Publ. Astron. Soc. Japan (2018) 00(0), 1–23
doi: 10.1093/pasj/xxx000

Weak lensing clusters from HSC survey
first-year data: Mitigating the dilution effect offoreground and cluster member galaxies
Takashi H AMANA1,2 , Masato S HIRASAKI1 and Yen-Ting L IN3
1

National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, JapanThe Graduate University for Advanced Studies, SOKENDAI, Mitaka, Tokyo 181-8588, Japan
3
Academia Sinica Institute of Astronomy and Astrophysics, P.O. Box 23–141, Taipei 10617,
Taiwan2

Received 2020 July 1; Accepted 2020 July 1

Abstract
We present a weak lensing cluster search using Hyper Suprime-Cam Subaru Strategic
Program (HSC survey) first-year data. We pay special attention to the dilution effect of cluster
member and foreground galaxies on weak lensing signals from clusters of galaxies; we adopt
the globally normalized weak lensing estimator which is least affected by cluster member galaxies, and we select source galaxies by using photometric redshift information to mitigate theeffect of foreground galaxies. We produce six samples of source galaxies with different low-z
galaxy cuts, construct weak lensing mass maps for each of the source sample, and search forhigh peaks in the mass maps that cover an effective survey area of ∼120 deg2 . We combine
six catalogs of high peaks into a sample of cluster candidates which contains 124 high peaks
with signal-to-noise ratios greater than five. We cross-match the peak sample with the public
optical cluster catalog constructed from the same HSC survey data to identify cluster counterparts of the peaks. We find that 107 out of 124 peaks have matched clusters within 5 arcmin
from peak positions. Among them, we define a sub-sample of 64 secure clusters that we use
to examine dilution effects on our weak lensing cluster search. We find that source samples
with the low-z galaxy cuts mitigate the dilution effect on weak lensing signals of high-z clusters(z & 0.3), and thus combining multiple peak catalogs from different source samples improves
the efficiency of weak lensing cluster searches.
Key words: galaxies: clusters: general — cosmology: observations — dark matter — large-scale structure of universe

1 Introduction
Clusters of galaxies have been playing important roles in
the modern cosmology: Their abundance and evolution
have been used to place constraints on cosmological parameters (Allen et al. 2011), and their baryonic components (galaxies and hot intra-cluster gas) have been used
to study physical processes of hierarchical structure formation in the universe (Kravtsov & Borgani 2012). In thosec 2018. Astronomical Society of Japan.

studies, a large sample of clusters of galaxies is the fundamental data, which has been constructed by identifying
their tracers such as optical galaxy concentrations, X-ray
emissions, Sunyaev-Zel’dovich effect (SZE), and dark matter concentrations via the weak lensing technique (Prattet al. 2019). Since all cluster mass-observable relations
have scatters, sample completeness in terms of the cluster
mass, which is the principal quantity to link an observation
to a theory, varies from method to method. Weak lensing

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cluster finding is unique in that it uses the matter concentration as the tracer regardless of physical state of baryonic components, enabling one to locate under-luminous(in optical/X-ray/SZE) clusters.
Observationally, there are two conflicting difficulties inconstructing a sizable cluster sample with weak lensing ina practical time scale; a wide survey area to locate rare
objects, and a deep imaging to achieve a sufficient number
density of source galaxies. Thanks to the development ofwide-field optical cameras with dedicated wide field surveys, weak lensing cluster finding has made rapid progressin the last two decades, (see Table 1 of Miyazaki et al.
2018a, and references therein). Recently, Miyazaki et al.
(2018a) have conducted a weak lensing cluster search in a∼160 deg2 area of Hyper Suprime-Cam Subaru Strategic
Program (hereafter, HSC survey, Aihara et al. 2018b) firstyear data (Aihara et al. 2018a; Mandelbaum et al. 2018a),and have reported a detection of 65 peaks1 with signalto-noise ratio (SN ) greater than 4.7 in weak lensing mass
maps. They have cross-matched the peaks with opticalcluster catalogs and found that 63 out of 65 peaks had
optical counterparts, demonstrating that a wide field survey with a sufficient depth (for their case i = 24.5 mag)is indeed able to yield a sizable and high purity cluster
sample.
In the near future, the size of weak lensing cluster sample will become much larger as many more wide-area weaklensing-oriented surveys will come: The final survey area of
HSC survey is 1400 deg2 (more than eight times of the firstyear data), and Legacy Survey of Space and Time (LSST,
Ivezić et al. 2019) and Euclid survey (Laureijs et al. 2012;
Racca et al. 2018) will cover a large portion of the sky witha sufficient depth. It is thus worth improving methods of
weak lensing cluster finding by making best use of multiband dataset that on-going/future surveys take. This isexactly the purpose of this paper.
In this paper, we focus on the dilution effect that webriefly explain below: Weak lensing effect by clusters distorts shapes of background galaxies in a coherent manner. Since the shape distortions by weak lensing (lensing shear), which are generally smaller than intrinsic ellipticities of galaxies, can not be extracted from individual
galaxies, a lensing analysis necessarily involves averagingof shear estimators among a sample of galaxies to derivelensing shears and to suppress the noise from intrinsic el1

In this paper, we use the term ”peak” to mean a local maximum on a weaklensing mass map with its height exceeding a given threshold (see Section
3.3 for details). We adopt the signal-to-noise ratio (SN ) of mass maps (seeSection 3.2 for its definition) to define the threshold, because the noise in
the mass map originating from intrinsic galaxy shapes is well characterized
by the random Gaussian field (van Waerbeke 2000), and thus it gives arough estimate of a significance of a peak.

lipticities (called the shape noise). If a galaxy sample used
for a weak lensing analysis contains not only backgroundlensed galaxies but also foreground and/or cluster member galaxies which are not affected by cluster lensing andthus have no lensing signal, the latter acts as contaminants
in weak lensing analyses and dilutes the lensing signals
by clusters (see Broadhurst et al. 2005; Limousin et al.
2007; Hoekstra 2007; Medezinski et al. 2007; Umetsu &Broadhurst 2008; Okabe et al. 2010, for observational studies of the dilution effects in analyses of cluster lensing). In
most of recent lensing analyses of individual clusters withknown redshifts, source galaxies are selected using multiband galaxy photometry data so that the contamination of
foreground and cluster member galaxies is minimized (seeMedezinski et al. 2018, and references therein). However,in weak lensing cluster findings, redshifts of clusters are
unknown in advance, and thus a galaxy sample was commonly selected by a simple magnitude-cut on a single band
photometry (for example, Miyazaki et al. 2002; Hamana
et al. 2015). Such a galaxy sample inevitably contains
foreground/cluster member galaxies and suffers from thedilution effect.
Weak lensing cluster finding is based on peak heights
in mass maps. The detection threshold is set by the peakheight SN considering the trade-off between completenessand purity (lowering the threshold SN leads to a larger
number of cluster detections at the cost of a higher false
detection rate). However, the peak heights of cluster lensing are indeed affected by the dilution effect. Its direct
impact is the decline in numbers of cluster detections.
Another impact is on theoretical models of weak lensing
mass map peaks; incorporating its effect into theoreticalmodels requires a realistic modeling of the dilution effectwhich is most likely dependent on cluster mass, redshift,and galaxy selection criteria (for example, the detection
band, magnitude-cut, and size-cut). Therefore it is fundamentally important to understand actual dilution effects
on weak lensing mass maps on a case-by-case basis.
The purpose of this paper is two-fold: The first is todevelop a weak lensing cluster finding method that mitigates the dilution effects by incorporating photometric redshift information of galaxies. We apply it to the HSC survey first-year data in which both the weak lensing shapecatalog and photometric redshift data are publicly available (Mandelbaum et al. 2018a; Tanaka et al. 2018). We
present a sample of weak lensing peaks located by ourfinding method. We identify their counterpart clusters
by cross-matching with the optical cluster catalog (Oguri
et al. 2018). Using the derived weak lensing cluster sample, we examine the dilution effects on actual weak lensing
mass maps in an empirical manner, which is our secondPublications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

purpose.
The structure of this paper is as follows. In Section 2,
we briefly summarize the HSC survey first-year shear catalog and the photometric redshift data used in this study. In
Section 3, we describe the methods to generate a sample of
weak lensing peaks, including the selection of source galaxies, the method to reconstruct weak lensing mass maps,
and the peak finding algorithm. In Section 4, we crossmatch the weak lensing peaks with a sample of opticalclusters to identify their cluster counterparts. Then we
examine fundamental properties of weak lensing clusters
detected by our method. In Section 5, we examine the dilution effects of foreground and cluster member galaxies on
our weak lensing peaks in an empirical manner using actual source galaxy samples and empirical models. Finally,
we summarize and discuss our results in Section 6. In
Appendix 1, we present results of cross-matching of oursample of weak lensing peaks with selected catalogs ofknown clusters. In Appendix 2, we describe systems ofneighboring peaks in our peak sample. In Appendix 3, we
present results of the cluster mass estimate of the weaklensing peak sample based on a model fitting to weak lensing shear profiles. In Appendix 4, we compare the globallynormalized SN estimator, which is adopted in this study,
with the locally normalized SN estimator adopted in someprevious studies (for example, Hamana et al. 2015).
Throughout this paper, unless otherwise stated, we
adopt the cosmological model with the cold dark matter(CDM) density Ωcdm = 0.233, the baryon density Ωb =0.046, the matter density Ωm = Ωcdm + Ωb = 0.279, thecosmological constant ΩΛ = 0.721, the spectral index ns =0.97, the normalization of the matter fluctuation σ8 = 0.82,
and the Hubble parameter h = 0.7, which are the bestfit cosmological parameters in the Wilkinson Microwave
Anisotropy Probe (WMAP) 9-year results (Hinshaw et al.
2013).

2 HSC survey data
In this section, we briefly describe those aspects of the
HSC survey first year products that are directly relevantto this study, see the following references for full details:
Aihara et al. (2018b) for an overview of the HSC survey and survey design, Aihara et al. (2018a) for the firstpublic data release, Miyazaki et al. (2018b); Komiyama
et al. (2018); Kawanomoto et al. (2018); Furusawa et al.
(2018) for the performance of the HSC instrument itself,
Bosch et al. (2018) for the optical imaging data processing pipeline used for the first-year data, Mandelbaum et al.
(2018a) for the first-year shape catalog, Mandelbaum et al.
(2018b) for the calibration of galaxy shape measurements

3with image simulations, Aihara et al. (2019) for the public data release of the first-year shape catalog, and Tanaka
et al. (2018) for photometric redshifts derived for the firstyear data.

2.1 HSC first-year shape catalog
We use the HSC first-year shape catalog (Mandelbaum
et al. 2018a), in which the shapes of galaxies are estimated on the i-band coadded image adopting the reGaussianization PSF correction method (Hirata & Seljak
2003). Only galaxies that pass given selection criteria are
included in the catalog. Among others, the four major
criteria, which are relevant to the following analyses, forgalaxies to be selected are,(1) full-color and full-depth cut: the object should be located in regions reaching approximately full survey
depth in all five (grizy) broad bands,(2) magnitude cut: the i-band cmodel magnitude (corrected
for extinction) should be brighter than 24.5 AB mag,(3) resolution cut:
the galaxy size normalized by
the PSF size, which varies from position to position on coadded images depending on observational condition, defined by the re-Gaussianization
method, should be larger than a given threshold of
ishape hsm regauss resolution ≥ 0.3,
(4) bright object mask cut: the object should not be locatedwithin the bright object masks.
See Table 4 of Mandelbaum et al. (2018a) for the full description of the selection criteria.
The HSC shape catalog contains all the basic parameters needed to perform weak lensing analyses in this study.
The following five sets of parameters for each galaxy are
directly relevant to this study (see Mandelbaum et al.
2018a, for a detail description of each item); (1) the twocomponent distortion, e = (e1 , e2 ), which represents theshape of each galaxy image, (2) shape weight, w, (3) intrinsic shape dispersion per component, erms , (4) multiplicative bias, m, and (5) additive bias, (c1 , c2 ).

