11institutetext: Argelander-Institut für Astronomie (AIfA), Universität Bonn,
Auf dem Hügel 71, 53121 Bonn, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of Physics, Nara
Women’s University, Kitauoyanishi-machi, Nara, 630-8506, Japan
33institutetext: Institut für Astronomie und Astrophysik Tübingen (IAAT),
Universität Tübingen, Sand 1, 72076 Tübingen, Germany 44institutetext:
Graduate School of Science, Division of Particle and Astrophysical Science,
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8602, Japan
55institutetext: Center for Frontier Science, Chiba University, 1-33 Yayoi-
cho, Inage-ku, Chiba 263-8522, Japan 66institutetext: Department of Physics,
Graduate School of Science, Chiba University, 1-33 Yayoi-Cho, Inage-Ku, Chiba
263-8522, Japan 77institutetext: Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU, WPI), University of Tokyo, Chiba
277-8582, Japan 88institutetext: Faculty of Physics, Ludwig-Maximilians-
Universität, Scheinerstr. 1, 81679, Munich, Germany 99institutetext: Physics
Program, Graduate School of Advanced Science and Engineering, Hiroshima
University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
1010institutetext: Hiroshima Astrophysical Science Center, Hiroshima
University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
1111institutetext: Core Research for Energetic Universe, Hiroshima University,
1-3-1, Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan
1212institutetext: Max Planck Institute for Extraterrestrial Physics,
Giessenbachstrasse 1, 85748 Garching, Germany 1313institutetext: University of
Hamburg, Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
1414institutetext: Tsung-Dao Lee Institute, and Key Laboratory for Particle
Physics, Astrophysics and Cosmology, Ministry of Education, Shanghai Jiao Tong
University, Shanghai 200240, China 1515institutetext: Department of Astronomy,
School of Physics and Astronomy, and Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China
1616institutetext: Academia Sinica Institute of Astronomy and Astrophysics
(ASIAA), 11F of AS/NTU Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt
Rd, Taipei10617, Taiwan 1717institutetext: Kobayashi-Maskawa Institute for the
Origin of Particles and the Universe (KMI), Nagoya University, Nagoya,
464-8602, Japan 1818institutetext: National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 1919institutetext: SOKENDAI (The
Graduate University for Advanced Studies), Mitaka, Tokyo, 181-8588, Japan

# The eROSITA Final Equatorial-Depth Survey (eFEDS)

X-ray properties of Subaru’s optically selected clusters

N. Ota 11 2 2 N. T. Nguyen-Dang 33 I. Mitsuishi 44 M. Oguri 556677 M. Klein 88
N. Okabe 9910101111 M. E. Ramos-Ceja 1212 T. H. Reiprich 11 F. Pacaud 11 E.
Bulbul 1212 M. Brüggen 1313 A. Liu 1212 K. Migkas 11 I. Chiu 141415151616 V.
Ghirardini 1212 S. Grandis 88 Y.-T. Lin 1616 H. Miyatake 771717 S. Miyazaki
18181919 J. S. Sanders 1212

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. We present the results of a systematic X-ray analysis of optically
rich galaxy clusters detected by the Subaru Hyper Suprime-Cam (HSC) survey in
the eROSITA Final Equatorial-Depth Survey (eFEDS) field.

Aims. Through a joint analysis of the SRG (Spectrum Roentgen Gamma)/eROSITA
and Subaru/HSC surveys, we aim to investigate the dynamical status of the
optically selected clusters and to derive the cluster scaling relations.

Methods. The sample consists of 43 optically selected galaxy clusters with a
richness $>40$ in the redshift range of 0.16–0.89. We systematically analyzed
the X-ray images and emission spectra using the eROSITA data. We identified
the brightest cluster galaxy (BCG) using the optical and far-infrared
databases. We evaluated the cluster’s dynamical status by measuring three
quantities: offset between the X-ray peak and BCG position, the gas
concentration parameter, and the number of galaxy-density peaks. We
investigated the luminosity-temperature and mass-luminosity relations based on
eROSITA X-ray spectra and HSC weak-lensing data analyses.

Results. Based on these three measurements, we estimated the fraction of
relaxed clusters to be $2(<39)$%, which is smaller than that of the X-ray-
selected cluster samples. After correcting for a selection bias due to the
richness cut, we obtained a shallow $L-T$ slope of $2.1\pm 0.5$, which is
consistent with the predictions of the self-similar model and the baseline
model incorporating a mass-concentration relation. The $L-M$ slope of $1.5\pm
0.3$ is in agreement with the above-cited theoretical models as well as the
data on the shear-selected clusters in the eFEDs field.

Conclusions. Our analysis of high-richness optical clusters yields a small
fraction of relaxed clusters and a shallow slope for the luminosity-
temperature relation. This suggests that the average X-ray properties of the
optical clusters are likely to be different from those observed in the X-ray
samples. Thus, the joint eROSITA and HSC observations are a powerful tool in
extending the analysis to a larger sample and understanding the selection
effect with a view to establish cluster scaling relations.

###### Key Words.:

Galaxies: clusters: intracluster medium; intergalactic medium; X-rays:
galaxies: clusters

##  1 Introduction

One of the most powerful constraints on current cosmological models comes from
observations of how the galaxy cluster population evolves over time. The
redshift evolution of the cluster mass function, particularly in the number of
high-mass clusters, is sensitive to cosmological parameters (Allen et al.,
2011). This makes observations of massive clusters in the distant universe
vital. In particular, understanding the mass-observable scaling relations of
these clusters is crucial to facilitating their use in cosmology (Giodini et
al., 2013). However, measurements at higher redshifts pose a considerable
challenge due to the small sample size and difficulties in accounting for
selection bias (Pratt et al., 2019), along with the overall need for deep
observations.

A combination of eROSITA and HSC observations has given us the unique
opportunity to conduct a systematic study of optically selected clusters. The
extended ROentgen Survey with an Imaging Telescope Array (eROSITA) on board
the Spectrum-Roentgen-Gamma (SRG) satellite performed scanning X-ray
observations of the $140\leavevmode\nobreak\ {\rm deg^{2}}$ eROSITA Final
Equatorial Depth Survey field (the eFEDS field), during the performance
verification phase (Predehl et al., 2021). The eFEDS survey has improved X-ray
data uniformity of optically selected clusters; the vignetted, average
exposure time is $1.3$ ks in the 0.5–2 keV band (Brunner et al., 2022). On the
other hand, the Hyper Suprime-Cam (HSC; Miyazaki et al., 2018) Subaru
Strategic Program (SSP) is a wide-field optical imaging survey (Aihara et al.,
2018a, b, 2022; Tanaka et al., 2018; Bosch et al., 2018), which has a
significant overlap with the eFEDS survey. From the HSC-SSP dataset, Oguri et
al. (2018) constructed a cluster catalog using the red-sequence cluster
finding algorithm, CAMIRA (Oguri, 2014). The catalog contains about 20,000
clusters at $0.1<z<1.3$ with richness $N>15$. The cluster mass-richness
relation is carefully calibrated using weak-lensing analysis (Okabe et al.,
2019; Murata et al., 2019). Therefore, by a joint analysis of the two surveys,
the scaling relations and their dependence on cluster dynamical status can be
studied over a wide range of redshifts and masses.

Recent X-ray follow-up observations of optically selected clusters have
reported that the X-ray properties of the optical clusters are marginally
different from those observed in the X-ray samples. Willis et al. (2021)
compared the XXL and CAMIRA catalogs, finding that 71/150 XXL clusters (67/270
CAMIRA clusters) are matched to the location of a CAMIRA cluster (an XXL
cluster). Of the unmatched CAMIRA clusters, the stacked XMM-Newton data
yielded a low, flat surface-brightness distribution, which is unlikely to
follow the conventional $\beta$-model (Cavaliere & Fusco-Femiano, 1976). From
the XMM-Newton data analysis of 37 CAMIRA clusters, Ota et al. (2020) found a
small fraction of relaxed clusters compared to X-ray cluster samples
suggesting that the optical cluster sample covers a larger range of the
cluster morphologies. They also derived the luminosity-temperature relation
and found that the slope is marginally shallower than those of X-ray-selected
samples and consistent with the self-similar model prediction of 2 (Kaiser,
1986). To obtain more conclusive results, we aim to improve the measurement
accuracy and sample uniformity.

In this paper, we study the dynamical status and scaling relations for a
subsample of high-richness, optically selected clusters in the eFEDS field.
This paper is structured as follows. Sectsions 2 and 3 present the sample
selection and optical analyses of the brightest cluster galaxies and weak-
lensing mass, respectively. Section 4 describes the X-ray measurements of the
cluster centroid and spectral properties. Section 5 derives the centroid
offset and the relations between mass and X-ray observables. Section 6
presents a discussion of the implication of our results.

