstruction of the matter distribution in the Northern part of the
eFEDS supercluster, where Ghirardini et al. (2021) found an interacting
cluster pair (eFEDS4746/4910) surrounded by a companion (eFEDS5418). The
masses of the clusters are found to be $2.73_{-1.48}^{+1.08}\times 10^{14}$ M⊙
(eFEDS J093513.3+004746), $2.63_{-1.18}^{+0.96}\times 10^{14}$ M⊙ (eFEDS
J093510.7+004910), and $2.43_{-1.30}^{+1.07}\times 10^{14}$ (eFEDS
J093501.1+005418), based on our analysis. Our result supports the scenario
drawn from the X-ray and radio observations, in which eFEDS4746 amd eFEDS4910
are undergoing a major merger. However, eFEDS5418 does not show any evidence
of being gravitationally bound to the pair. We also found no impact of the
halo concentration on the mass modeling and kinematic description,
irrespective of the method employed (i.e., from a scaling relation or being a
free parameter of the lensing model). A spectroscopic follow-up of the cluster
members can offer a complementary view of the merger dynamics and corroborate
our results, which are based solely on photometric redshifts.

## Acknowledgements

I want to thank an anonymous referee for his/her valuable comments that helped
improve this paper. My thanks also go to Prof. Yen-Ting Lin for revising this
manuscript and to ASIAA, where this work was completed, for the hospitality.

This work was conceived and developed during the most acute phase of the
COVID-19 pandemic. Although many people on the front line had to be exposed
daily to the invisible danger in an epoch where vaccines did not exist, I had
the privilege of working safely from home. This work is dedicated to those
brave people. On a similar note, this work represents a milestone in my
career. It is also dedicated to my former advisor Eduardo Cypriano, who
patiently taught and guided me in the early stages of my career. I am
incredibly grateful for his support.

To accomplish the results presented here, I used the R software. For more
details, please visit https://www.r-project.org/

This work make used of data from the Hyper Suprime-Cam (HSC) collaboration.
The Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) is led by the
astronomical communities of Japan and Taiwan, and Princeton University. The
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
The survey was made possible by funding contributed by the Ministry of
Education, Culture, Sports, Science and Technology (MEXT), the Japan Society
for the Promotion of Science (JSPS), (Japan Science and Technology Agency
(JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and
Princeton University. For more details, please visit https://hsc-
release.mtk.nao.ac.jp/doc/index.php/acknowledging-hsc-2/

## Data Availability

All data used in this work is already publicity available. However, the
processed data can be shared under request to the author.