2.2 Photometric redshifts
Using the HSC five-band photometry, photometric redshift(hereafter photo-z) was estimated with six independentcodes, described in detail in Tanaka et al. (2018). In thisstudy, we adopt Ephor AB photo-z data which were derived from the PSF-matched aperture photometry (called
the afterburner photometry) using a neural network code,
Ephor2 . The data-set contains not only the point estimate2

https://hsc-release.mtk.nao.ac.jp/doc/index.php/photometric-redshifts/.

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take zmax = 3. Since we do not know redshifts of clusters
to be located in mass maps in advance, we take multiple
choices of zmin ; to be specific, we take zmin = 0, 0.2, 0.3,
0.4, 0.5, and 0.6.
The summation of P (z) over selected galaxies gives areasonably reliable estimate of redshift distribution of thesource sample3 . Taking the lensing weight (wi ) into account, we havens (z) =X

wi Pi (z).

(2)

iThe effective redshift distributions derived by this method
are shown in Fig. 1 in comparison with the full galaxy
sample. It is seen in the Figure that the P-cut method
works well to suppress the probability that source samples
include galaxies being at outside the given redshift ranges.
The mean source galaxy number densities for each sample
are summarized in Table 1.
Fig. 1. Bottom panel: Estimates of redshift distribution of the source samples computed by summing up the redshift probability distribution, P (z),over selected source galaxies [see equation (2)]. The normalization is takenRso that dzns (z) = 1 for the “no-cut” case (i.e. the full galaxy sample)
shown in black histogram. Top panel: Ratio of the redshift distribution for asource sample to that of “no-cut” case.

but also the probability distribution function of the redshift for each galaxy, that we use to select source galaxies(see Section 3.1).

3 Weak lensing mass maps and high SN
peaks
In this section, we describe our procedure for constructing
a sample of high SN peaks located in weak lensing mass
maps.

3.1 Source galaxy selection
We use the photo-z information to select source galaxies
which are used in constructing weak lensing mass maps
(detailed in the next subsection). We adopt the P-cut
method proposed by Oguri (2014) that uses the full probability distribution function of redshift, denoted by P (z),for each galaxy estimated by the ephor method; we define
samples of source galaxies that satisfy
Pint ≡

Z zmax

P (z) dz > Pth ,

(1)

zmin

with the threshold integrated probability of Pth = 0.95.
Our main aim here is to mitigate the dilution effects offoreground and cluster member galaxies, and thus a choiceof zmax is not crucial as long as it does not so much reduce
the number density of source galaxies. In this study, we

3.2 Weak lensing mass reconstruction
The weak lensing mass map which is the smoothed lensing
convergence field (κ) is evaluated from the tangential shear
data by (Schneider 1996)
K(θ) =

Z

d2 φ γt (φ : θ)Q(|φ|),(3)

where γt (φ : θ) is the tangential component of the shear at
position φ relative to the point θ, and Q is the filter function for which we adopt the truncated Gaussian function(for κ field) (Hamana et al. 2012),
Q(θ) =θ2
1
1− 1+ 2
2
πθ
θG





θ2
exp − 2
θG



,

(4)

for θ < θo and Q = 0 elsewhere. The filter parameters
should be chosen so that signals (high peaks in weaklensing mass maps) from expected target clusters (i.e.
Mvir > 1014 h−1 M⊙ at 0.1 < z < 0.6) become largest (seeHamana et al. 2004). We take θG = 1.5 arcmin and θo = 15
arcmin.
In our actual computation, K is evaluated on regulargrid points with a grid spacing of 0.15 arcmin. Since galaxy
positions are given in the sky coordinates, we use the tangent plane projection to define the grid. On and aroundregions where no source galaxy is available due to imaging
data being affected by bright stars or large nearby galaxies, K may not be accurately evaluated. We define “dataregion”, “masked-region” and “edge-region” by using the
distribution of source galaxies as follows: First, for eachgrid point, we check if there is a galaxy within 0.75 arcmin (about three times the mean galaxy separation) from3

Notice that the stacking photo-z P (z) is not a mathematically sound way
to infer the true redshift distribution (see Section 5.2 of Hikage et al. 2019).

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Table 1. Summary of source galaxy samples: Total area of data-region (see Section 3.2 for its definition), The effective number density
2of source galaxies defined by equation (1) of Heymans et al. (2012) (n̄g ), the averaged shape noise (hσshapei1/2 ), and numbers ofpeaks with SN ≥ 5. The last column lists the numbers of merged peaks with zopt = zmin (i.e., a peak’s SNmax comes from that source
sample) with numbers in the parentheses being those not existing in zmin = 0 sample with SN (zmin = 0) ≥ 5.

zmin
0.0
0.2
0.3
0.4
0.5
0.6

Area
[deg−2 ]120.01119.51118.90118.08117.50116.63n̄g[arcmin−2 ]
19.3
17.2
15.2
13.6
12.7
11.4

2
hσshapei1/2

0.01580.01670.01790.01900.01980.0209

Table 2. The effective survey area of each field. This is for thesource sample with zmin = 0. Total areas of other source sample
are summarized in Table 1.
a2

Field nameData-region area [deg ]
XMM
26.30
GAMA09H
28.52
WIDE12H
11.45
GAMA15H
27.50
HECTOMAP
9.48
VVDS
16.77
total
120.01aArea after removing regions affected by bright objects(masked-region) and edge-region in unit of degree2 . See
Section 3.2 for the definitions of those regions.

Npeak
SN ≥ 5
68
71
70
75
73
69

Npeak [merged] at zopt
SNmax > 5
24 (-)
14 (3)
18 (9)
22 (13)15 (8)
31 (23)over galaxies;
K(θ) =

1 X
γ̂t,i Q(|φi |),
n̄g

(5)

i

where the summation is taken over galaxies within θo froma grid point at θ, γ̂t,i is an estimate of tangential shear ofi-th galaxy at the angular position φi from the grid point,and n̄g is the mean galaxy number density (see Section 5.1
and Appendix 4 for discussion on our choice of the global
normalization, and see also Schmidt & Rozo 2011 for a
related study). The noise on mass maps coming from intrinsic shapes of galaxies is evaluated on each grid point(Schneider 1996),
1 X 2 2γ̂i Q (|φi |).
2n̄2g2σshape
(θ) =

(6)

i

the grid point. If there is no galaxy, then the grid point
is flagged as “no-galaxy”. After performing the procedure
for all the grid points, all the “no-galaxy” grid points plus
all the grid points within 0.75 arcmin from all the “nogalaxy” grid points are defined as the “masked-region”.
All the masked-regions are excluded from our weak lensing analysis. All the grid points located within 1.5 arcmin(we take this value by setting it equal to θG ) from any ofmasked-region grid points are defined as the “edge-region”.
All the rest of grid points are defined as the “data-region”.
Since the sky distribution of galaxies differs among different source samples, we carry out this procedure for every
source sample. The total survey areas (data-region) of eachsource sample are summarized in Table 1, and areas of six
fields for zmin = 0 sample are summarized in Table 2. The
difference in the total areas among different source samples
is not large but 3 percent at largest. The total areas of theedge-region are ∼ 30 degree2 , accounting for ∼ 20 percentof the data- plus edge-region.
On grid points, K is evaluated using equation (3), but
the integral in that equation is replaced with a summation

We define the signal-to-noise ratio (SN ) of weak lensing
mass map by
SN (θ) =K(θ),
2hσshape
i1/2

(7)

2
where hσshape
i is the mean value over all the grids in thedata-region.
Taking the lensing weight, which we normalized so thatthe total weight equals the total number of galaxies (i.e.,P
w = Ng ), and measurement biases into account, equai ition (5) is modified to (Mandelbaum et al. 2018a),1
K(θ) =n̄gPiwi (et,i /2R − ĉt )Q(|φi |),1 + m̂

(8)

where et is the tangential component of distortion takenfrom the HSC shape catalog. Sample averaged multiplicative bias, responsibity factor and additive bias are given as
follows,

P

wm
i i i
,
m̂ = Pw
i i
R = 1−

P

wi e2
w
i i

P rms,i ,

i

(9)
(10)

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and

P
wc
i i t,iĉt = P
,
i

(11)

wi

where ct,i is the tangential component the additive bias foreach galaxy. Similarly, the expression for the shape noise,equation (6), is modified to,2
σshape
(θ) =

1
2n̄2g

P

i

wi2 (et,i /2R − ĉt )2 Q2 (|φi |).
(1 + m̂)2(12)

3.3 Peak finding and merging multiple peak catalogs
We first apply the weak lensing mass reconstruction toeach sample of source galaxies. We define a peak in thegenerated mass maps as the grid point with SN value being
higher than all the surrounding eight grid points. We first
select peaks with SN ≥ 4 located in the data-region. If√
there is a pair of peaks with separation smaller than 2 ×θG ≃ 2.1 arcmin, the lower SN peak of the pair is discarded
to avoid multiple peaks from a single cluster (due to, forexample, substructures of clusters).
The numbers of peaks with SN ≥ 5 for six source samples are summarized in Table 1. Note that only peaks located in the data-region are included in the peak catalogs.
In the same Table, the mean shape noise values measured
from each sample are summarized, which scale with the2galaxy number density approximately as hσshapei ∝ n̄−1g
as expected (Schneider 1996). It should be noticed that
although the shape noise becomes larger for higher zmin
samples, the number of peak detection does not always
decrease. This may indicate that our source sample selection with a low-z cut indeed mitigates the dilution effects,
that we will go into detail in Section 5.
We combine the six catalogs of high peaks (SN ≥ 4)from different source samples by matching peak positions
to a tolerance of 2 × θG = 3 arcmin. Most of peaks havemultiple matches. Matched peaks from different source
samples are merged and are considered as peaks from thesame cluster, and the highest SN among matched peaks
is taken as its peak SN that we denote SNmax and we define its source sample’s zmin as zopt . There are 124 merged
peaks with SNmax ≥ 5, which we take as our primary sample of cluster candidates. In Table 3, basic information of
those 124 merged peaks are summarized.
In the last column of Table 1, we present numbers ofmerged peaks for each zopt = zmin with numbers in theparentheses showing those that do not exist in the zmin = 0
sample. We see that zopt is distributed rather broadly witha noticeable number at the highest zmin sample. It is foundthat 56 out of 124 merged peaks have SN (zmin = 0) < 5(to be specific, SN s of those peaks measured in mass maps
from zmin = 0 source sample are smaller than 5). This may

be an indication that the dilution effects indeed have nonnegligible influence on peak SN s in mass maps of zmin = 0.

4 Cross-matching with CAMIRA-HSC
clusters
We cross-match our merged peak catalog with the
CAMIRA (Cluster-finding Algorithm based on Multi-band
Identification of Red-sequence gAlaxies, Oguri 2014) HSC
cluster sample to identify clusters of galaxies from which
weak lensing peak signals originate. CAMIRA-HSC cluster sample is based on the same HSC S16A data set (Oguri
et al. 2018) used in our study, and thus covers our survey
fields uniformly except for regions affected by blight objects. We take this optically-selected cluster catalog as ourprimary reference sample, because it covers a sufficiently
wide redshift range (0.1 < z < 1.1) and cluster mass range(the richness Nmem > 15, where richness is defined as theeffective number of member galaxies above stellar mass
greater than 1010.2 M⊙ ). For each cluster, the sky coordinates and cluster redshifts based on the red sequence ofcluster member galaxies are estimated (see details of cluster finding algorithm and definitions of those quantities,
Oguri 2014; Oguri et al. 2018), that we use in the following analysis. See Appendix 1 for results of cross-matchingwith other selected cluster samples.
We cross-match our merged peak catalog withCAMIRA-HSC clusters4 with their positions to a tolerance of 5 arcmin. We summarize the results in Table 1, in
which the angular separation between a peak position anda matched CAMIRA-HSC cluster position is given (θsep ).
Since the smoothing scale of weak lensing mass map isθG = 1.5 arcmin, the tolerance radius could be large enoughto identify clusters of galaxies from which the weak lensing peaks originate. However, 17 out of 124 peaks have
no CAMIRA-HSC cluster counterpart (see Appendix 1.1
for some details of those peaks). Among the rest of 107
peaks, 25 peaks have multiple matches (mostly matchingwith two CAMIRA-HSC clusters, but 3 out of 25 peaks
have three matches). There are some possible reasons for
those systems: Some of such peaks could be due to physically interacting nearby cluster systems, but others could
be generated not from a single system but from a line-ofsight projection of multiple clusters (Hamana et al. 2004).
In this paper, we are not going into details of such multiplematch peaks.
Among 82 peaks matching with a single CAMIRA-HSC
4

There are some different CAMIRA-HSC catalogs based on different HSC
data sets. We use the HSC wide cluster catalog based on HSC S16A datawith updated star mask called ’Arcturus’ (Mandelbaum et al. 2018a). The
catalog is available from https://www.slac.stanford.edu/˜oguri/cluster/.