The cosmological parameters used throughout this paper are $\Omega_{\rm
m}=0.3$, $\Omega_{\Lambda}=0.7$ and $h=0.7$. We use the abundance table from
Asplund et al. (2009) in the X-ray spectral modeling. The quoted errors
represent the $1\sigma$ statistical uncertainties, unless stated otherwise.

##  2 Sample

We selected a sample of optically selected clusters based on the CAMIRA S20a
v2 cluster catalog, an updated version of the HSC CAMIRA cluster catalog
presented in Oguri et al. (2018). Among the 21,250 optical clusters discovered
in the Subaru HSC survey fields, 997 objects with richness $N>15$ and redshift
$0.10<z<1.34$ lie in the eFEDS field. The richness range $N>15$ ($N>40$)
corresponds to the cluster mass $M_{500}\gtrsim 5\times
10^{13}\leavevmode\nobreak\ {\rm M_{\odot}}$ ($M_{500}\gtrsim 2\times
10^{14}\leavevmode\nobreak\ {\rm M_{\odot}}$) at $z>0.1$ (Okabe et al., 2019).
We cross-matched the CAMIRA catalog with the eFEDS X-ray cluster catalog (Liu
et al., 2022) to find 211 optical clusters that have at least one spatially-
extended X-ray source within the scale radius $R_{500}$ from the optical
centers and the redshift difference of $|\Delta z|<0.02$. Here, $R_{500}$ was
estimated from the mass-richness relation (Okabe et al., 2019) and the cluster
richness (Oguri et al., 2018). Table 1 summarizes the result of the cross-
matching of two catalogs.

Table 1: Cross-matching of optical and X-ray cluster catalogs $\hat{N}_{\mathrm{mem}}$ | CAMIRA | eFEDS | Fraction (%)
---|---|---|---
15–20 | 524 | 27 | 5
20–30 | 335 | 62 | 19
30–40 | 97 | 50 | 52
$>40$ | 41 | 32 | 78
Total | 997 | 171 | 17

In general, the X-ray detectability tends to be lower for lower richness and
higher redshift; however, the observed fraction of 17% is lower than expected
from the known cluster mass-observable relations (Okabe et al., 2019) and
eROSITA’s sensitivity, when assuming the typical $\beta$-model brightness
distribution of regular clusters (Ota & Mitsuda, 2004). This result can be
attributed to multiple factors such as a large fraction of irregular clusters
with low surface brightness (Ota et al., 2020), the BCG miscentering effect
(Oguri et al., 2018), and misclassification as point sources (Bulbul et al.,
2022), thus requiring further investigation.

To study the optical clusters’ dynamical status and scaling relations based on
the eROSITA analysis of individual objects, we focus on high-richness clusters
with $N>40$ and $0.16<z<0.89$ in this paper. We note that there were 41
clusters with $N>40$ in the CAMIRA catalog (Table 1); however, since several
CAMIRA clusters have two galaxy-density peaks separated by $\gtrsim R_{500}$,
we treat them as two individual clusters in the following analysis. There are
seven clusters identified in this way: HSC J085621+014649, HSC J085629+014157,
HSC J092050+024514, HSC J092246+034241, HSC J093512+004738, HSC
J093501+005415, and HSC J093523+023222. Their optical properties were obtained
by the forced run of the CAMIRA optical cluster finder (Klein et al., 2022;
Oguri et al., 2018). This yields a sample size of 43. Table 2 gives the sample
list. Fig. 1 shows examples of HSC images with superposed X-ray contours. A
statistical analysis of the whole cluster sample will be presented in a
separate paper.

Figure 1: Examples of Subaru riz-composite images of the optically selected
clusters with overlaid the eROSITA X-ray intensity maps, HSC J084528+032739
(left) and HSC J092557+004122 (right). The red contours are linearly spaced by
half of the average height of galaxy density maps over all CAMIRA clusters at
the same redshift. The white contours for X-ray emission are ten levels
logarithmically spaced from $[10-1000]\leavevmode\nobreak\ {\rm
cts\,s^{-1}deg^{-2}}$. In each panel, the X-ray centroid and the initial
CAMIRA coordinates are marked by the yellow and pink cross, respectively. The
green circle denotes the position of the confirmed BCG. The magenta circle
indicates $R_{500}$.  Table 2: Sample list.