## References

  * Aihara et al. (2019) Aihara H., et al., 2019, PASJ, 71, 114
  * Beers et al. (1982) Beers T. C., Geller M. J., Huchra J. P., 1982, ApJ, 257, 23
  * Bertin & Arnouts (1996) Bertin E., Arnouts S., 1996, A&AS, 117, 393
  * Bridle et al. (1998) Bridle S. L., Hobson M. P., Lasenby A. N., Saunders R., 1998, MNRAS, 299, 895
  * Chadayammuri et al. (2022) Chadayammuri U., ZuHone J., Nulsen P., Nagai D., Felix S., Andrade-Santos F., King L., Russell H., 2022, MNRAS, 509, 1201
  * Cho et al. (2021) Cho H., Jee M. J., Smith R., Finner K., Lee W., 2021, arXiv e-prints, p. arXiv:2109.06879
  * Chon et al. (2015) Chon G., Böhringer H., Zaroubi S., 2015, A&A, 575, L14
  * Dawson (2013) Dawson W. A., 2013, ApJ, 772, 131
  * Dietrich et al. (2012) Dietrich J. P., Böhnert A., Lombardi M., Hilbert S., Hartlap J., 2012, MNRAS, 419, 3547
  * Doubrawa et al. (2020) Doubrawa L., Machado R. E. G., Laganá T. F., Lima Neto G. B., Monteiro-Oliveira R., Cypriano E. S., 2020, MNRAS, 495, 2022
  * Duffy et al. (2008) Duffy A. R., Schaye J., Kay S. T., Dalla Vecchia C., 2008, MNRAS, 390, L64
  * Fischer et al. (2021) Fischer M. S., Brüggen M., Schmidt-Hoberg K., Dolag K., Ragagnin A., Robertson A., 2021, MNRAS,
  * Ghirardini et al. (2021) Ghirardini V., et al., 2021, A&A, 647, A4
  * Ha et al. (2018) Ha J.-H., Ryu D., Kang H., 2018, ApJ, 857, 26
  * Harvey et al. (2015) Harvey D., Massey R., Kitching T., Taylor A., Tittley E., 2015, Science, 347, 1462
  * Hernández-Lang et al. (2021) Hernández-Lang D., et al., 2021, arXiv e-prints, p. arXiv:2111.15443
  * Hetterscheidt et al. (2005) Hetterscheidt M., Erben T., Schneider P., Maoli R., van Waerbeke L., Mellier Y., 2005, A&A, 442, 43
  * Kass & Raftery (1995) Kass R. E., Raftery A. E., 1995, Journal of the American Statistical Association, 90, 773
  * Kelkar et al. (2020) Kelkar K., et al., 2020, MNRAS, 496, 442
  * Keshet et al. (2021) Keshet U., Raveh I., Naor Y., 2021, MNRAS, 508, 3455
  * Knowles et al. (2021) Knowles K., et al., 2021, arXiv e-prints, p. arXiv:2111.05673
  * Kuchner et al. (2022) Kuchner U., et al., 2022, MNRAS, 510, 581
  * Leauthaud et al. (2007) Leauthaud A., et al., 2007, ApJS, 172, 219
  * Liu & Haiman (2016) Liu J., Haiman Z., 2016, Phys. Rev. D, 94, 043533
  * Machado & Lima Neto (2013) Machado R. E. G., Lima Neto G. B., 2013, MNRAS, 430, 3249
  * Machado et al. (2015) Machado R. E. G., Monteiro-Oliveira R., Lima Neto G. B., Cypriano E. S., 2015, MNRAS, 451, 3309
  * Machado et al. (2022) Machado R. E. G., Laganá T. F., Souza G. S., Caproni A., Antas A. S. R., Mello-Terencio E. A., 2022, arXiv e-prints, p. arXiv:2206.14127
  * Marshall et al. (2002) Marshall P. J., Hobson M. P., Gull S. F., Bridle S. L., 2002, MNRAS, 335, 1037
  * Martel et al. (2014) Martel H., Robichaud F., Barai P., 2014, ApJ, 786, 79
  * Martin et al. (2011) Martin A. D., Quinn K. M., Park J. H., 2011, Journal of Statistical Software, 42, 22
  * Medezinski et al. (2010) Medezinski E., Broadhurst T., Umetsu K., Oguri M., Rephaeli Y., Benítez N., 2010, MNRAS, 405, 257
  * Medezinski et al. (2018) Medezinski E., et al., 2018, PASJ, 70, 30
  * Merloni et al. (2012) Merloni A., et al., 2012, arXiv e-prints, p. arXiv:1209.3114
  * Molnar et al. (2020) Molnar S. M., Ueda S., Umetsu K., 2020, ApJ, 900, 151