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Fig. 2. Black histogram shows the redshift distribution of the weak lensing
secure clusters (64 out of 124 merged peaks), whereas the red hatchedhistogram shows the same clusters but having SN ≥ 5 in weak lensing
mass maps from the source sample of zmin = 0 (36 out of 68 peaks locatedin zmin = 0 mass maps with SN ≥ 5).

cluster, 64 peaks have CAMIRA-HSC cluster counterparts
within 2 arcmin from peak positions. Although it is possible that some of those peaks are affected by line-of-sightprojections of small clusters (below the richness thresholdof CAMIRA algorithm), it is highly likely that the major
lensing contribution comes from the matched CAMIRAHSC clusters. We have also visually inspected those systems with HSC riz-color image, and found good correlations between weak lensing mass over-densities and galaxy
concentrations for all the cases. We thus define those 64
peaks as the secured sample of WL clusters, with redshift(that we denote zcl ) taken from the matched CAMIRAHSC cluster, which we will use to investigate the dilution
effects in the following section. The redshift distribution of
those secured weak lensing clusters is shown in Figure 2. In
the same plot, we also show the distribution of those weaklensing secure clusters that have SN ≥ 5 in weak lensing
mass maps from the source sample of zmin = 0. Comparing
the two distributions, we see that a large part of clusters
at zcl > 0.4 have peak SN s below our threshold of SN = 5
in the mass maps of zmin = 0, and pass the threshold in
mass maps of zmin ≥ 0.2.
We derive the cluster masses of the weak lensing secureclusters by fitting the NFW model to measured weak lensing shear profiles based on the standard likelihood analysis (see Appendix 3 for details). Derived cluster masses
are plotted on the cluster mass–redshift plane in Figure 3,

7

Fig. 3. Distribution of weak lensing secure clusters on the M200c − zcl
plane. The cluster masses defined by the spherical over-density mass
M200c are derived by fitting the NFW model to measured weak lensing
shear profiles based on the standard likelihood analysis (see Appendix 3),
and filled squares and error bars show the peak and 68.3% confidence interval of the posterior distributions. Red (black) symbols are for clusters withthe peak SN ≥ 5 (< 5) in weak lensing mass maps from the source sampleof zmin = 0.

where red (black) symbols are for clusters with the peakSN ≥ 5(< 5) in weak lensing mass maps from the source
sample of zmin = 0. From this Figure, we find that clusters below the peak height threshold (SN = 5) in the mass
maps of zmin = 0 are mostly relatively lower mass clusters
at zcl & 0.4. This is a natural result of the following two
facts that (1) for a fixed cluster redshift, the peak height ishigher for more massive clusters (Hamana et al. 2004), and(2) the dilution effect of foreground galaxies is stronger for
higher redshift clusters and for lower zmin galaxy samples(see the next section).

5 Dilution effects on weak lensing peaks
from clusters
The dilution effects on weak lensing high peaks originatingfrom clusters are caused by foreground and cluster member galaxies (for observational studies of the dilution effects
in analyses of cluster lensing, see for example Broadhurst
et al. 2005; Okabe et al. 2010; Medezinski et al. 2018).
Let us first make a rough estimate of proportions of thosegalaxies in our source galaxy samples. We see from the estimated redshift distributions of source samples shown in
Figure 1 that a proportion of foreground to background8

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 4. Top-panels: Stacked galaxy number density profiles as a function of angular radius from peak positions. Samples of the weak lensing secure clusters(defined in Section 4) with cluster redshifts within a given redshift range (denoted in each panel) are used. Different colors are for different source galaxy
samples characterized by zmin (see Section 3.1). Bottom-panels: The same stacked galaxy number density profiles shown in the top-panels but normalized
by the mean number density of each source galaxy sample.

galaxies depends strongly on both cluster redshifts andsource samples, and it can be more than 20 percent for
high-z clusters in low-zmin source samples. We estimate
the proportion of cluster member galaxies by measuring
stacked galaxy number density profiles of sub-samples of
weak lensing secure clusters selected based on cluster redshifts. The measurement is done for every source galaxy
sample and the results are presented in Figure 4 for four
redshift ranges. We find that, at cluster central regions, aconsiderable number of cluster member galaxies are contained in source samples with zmin < zcl except for the caseof the lowest cluster redshift range. The excess mostly
disappears in source samples with zmin > zcl . However,
we note that the degree of the excess and its suppression
largely vary from cluster to cluster.
We have adopted two means to mitigate the dilution effects: One is to take the globally normalized SN estimator,
equation (7) with equations (5) and (6), and the other is
to combine multiple peak catalogs from weak lensing mass
maps of source samples with different zmin . In the following sub-sections, we will first describe the former, thenwe will examine the effectiveness of the latter using actual
source galaxy samples.
Another important point seen in Figure 4 is that the
deficiency of source galaxies in cluster central regions forthe lowest redshift cluster sample and for the other samples
with zmin > zcl . There are two possible causes of this: One
is the masking effect of bright cluster galaxies that screen

background galaxies behind them. The other is the lensing magnification effect that enlarges a sky area behindclusters resulting in a decrease in the local galaxy number density (for more details, see Bartelmann & Schneider
2001; and see Chiu et al. 2019 for a measurement of lensing
magnification effect in the HSC data). We are not going
into further details of those two effects because it is beyond
the scope of this paper, but we examine their influence on
the peak height using empirical models in Section 5.3.

5.1 The globally normalized SN estimator
Here, we explain how the globally normalized SN estimator defined by equation (7) can mitigate the dilution effectof cluster member galaxies. We examine actual advantageof this estimator over the locally normalized estimator in
Appendix 4.
Let us assume the following simple model of a galaxy
distribution which consists of three populations; lensed
background galaxies (bg), unlensed foreground galaxies(f g), and unlensed cluster member galaxies (cl), with number densities of nbg , nf g , and ncl (θ), respectively. Notethat we have assumed that only ncl (θ) has a non-uniform
sky distribution associated with clusters of galaxies. As isseen in Figure 4, ncl (θ) can be comparable to nbg + nf g atcluster central regions. However, since the cluster population is very rare in the sky, in what follows, we assume thatthe globally averaged ncl (θ) is much smaller than nbg +nf g ,Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

9and we take n̄g = nbg + nf g . Then the globally normalized
estimator, equation (5), can be formally written by
KG (θ) =1 X
γ̂t,i Qi
n̄g
i

1
=
n̄g
=

X

γ̂t,i Qi +X

γ̂t,i Qi +γ̂t,i Qi

i∈cl

i∈f g

i∈bg

X

X
1
γ̂t,i Qi ,nf g + nbg

!
(13)

i∈bg

where from the second to third line, we have used the factthat the foreground and cluster member galaxies have nolensing signal. Denoting the galaxy intrinsic ellipticity byeint and its shear converted one by ê = eint /2R, the estimator of the shape noise, equation (6), can be written, inthe same manner, by
2
σshape,G(θ) =1
2(nf g + nbg )2
×

X 2

i∈bg

X 2

êi Q2i +êi Q2i +X 2êi Q2i

i∈cl

i∈f g

!

, (14)

where we have ignored the contribution from lensing shear.
Taking the average over a survey field, we have,1
2(nf g + nbg )22
hσshape,G
i≃

×

*

X 2

êi Q2i

i∈bg

+

+

*

X 2

êi Q2i

i∈f g

+!

,

(15)

where we have again assumed that on global average thecontribution from the cluster member population is smalland have ignored it. Using those expressions, the globallynormalized SN defined by equation (7), can be written by
SNG (θ) =KG (θ)

2hσshape,G
i1/2

≃

P

√ P
2 bg γ̂t,i Qi

ê2 Q2 +bg i iPê2 Q2
fg i i1/2 .

(16)

Note that in the above expression, there is no contribution from cluster member population. Therefore, the globally normalized estimator is, to a good approximation, freefrom the dilution effect of the cluster member galaxies.

Fig. 5. Shown is dependence of any of hDls /Ds iz (blue long-dashed−1
lines), σshape(black dashed lines) or SNpeak (red solid lines) on zmin .
All the quantities (signified by Y (zmin )) are normalized by their values atzmin = 0. Different panels for different cluster redshifts, zcl , which are denoted in each panel.

Focusing on contributions from foreground and background galaxies, from equation (13), the peak signal canbe approximately written by
KG (θ) =X
nbg hγ̂t iz
1
γ̂t,i Qi ∝,nf g + nbg
nf g + nbg

where hγ̂t iz is the source redshift distribution weighted
mean tangential shear. Since the source redshift dependence of the tangential shear enters only through the distance ratio, Dls /Ds , we can re-write equation (17) by
KG (θ) ∝

D

Dls
Ds

Foreground galaxies have two effects on weak lensing peakSN s from clusters. One is to dilute the lensing signal, andthe other is to make the shape noise level on mass maps
smaller. Below we will first derive relevant expressions forthese effects, and evaluate the dilution effect of foregroundgalaxies using the actual redshift distributions of source
galaxies. Then, we will compare it with the real data measured using the secure weak lensing cluster sample.

E

=
z

R∞
zcl

dz ns (z)Dls (zcl , z)/Ds (z)R∞0dz ns (z),(18)

where ns (z) is the redshift distribution of source galaxies.
In the same manner, from equation (15) we have2
hσshape,Gi∝

5.2 Dilution effect of foreground galaxies

(17)

i∈bg

(nf g + nbg )hê2 ihê2 i,∝2(nf g + nbg )2
n̄g

(19)

where we have ignored a possible redshift dependence ofhê2 i. This is the well known scaling relation between the
shape noise and galaxy number density (Schneider 1996).
The weak lensing peak SN from clusters is related tothe source redshift weighted distance ratio and the shapenoise via
hDls /Ds iz.
(20)
SN ∝ 2hσshape,G i1/2
Since both the distance ratio and the shape noise depend

10

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 6. Shown is the SN (zmin ) of weak lensing secure cluster normalized
by its SN (zmin = 0). Weak lensing secure clusters are divided into four
sub-samples based on the cluster redshifts (denoted in panels). The horizontal axis is zmin of source galaxy samples. For each sub-sample andeach source galaxy sample, the mean and its 1-σ error among the clusters(the number of clusters in each sub-sample is given in each panel) are plotted. For comparison, the red lines show the expected SN ratios plotted in
Figure 5 (red lines) for a cluster at a central redshift in respective redshift
ranges.

on source galaxy samples, so does the peak SN from acluster. In our case, the source galaxy selection is characterized by zmin , and thus we evaluate the dependence of
those quantities on zmin using the redshift distributions of
our source samples defined by equation (2). The results areshown in Figure 5 for cluster redshifts of zcl = 0.15, 0.25,
0.35, and 0.45. Findings from that figure are as follows:
The shape noise, which is not dependent on zcl , monotonically increases with zmin as expected. The source redshiftweighted distance ratio increases with zmin . Since a fraction of foreground galaxies is larger for the higher redshift
clusters, the higher the cluster redshift, the larger the distance ratio. Those two effects compete; For clusters with
redshifts lower than 0.3, their weak lensing peak SN decreases with zmin . However, for higher redshift clusters,
SN stays almost constant or slightly increases with zmin .
We examine the actual dependence of peak SN s on zmin
using the weak lensing secure clusters. In doing so, we divide the secure clusters into four sub-samples based on thecluster redshift (to be specific, 0.1 < zcl < 0.2, 0.2 < zcl < 0.3,
0.3 < zcl < 0.4, and 0.4 < zcl < 0.5). For each sub-sample,

we evaluate the mean of SN (zmin )/SN (zmin = 0) and its
standard error among sample clusters. The results areshown in Figure 6. We find that the measured ratios ofSN (zmin )/SN (zmin = 0) is systematically larger than the
expectations shown by the red lines (which are same asones plotted in Figure 5), especially for lower-z clusters(zcl < 0.3). The reason of this is unclear; a possible cause is
the intrinsic alignment of galaxies: Because the major axisof galaxies surrounding a cluster tend to point towards thecluster center due to the intrinsic alignment effects (originating from, e.g., the gravitational tidal stretching, see fora review Joachimi et al. 2015, and references therein), it
reduces the peak SN value. Peak SN s of zmin = 0 maps
are likely affected by this effect, and consequently they
are likely biased low. If this is the case, it accounts forthe systematically larger SN (zmin )/SN (zmin = 0) foundin the measured results, though this argument is rather
phenomenological. Aside from this systematic difference,
the measured ratios are in reasonable agreement with expectations in their amplitudes and in its increasing trend
toward higher-z clusters. From the above findings, we conclude that combining multiple peak catalogs from source
samples with different zmin can mitigate the dilution effectof foreground galaxies, especially on high-z clusters.