Cluster | $z$ |  $\hat{N}_{\mathrm{mem}}$aa$a$Richneess. | $R_{500}$ | BCG position | X-ray centroid | X-ray peak |  $D_{\mathrm{XC}}$bb$b$Centroid offset. |  $D_{\mathrm{XP}}$cc$c$Peak offset. The error range estimated by changing the smoothing scale of the X-ray image is given in the parenthesis (see Sects. 4.2 and 5.2).
---|---|---|---|---|---|---|---|---
|  |  | (Mpc/”) | RA, Dec (deg) | RA, Dec (deg) | RA, Dec (deg) | (kpc) | (kpc)
HSC J083655+025855 | 0.189 | 42.5 | 0.852 / 269 | 129.2144 , 3.0004 | 129.2141 , 2.9992 | 129.2163 , 2.9988 | 13 | 27 (27,37)
HSC J083932-014128 | 0.255 | 45.7 | 0.866 / 217 | 129.8891 , -1.6791 | 129.8844 , -1.6831 | 129.8916 , -1.6790 | 87 | 35 (30,35)
HSC J084222+013826 | 0.418 | 75.8 | 1.050 / 188 | 130.5912 , 1.6406 | 130.5975 , 1.6463 | 130.5919 , 1.6385 | 167 | 44 (44,657)
HSC J084245-000936 | 0.420 | 48.5 | 0.848 / 152 | 130.6896 , -0.1601 | 130.6991 , -0.1635 | 130.7021 , -0.1411 | 202 | 453 (453,453)
HSC J084435+031020 | 0.735 | 40.9 | 0.709 / 96 | 131.1453 , 3.1723 | 131.1449 , 3.1697 | 131.1443 , 3.1703 | 68 | 57 (48,75)
HSC J084441+021656 | 0.649 | 71.3 | 0.949 / 135 | 131.1710 , 2.2823 | 131.1582 , 2.2847 | 131.1743 , 2.2833 | 325 | 84 (65,94)
HSC J084502+012631 | 0.415 | 40.5 | 0.779 / 141 | 131.2596 , 1.4420 | 131.2592 , 1.4554 | 131.2533 , 1.4626 | 265 | 426 (238,426)
HSC J084528+032739 | 0.320 | 103.6 | 1.255 / 268 | 131.3657 , 3.4608 | 131.3675 , 3.4589 | 131.3691 , 3.4588 | 44 | 65 (65,69)
HSC J084548+020640 | 0.582 | 44.8 | 0.777 / 116 | 131.4498 , 2.1054 | 131.4508 , 2.1130 | 131.4619 , 2.1203 | 181 | 454 (421,454)
HSC J084656+013836 | 0.597 | 41.1 | 0.743 / 110 | 131.7340 , 1.6432 | 131.7481 , 1.6482 | 131.7306 , 1.6670 | 359 | 575 (575,644)
HSC J084824+041206 | 0.873 | 60.1 | 0.816 / 104 | 132.0988 , 4.2017 | 132.0949 , 4.2020 | 132.0995 , 4.2023 | 107 | 25 (8,33)
HSC J084939-005121 | 0.616 | 53.4 | 0.836 / 122 | 132.4136 , -0.8560 | 132.4156 , -0.8572 | 132.4165 , -0.8583 | 57 | 91 (91,97)
HSC J085019+020011 | 0.419 | 44.8 | 0.817 / 146 | 132.5635 , 2.0099 | 132.5762 , 1.9969 | 132.5776 , 2.0078 | 362 | 283 (275,284)
HSC J085056-000931 | 0.890 | 44.4 | 0.703 / 89 | 132.7330 , -0.1586 | 132.7328 , -0.1586 | 132.7301 , -0.1605 | 4 | 96 (86,103)
HSC J085232+002551 | 0.280 | 46.6 | 0.868 / 203 | 133.1256 , 0.4205 | 133.1255 , 0.4085 | 133.1270 , 0.4066 | 183 | 213 (213,213)
HSC J085741+031135 | 0.188 | 73.6 | 1.108 / 351 | 134.4751 , 3.1764 | 134.4669 , 3.1810 | 134.4642 , 3.1797 | 105 | 128 (125,128)
HSC J090143-014019 | 0.302 | 49.6 | 0.888 / 197 | 135.3779 , -1.6548 | 135.3775 , -1.6602 | 135.3715 , -1.6685 | 87 | 242 (88,441)
HSC J090330-013622 | 0.440 | 50.8 | 0.862 / 150 | 135.8756 , -1.6062 | 135.8718 , -1.6070 | 135.8570 , -1.6010 | 81 | 395 (389,395)
HSC J090451+033310 | 0.807 | 50.6 | 0.768 / 100 | 136.2176 , 3.5537 | 136.2229 , 3.5593 | 136.2183 , 3.5606 | 207 | 189 (178,246)
HSC J090541+013226 | 0.636 | 40.4 | 0.727 / 105 | 136.4216 , 1.5406 | 136.4212 , 1.5366 | 136.4150 , 1.5191 | 99 | 555 (26,555)
HSC J090754+005732 | 0.693 | 47.5 | 0.772 / 107 | 136.9765 , 0.9590 | 136.9794 , 0.9589 | 136.9765 , 0.9561 | 74 | 72 (72,212)
HSC J090914-001220 | 0.305 | 50.7 | 0.897 / 198 | 137.3074 , -0.2056 | 137.3120 , -0.2067 | 137.3002 , -0.2079 | 77 | 122 (122,122)
HSC J090917-010134 | 0.823 | 49.9 | 0.758 / 98 | 137.3190 , -1.0261 | 137.3137 , -1.0191 | 137.3168 , -1.0188 | 239 | 205 (198,216)
HSC J091352-004535 | 0.264 | 49.1 | 0.893 / 218 | 138.4674 , -0.7597 | 138.4687 , -0.7580 | 138.4775 , -0.7622 | 31 | 151 (147,151)
HSC J091606-002338 | 0.297 | 113.9 | 1.322 / 297 | 139.0385 , -0.4045 | 139.0466 , -0.3970 | 139.0427 , -0.3974 | 176 | 131 (131,131)
HSC J091843+021231 | 0.268 | 57.6 | 0.964 / 233 | 139.7092 , 2.2009 | 139.7056 , 2.2142 | 139.7055 , 2.2014 | 204 | 54 (54,60)
HSC J092024+013444 | 0.698 | 44.3 | 0.746 / 103 | 140.0981 , 1.5788 | 140.0974 , 1.5783 | 140.0990 , 1.5784 | 21 | 27 (26,43)
HSC J092121+031713 | 0.345 | 95.4 | 1.198 / 243 | 140.3380 , 3.2870 | 140.3392 , 3.2891 | 140.3417 , 3.2948 | 42 | 152 (152,152)
HSC J092211+034641 | 0.252 | 65.9 | 1.032 / 261 | 140.5319 , 3.7663 | 140.5369 , 3.7667 | 140.5467 , 3.7763 | 71 | 252 (243,257)
HSC J092557+004122 | 0.260 | 57.8 | 0.967 / 239 | 141.4748 , 0.7199 | 141.4845 , 0.7133 | 141.4919 , 0.7071 | 169 | 310 (310,310)
HSC J092847+005132 | 0.310 | 41.4 | 0.813 / 177 | 142.1952 , 0.8587 | 142.2048 , 0.8775 | 142.2499 , 0.8913 | 345 | 1043 (540,1043)
HSC J092942+022843 | 0.537 | 43.6 | 0.778 / 121 | 142.4255 , 2.5061 | 142.4232 , 2.4956 | 142.4344 , 2.5122 | 243 | 246 (75,246)
HSC J093025+021726 | 0.532 | 66.0 | 0.949 / 149 | 142.5929 , 2.3068 | 142.6030 , 2.2916 | 142.6045 , 2.2907 | 414 | 450 (450,456)
HSC J093049-003651 | 0.310 | 43.4 | 0.832 / 181 | 142.7055 , -0.6159 | 142.7052 , -0.6197 | 142.7051 , -0.6175 | 62 | 27 (27,369)
HSC J093431-002308 | 0.342 | 45.5 | 0.842 / 172 | 143.6300 , -0.3855 | 143.6353 , -0.3870 | 143.6196 , -0.3844 | 96 | 183 (183,184)
HSC J094025+022840 | 0.164 | 66.4 | 1.062 / 377 | 145.1024 , 2.4776 | 145.1188 , 2.4822 | 145.1902 , 2.4350 | 171 | 985 (520,985)
HSC J085621+014649 | 0.769 | 44.1 | 0.728 / 97 | 134.0858 , 1.7804 | 134.0865 , 1.7794 | 134.0886 , 1.7810 | 34 | 77 (42,77)
HSC J085629+014157 | 0.692 | 42.6 | 0.733 / 102 | 134.1288 , 1.7105 | 134.1294 , 1.7077 | 134.1136 , 1.7099 | 74 | 389 (388,389)
HSC J092050+024514 | 0.284 | 47.5 | 0.875 / 203 | 140.2071 , 2.7539 | 140.2084 , 2.7565 | 140.2067 , 2.7506 | 45 | 50 (50,50)
HSC J092246+034241 | 0.257 | 55.3 | 0.948 / 236 | 140.6929 , 3.7113 | 140.6893 , 3.7079 | 140.6911 , 3.7151 | 71 | 60 (58,133)
HSC J093512+004738 | 0.352 | 96.2 | 1.200 / 240 | 143.8018 , 0.7939 | 143.8069 , 0.7977 | 143.8083 , 0.8010 | 112 | 172 (172,177)
HSC J093501+005415 | 0.374 | 53.9 | 0.905 / 174 | 143.7528 , 0.9041 | 143.7553 , 0.9036 | 143.7591 , 0.9030 | 46 | 117 (109,117)
HSC J093523+023222 | 0.513 | 84.5 | 1.074 / 172 | 143.8439 , 2.5569 | 143.8414 , 2.5451 | 143.8383 , 2.5436 | 267 | 321 (321,329)

111

##  3 Optical data analysis

###  3.1 BCG identification

The CAMIRA algorithm (Oguri, 2014) first calculates the probability of each
galaxy in the field being on the red sequence. Consequently, the galaxy
clusters are defined as the overdensity of this type of galaxy population. At
the same time, the selection method also returns the brightest cluster
galaxy’s (BCG) position for each cluster. This cluster’s optical center is the
position of the most luminous red-sequence galaxy close to the galaxy density
peak. However, ultra-bright BCGs may not be correctly identified as BCGs
because they are saturated in the HSC images, and the color estimates tend to
be inaccurate. These reasons suggest that an additional confirmation of the
BCG’s position is necessary.

To determine the BCGs of all the clusters, we used the optical/NIR data from
SDSS (York et al., 2000), Pan-STARRS (Kaiser et al., 2002, 2010), 2MASS
(Skrutskie et al., 2006), and WISE (Wright et al., 2010). Each survey has a
different sensitivity and filter response. In particular, SDSS has an i-band
saturation level of 14 mag for point sources, which is brighter than the 18
mag of the HSC. In fact, for nearly half of the clusters, the brightest
galaxies exceeded the HSC’s saturation level and were re-identified in the
following way. We searched for the brightest galaxy that locates within an
$R_{500}$ radius from the cluster’s optical center, which is taken from the
CAMIRA catalog. For the optical data, we used the $r$-band magnitudes to
determine the brightest galaxy, whilst the $K_{s}$ band and $W_{1}$ values
were used in the IR regime. The redshifts of the galaxies were either taken
from the SDSS catalog or the NASA/IPAC Extragalactic Database
(NED)222https://ned.ipac.caltech.edu/. The redshift constraints of the BCGs
search is $\Delta
z=|z_{\mathrm{galaxy}}-z_{\mathrm{cluster}}|=0.01\times(1+z_{\mathrm{cluster}})$
for the spectroscopic redshifts and $\Delta
z=0.02\times({1+z_{\mathrm{cluster}}})$ for the photometric ones,
respectively. Here, we considered uncertainties of redshift measurements and
redshift tolerance of $3\sigma/c\sim 0.01$ for the typical velocity dispersion
of rich clusters, $\sigma\sim 1000\leavevmode\nobreak\ {\rm km\,s^{-1}}$
(Fadda et al., 1996).

Two additional corrections have been made before the photometric data are used
for the BCGs search. First, we corrected for the galactic extinction, using
the Schlegel map (Schlegel et al., 1998), assuming an extinction law
(Fitzpatrick, 1999) with $R_{V}$= 3.1. Second, appropriate $K$-correction
methods are applied to obtain the magnitudes as in the rest frames of
individual galaxies. We used the $K$-correction code version 2012
(Chilingarian et al., 2010; Chilingarian & Zolotukhin, 2012) for the SDSS,
Pan-STARRS, and 2MASS data because this method performs well with bright
sources and does not require the compulsory input of multiple filter bands.
However, this method does not provide the coefficients for WISE filters;
therefore, we applied the $K$-correction code from Blanton & Roweis (2007) on
the WISE photometric data instead.