  * Monteiro-Oliveira et al. (2017a) Monteiro-Oliveira R., Cypriano E. S., Machado R. E. G., Lima Neto G. B., Ribeiro A. L. B., Sodré L., Dupke R., 2017a, MNRAS, 466, 2614
  * Monteiro-Oliveira et al. (2017b) Monteiro-Oliveira R., Lima Neto G. B., Cypriano E. S., Machado R. E. G., Capelato H. V., Laganá T. F., Durret F., Bagchi J., 2017b, MNRAS, 468, 4566
  * Monteiro-Oliveira et al. (2018) Monteiro-Oliveira R., Cypriano E. S., Vitorelli A. Z., Ribeiro A. L. B., Sodré L., Dupke R., Mendes de Oliveira C., 2018, MNRAS, 481, 1097
  * Monteiro-Oliveira et al. (2020) Monteiro-Oliveira R., Doubrawa L., Machado R. E. G., Lima Neto G. B., Castejon M., Cypriano E. S., 2020, MNRAS, 495, 2007
  * Monteiro-Oliveira et al. (2021) Monteiro-Oliveira R., Soja A. C., Ribeiro A. L. B., Bagchi J., Sankhyayan S., Candido T. O., Flores R. R., 2021, MNRAS, 501, 756
  * Monteiro-Oliveira et al. (2022) Monteiro-Oliveira R., Morell D. F., Sampaio V. M., Ribeiro A. L. B., de Carvalho R. R., 2022, MNRAS, 509, 3470
  * Moura et al. (2021) Moura M. T., Machado R. E. G., Monteiro-Oliveira R., 2021, MNRAS, 500, 1858
  * Navarro et al. (1996) Navarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462, 563
  * Nishizawa et al. (2020) Nishizawa A. J., Hsieh B.-C., Tanaka M., Takata T., 2020, arXiv e-prints, p. arXiv:2003.01511
  * Nychka et al. (2014) Nychka D., Furrer R., Sain S., 2014, Fields: Tools for spatial data. http://CRAN.R-project.org/package=fields
  * Pandge et al. (2019) Pandge M. B., Monteiro-Oliveira R., Bagchi J., Simionescu A., Limousin M., Raychaudhury S., 2019, MNRAS, 482, 5093
  * Plummer et al. (2006) Plummer M., Best N., Cowles K., Vines K., 2006, R News, 6, 7
  * Predehl et al. (2021) Predehl P., et al., 2021, A&A, 647, A1
  * R Core Team (2014) R Core Team 2014, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, http://www.R-project.org/
  * Sarazin (2004) Sarazin C. L., 2004, Journal of Korean Astronomical Society, 37, 433
  * Schirmer (2004) Schirmer M., 2004, PhD thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn
  * Schneider (1996) Schneider P., 1996, MNRAS, 283, 837
  * Soja et al. (2018) Soja A. C., Sodré L., Monteiro-Oliveira R., Cypriano E. S., Lima Neto G. B., 2018, MNRAS, 477, 3279
  * Stott et al. (2009) Stott J. P., Pimbblet K. A., Edge A. C., Smith G. P., Wardlow J. L., 2009, MNRAS, 394, 2098
  * Tam et al. (2020) Tam S.-I., et al., 2020, MNRAS, 496, 4032
  * Tanaka et al. (2018) Tanaka M., et al., 2018, PASJ, 70, S9
  * Torri et al. (2004) Torri E., Meneghetti M., Bartelmann M., Moscardini L., Rasia E., Tormen G., 2004, MNRAS, 349, 476
  * Ueda et al. (2020) Ueda S., Ichinohe Y., Molnar S. M., Umetsu K., Kitayama T., 2020, ApJ, 892, 100
  * Ueda et al. (2021) Ueda S., Umetsu K., Ng F., Ichinohe Y., Kitayama T., Molnar S. M., 2021, ApJ, 922, 81
  * Umetsu (2020) Umetsu K., 2020, A&ARv, 28, 7
  * Visvanathan & Sandage (1977) Visvanathan N., Sandage A., 1977, ApJ, 216, 214
  * Wei et al. (2018) Wei C., Li G., Kang X., Liu X., Fan Z., Yuan S., Pan C., 2018, MNRAS, 478, 2987
  * Wen & Han (2013) Wen Z. L., Han J. L., 2013, MNRAS, 436, 275
  * Wittman et al. (2018) Wittman D., Golovich N., Dawson W. A., 2018, ApJ, 869, 104
  * Zhang et al. (2020) Zhang C., Churazov E., Dolag K., Forman W. R., Zhuravleva I., 2020, MNRAS, 494, 4539
  * van Weeren et al. (2019) van Weeren R. J., de Gasperin F., Akamatsu H., Brüggen M., Feretti L., Kang H., Stroe A., Zandanel F., 2019, Space Sci. Rev., 215, 16