5.3 Impact of the source galaxy deficiency on peakheights
Here we examine the impact on the peak SN from source
galaxy deficiency at cluster central regions seen in thestacked galaxy number density profiles shown in Figure
4. We note, however, that deficiency profiles vary greatly
from cluster to cluster as is shown in Figure 7.
We use the empirical models of dark matter halo andsimple model of the galaxy deficiency profiles, which we describe below. In the presence the source galaxy deficiency,
the theoretical expression for the lensing signal from clusters, equation (3), is modified toK(θ) =

Z

d2 φ [1 + fm (φ)]γt (φ : θ)Q(|φ|),(21)

where fm (φ) is the source galaxy deficiency profile, forwhich we adopt the following parametric function, 

fm (φ) = max a





φ
π
− tan−1 ( ) , −1 ,
2
φ0

(22)

where a and φ0 are the amplitude and scale parameters,
respectively. We fit the measured deficiency profiles shownin Figure 4 with this function and derive typical values of
those parameters that we take in the following analysis.
We consider three models of fm (φ) shown in the top-panelof Figure 8: The model with a = −0.25 mimics the stacked

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 7. Thin colored lines show azimuthally averaged galaxy number density
normalized by the mean number density as a function of the angular separation from peak positions. Each colored line is for an individual cluster.
Sub-samples of weak lensing secure clusters are shown. The upper/lower
panel is for the clusters at 0.1 < zcl < 0.2/0.3 < zcl < 0.4 with the source
galaxy sample of zmin = 0.3/0.4. The thick black lines show the parametricmodel, equation (22) but plotted is [1 + fm (θ)], for four sets of parameters
denoted in the upper panel.

deficiency profile of a case 0.3 < zcl < 0.4 with zmin > zcl ,and the model with a = −0.5 mimics ones of 0.1 < zcl < 0.2(see Figure 4), whereas the model with a = −0.75 represents extreme cases from individual clusters shown in
Figure 7. Since details of the models needed to computeequation (21) are described in Hamana et al. (2012), here
we summarize the main ingredients and relevant references:• Dark matter halos of clusters are modeled by the truncated NFW model (Baltz et al. 2009) with the massconcentration relation by Macciò et al. (2008) and Duffyet al. (2008).
• Redshift distributions of source galaxies of our source
samples are estimated by equation (2), which are dependent on zmin , and are presented in Figure 1.
Results are presented in Figure 8. Note, however, thatsince the deficiency model is a crude approximation andhalo mass dependence of the source deficiency is not taken
into account, the results should be considered as a roughestimate of the impact of source deficiency on a peak SN .
In the bottom-panels, ratios between expected peak SN s
with and without taking the source deficiency into accountas a function of halo mass are shown for three deficiency

11

Fig. 8. Top-panel: Three models of the source galaxy deficiency profile areshown. See equation (22) for the functional form of the model. Bottompanels: Shown is ratios between expected peak SN s of an NFW halo withand without taking the source galaxy deficiency into account as a function of
halo mass (the virial mass is taken here). The left-panel is for the casezhalo = 0.15 with the galaxy redshift distribution taken from the source
sample with zmin = 0.2, whereas the right-panel is for zhalo = 0.35 withzmin = 0.4. Different lines are for different deficiency models shown in the
top-panel.

models presented in the top-panel. The left-panel is for thecase zhalo = 0.15, whereas the right-panel is for zhalo = 0.35.
We find that in both the cases, the suppression of SN due
to the source deficiency is 2–5 percent for models with a =−0.25, 8–10 percent for a = −0.5 and 16–20 percent for theextreme case of a = −0.75. We thus conclude that a typicalimpact of source deficiency on a peak SN is a suppressionof a few to ∼ 10 percent. However it can be ∼ 20 percent
for individual cases. It is also seen in the Figure that fora given source deficiency model, the suppression decreases
with increasing halo mass. The reason for this is that a
relative contribution to a peak SN from galaxies within afixed aperture is smaller for more massive halos.

6 Summary and discussions
We have presented a weak lensing cluster search using HSC
first-year data. We generated six samples of source galaxies
with different zmin -cuts (we took zmin = 0, 0.2, 0.3, 0.4, 0.5
and 0.6), and made weak lensing mass maps for each source
sample from which we searched for high peaks. From eachsource sample, we detected a sample of 68–75 weak lensing
peaks with SN ≥ 5. We compiled the six peak samples
into a sample of merged peaks. We obtained a sample of12

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

124 weak lensing merged peaks with SNmax ≥ 5 which are
candidates of clusters of galaxies.
We cross-matched our peak sample with CAMIRA-HSC
clusters (Oguri et al. 2018) to identify cluster counterparts of the peaks. We found that 107 out of 124 merged
peaks have matched CAMIRA-HSC clusters within 5 arcmin from the peak positions. Among the 107 matchedpeaks, 25 peaks have multiple matches, which might begenerated by line-of-sight projections of multiple clusters.
Among the remaining 82 peaks matching with a single
CAMIRA-HSC cluster, 64 peaks have CAMIRA-HSC cluster counterparts within 2 arcmin from peak positions. We
confirmed by visual inspection of HSC images and foundthat, for all 64 peaks, there exist good correlations between weak lensing mass over-densities and galaxy concentrations. We thus defined those peaks as the sample
of weak lensing secure clusters, and used them to examine
the dilution effects on our weak lensing peak finding.
We have paid particular attention to the dilution effectof cluster member and foreground galaxies on weak lensing
peak SN s, and have adopted two means to mitigate its impact, namely the globally normalized estimator [equations(5), and (6)], and the source galaxy selection with different
zmin -cuts using the full probability distribution function ofgalaxy photo-zs.
We have demonstrated, using the simple model ofgalaxy populations introduced in Section 5.1, that the peak
SN defined by the globally normalized estimators is, to agood approximation, not affected by the dilution effect of
the cluster member galaxies. This is in marked contract
to the locally normalized SN which is indeed affected by
the cluster member galaxies as demonstrated in Appendix
4. We compared the peak heights of the globally normalized SNG with ones of the locally normalized SNL using
our weak lensing mass maps, and found that for the peaksamples with SNG ≥ 5, SNG s are, on average, about 10
percent larger than the corresponding SNL s.
In Section 5.2, we have examined the dilution effectof foreground galaxies and have demonstrated the ability of our source galaxy selection to mitigate it. We
used the probability distribution function of photo-z, andadopted P-cut method (Oguri 2014) to remove galaxies
at z < zmin which are foreground galaxies of clusters at
zcl > zmin . This galaxy selection has two competing influences on weak lensing peak SN s from clusters: One isto reduce the dilution effect of foreground galaxies, andthe other is to increase the shape noise level as the zmin cut reduces the number density of source galaxies. We
examined the expected impact on peak SN heights fromthose two factors using the estimated redshift distributionof the source samples, and found that for high/low-z clus-

ters (z & 0.3/z . 0.2), the former/latter is more effective
than the latter/former, leading to a gain/decline in peakSN s with increasing zmin .
We examined the actual dependence of peak SN s on
the source selection using the weak lensing secure clusters. We measured the ratios of SN (zmin )/SN (zmin = 0)
for four sub-samples of secure clusters divided based on
the cluster redshift (shown in Figure 6). We found thatthe measured results were in reasonable agreement withthe expectations in their amplitudes and in their increasing trend toward higher-z clusters, except for the systematic offset of about +5 to +10 percent which could be
due to the intrinsic alignment of cluster neighbor galaxies. From the above findings, along with the fact thatthe number of merged peak sample (124 for SNmax ≥ 5) isnearly twice of the numbers from individual source samples(68–75 for SN ≥ 5), we conclude that combining multiple
peak samples from source samples with different zmin indeed improve the efficiency of weak lensing cluster search,especially for high-z clusters.
We have also examined the effect of source galaxy deficiency on weak lensing peak heights. The source deficiency
was clearly observed in stacked galaxy number density profiles of secure clusters at cluster central regions for the cluster sample of 0.1 < zcl < 0.2 and for the other samples withzmin > zcl (Figure 4). This can be due to the masking effectof bright cluster galaxies and/or the lensing magnification
effect. Using a simple model consisting of spatial profilesof dark matter halo density and source deficiency, we makepredictions for the source deficiency effect on the peak SN .
We found that for realistic models of source deficiency, apeak SN is suppressed by a few to ∼ 10 percent.
Since we have focused on the dilution effect, there aresome important tasks/issues related to weak lensing cluster
search which have not been examined in this paper: The
three major matters among others are:
1. The purity of the sample of 124 weak lensing cluster
candidates: For each of 64 weak lensing secure clusters,
we have found a good correlation between weak lensing
mass over-density and galaxy concentration, and haveconcluded that those weak lensing signals have a physical relationship with the counterpart CAMIRA-HSC
cluster. The remaining 60 peaks fall into the following
three categories:(a) 18 cases: peaks matching with a single CAMIRAHSC clusters but their separations are larger than 2
arcmin. Physical connections between those weak lensing mass over-densities and clusters are not clear, which
are a subject of a future study.
(b) 25 cases: peaks having multiple CAMIRA-HSC clus-

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

ters within 5 arcmin. In 23 out of the 25 cases, matchedclusters of the same peak are separated in the redshift
direction by ∆z > 0.1. Thus those peaks are likely
affected by line-of-sight projections of physically unrelated clusters, though detail investigations of each peak
are required to reveal their real nature.
(c) 17 cases: Peaks have no matched CAMIRA-HSC
cluster, for which we searched for possible counterpart clusters in a known cluster database taken froma compilation by NASA/IPAC Extragalactic Database(NED5 ). The results are presented in Appendix 1.1. In
10 out of 17 peaks, possible counterpart clusters arefound (see Table 4). In Figure 9, we show HSC riz
composite images of the remaining 7 peaks (that haveno counterpart cluster found), in which good correlations between the weak lensing mass over-density andgalaxy concentration are seen in some of those systems.
Clearly, the above information is not enough to evaluate the purity of our sample; further followup studies
combining information from other wavelength data (forexample, X-ray and Sunyaev-Zel’dovich effect) are required.
2. Weak lensing mass estimate of our cluster candidates:
Although cluster mass derived from weak lensing analysis can add valuable information to our sample, an accurate determination of cluster redshift as well as carefully taking account of line-of-sight projections of uncorrelated objects are required to estimate weak lensing
mass accurately. We have derived weak lensing cluster masses only for weak lensing secure clusters whichhave good correlations between the weak lensing mass
peak and galaxy over-density (see Appendix 3). Since
the remaining weak lensing peaks have either multiple CAMIRA-HSC cluster counterparts or a less correlated/no CAMIRA-HSC cluster counterpart, further
detailed studies of individual systems are needed to derive their cluster masses, which we leave for a future
study.
3. Masking effect of bright cluster galaxies and lensing
magnification effect on weak lensing peak finding: As wediscussed in the above, we have seen an observational
indication of those effects as the deficiency of source
galaxies at cluster central regions, and have examined
their impact on the peak SN in Section 5.3. Since thoseeffects are unavoidable in weak lensing cluster search, afurther detail study of those effects is important for cosmological applications of weak lensing selected clusters.
It is, however, beyond the scope of this paper, and weleave it for a future study.
5

http://ned.ipac.caltech.edu/

13

Weak lensing mass maps contain a wealth of cosmological information beyond those obtained by analyses of thecosmic shear power spectrum or two-point correlation function (see, for example, Dietrich & Hartlap 2010; Yang et al.
2011; Petri et al. 2013; Shirasaki et al. 2017). However,in this study, we showed that if a source galaxy sample
is selected by, for example, a simple magnitude-cut, thedilution effects may alter SN s of high peaks in a nonnegligible amount, and thus may modify statistical properties of weak lensing mass maps. Therefore, when one
uses weak lensing mass maps for a cosmological application, the dilution and the source deficiency effects must be
taken into account. We note that the effects are dependent on the source sample that one takes, and thus should
be examined on a case-by-case basis. At the same time,developing source galaxy selection methods that can mitigate the dilution effects is another important subject inthat research field.