Finally, we assessed the BCG search result through a careful visual
inspection. In the case that multiple BCG candidates are suggested for one
cluster, we assigned the BCG to the elliptical galaxy with a more extended
envelope. If the search from different surveys returns different BCGs, we gave
more weight to the BCG results from the optical observations. There were two
cases (HSC J091843+021231 and HSC J092041+024660), where we needed to check
the information of the obvious BCG candidates manually. These were the
seemingly best choices but were not selected by the BCG finding program.
Indeed, these galaxies do not have photometric imprints in any of the
aforementioned surveys. We found the redshift of these visible BCG candidates
on NED and their brightness in GAMA (Wright et al., 2016). We thus confirmed
that they are the BCGs of their host clusters because they also follow the
spatial and redshift constraints.

###  3.2 Weak-lensing mass measurement

Next, we describe weak-lensing (WL) analyses for the CAMIRA clusters in the
eFEDS field. We used the latest shape catalog and the weak-lensing mass
calibration from the three-year HSC data (S19A; Li et al., 2022). The galaxy
shapes are measured by the re-Gaussianization method (Hirata & Seljak, 2003)
implemented in the HSC pipeline (see details in Mandelbaum et al., 2018a, b).
The same shape catalog is used in the WL mass measurements for the eFEDS
clusters (Chiu et al., 2022). We adopted the full-color and full-depth
criteria for precise shape measurements and photometric redshift estimations.

The dimensional, reduced tangential shear, $\Delta\Sigma_{+}$, is computed by
averaging the tangential component of a galaxy ellipticity $e_{+}=-(e_{1}\cos
2\varphi+e_{2}\sin 2\varphi)$ where $\varphi$ is the angle measured in sky
coordinates from the RA direction to the line between the source galaxy and
the lens. The formulation is specified by:

$\displaystyle\Delta\Sigma_{+}(R_{k})=\frac{\sum_{i}e_{+,i}w_{i}\langle\Sigma_{{\rm
cr}}(z_{l},z_{s,i})^{-1}\rangle^{-1}}{2\mathcal{R}(R_{k})(1+K(R_{k}))\sum_{i}w_{i}},$
(1)

(e.g., Miyaoka et al., 2018; Medezinski et al., 2018a; Okabe et al., 2019;
Miyatake et al., 2019; Murata et al., 2019; Umetsu et al., 2020; Okabe et al.,
2021; Chiu et al., 2022). Here, the subscripts, $i$ and $k$, denote the $i$-th
galaxy located in the $k-$th radial bin, and $z_{l}$ and $z_{s}$ are the
cluster and source redshift, respectively. The inverse of the mean critical
surface mass density, $\langle\Sigma_{{\rm cr}}(z_{l},z_{s})^{-1}\rangle$, is
calculated by weighting the critical surface mass density, $\Sigma_{{\rm
cr}}=c^{2}D_{s}/4\pi GD_{l}D_{ls}$, by the probability function of the
photometric redshift, $P(z)$:

$\displaystyle\langle\Sigma_{{\rm
cr}}(z_{l},z_{s})^{-1}\rangle=\frac{\int^{\infty}_{z_{l}}\Sigma_{{\rm
cr}}^{-1}(z_{l},z_{s})P(z_{s})dz_{s}}{\int^{\infty}_{0}P(z_{s})dz_{s}},$ (2)

Here, $D_{l}$, $D_{s}$, and $D_{ls}$ are the angular diameter distances from
the observer to the cluster, to the sources, and from the lens to the sources,
respectively. The photometric redshift is estimated by the machine learning
method (MLZ; Carrasco Kind & Brunner, 2014) calibrated with spectroscopic data
(Nishizawa et al., 2020). The dimensional weighting function is expressed as

$\displaystyle w=\frac{1}{e_{\rm rms}^{2}+\sigma_{e}^{2}}\langle\Sigma_{{\rm
cr}}^{-1}\rangle^{2},$ (3)

where $e_{\rm rms}$ and $\sigma_{e}$ are the root mean square of intrinsic
ellipticity and the measurement error per component ($e_{\alpha}$; $\alpha=1$
or $2$), respectively. The shear responsivity, $\mathcal{R}$, and the
calibration factor, $K$, are obtained by
$\mathcal{R}=1-\sum_{ij}w_{i,j}e_{{\rm rms},i}^{2}/\sum_{ij}w_{i,j}$ and
$K=\sum_{ij}m_{i}w_{i,j}/\sum_{ij}w_{i,j}$, with the multiplicative shear
calibration factor $m$ (Mandelbaum et al., 2018a, b), respectively. We also
conservatively subtracted an additional, negligible offset term for
calibration. The radius position, $R_{k}$, is defined by the weighted harmonic
mean (Okabe & Smith, 2016). We selected background galaxies behind each
cluster using the photometric selection (p-cut) following Medezinski et al.
(2018b):

$\displaystyle\int^{\infty}_{z_{l}+0.2}P(z)dz>p_{\rm cut},$ (4)

where we allowed for a 2% contamination level with $p_{\rm cut}=0.98$.

In order to measure individual cluster masses, we used the NFW profile
(Navarro et al., 1996, 1997). The three-dimensional mass density profile of
the NFW profile is expressed as

$\rho_{\rm NFW}(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}},$ (5)

where $r_{s}$ is the scale radius and $\rho_{s}$ is the central density
parameter. The NFW model is also specified by the spherical mass,
$M_{\Delta}=4\pi\Delta\rho_{\rm cr}r_{\Delta}^{3}/3$, and the halo
concentration, $c_{\Delta}=r_{\Delta}/r_{s}$. Here, $r_{\Delta}$ is the
overdensity radius. We treat $M_{\Delta}$ and $c_{\Delta}$ as free parameters
and adopt $\Delta=500$. We computed the model of the dimensional, reduced
tangential shear, $f_{\rm model}$, at the projected radius $R$ by integrating
the mass density profile along the line of sight (Okabe et al., 2019; Umetsu,
2020);

$\displaystyle f_{\rm
model}(R)=\frac{\bar{\Sigma}(<R)-\Sigma(R)}{1-\mathcal{L}_{z}\Sigma(R)},$ (6)

where $\Sigma(R)$ is the local surface mass density at $R$, $\bar{\Sigma}(<R)$
is the average surface mass density within $R$, and
$\mathcal{L}_{z}=\sum_{i}\langle\Sigma_{{\rm cr},i}^{-1}\rangle
w_{i}/\sum_{i}w_{i}$. We chose the X-ray-defined centers and adopt an adaptive
radial-bin choice (Okabe & Smith, 2016) for cluster mass estimation.

The log-likelihood of the weak-lensing analysis is expressed as

$\displaystyle-2\ln{\mathcal{L}}_{\rm WL}=\ln(\det(C_{km}))+$ (7)
$\displaystyle\sum_{k,m}(\Delta\Sigma_{+,k}-f_{{\rm
model}}(R_{k}))C_{km}^{-1}(\Delta\Sigma_{+,m}-f_{{\rm model}}(R_{m})),$

where $k$ and $m$ denote the $k-$th and $m-$th radial bins. We considered
three components in the covariance matrix $C=C_{g}+C_{s}+C_{\rm LSS}$; the
shape noise $C_{g}$, the errors of the source redshifts, $C_{s}$ and the
uncorrelated large-scale structure (LSS), $C_{\rm LSS}$, of which the elements
are correlated with each other (Schneider et al., 1998; Hoekstra, 2003). The
details of calculations are described in Sect. 3 of Okabe & Smith (2016). We
measured WL masses in 38 out of the 43 CAMIRA clusters that satisfy the full-
color and full-depth conditions (Table 3). The signal-to-noise ratios,
$S/N=(\Delta\Sigma_{+,k}C_{km}^{-1}\Delta\Sigma_{+,m})^{1/2}$, of individual
clusters were small; $S/N<2$ for 3, $2\leq S/N<4$ for 20, and $4<S/N$ for 17
clusters, resulting in uncertainties in $M_{500}$. Here, for consistency with
other X-ray observables, the center coordinates in the WL measurements were
assumed to be equal to the X-ray centroids.