Acknowledgments
We would like to thank Masamune Oguri for useful comments
on an earlier manuscript, and Satoshi Miyazaki for useful discussions. We would like to thank Nick Kaiser for making thesoftware imcat publicly available, and ds9 developers for ds9
publicly available. We have heavily used those softwares in thisstudy. We would like to thank HSC data analysis software teamfor their effort to develop data processing software suite, and
HSC data archive team for their effort to build and to maintainthe HSC data archive system.
This work was supported in part by JSPS KAKENHI Grant
Number JP17K05457. MS is supported by JSPS Overseas
Research Fellowships.
Data analysis were in part carried out on PC cluster at Center for Computational Astrophysics, National
Astronomical Observatory of Japan.
Numerical computations were in part carried out on Cray XC30 and XC50 atCenter for Computational Astrophysics, National Astronomical
Observatory of Japan, and also on Cray XC40 at YITP in Kyoto
University.
The Hyper Suprime-Cam (HSC) collaboration includes theastronomical 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 Mathematicsof the Universe (Kavli IPMU), the University of Tokyo, theHigh Energy Accelerator Research Organization (KEK), theAcademia Sinica Institute for Astronomy and Astrophysics
in Taiwan (ASIAA), and Princeton University. Funding wascontributed by the FIRST program from Japanese Cabinet
Office, the Ministry of Education, Culture, Sports, Science andTechnology (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. This paper makes use ofsoftware developed for the Large Synoptic Survey Telescope.

14

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

We thank the LSST Project for making their code available as
free software at http://dm.lsst.org
The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy,
the University of Hawaii, the Pan-STARRS Project Office,the Max-Planck Society and its participating institutes, theMax Planck Institute for Astronomy, Heidelberg and the
Max Planck Institute for Extraterrestrial Physics, Garching,
The Johns Hopkins University, Durham University, theUniversity of Edinburgh, Queen’s University Belfast, theHarvard-Smithsonian Center for Astrophysics, the Las Cumbres
Observatory Global Telescope Network Incorporated, theNational Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space
Administration under Grant No. NNX08AR22G issued throughthe Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation under Grant No.
AST-1238877, the University of Maryland, and Eotvos Lorand
University (ELTE) and the Los Alamos National Laboratory.
Based in part on data collected at the Subaru Telescope andretrieved from the HSC data archive system, which is operated
by Subaru Telescope and Astronomy Data Center at National
Astronomical Observatory of Japan.

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84, 043529

Appendix 1 Cross-matching with selectedcluster catalogs
A.1.1 Cluster counterparts of weak lensing peaks
with no CAMIRA-HSC cluster counterpart
Among 124 weak lensing merged peaks, 17 peaks have no
CAMIRA-HSC cluster counterparts within a 5 arcmin radius from the peak positions. However, note that oneof them, HWL16a-121, is a known cluster (Abell 2457)
at z = 0.059 which is outside the redshift coverage of
CAMIRA algorithm (Oguri et al. 2018).
We search for cluster counterparts of those peaks
in a known cluster database taken from a compilation
by NASA/IPAC Extragalactic Database (NED). Clusters
matched within 5 arcmin radius from the peak positions
are summarized in Table 4. In 10 out of 17 peaks, possible
counterpart clusters are found. For the remaining 7 peaks,
we present HSC riz-band composite images in Figure 9[panels (a)–(g)]. In that Figure, we find apparent galaxy
concentrations near the weak lensing peaks of HWL16a001, 011, 096, 118, and 122. It follows from this that thosepeaks are not necessarily false signals, but undiscovered
counterpart clusters may exist.
In summary, combining results of cross-matching withCAMIRA-HSC cluster catalog and with known cluster
database, we have found possible counterpart clusters for
117 out of 124 weak lensing peaks. However, since ourmatching is based on a simple positional correlation, someof matches can be chance projections. Future followup
studies of individual peaks on a case-by-case basis are required to reveal physical connections between weak lensing
peaks and matched clusters.

15

A.1.2 Cross-matching with XXL clusters
Adami et al. (2018) have presented a sample of 365 clustersof galaxies detected in the XXL Survey, which is a widefield and deep X-ray imaging survey conducted with XMMNewton (Pierre et al. 2016). The XXL survey consists of
two survey fields, each covering ∼ 25 deg2 area, and its
north field (XXL-N) largely overlaps with our XMM field(see Figure 1 of Umetsu et al. 2020). Since the selection
function of XXL clusters with respect to the cluster mass
and redshift well covers that of our clusters (see, e.g., Fig 12
of Miyazaki et al. 2018a), the XXL cluster sample provides
good reference data to test the completeness of our weaklensing clusters.
Among 23 weak lensing merged peaks in XMM field(HWL16a-001–023), 14 peaks are located on the XXL survey footprints (see Figure 1 of Pacaud et al. 2016). 11 out
of 14 peaks have XXL cluster counterparts (see Table 3).
The peaks with no XXL cluster counterpart are HWL16a015, 018, and 019, for which brief descriptions are given
below, though future detail investigations of each peak arerequired to reveal their real nature:• HWL16a-015 matches with XLSSC 074 (Clerc et al.
2014) which is not contained in XXL 365 cluster sample (Adami et al. 2018). There are three known clusters
within 5 arcmin from the peak position; see Appendix
1.1 for details.
• HWL16a-018 has no CAMIRA-HSC cluster counterpart.
There are four known clusters within 5 arcmin from thepeak position; see Appendix 1.1 for details.
• HWL16a-019 has one CAMIRA-HSC cluster counterpart (CAMIRA-ID 388, zcl = 1.011), but the separation
between them is 5.0 arcmin. Therefore the physical connection between the weak lensing peak and CAMIRAHSC 388 is uncertain. There is one known cluster,
CFHTLS W1-2587 (zcl = 0.30, Durret et al. 2011), withthe angular separation of 1.2 arcmin.
We note that both HWL16a-021 and HWL16a-022
match with the same XXL cluster, XLSSC 151 at z = 0.189.
Also, both peaks match with the same CAMIRA-HSC cluster (CAMIRA-ID 355, the estimated redshift of z = 0.276).
In fact, those two are a close pair of peaks with their separation of ∼ 3 arcmin (see Figure 9 (i)), though they are
identified as two individual peaks under our peak identification criteria (described in section 3.3). It is seenin Figure 9 (i) that the X-ray cluster XLSSC 151 is at
the peak position of HWL16a-021, whereas CAMIRA-HSC
cluster 355 matches better with HWL16a-022. Considering
the difference in redshifts of those two clusters, it is likelythat the twin peaks of the weak lensing SN map arise froma chance line-of-sight projection of two physically sepa-

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Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0(a) HWL16a-001(b) HWL16a-011(c) HWL16a-061(d) HWL16a-096(e) HWL16a-118(f) HWL16a-122

Fig. 9. HSC riz-band composite images with a side length of 10 arcmin. The yellow contour shows the weak lensing SN , the contour lines start from SN = 2
with the interval of 1. The first 7 panels [(a)–(g)] show weak lensing peaks having no CAMIRA-HSC counterpart being matched within 5 arcmin radius fromthe peak position. The last 7 panels [(h)–(n)] show systems of neighboring peaks having a common CAMIRA-HSC counterpart. Positions of CAMIRA-HSC
cluster (based on HSC S16A data with updated star mask, Oguri et al. 2018) and XXL clusters (Adami et al. 2018) are marked with plus symbols.

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

17

XLSSC117CAMIRA-149

XLSSC111(g) HWL16a-123(h) HWL16a-007 (lower peak), and 008 (upper peak)

CAMIRA-870

CAMIRA-355

XLSSC151XLSSC152(i) HWL16a-021 (lower peak), and 022 (upper peak)(j) HWL16a-044 (lower peak), and 045 (upper peak)

CAMIRA-1161

CAMIRA-1155

CAMIRA-1103

CAMIRA-1157

CAMIRA-1105(k) HWL16a-062 (right-peak), and 063 (left peak)Fig. 9. (Continued)(l) HWL16a-068 (upper peak), and 069 (lower peak)18

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

CAMIRA-1628
CAMIRA-1344

CAMIRA-1630

CAMIRA-1339
CAMIRA-1341(m) HWL16a-085 (lower peak), and 086 (upper-left peak)(n) HWL16a-105 (lower peak), and 106 (upper peak)Fig. 9. (Continued)

rated clusters. If this is the case, HWL16a-022 is anotherweak lensing peak having no XXL cluster counterpart.

A.1.3 Cross-matching with weak lensing peaks in
Miyazaki et al. (2018a)
Miyazaki et al. (2018a, M18 hereafter) presented a sample
of weak lensing peaks detected in mass maps constructedfrom the HSC first-year shape catalog (Mandelbaum et al.
2018a) that we also used in this study. Although their
method of weak lensing mass map construction is very similar to that of this study, differences in source galaxy selection and criteria of peak identification result in different
peak samples. Their peak sample contains 65 peaks with
SN > 4.7. We cross-match their peaks with our extendedsample (peaks with SN ≥ 4 are included) by their peak
positions to a tolerance of θG = 1.5 arcmin. Among their
65 peaks, 51 peaks match with our final merged peaks(SN ≥ 5, see Table 3). The remaining 14 peaks fall into
the following five categories:
1. [M18 rank 51]: A corresponding peak exists in our final
merged sample (HWL16a-054), but their separation is
2.6 arcmin which exceeds the tolerance length.
2. [M18 rank 49, 55, and 60]: A matched peak exists in our
extended-sample with 5 > SNmax ≥ 4, but its SNmax is
below our threshold.
3. [M18 rank 7, 27, 31, 41 and 43]: A corresponding peakexists in our extended-sample with SNmax ≥ 4, but islocated in the edge-region.
4. [M18 rank 37 and 38]: No corresponding peak exists
in our extended-sample. Note that both the peaks arelocated in our edge-region.

5. [M18 rank 15, 45 and 46]: Those peaks are located in
our masked-regions where we have not performed weaklensing analysis.
In summary, among 65 peaks in M18 sample, all the 55
peaks located in our data-regions have counterpart peaks
in our extended-sample (including M18 rank 51–HWL16a054).