We also carried out the NFW model fitting with a free central position using
two-dimensional shear pattern (Oguri et al., 2010; Okabe et al., 2011). The
log-likelihood is expressed as

$\displaystyle-2\ln\mathcal{L}_{\rm
WL}=\sum_{\alpha,\beta=1}^{2}\sum_{k,m}\left[\Delta\Sigma_{\alpha,k}-f_{{\rm
model},\alpha}\left(\bm{R}_{k}\right)\right]\bm{C}^{-1}_{\alpha\beta,km}$
$\displaystyle\times\left[\Delta\Sigma_{\beta,m}-f_{{\rm
model},\beta}\left(\bm{R}_{m}\right)\right]+\ln(\det(C_{\alpha\beta,km})).$
(8)

Here, the subscripts $\alpha$ and $\beta$ denote each shear component. We used
the box size of $1\arcmin.5\times 1\arcmin.5$ for the shear pattern. We
constrained the central positions for 23 clusters with good posterior
distributions and compared them with X-ray centers, CAMIRA centers, and galaxy
map peaks (Sect. 5.1).

##  4 X-ray data analysis

###  4.1 Data reduction

The eFEDs data of seven telescope modules (TMs) were reduced in a standard
manner by using the eROSITA Standard Analysis Software System (eSASS) version
eSASSusers_201009 (Brunner et al., 2022). We extracted cleaned event files by
applying a flag to reject bad events and selecting all valid patterns, namely
single, double, triple, and quadruple events. The point sources were removed
by referring to the main eFEDS X-ray source
catalog333https://erosita.mpe.mpg.de/edr/eROSITAObservations/Catalogues/
(Brunner et al., 2022). We checked that the faint sources with low detection
likelihood in the supplemental eFEDS catalog do not affect our X-ray analysis.

###  4.2 Centroid and peak determination

We extracted the X-ray image of each cluster from merged event files of seven
TMs in the 0.5 – 2 keV band and corrected them for exposure and vignetting.
The pixel size is 1″.5. The X-ray centroid was determined from the mean of the
photon distribution within a circle of radius $R_{500}$ using the algorithm
described in Ota et al. (2020); Starting with the BCG coordinates, we iterated
the centroid search until its position converged within 1″. The X-ray peak
within $R_{500}$ was measured using the 0.5 – 2 keV image smoothed with a
$\sigma=3$ (pixels) Gaussian function. We note that we calculated the two-
dimensional (2D) PSF image of the survey mode at each cluster coordinates
using ermldet in the eSASS package and confirmed that it is almost symmetric
and does not affect the present measurement.

###  4.3 Spectral analysis

To measure the gas temperature and bolometric luminosity, we extracted the
spectra from a circular region of a radius $R_{500}$ centered on the X-ray
centroid. Here, it is reasonable to assume that the cluster center is
represented by the X-ray centroid (Sect. 5.1, Fig. 3). The TM1,2,3,4,6/TM5,7
spectra in the 0.3 – 10 keV/1 – 10 keV band were simultaneously fit using
XSPEC 12.1.1. For TM5 and TM7, the energies below 1 keV were excluded due to
the light-leak contamination (Predehl et al., 2021). The spectral model
consists of cluster emission and background components. For the cluster
component, we assumed the APEC thin-thermal plasma model (Smith et al., 2001;
Foster et al., 2012), with the Galactic absorption model tbabs (Wilms et al.,
2000). The redshift and metal abundance were fixed at the optical value (Table
2) and 0.3 solar, respectively. The hydrogen column density $N_{\rm H}$ in the
tbabs model was fixed at a value taken from Willingale et al. (2013). For the
background components, the Galactic emission and cosmic X-ray background were
determined by fitting an annulus with inner and outer radii of 2.5 and 4 Mpc
from the cluster. The instrumental background was estimated based on the
filter wheel closed (FWC) data444The FWC spectral model version 1.0
(https://erosita.mpe.mpg.de/edr/eROSITAObservations/EDRFWC/). Table 3 lists
the resultant gas temperature, $kT,$ and bolometric luminosity, $L_{\rm X}$.
We note that for three clusters (HSC J084548+020640, HSC J084656+01383, and
HSC J090754+005732), we fixed $kT$ at a value expected from the $N-T$ relation
(Oguri et al., 2018) and the richness due to the large statistical
uncertainty.

Table 3: Results of X-ray spectral analysis and lensing mass measurements

Cluster | $N_{\rm H}$ |  $kT$aa$a$Gas temperature; |  $L_{\rm X}$bb$b$Bolometric luminosity within the scale radius $R_{500}$ |  $M_{500}$cc$c$ Weak-lensing mass within $R_{500}$.
---|---|---|---|---
| ($10^{20}\leavevmode\nobreak\ {\rm cm^{-2}}$) | (keV) | ($10^{44}\leavevmode\nobreak\ {\rm erg\,s^{-1}}$) | ($10^{14}\leavevmode\nobreak\ {\rm M_{\odot}}$)
HSC J083655+025855 | 3.54 | $2.39^{+0.36}_{-0.28}$ | $1.31^{+0.09}_{-0.08}$ | $4.57^{+1.70}_{-1.43}$
HSC J083932-014128 | 3.09 | $3.71^{+0.71}_{-0.55}$ | $3.73^{+0.30}_{-0.26}$ | –
HSC J084222+013826 | 5.52 | $1.95^{+0.48}_{-0.35}$ | $2.55^{+0.25}_{-0.25}$ | $9.28^{+3.05}_{-2.57}$
HSC J084245-000936 | 3.33 | $3.34^{+3.04}_{-1.17}$ | $1.33^{+0.41}_{-0.24}$ | $5.66^{+2.51}_{-2.00}$
HSC J084435+031020 | 3.71 | $2.37^{+1.21}_{-0.53}$ | $3.43^{+0.53}_{-0.44}$ | $7.27^{+9.67}_{-7.26}$
HSC J084441+021656 | 5.17 | $2.42^{+0.66}_{-0.45}$ | $5.98^{+0.60}_{-0.55}$ | $15.92^{+18.92}_{-9.18}$
HSC J084502+012631 | 4.58 | $4.37^{+3.40}_{-1.54}$ | $2.18^{+0.57}_{-0.37}$ | $2.67^{+1.63}_{-1.25}$
HSC J084528+032739 | 3.63 | $7.72^{+3.52}_{-1.43}$ | $8.69^{+1.18}_{-0.75}$ | $5.32^{+1.64}_{-1.37}$
HSC J084548+020640 | 4.82 |  $3.69$(fix) | $1.45^{+0.30}_{-0.28}$ | $2.69^{+3.16}_{-2.26}$
HSC J084656+013836 | 4.20 |  $3.54$(fix) | $0.44^{+0.22}_{-0.21}$ | $5.22^{+4.78}_{-2.83}$
HSC J084824+041206 | 4.12 | $4.09^{+3.58}_{-1.44}$ | $6.15^{+1.67}_{-0.94}$ | –
HSC J084939-005121 | 2.60 | $6.29^{+10.99}_{-2.96}$ | $5.02^{+2.17}_{-1.20}$ | $2.81^{+3.33}_{-2.87}$
HSC J085019+020011 | 3.24 | $2.76^{+4.47}_{-1.12}$ | $0.89^{+0.41}_{-0.20}$ | $1.68^{+1.52}_{-1.27}$
HSC J085056-000931 | 3.21 | $1.44^{+0.57}_{-0.24}$ | $3.25^{+0.69}_{-0.61}$ | $7.37^{+15.94}_{-10.67}$
HSC J085232+002551 | 4.13 | $2.67^{+1.27}_{-0.64}$ | $1.11^{+0.20}_{-0.14}$ | $2.69^{+1.17}_{-0.91}$
HSC J085741+031135 | 4.15 | $5.94^{+1.30}_{-0.88}$ | $5.25^{+0.43}_{-0.35}$ | $3.21^{+1.57}_{-1.29}$
HSC J090143-014019 | 2.28 | $5.39^{+2.37}_{-1.31}$ | $5.00^{+0.81}_{-0.58}$ | –
HSC J090330-013622 | 2.26 | $5.00^{+7.49}_{-1.90}$ | $3.37^{+1.43}_{-0.62}$ | –
HSC J090451+033310 | 3.60 | $5.06^{+4.35}_{-1.69}$ | $6.51^{+1.80}_{-1.00}$ | –
HSC J090541+013226 | 4.01 | $3.88^{+2.90}_{-1.56}$ | $2.90^{+0.72}_{-0.53}$ | –
HSC J090754+005732 | 3.39 |  $3.80$ (fix) | $3.14^{+0.43}_{-0.41}$ | $0.16^{+3.12}_{-2.04}$
HSC J090914-001220 | 3.00 | $3.22^{+0.97}_{-0.77}$ | $1.82^{+0.23}_{-0.20}$ | $1.99^{+1.32}_{-0.93}$
HSC J090917-010134 | 2.84 | $2.98^{+1.61}_{-0.72}$ | $5.93^{+0.99}_{-0.70}$ | –
HSC J091352-004535 | 3.20 | $2.21^{+0.84}_{-0.44}$ | $0.85^{+0.12}_{-0.10}$ | $3.50^{+1.17}_{-1.00}$
HSC J091606-002338 | 3.20 | $4.39^{+1.10}_{-0.79}$ | $5.40^{+0.56}_{-0.45}$ | $8.65^{+3.26}_{-2.52}$
HSC J091843+021231 | 2.74 | $2.57^{+0.49}_{-0.34}$ | $2.12^{+0.17}_{-0.15}$ | $2.80^{+1.47}_{-1.09}$
HSC J092024+013444 | 2.90 | $2.95^{+0.70}_{-0.43}$ | $10.09^{+0.82}_{-0.66}$ | $10.84^{+8.61}_{-6.84}$
HSC J092121+031713 | 3.87 | $6.19^{+1.52}_{-1.09}$ | $10.38^{+0.96}_{-0.82}$ | $5.12^{+1.69}_{-1.37}$
HSC J092211+034641 | 3.98 | $5.33^{+1.34}_{-1.06}$ | $4.53^{+0.45}_{-0.41}$ | $3.82^{+1.52}_{-1.27}$
HSC J092557+004122 | 3.43 | $3.25^{+1.27}_{-0.90}$ | $1.09^{+0.17}_{-0.15}$ | $2.03^{+1.26}_{-0.98}$
HSC J092847+005132 | 4.27 | $1.74^{+0.57}_{-0.27}$ | $0.46^{+0.08}_{-0.07}$ | $1.08^{+1.04}_{-0.86}$
HSC J092942+022843 | 4.85 | $3.53^{+4.22}_{-1.35}$ | $1.69^{+0.61}_{-0.33}$ | $1.86^{+1.95}_{-1.38}$
HSC J093025+021726 | 5.06 | $6.98^{+12.14}_{-1.97}$ | $8.27^{+3.30}_{-1.09}$ | $5.11^{+3.40}_{-2.45}$
HSC J093049-003651 | 3.09 | $2.43^{+0.73}_{-0.40}$ | $1.16^{+0.15}_{-0.11}$ | $3.21^{+1.42}_{-1.19}$
HSC J093431-002308 | 3.15 | $2.78^{+1.23}_{-0.66}$ | $1.59^{+0.26}_{-0.19}$ | $9.13^{+4.05}_{-2.83}$
HSC J094025+022840 | 3.72 | $2.18^{+1.29}_{-0.52}$ | $1.31^{+0.30}_{-0.21}$ | $7.10^{+2.21}_{-1.75}$
HSC J085621+014649 | 3.87 | $4.49^{+1.60}_{-0.91}$ | $15.93^{+2.06}_{-1.42}$ | –
HSC J085629+014157 | 3.97 | $3.21^{+1.06}_{-0.69}$ | $6.62^{+0.78}_{-0.65}$ | $1.46^{+4.00}_{-3.20}$
HSC J092050+024514 | 3.50 | $3.05^{+0.56}_{-0.42}$ | $2.84^{+0.22}_{-0.19}$ | $2.84^{+2.65}_{-1.38}$
HSC J092246+034241 | 3.93 | $1.61^{+0.38}_{-0.25}$ | $0.42^{+0.06}_{-0.06}$ | $1.13^{+0.93}_{-0.78}$
HSC J093512+004738 | 4.34 | $5.90^{+1.77}_{-1.12}$ | $10.40^{+1.15}_{-0.89}$ | $12.39^{+3.74}_{-2.89}$
HSC J093501+005415 | 4.46 | $3.12^{+1.07}_{-0.61}$ | $2.73^{+0.36}_{-0.26}$ | $3.16^{+1.52}_{-1.38}$
HSC J093523+023222 | 3.43 | $3.95^{+0.78}_{-0.56}$ | $14.71^{+1.13}_{-0.92}$ | $5.41^{+3.50}_{-3.50}$
555 ;

##  5 Results

###  5.1 Centroid offsets

We calculated the spatial separation from the cluster’s X-ray centroid to the
BCG’s position and report it as the BCG-and-X-ray centroid offset,
$D_{\mathrm{XC}}$. In addition, we estimate the distance between the cluster’s
X-ray peak to its BCG $D_{\mathrm{XP}}$. Figure 2 demonstrates the
distributions of $D_{\rm XC}$ and $D_{\rm XP}$ for 43 clusters in our sample.
The median values of these offsets in the unit of kpc and $R_{500}$ are shown
in Table 4.

Table 4: Median values of centroid offset and peak offset

$D_{\mathrm{XC}}$ | $D_{\mathrm{XC}}$ | $D_{\mathrm{XP}}$ | $D_{\mathrm{XP}}$ | Fractionaa$a$Fraction of relaxed clusters based on the BCG-X-ray peak offset (Sect. 5.2).
---|---|---|---|---
(kpc) | ($R_{\mathrm{500}}$) | (kpc) | ($R_{\mathrm{500}}$) |
100 | 0.10 | 152 | 0.14 | 2(¡16)%
666

Next, we compare the projected distance between 2D weak-lensing mass and three
kinds of cluster centers, namely X-ray centroid, galaxy map peak, and the
CAMIRA coordinates for 23 clusters. We estimated the statistical errors of the
probability distributions by the bootstrap resampling method. As shown in Fig.
3, all the histograms are well described by double Gaussian distributions:

$p(r)=f_{\rm cen}p_{1}(r)+(1-f_{\rm cen})p_{2}(r),$ (9)

where $p_{i}(r)=(r/\sigma_{i}^{2})\exp\left[-r^{2}/2\sigma_{i}^{2}\right]$ and
$f_{\rm cen}$ is the fraction of the central component (Oguri et al., 2010,
2018). The first standard deviation, $\sigma_{1}$, for the X-ray centroid is
smaller than those of the CAMIRA and galaxy ones, suggesting that the X-ray
centroids are closer to the WL mass centers. Although the values of
$\sigma_{1}$ for the CAMIRA centers are the largest and the histogram is
slightly skewed from the double Gaussian distributions, the amplitude of the
CAMIRA centers at $r\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}60$ kpc
is comparable to that of the X-ray centers. Although the value of $\sigma_{1}$
for the galaxy peak is the second smallest among the first components of the
three centers, the amplitude at $r\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}100$ kpc is $\sim 60-70$ % of the
others. The second standard deviation, $\sigma_{2}$, of the galaxy peak is the
smallest among the three centers.

###  5.2 Morphological classification

In this subsection, we use three different methods to classify the morphology
of galaxy clusters: i) BCG-X-ray offset, ii) concentration parameter, and iii)
galaxy peak-finding method.

First, the BCG-X-ray offset: We divided the clusters into two groups following
the criteria used in Sanderson et al. (2009), namely, “relaxed” clusters with
a small peak offset ($D_{\rm XP}<0.02R_{500}$) and “disturbed” clusters with a
large offset ($D_{\rm XP}>0.02R_{500}$). There is only one cluster that
posseses a small offset of $D_{\mathrm{XP}}\leq 0.02R_{500}$, which
corresponds to a very small fraction of relaxed clusters $2.3\pm
2.2\,(^{+0.0}_{-2.3})$% or ¡ 5%. Here, the statistical and systematic errors
are calculated via the bootstrap resampling method and by varying the
smoothing scales between 2 and 4 pixels, respectively.

The accuracy of the X-ray peak position depends on the statistical quality of
the X-ray observations, which broadly varies among clusters. In fact, based on
the XMM-Newton observations of the CAMIRA clusters, Ota et al. (2020) reported
that the clusters with low photon statistics tend to show larger centroid and
peak offsets. We thus calculated the standard error of the peak offset $\delta
D_{\rm XP}$ based on a comparison of X-ray images of each cluster with
different smoothing scales ($\sigma=2,3,4$ pixels). As a result, $\delta
D_{\rm XP}$ ranges from 0% to 55 % (the mean is 5%), which is smaller than
that of the XMM-Newton study (Ota et al., 2020).

At high redshifts, however, $0.02R_{500}$ is comparable to the accuracy of the
attitude determination (Predehl et al., 2021). With regard to how much the
results would change based on the threshold, when we set the threshold value
to $0.05R_{500}$, the percentage of relaxed clusters was $9$%. This example
suggests that the systematic uncertainty of the relaxed fraction is not
negligible. Accordingly, we assigned the typical positional accuracy of 4″.7
(Brunner et al., 2022) to the systematic error of $D_{\rm XP}$. This increases
the number (or fraction) of relaxed clusters that fall in $D_{\mathrm{XP}}\leq
0.02R_{500}$ to 7 (or 16%). Consequently, we estimate the fraction of relaxed
clusters as $2(<16)$% from the BCG-X-ray offset (Table 4).