Appendix 2 Systems of neighboring weaklensing peaks
In weak lensing mass maps, there are systems of neighboring peaks; those are either two isolated clusters or one
cluster having a significant substructure or under a merging process. Distinguishing clusters’ dynamical states with
only the weak lensing information is practically impossible. Nevertheless, we have adopted a simple criterion
that neighboring peaks with their separation larger than√
2 × θG ≃ 2.1 arcmin are regarded as two isolated peaks(see Section 3.3). Consequently, there are systems of neighboring peaks, whose dynamical states are ambiguous, in
our final peak catalog (Table 3).
Here we describe those systems of neighboring peaks
having a common CAMIRA-HSC cluster within 5 arcmin
from both the peaks. There are seven such systems, whose
HSC riz composite images are shown in Figure 9 [panels(h)–(n)]. Below we give short descriptions of them:• HWL16a-007 and 008 [Figure 9 (h)]: Although bothpeaks have a common CAMIRA-HSC counterpart (ID149, zcl = 0.287), they match with the different XXL
clusters, XLSSC 111 (z = 0.299) and XLSSC 117 (z =0.300). Thus those are likely two isolated clusters atPublications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0very close redshifts.
• HWL16a-021 and 022 [Figure 9 (i)]: See Appendix 1.2.
• HWL16a-044 and 045 [Figure 9 (j)]: CAMIRA-HSC
cluster (ID-870, zcl = 0.260) matches better with
HWL16a-045. Not enough information is available toinfer the physical connection between those two peaks.
• HWL16a-062 and 063 [Figure 9 (k)]: CAMIRA-HSC
cluster (ID-1103, zcl = 0.144) matches better with
HWL16a-062. Another cluster (ID-1105, zcl = 1.105)
is probably a non-related high-z cluster as no associated lensing signal appears. Other than that, no known
clusters can be associated with it. However, a good correlation between HWL16a-063 and an apparent concentration of bright galaxies is clearly seen.
• HWL16a-068 and 069 [Figure 9 (l)]: Both the peaks
match with CAMIRA-HSC clusters (ID-1155, zcl =0.322) and (ID-1157, zcl = 0.515). Because of the difference in the cluster redshifts, the lensing signals likely
originate from line-of-sight projections of the two clusters at different redshifts.
• HWL16a-085 and 086 [Figure 9 (m)]: HWL16a-085
matches better with CAMIRA-HSC cluster (ID-1339,
zcl = 0.536), whereas HWL16a-086 matches better withCAMIRA-HSC cluster (ID-1344, zcl = 0.149). Anothercluster (ID-1341, zcl = 0.884) is probably a non-related
high-z cluster, as no associated lensing signal appears.
Thus the two peaks originate from two isolated clusters.
• HWL16a-105 and 106 [Figure 9 (n)]: CAMIRA-HSC
cluster (ID-1628, zcl = 0.100) matches better with
HWL16a-106. No apparent lensing signal associatedwith another cluster (ID-1680, zcl = 0.357) appears. Not
enough information is available to infer the physical connection between those two peaks.

Appendix 3 Cluster mass estimate
Here we present the method and results of cluster massestimate of the weak lensing peaks that meet the following
two conditions:
1. Those peaks should be classified as weak lensing secureclusters (see Section 4) to ensure the presence of a securecluster counterpart, and to avoid systems with line-ofsight projection. The latter is required as our clustermodel (described below) assumes a single dark matter
halo.
2. Cluster redshift should be lower than 0.7 to have a sufficient number density of background galaxies for themeasurement of weak lensing shear profile (described
below).

Table 5).
We derive cluster masses by fitting the NFW model(Navarro et al. 1997) to measured weak lensing shear profiles based on the standard likelihood analysis. We employ
the weak lensing mass estimate procedure of Umetsu et al.
(2020) who used the same HSC first-year shear catalog as
one used in this study, allowing us to closely follow theirprocedure. Since details of the procedure are described in
Umetsu et al. (2020, and see the references therein), below
we describe those aspects that are directly relevant to thisstudy.
For each cluster, we select background galaxies using the P -cut method (see Section 3.4 of Umetsu et al.
2020, and see also Medezinski et al. 2018) with the cluster redshift taken from the estimated redshift of matched
CAMIRA-HSC clusters6 , and we measure the azimuthally
averaged tangential shear (γt ) which relates to the excess
surface mass density ∆Σ as (Kaiser 1995)
γt (R) =Σ̄(< R) − Σ(R)
∆Σ(R)
≡
,
Σcr (zcl , zs )
Σcr (zcl , zs )(A1)

where Σ(R) is the azimuthally averaged surface mass density at R, Σ̄(< R) denotes the average surface mass density interior to R, and Σcr (zcl , zs ) is the critical surface
mass density. We take the peak positions as the cluster
centers, and we measure γt (R) in 5 radial bins of equal
logarithmic spacing of ∆ log R = 0.25 with bin centers of
Rc (i) = 0.3 × 10i∆ log R [h−1 Mpc] where i runs from 0 to
4. We use the photo-z probability distribution functions(PDFs) of background galaxies to evaluate Σcr (zcl ,zs ) following Umetsu et al. (2020, Section 3.2). The resulting∆Σ(R) signals are shown in Figure 10.
We adopt the NFW model to make the model prediction of the weak lensing shear profile by a cluster. The
spherical NFW density profile is specified by two parameters, the characteristic density parameter (ρs ), and thescale radius (rs ), as ρNFW (r) = ρs /[r/rs (1 + r/rs )2 ]. We
define the halo mass by the over-density mass (M∆ ) which
is given by integrating the halo density profile out to thecorresponding over-density radius (r∆ ) at which the meaninterior density is ∆ × ρcr (z). The corresponding concentration parameter is defined by c∆ = r∆ /rs . For a given setof (M∆ ,c∆ ), which is of our primary interest, the NFW parameters (ρs ,rs ) are uniquely determined, and thus ∆Σ(R)is as well. Therefore we take (M∆ , c∆ ) as fitting parameters in the likelihood analysis. We consider two cases,∆ = 200 and ∆ = 500.
We employ the standard likelihood analysis for deriving
constraints on the model parameters. The log-likelihood is
6

61 weak lensing secure clusters meet those conditions (see19

For HWL16a-002, the redshift of matched XXL cluster (XLSSC 114) istaken as it is based on spectroscopic redshifts (Adami et al. 2018).

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Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 10. Radial profiles of the excess surface mass density of individual clusters (the weak lensing peak ID is denoted in upper-right of each panel).
Measurement results are plotted with filled squares with error bars; inner error bars show the shape noise components (σshape ) only, whereas outer error bars show the total errors being composed of shape noise, cosmic shear covariance due to large-scale structures, and intrinsic scatter components (thediagonal components of Covshape + Covlss + Covint , see Section 3.3 of Umetsu et al. 2020 and references therein). The best-fit NFW model in M200c -c200c
space is shown by the solid line. Here all the results are based on the cosmological parameters inferred from the WMAP 9-year results (Hinshaw et al. 2013).

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 10. (Continued)

21

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Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

given by,
−2 ln L(p) =Xi,j[di − mi (p)]Cov−1
ij [dj − mj (p)],(A2)

where the data vector di = ∆Σ(Ri ), and mi (p) is the model
prediction with the model parameters p = (M∆ , c∆ ). The
covariance matrix (Cov) is composed of the three components (see Umetsu et al. 2020, and references therein for
detailed descriptions): The statistical uncertainty due tothe galaxy shape noise (Covshape ), the cosmic shear covariance due to uncorrelated large-scale structures projectedalong the line of sight (Hoekstra 2003) (Covlss ), and the
intrinsic variation of the cluster lensing signals at the fixedmodel parameters due to e.g., cluster asphericity, and thepresence of correlated halos (Covint ) (Gruen et al. 2015;
Miyatake et al. 2019).
We compute the log-likelihood function over
the two-parameter space in the ranges of 0.01 <M∆ [×1014 h−1 M⊙ ] < 30 and 0.01 < c∆ < 30, andmarginalize it to derive one-parameter posterior distributions.
Peaks and 68.3% confidence intervals of
marginalized posterior distributions of c200c , M200c , andM500c are summarized in Table 5. We present two setsof results based on the cosmological parameters from the
WMAP 9-year results (Hinshaw et al. 2013), and fromthe Planck 2018 results (Planck Collaboration et al. 2018,Ωm = 0.32, Ωb = 0.049, ΩΛ = 0.68, ns = 0.96, σ8 = 0.83,
and h = 0.67). The differences in the derived values
between two cosmological models are much smaller thanthe derived 68.3% confidence intervals. Note that thosedifferences are not systematic but rather random. The
reason for this is that since we take the comoving angulardistance, which depends on the cosmology, for binning ofγt (R) measurement, the corresponding angular ranges ofbins vary between two analyses and thus measured signals
and errors do as well. Note that ”N/A” in the resultsof c200c means either the upper/lower bound of 68.3%
confidence interval or the minimum of the marginalized
likelihood function is not enclosed within the parameter
range of c200c . This is due to the limited coverage in
R with relatively large error bars. In Figure 10, we
compare the measured excess surface mass density profiles
with the best-fit NFW model (based on WMAP 9-year
cosmological model) in the M200c -c200c space, from which
the reader can judge the goodness of fits.

Appendix 4 The locally normalized SN
estimator
In this study, we have adopted the globally normalized SN
estimator defined by equation (7) with equations (5) and(6). In some studies (for example, Hamana et al. 2015),
however, the peak SN (θ) is defined by the locally normal2
ized estimators, for which K(θ) and σshape(θ) are normalized by the local galaxy number density, ng (θ), insteadof the mean density n̄g . Here we compare those two estimators using a simple model, and using the actual weaklensing data. See Schmidt & Rozo (2011) for a related
study on those estimators.
Following the same manner as introduced in Section 5.1,
the local estimators are given by,
KL (θ) =1
nf g + nbg + ncl (θ)

Xγ̂t,i Qi ,

(A3)

i∈bg

and
2
σshape,L(θ) =1
2[nf g + nbg + ncl (θ)]2×X 2

êi Q2i +

i∈bg

X 2

êi Q2i +

i∈f g

X 2

êi Q2ii∈cl!

. (A4)

Notice that contributions from cluster member population
can not be ignored at the cluster central regions where weare interested in. Thus we have,√ P
2 bg γ̂t,i Qi. (A5)SNL (θ) = 
P 2 2 P 2 2 P 2 2 1/2
Q
ê
+
Q
ê
+
Q
ê
cl i i
fg i i
bg i i

Therefore, the locally defined SN is affected by the cluster
member population and can be smaller than the globallydefined SN [see equation (16)], though it depends on thelocal proportion of cluster member galaxies to backgroundand foreground galaxies.
We examine the actual differences between the globallynormalized and the locally normalized SN values using
our source galaxy samples. We have generated the locallynormalized SN maps for the six source samples used inthis study. We evaluate locally normalized SNL values at
positions of high peaks (SNG ≥ 5) located in the globallynormalized SN maps. This SNG –SNL comparison is done
for six sets of SN maps. Results are shown in Figure 12,
in which we find that SNL tends to be smaller than SNG ,and that this trend is more clearly seen in lower zmin cases
as expected. We find that SNL is smaller than about 10
percent on average than SNG for weak lensing maps usedin this study.
We note that one may take an averaged local shape2i) to define the SN , instead of thenoise (that is hσshape,Llocally defined one. In this case, deriving its expression using the above manner is not straightforward, because it isnecessary to take into account the covariance between numerator and denominator in equation (A4) (see Schmidt &Rozo 2011 for an approximative approach to this). Instead,2
we evaluate hσshape,Li with actual weak lensing data used

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Fig. 11. Peak SNG values in the globally normalized SN maps are compared with SNL values at the same positions in the locally normalized SN
maps. Plus marks are for high peaks (SNG ≥ 5) located in the globallynormalized SN maps. Different panels are for different source samples withzmin being shown in each panel.

Fig. 12. Shown is the local galaxy number density at cluster regions, which
is defined by the mean number density within an angular radius of 15 arcmin from peak positions, normalized by the global mean galaxy number
density. Weak lensing secure clusters are used, and are divided into four
sub-samples based on the cluster redshifts (denoted in panels). The horizontal axis is zmin of source galaxy samples. For each sub-sample and eachsource galaxy sample, the mean and its 1-σ error among the clusters (thenumber of clusters in each sub-sample is given in each panel) are plotted.

23

2
in this study and compare it with hσshape,Gi. We find2
1/2
that the two are very close; hσshape,L i
is only slightly2
smaller than hσshape,G
i1/2 (to be specific, the fractional difference is smaller than 0.5 percent). Therefore, replacing22
σshape,L(θ) with hσshape,Li does not mitigate the dilution
effect, but an additional ncl (θ) term in the normalization ofKL (θ) suppresses the peak signal [compare equations (13)with (A3)]. We measure ncl from our data; in doing this,we have defined the local galaxy number density at cluster
regions by the mean number density within a circular areawith an angular radius of 15 arcmin from peak positions.7
Results are shown in Figure 12 for four cluster redshift
ranges and six source samples, in which we find that forzmin from 0 to 0.3, the galaxy density excess is 5-10 percent; while for higher zmin , it is consistent with zero forhigher redshift clusters (0.3 < zcl < 0.5), but the excess isstill 5-10 percent for lower redshift clusters. It follows fromthese results that for low-zmin source samples, a peak SN
from the globally normalized estimator can be 5-10 percent
larger than one from the locally normalized estimator.
We note that the decreasing trend of the number excess at higher zmin seen in higher redshift clusters is expected, as zmin -cut may exclude cluster member galaxies
of clusters at zcl < zmin . However, the trend is not seenin the lower redshift clusters. The reason for this is notunderstood well; possible causes are the line-of-sight projection of undiscovered clusters at higher redshifts, anderrors in photo-z (cluster member galaxies at a low-z are
mis-estimated as higher-z galaxies). We are not going into
this issue in this study but leave it for a future study.

7

We note that the local galaxy number density is not uniquely defined,
because it is necessary to define a local scale, or an averaging scheme.
Thus the estimated values given there are not general but are specific to
our definition of the local galaxy number density.

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Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Table 3. Summary of weak lensing merged peaks. First column is the merged peak ID, the second to fifth columns are for informationof weak lensing peaks (see Section 3.3), the sixth to ninth columns are for information of matched CAMIRA-HSC clusters (see Section4), and the last columns is for matched known clusters in Adami et al. (2018) (XXL clusters, see Appendix 1.2), Abell et al. (1989) (Abellclusters), and Miyazaki et al. (2018a, cited by M18 in this table) (weak lensing peaks, see Appendix 1.3).

Weak lensing
ID

SNmax

zopt

HWL16a-001
HWL16a-002

5.19
7.23

0.0
0.6

HWL16a-003
HWL16a-004

5.22
5.87

0.5
0.6

31.2073
31.3577−3.0587−5.7178

HWL16a-005
HWL16a-006
HWL16a-007
HWL16a-008

5.24
5.46
6.53
5.46

0.5
0.3
0.3
0.3

31.4584
32.0082
33.111233.1188−3.3714−3.4128−5.6214−5.5365

HWL16a-009
HWL16a-010

6.11
5.97

0.0
0.2

33.362533.4777−2.9126−2.8852

HWL16a-011
HWL16a-012
HWL16a-013
HWL16a-014

5.12
6.77
5.31
6.38

0.6
0.6
0.6
0.2

33.8206
35.4434
36.122936.3758−2.7029−3.7668−4.2378−4.2496

HWL16a-015
HWL16a-016
HWL16a-017
HWL16a-018
HWL16a-019
HWL16a-020

6.04
7.95
5.45
5.26
5.08
9.68

0.6
0.4
0.3
0.2
0.0
0.3

37.051237.3963
37.5572
37.779637.8096
37.9163

HWL16a-021

6.06

0.2

HWL16a-022
HWL16a-023
HWL16a-024
HWL16a-025

6.59
5.30
5.11
5.64

HWL16a-026
HWL16a-027

RA
Dec
J2000.0 [◦ ]30.3800 −5.5078
30.4273 −5.0219

CAMIRA-HSC
ID
zcl
Nmem

Note

31

0.809

21.4

θsep
[′ ]
1.7

65
74
71
76
94
149
149

0.553
0.290
0.697
0.167
0.204
0.287
0.287

29.2
43.5
24.1
21.6
15.7
64.0
64.0

1.3
0.9
3.8
1.3
2.7
1.7
4.0

165
174
176
252
285
293

0.150
0.274
1.018
0.430
0.264
0.155

40.4
16.1
29.6
68.3
15.1
18.1

2.4
2.8
4.2
0.3
0.9
0.7

−5.5929−3.6121−5.6526−5.5840−4.4738−4.8799

324
327
338
343

0.312
0.500
1.011
0.187

57.1
22.3
16.3
116.8

0.4
1.8
5.0
0.4

38.1182−4.7890

355

0.276

33.1

3.5

0.3
0.3
0.5
0.0

38.1580
38.3915129.3206130.3706−4.7513−5.5027
1.60690.4379

7.93
5.11

0.5
0.4

130.5895
131.0585

1.64731.0644

HWL16a-028
HWL16a-029

6.98
5.27

0.6
0.6

133.1296
135.9841

0.40411.4088HWL16a-030

5.16

0.3

136.0810

0.5865

355
362
401
433
430
435
438
449
448
505
591
592
594
593

0.276
0.420
0.360
0.413
0.454
0.215
0.424
0.323
0.661
0.270
0.817
0.665
0.399
0.303

33.1
46.9
36.5
19.0
24.4
21.3
78.9
29.1
16.8
43.2
16.1
15.6
16.1
26.8

0.4
0.4
0.8
0.4
4.0
4.1
0.4
1.5
1.8
1.6
2.0
2.8
2.7
4.9

M18 rank 29
XLSSC 114 (z = 0.234)
M18 rank 13
XLSSC 106 (z = 0.300)

XLSSC 111 (z = 0.300)
XLSSC 117 (z = 0.298)
M18 rank 58
M18 rank 16
M18 rank 63

XLSSC 006 (z = 0.429)
XLSSC 044 (z = 0.263)
XLSSC 041 (z = 0.142)
M18 rank 36
M18 rank 9
XLSSC 169 (z = 0.498)

XLSSC 091 (z = 0.186)
Abell 362 (z = 0.184)
M18 rank 2
XLSSC 151 (z = 0.189)
XLSSC 152 (z = 0.205)
M18 rank 28
XLSSC 151 (z = 0.189)
XLSSC 105 (z = 0.432)
M18 rank 22

M18 rank 18

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

25

Table 3. (Continued)

Weak lensing
ID

SNmax

zopt

HWL16a-031
HWL16a-032
HWL16a-033
HWL16a-034

5.18
8.31
5.55
7.72

0.2
0.0
0.0
0.4

HWL16a-035
HWL16a-036
HWL16a-037
HWL16a-038
HWL16a-039
HWL16a-040
HWL16a-041
HWL16a-042

5.29
5.99
5.18
5.03
5.88
5.97
6.01
5.53

0.4
0.6
0.6
0.3
0.0
0.6
0.3
0.0

139.3198
139.3405
140.0954
140.1431
140.4154
140.5592
140.6790
177.1051

0.99853.82811.57480.7907−0.2491−0.1323
2.1327−0.6610

HWL16a-043
HWL16a-044
HWL16a-045
HWL16a-046

5.49
6.14
5.26
8.07

0.6
0.6
0.4
0.5

177.1322
177.2646
177.2946
177.5842

0.00880.2836
0.3636−0.6009

HWL16a-047
HWL16a-048
HWL16a-049

7.05
6.11
5.22

0.4
0.4
0.4

178.0615
178.0989
178.6288

0.5187−0.5111−0.1237

HWL16a-050
HWL16a-051
HWL16a-052
HWL16a-053

5.05
7.75
5.25
6.36

0.4
0.0
0.4
0.3

178.8288
179.0517
179.6138
180.4286

0.8712−0.3490−0.0412−0.1839

HWL16a-054

5.01

0.6

180.4536−0.4986

HWL16a-055

5.39

0.4

180.6834

0.9709HWL16a-056
HWL16a-057
HWL16a-058
HWL16a-059
HWL16a-060
HWL16a-061
HWL16a-062

6.88
5.32
5.25
5.52
7.33
5.26
5.42

0.6
0.4
0.6
0.5
0.4
0.4
0.2

181.3878
210.7874
211.2955
211.7872
211.9925
212.3195213.6054−0.6432−0.3084−0.1472−0.2717−0.4857−0.1997−0.3669

HWL16a-063
HWL16a-064

5.07
5.44

0.4
0.0

213.7248
213.7770−0.3423−0.4892

HWL16a-065
HWL16a-066
HWL16a-067

7.16
5.63
5.13

0.2
0.5
0.0

213.8891
214.6762
214.8009−0.0527−0.0506
0.2418RA
Dec
J2000.0 [◦ ]136.8921 −0.0580
138.4612 −0.7631
138.5051
1.6655139.0387 −0.3966

CAMIRA-HSC
ID
zcl
Nmem

Note

628
686
691
716

1.062
0.285
0.380
0.315

21.4
36.1
15.6
78.5

θsep
[′ ]
0.4
1.0
4.6
1.4

733
734
770
771
784
793
798
864
861
860
870
870
878

0.344
0.420
0.697
0.463
0.310
0.794
0.194
0.419
0.898
1.080
0.260
0.260
0.135

24.7
22.9
55.0
25.2
29.1
17.5
24.8
25.9
15.9
27.6
16.5
16.5
51.8

1.5
0.4
0.3
1.3
1.7
0.9
0.2
3.5
4.3
5.0
4.1
1.1
0.6

892
893
909
916
912
922
928
942
966

0.472
0.311
0.246
0.548
0.892
0.481
0.254
0.252
0.167

61.0
15.7
30.6
19.0
17.7
21.9
66.9
29.5
45.8

0.3
2.2
3.4
3.6
5.0
1.1
1.5
1.7
1.2

968
967
978
982
994
1037
1046
1057
1062
1103
1105
1105
1112

0.322
0.162
0.568
0.434
0.470
0.450
0.248
0.561
0.469
0.144
1.024
1.024
0.144

24.5
24.0
33.7
23.2
40.7
35.1
27.5
48.3
34.0
60.0
15.4
15.4
38.8

0.7
0.9
0.7
3.3
0.7
0.0
0.7
1.3
0.9
1.1
3.6
4.4
0.5

-

-

-

-

M18 rank 4
M18 rank 20
Abell 776 (z = 0.336)
M18 rank 8

M18 rank 39
M18 rank 30
M18 rank 19

Abell 1392 (z = 0.139)
M18 rank 10
M18 rank 17
M18 rank 32

M18 rank 5
Abell 1445 (z = 0.169)
M18 rank 12

M18 rank 35
M18 rank 54

Abell 1882 (z = 0.137)
M18 rank 26
M18 rank 23

26

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Table 3. (Continued)

Weak lensing
ID

SNmax

zopt

HWL16a-068

5.31

0.0

RA
Dec
J2000.0 [◦ ]215.0330
0.9984

HWL16a-069

7.18

0.6

215.0729

0.9557

HWL16a-070
HWL16a-071
HWL16a-072

6.22
5.38
5.11

0.6
0.5
0.6

215.2574
215.9165
216.0089

0.36650.44910.1418HWL16a-073
HWL16a-074
HWL16a-075
HWL16a-076
HWL16a-077
HWL16a-078
HWL16a-079
HWL16a-080
HWL16a-081
HWL16a-082

7.02
5.01
5.65
9.36
5.46
5.28
5.05
6.14
5.66
5.49

0.5
0.0
0.3
0.5
0.6
0.0
0.0
0.5
0.4
0.4

216.6510
216.6535
216.6760
216.7785
216.8310
216.8484
216.8659
217.6808
218.8457
218.8858

0.8016−0.0903
0.16430.72670.9541−0.2403−0.19780.8093−0.3931−1.1228

HWL16a-083
HWL16a-084
HWL16a-085

5.82
5.42
5.82

0.0
0.3
0.0

219.2131
220.0846
220.4015−0.7026−0.6101−0.9068

HWL16a-086

5.23

0.5

220.4589−0.8247

HWL16a-087
HWL16a-088
HWL16a-089
HWL16a-090
HWL16a-091
HWL16a-092
HWL16a-093
HWL16a-094
HWL16a-095
HWL16a-096
HWL16a-097
HWL16a-098
HWL16a-099

5.02
6.01
5.71
5.39
5.91
5.05
6.80
6.49
7.66
5.02
5.05
5.58
5.76

0.0
0.4
0.0
0.6
0.6
0.3
0.6
0.2
0.6
0.2
0.0
0.4
0.2

220.5909
220.7952
221.0371
221.1442
221.1917
221.2090
221.3335
223.0801
223.0929
223.9242
224.2746
224.6567
244.4326

0.33671.04520.17430.2464−0.66940.20150.11160.1689−0.9713−0.3384
0.11640.485842.5427