We should note that this modest fraction of relaxed clusters is potentially
impacted by other effects. First, the selection of the BCG itself faces
undeniable obstacles. The lack of redshift information and the use of
photometric redshifts with large errors could lead to a poor choice of BCG. In
some cases, the true BCG may not be detected due to being overbright for the
optical instruments.

Second, there is the concentration parameter: if we refer to the concentration
parameter of the X-ray surface brightness, $C_{{\rm SB},R_{500}}$, and apply
$C_{{\rm SB},R_{500}}>0.37$ to identify relaxed clusters (Lovisari et al.,
2017), the fraction of relaxed clusters is estimated as 39% for 33 optical
clusters with X-ray counterparts in the eFEDS catalog within $R_{500}$ from
the X-ray centroid and $\Delta z<0.02$. Here, $C_{{\rm SB},R_{500}}$ is
defined as $C_{{\rm SB},R_{500}}=S_{B}(<0.1R_{500})/S_{B}(<R_{500})$ (Maughan
et al., 2012) and we quoted the value measured by Ghirardini et al. (2022),
based on the X-ray morphological study of the eFEDS clusters. Because all ten
unmatched clusters are disturbed clusters, according to the BCG-X-ray offset
measurements, we consider the above estimate to be an upper limit on the
relaxed fraction.

Third, the galaxy peak-finding method: Since our optically selected sample
covers a wide range of morphologies, we expect a large percentage of merging
clusters. These complex systems could be classified using the peak-finding
method (Okabe et al., 2019). Thus, we also checked the fraction of merging
clusters by finding peaks of member-galaxy distribution (Ramos-Ceja et al.,
2022). The threshold corresponds to the peak height of the richness $N=15$ at
each redshift. In the eFEDS field, the fraction of clusters with single over
multiple peaks is 27/73% for the high-richness CAMIRA clusters, while the
fraction of single over multiple peaks is 83/17% for the eFEDS clusters.

To be conservative, we quote the results from three kinds of measurements and
estimate the relaxed fraction of our sample to be $2(<39)$%.

Figure 2: Distributions of centroid offset (upper panels) and peak offset
(lower panels) in units of kpc (left panels) and $R_{500}$ (right panels). In
each panel, the dashed line indicates the median. In the upper-left panel, the
solid and dotted curves show the best-fit double-Gaussian models for the
present sample and the positional offset between the CAMIRA and XXL clusters
(Oguri et al., 2018), respectively.

Figure 3: Histograms of the projected distance between 2D weak-lensing mass
center and X-ray centroid (left), CAMIRA center (middle), galaxy map peak
(right) for 23 CAMIRA clusters. The error bars indicate the statistical
uncertainties. The best-fit model (red) consisting of the first and second
Gaussian components is indicated by the cyan dashed, and green dotted curves,
respectively.

###  5.3 Scaling relations

To derive the temperature- and mass-observable relations of the high-richness
clusters, we fit the data to the power-law model (Eq. 10) via the Bayesian
regression method (Kelly, 2007):

$\log{y}=a+b\log{x}.$ (10)

The quantities $a$, $b$, and the intrinsic scatter are treated as free
parameters. According to the self-similar model and Hubble’s law,
$E(z)=[\Omega_{M}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$ describes the redshift
evolution of the scaling relation. The luminosity of the cluster in the
hydrostatic equilibrium follows $E(z)^{-1}L\propto T^{2}$ and
$E(z)^{-1}L\propto[E(z)M]^{4/3}$. In Fig. 4, we corrected the redshift
evolution by applying the self-similar model and plotted $E(z)^{-1}L$ against
$T$ or $E(z)M$ since no clear consensus has been reached on the evolution of
the scaling relations (Giodini et al., 2013). Table 5 lists the best-fit
parameters for the $N-T$ and $L-T$ relations of 43 clusters and the $N-M$ and
$L-M$ relations for 38 clusters with the weak-lensing measurements (Sect.
3.2). The correlation coefficient is 0.62 – 0.70 for the four relations.

As described in Akino et al. (2022), a multi-variate analysis is needed to
properly correct the selection bias and the dilution effect and incorporate
the weak-lensing mass calibration. Thus, to correct for the selection bias due
to the richness cut of $N>40$, we simultaneously fit two kinds of relations,
$N-T$ and $L-T$ or $N-M$ and $L-M$, by the hierarchical Bayesian regression
method (HiBRECS; Akino et al., 2022) (Fig. 4). Because the slopes of $N-T$ and
$N-M$ were not well constrained due to the large data scatter, we fixed them
at 1.05 and 0.70, respectively, that were deduced from the best-fit mass-
richness relation (Okabe et al., 2019). We also assumed the intrinsic scatter
of the weak-lensing mass to be $\ln{\sigma_{M}}=-1.54$ (Umetsu et al., 2020).
Table 5 summarizes the best-fit scaling relations obtained from three types of
fitting codes.

From Table 5, we find that the fitting results with the Kelly and 1D HiBRECS
codes agree well. The Kelly code gave a marginally shallower $N-T$ slope of
$0.61\pm 0.21,$ as compared to the expectation from $M\propto T^{3/2}$ and
$M\propto N^{1.4}$(Okabe et al., 2019). In the present analysis, the impact of
selection bias correction was small; the 1D and 2D HiBRECS analyses gave
consistent results within the error bars. Therefore, in Sect. 6, we quote the
above results based on the 2D HiBRECS code, which can properly handle the bias
correction and the mass calibration.

Figure 4: Scaling relations of the high-richness, optically selected clusters
(circles). In the upper panels, the simultaneous fit of $T-N$ and $T-L$
relations are shown. In the lower panels, $M-N$ and $M-L$ relations are shown.
In each panel, the best-fit power-law model and the $1\sigma$ uncertainty
around the mean relation are indicated with the solid and dashed lines,
respectively.  Table 5: Best-fit scaling relations of the optically selected
clusters obtained from three types of fitting codes

| Kelly | 1D HiBRECS | 2D HiBRECS
---|---|---|---
Relation | $a$ | $b$ | $\sigma$ | $a$ | $b$ | $\sigma$ | $a$ | $b$ | $\sigma$
$N-T$ | $1.44\pm 0.10$ | $0.61\pm 0.21$ | 0.095 | $1.19\pm 0.02$ | 1.05 (fix) | 0.24 | $1.07\pm 0.04$ | 1.05 (fix) | 0.30 (fix)
$E(z)^{-1}L-T$ | $1.38\pm 0.24$ | $1.87\pm 0.45$ | 0.28 | $1.48\pm 0.21$ | $1.70\pm 0.40$ | 0.63 | $1.23\pm 0.28$ | $2.08\pm 0.46$ | 0.63
$N-E(z)M$ | $1.43\pm 0.19$ | $0.45\pm 0.24$ | 0.10 | $1.22\pm 0.03$ | 0.70 (fix) | 0.25 | $1.29\pm 0.06$ | 0.70 (fix) | 0.16
$E(z)^{-1}L-E(z)M$ | $1.44\pm 0.32$ | $1.22\pm 0.43$ | 0.32 | $1.50\pm 0.29$ | $1.27\pm 0.25$ | 0.65 | $1.24\pm 0.25$ | $1.52\pm 0.34$ | 0.65
$N-T$ | – | – | – | – | – | – | $1.07\pm 0.03$ | 1.05 (fix) | 0.30 (fix)
$L-T$ | $1.45\pm 0.27$ | $1.94\pm 0.50$ | 0.31 | $1.57\pm 0.24$ | $1.72\pm 0.44$ | 0.68 | $1.41\pm 0.30$ | $1.97\pm 0.50$ | 0.70
$N-M$ | $1.36\pm 0.22$ | $0.61\pm 0.33$ | 0.094 | $1.31\pm 0.03$ | 0.70 (fix) | 0.11 | $1.24\pm 0.08$ | 0.70 (fix) | 0.27
$L-M$ | $1.56\pm 0.34$ | $1.33\pm 0.49$ | 0.33 | $1.62\pm 0.30$ | $1.24\pm 0.45$ | 0.72 | $1.39\pm 0.32$ | $1.58\pm 0.44$ | 0.64

777$T$, $L$, $M$ are in the units of keV, $10^{42}\leavevmode\nobreak\ {\rm
erg\,s^{-1}}$, $10^{14}M_{\odot}$, respectively. The best-fit values of $a$,
$b$ in Eq. 10, and the intrinsic scatter of $y$ are listed. The results with
and without $E(z)$-correction are shown in 1–4 and 5–8 rows, respectively. The
2D HiBRECS code takes into account the correction for the selection bias and
mass calibration, but the other two codes do not (Sects. 5.3 and 6.2).

##  6 Discussion

###  6.1 BCG-X-ray offset and cluster dynamical status

As mentioned in Sect. 5, only $2(<16)$% clusters out of our cluster sample
possess a small X-ray peak offset $D_{\mathrm{XC}}\leq 0.02R_{500}$. Compared
with the previous XMM-Newton measurements of 17 optical clusters with $N>20$,
$29\pm 11(\pm 13)$% (Ota et al., 2020), the two results agree within the
errors.

Rossetti et al. (2016) investigated the dynamical states of 132 galaxy
clusters of a Sunyaev-Zel’dovich (SZ) selected a sample at a median redshift
of $z=0.16$ to find that $52\pm 4$% of the sample has $D_{\mathrm{XP}}\leq$
$0.02R_{500}$. The SZ selection method tends to pick out dynamically-active
systems, which leads to a relatively lower percentage of small peak shift. On
the other hand, the relaxed ratio is higher in the X-ray selected sample, at
$\approx$ 74%, indicating that clusters collected in an X-ray flux-limited
survey are subject to the cool-core bias. This accounts for a large fraction
of X-ray bright, hence, dynamically relaxed clusters with a small peak shift.
In addition, Migkas et al. (2021) found that approximately 44% of the X-ray
flux-limited eeHIFLUGCS sample have a small peak shift of
$D_{XP}<0.02R_{500}$. Therefore, our sample’s relaxed fraction is considerably
small compared to the above SZ and X-ray samples. We note that Ghirardini et
al. (2022) reported that the eFEDS-selected cluster sample is not biased
toward cool-core clusters, but that it does contain a similar fraction of
cool-cores as SZ surveys.

As shown in the upper-left panel of Fig. 2, we assessed the distribution of
the centroid offset by the double Gaussian model (Eq. 9) since there is a tail
toward large $D_{\rm XC}$. The fitting yields $\sigma_{1}=28\pm 2$ kpc,
$\sigma_{2}=95\pm 6$ kpc, and $f_{\rm cen}=0.46\pm 0.04$. In comparison with
the positional offset between the CAMIRA and XXL clusters (Oguri et al.,
2018), the present sample has a marginally smaller fraction of well-centered
clusters and a lower tail component.

Pasini et al. (2022) showed that BCGs with radio-loud active galactic nucleus
(AGN) are more likely to lie close to the cluster center than radio-quiet BCGs
and that the relations between the AGN and the intracluster medium (ICM) hold
regardless of the dynamical state of the cluster (see also Pasini et al.,
2020).

###  6.2 Scaling relations

In what follows, we discuss the $L-T$ and $L-M$ relations based on a
comparison with previous observations and theoretical models. The $L-T$
relation shows a large intrinsic scatter of $\sigma_{L|T}=0.63,$ as noted by
the previous X-ray studies (e.g., Ota et al., 2006; Pratt et al., 2009). The
observed $L-T$ slope of $2.08\pm 0.46$ agrees with $2.2\pm 0.6\,(\pm 0.2)$
derived from the XMM-Newton observations of the CAMIRA clusters (Ota et al.,
2020). In contrast, a steeper slope of $\sim 3$ has been reported by many
X-ray observations in the past (Giodini et al., 2013). From the analysis of
265 eFEDS clusters, Bahar et al. (2022) obtained the best-fit $L_{\rm bol}-T$
slope of $3.01^{+0.13}_{-0.12}$, suggesting a strong deviation from the self-
similar model, $L\propto T^{2}$. Because of the small fraction of relaxed
clusters, we consider the present measurement is less affected by the cool-
core emission.

The scatter of $L-M$ relation is comparably large, $\sigma_{L|M}=0.65$, which
confirms previous reports (e.g., Pratt et al., 2009). The $L-M$ slope of
$1.52\pm 0.34$ is consistent with $1.51\pm 0.09$ derived for 232 clusters at
$z=0.05-1.46$ based on a compilation of 14 published X-ray data sets (Reichert
et al., 2011). The best-fit $L-M$ relation of the present optical sample also
agrees with the result on 25 shear-selected clusters in the eFEDS field
(Ramos-Ceja et al., 2022) and the luminosity-mass-and-redshift relation of the
eFEDS cluster sample with the HSC weak-lensing mass calibration (Chiu et al.,
2022) within the measurement errors though the fitting functions are not
exactly the same.

Finally, we discuss the interpretation of our results by referring to the
baseline scaling relations. The self-similar model (Kaiser, 1986) predicts
simple relations between X-ray properties of ICM and mass in the absence of
baryonic physics such as AGN feedback and radiative cooling. Moreover, these
baseline relations were derived without considering the fact that less massive
clusters tend to be more concentrated and have higher characteristic densities
in the hierarchical structure formation in a CDM universe (Navarro et al.,
1997). On the other hand, Fujita & Aung (2019) constructed the new baseline
luminosity-temperature and mass relations by considering the mass-
concentration relationship. This is again the case when additional physics,
such as feedback and radiative cooling, do not work. They showed that the
baseline relations should be shallower than the conventional self-similar
model and follow $L\propto T^{1.6-1.8}$ and $L\propto M^{1.1-1.2}$. They also
suggested that the $L-T$ relation of high-mass clusters should be close to the
baseline model because the feedback from stars and AGN is less effective.

To carry out a comparison with their model, we derived the scaling relations
without $E(z)$-correction in the same manner as in Fujita & Aung (2019). From
rows 5-8 of Table 5, the fitted slopes of $L-T$ and $L-M$ relations agree with
the baseline models within the $1\sigma$ errors. Here, the baseline relations
show little difference between merging and non-merging clusters (Fig. 3 of
Fujita & Aung, 2019), while disturbed clusters dominate our sample. The nature
of the fundamental plane can explain the above trend because the $L-T$
relation is close to the edge-on view of the fundamental plane and the
clusters do not substantially deviate from the thin plane, even during a
merging process (Fujita et al., 2018). Therefore, the small $L-T$ slope
observed in the high-richness massive clusters agrees with the prediction of
the revised baseline model by Fujita & Aung (2019). We plan to extend the
analysis using the eROSITA and HSC surveys to test the shallow scaling
relations.

##  7 Summary

Based on a joint analysis of the eROSITA/eFEDS and Subaru/HSC surveys, we
studied the X-ray properties of 43 optically selected clusters with a high
richness of $>40$ at $0.16<z<0.89$. Our major findings are as follows:

  1. 1.

We studied the cluster dynamical status by the X-ray-BCG offset and the gas
concentration parameter and measured the morphology of member-galaxy
distributions by the peak-finding method. As a result, we estimated the
fraction of relaxed clusters to be $2(<39)$%, which is smaller than that of
the X-ray-selected cluster samples.

  2. 2.

We performed the X-ray spectral analysis and weak-lensing mass measurement,
deriving the scaling relations using the hierarchical Bayesian regression
method. The luminosity-temperature relation is shallow; the slope is
consistent with the predictions of the self-similar model and the baseline
model incorporating the mass-concentration relation. The luminosity-mass
relation also agrees with the two theoretical models cited in this work as
well as what has been observed for the shear-selected clusters in the eFEDS
field within the measurement errors.

Our joint eROSITA and HSC study showed that the average X-ray properties of
high-richness clusters are likely to be different from those found in the
X-ray cluster samples. To improve the sample size, we plan to incorporate more
than 900 objects with $15<N<40$ (Table 2) and extend the analysis to the
eROSITA all-sky survey (eRASS) data that overlap with the HSC footprint. These
works enable us to study the mass-observable relations and the redshift
evolution of the optical clusters. Furthermore, comparisons with the X-ray and
shear-selected samples will improve our knowledge about the selection effect
and cluster evolution.

###### Acknowledgements.

This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE). The development and
construction of the eROSITA X-ray instrument was led by MPE, with
contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU
Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. The eROSITA data shown here were processed
using the eSASS/NRTA software system developed by the German eROSITA
consortium. The Hyper Suprime-Cam (HSC) collaboration includes the
astronomical communities of Japan and Taiwan, and Princeton University. The
HSC 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.
Funding was contributed by the FIRST program from Japanese Cabinet Office, 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. We thank Y. Fujita for valuable
discussions and the anonymous referee for comments. This work was supported in
part by the Fund for the Promotion of Joint International Research, JSPS
KAKENHI Grant Number 16KK0101, 20K04027(NO), 20H05856, 20H00181, 19KK0076,
22H01260 (MO).

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