HWL16a-100

5.78

0.0

245.0550

42.5052

HWL16a-101

8.24

0.3

245.3758

42.7648

HWL16a-102
HWL16a-103
HWL16a-104

5.04
5.58
6.31

0.4
0.0
0.6

246.1339
246.5173333.0522

43.320343.7147−0.1334

CAMIRA-HSC
ID
zcl
Nmem

Note

1155
1157
1157
1155
1161
1164
1191
1195
1192
1220
1222
1226
1231
1229
1244
1273
1274
1277
1288
1322
1339
1341
1344
1341
1347
1351
1363
1366
1371
1417
1418
1443
1454
1530
1528
1540
1543
1547

0.322
0.515
0.515
0.322
0.168
0.645
0.534
0.539
0.319
0.604
0.531
0.296
0.294
0.164
0.312
0.283
0.756
0.260
0.198
0.549
0.536
0.884
0.149
0.884
0.166
0.528
0.295
0.523
0.286
0.592
0.304
0.220
0.395
0.285
0.598
0.141
0.800
0.152

16.6
46.3
46.3
16.6
15.3
47.6
29.4
39.1
17.8
18.8
25.3
24.2
36.6
19.3
35.4
27.3
15.6
24.4
18.4
39.6
52.9
17.3
29.3
17.3
21.0
40.2
54.2
36.6
32.7
26.1
38.4
25.9
18.1
30.8
17.9
27.4
22.7
33.7

θsep
[′ ]
2.8
3.9
0.5
1.2
4.7
1.4
0.5
0.8
2.9
3.8
2.2
0.4
2.0
4.2
0.6
1.3
3.5
4.1
4.1
1.4
1.0
2.5
1.6
4.9
2.6
0.6
1.4
0.4
0.2
0.3
0.2
0.6
0.4
1.2
4.1
0.9
4.9
0.6

1557
1561
1622

0.287
0.260
0.350

26.1
17.1
30.9

0.8
1.1
0.4

M18 rank 33

M18 rank 62

M18 rank 14

M18 rank 40
M18 rank 47

M18 rank 25
M18 rank 53

M18 rank 64
M18 rank 48
M18 rank 11

M18 rank 57
M18 rank 34
M18 rank 42

M18 rank 24
Abell 2183 (z = 0.136)
M18 rank 1

M18 rank 44

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

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Table 3. (Continued)

Weak lensing
ID

SNmax

zopt

HWL16a-105

5.36

0.3

HWL16a-106

5.55

0.3

333.3714−0.1542

HWL16a-107
HWL16a-108

5.57
5.14

0.6
0.2

333.5929
333.6801

0.7956
1.1171

HWL16a-109
HWL16a-110
HWL16a-111

5.32
5.91
5.43

0.5
0.3
0.4

333.7900
335.2140
335.4040

1.04220.9704
1.3854

HWL16a-112
HWL16a-113

8.34
6.75

0.2
0.2

336.0366
336.2291

0.3331−0.3668

HWL16a-114
HWL16a-115
HWL16a-116
HWL16a-117
HWL16a-118
HWL16a-119
HWL16a-120
HWL16a-121

5.63
6.58
5.31
6.23
5.66
5.39
5.13
5.39

0.4
0.5
0.0
0.6
0.5
0.6
0.6
0.2

336.4066
336.4217
336.9540
337.1293
338.0182
338.0183
338.5233
338.9150−0.3068
1.07300.11411.71350.02310.22881.69971.4837

HWL16a-122
HWL16a-123
HWL16a-124

6.47
5.54
5.65

0.6
0.6
0.0

339.1320
339.3176
339.7643

1.5266−0.36290.6652RA
Dec◦
J2000.0 [ ]333.3515 −0.2017

CAMIRA-HSC
ID
zcl
Nmem

Note

1628
1630
1628
1630
1635
1637
1638
1639
1664
1670
1669
1683
1688
1687
1694
1696
1707
1709
1724
1733
-

0.100
0.357
0.100
0.357
0.308
0.469
0.759
0.702
0.323
0.790
0.324
0.154
0.308
0.140
0.402
0.281
0.410
0.338
1.013
0.246
-

19.7
33.3
19.7
33.3
26.7
25.2
18.5
26.7
22.1
22.8
16.9
44.3
33.1
19.5
17.8
49.9
17.3
31.5
18.6
16.0
-

θsep
[′ ]
3.8
4.3
1.4
3.9
0.6
2.2
4.3
1.2
0.1
2.4
3.3
0.5
0.8
1.0
0.2
0.8
2.0
0.6
3.2
3.8
-

1748
1749

0.264
0.200

19.4
19.6

1.2
3.5

M18 rank 50

M18 rank 56
M18 rank 3
M18 rank 21

M18 rank 52
M18 rank 61
M18 rank 59

Abell 2457 (z = 0.059)
M18 rank 6

M18 rank 65

28

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Table 4. Summary of known cluster counterparts of 17 weak lensing merged peaks which have no CAMIRA-HSC cluster within 5
arcmin radius from the peak positions. A known cluster database taken from a compilation by NASA/IPAC Extragalactic Database(NED) was used for this counterpart search.

ID

Cluster namezcl

HWL16a-001
HWL16a-011
HWL16a-015

CFHTLS W1-2593
CFHTLS W1-2588
CFHT-W CL J022757.5−053537
CFHT-W CL J023111.0−0536
CFHTLS W1-2864
CFHTLS W1-2588
CFHTLS W1-2589
SDSS CE J213.904556−00.069648
WHL J141527.6−000319
SDSS CE J213.843536−00.001681
SDSS CE J213.919922+00.023597
SDSS CE J214.633743−00.016635
SDSS CE J214.788757+00.220532
GMBCG J216.67104−00.08426
SDSS CE J216.649841−00.110289
GMBCG J216.63912−00.10900
SDSS CE J216.635178−00.044207
GMBCG J216.67010−00.03407
SDSS CE J16.867157−00.209108
SDSS CE J216.868240−00.171960
SDSS CE J216.852905−00.249845
SDSS CE J221.044815+00.172764
GMBCG J221.00862+00.12188
FAC2011 CL 0061
GMBCG J221.15835+00.19581
WHL J144437.5+001402
SDSS CE J221.230865+00.138749
MaxBCG J221.20075+00.12862
SDSS CE J221.138031+00.233130
WHL J223540.8+012906
MCXC J2235.6+0128
ABELL 2457
-

0.30
0.68
0.32
0.67
0.60
0.68
1.00
0.29
0.15
0.29
0.33
0.23
0.44
0.39
0.27
0.25
0.42
0.40
0.18
0.35
0.18
0.30
0.29
0.62
0.43
0.31
0.27
0.29
0.30
0.058
0.060
0.059
-

HWL16a-018

HWL16a-061
HWL16a-065

HWL16a-066
HWL16a-067
HWL16a-074

HWL16a-079

HWL16a-089
HWL16a-092

HWL16a-096
HWL16a-118
HWL16a-121

HWL16a-122
HWL16a-123aθsep a[arcmin]
1.4
4.1
4.9
1.9
2.8
2.9
3.2
1.4
1.5
4.1
4.9
3.3
1.5
1.1
1.2
1.4
3.0
3.5
0.7
1.6
3.2
0.2
3.6
1.2
3.1
3.7
4.0
4.4
4.7
0.3
0.8
1.5
-

Ref
Durret et al. (2011)
Durret et al. (2011)
Wen & Han (2011)
Wen & Han (2011)
Durret et al. (2011)
Durret et al. (2011)
Durret et al. (2011)
Goto et al. (2002)
Wen et al. (2009)
Goto et al. (2002)
Goto et al. (2002)
Goto et al. (2002)
Goto et al. (2002)
Hao et al. (2010)
Goto et al. (2002)
Hao et al. (2010)
Goto et al. (2002)
Hao et al. (2010)
Goto et al. (2002)
Goto et al. (2002)
Goto et al. (2002)
Goto et al. (2002)
Hao et al. (2010)
Farrens et al. (2011)
Hao et al. (2010)
Wen et al. (2009)
Goto et al. (2002)
Koester et al. (2007)
Goto et al. (2002)
Wen et al. (2009)
Piffaretti et al. (2011)
Abell et al. (1989)-

The angular separation between the weak lensing peak position and the cluster position.

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

29

Table 5. Results of the likelihood analysis of the surface mass density profile of individual clusters based on NFW model (see Appendix
3). Cluster redshifts (zcl ) are taken from the estimated redshift of matched CAMIRA clusters except for HWL16a-002 for which theredshift of matched XXL cluster (XLSSC 114) is taken (marked by ∗ ). Peaks and 68.3% confidence intervals of marginalized posterior
distributions of c200c , M200c , M500c are summarized. The results based on the cosmological parameters from the WMAP 9-year(Hinshaw et al. 2013, Ωm = 0.279, Ωb = 0.046, ΩΛ = 0.721, ns = 0.97, σ8 = 0.82, and h = 0.7), and Planck 2018 results (Planck
Collaboration et al. 2018, Ωm = 0.32, Ωb = 0.049, ΩΛ = 0.68, ns = 0.96, σ8 = 0.83, and h = 0.67) are presented. ”N/A” in the resultsof c200c means either the upper/lower bound of 68.3% confidence interval or the minimum of the marginalized log-likelihood function is
not enclosed within the parameter range of 0.01 ≤ c∆ ≤ 30.

ID

zcl

HWL16a-002
HWL16a-003
HWL16a-005
HWL16a-007
HWL16a-012
HWL16a-013
HWL16a-014
HWL16a-016
HWL16a-017

0.234∗
0.553
0.167
0.287
0.430
0.264
0.155
0.312
0.500

WMAP9 cosmology
c200c
M200c
[1014 h−1 M⊙ ]5.34+5.19
1.91+0.74
−2.23
−0.48
2.45+5.94
1.10+0.69
−1.85
−0.54
+N/A
9.89−7.45
0.89+0.41
−0.36
+1.96
0.50+0.70
3.99
−0.43
−1.66
+3.22
3.01−1.54
2.03+1.19−0.57
2.14+2.43
1.11+0.75
−1.15
−0.39
2.07+2.41
1.05+0.69
−1.15
−0.40
+1.56
1.20+1.03
3.22
−0.63
−1.39
+2.09
0.98−0.88
1.00+0.81−0.51

HWL16a-020
HWL16a-022
HWL16a-023
HWL16a-024
HWL16a-026
HWL16a-028
HWL16a-032
HWL16a-034
HWL16a-035
HWL16a-036
HWL16a-037

0.186
0.276
0.420
0.360
0.424
0.270
0.285
0.315
0.344
0.420
0.697

5.39+3.74−1.95
3.42+4.07−1.71
5.19+13.79−3.00
2.57+3.43−1.48
1.20+0.89−0.63
2.93+2.59−1.36
2.87+2.40−1.27
3.22+1.67−1.08
0.73+1.02−0.55
3.09+2.55−1.36
0.32+1.35−N/A

3.06+1.29−0.76
1.38+0.66−0.42
1.12+0.50−0.40
1.29+0.79−0.55
6.20+2.60−2.07
1.63+1.13−0.54
1.61+1.15−0.55
5.69+1.87−1.62
0.80+1.24−0.49
1.55+1.15−0.56
2.08+2.58−1.10

2.41+0.79−0.55
1.09+0.41−0.33
0.89+0.36−0.31
0.99+0.51−0.41
3.34+1.10−0.98
1.25+0.60−0.39
1.22+0.60−0.39
3.94+1.03−0.95
0.54+0.46−0.32
1.17+0.64−0.40
1.57+1.22−0.82

5.14+3.64−1.87
3.06+3.62−1.55
5.10+14.04−2.97
2.24+3.00−1.34
1.21+0.90−0.63
2.95+2.66−1.37
2.83+2.52−1.29
3.10+1.66−1.06
0.67+0.96−0.53
2.97+2.49−1.32
0.40+1.39−N/A

3.06+1.39−0.79
1.35+0.69−0.44
1.13+0.50−0.40
1.28+0.83−0.56
6.04+2.63−2.09
1.64+1.13−0.53
1.55+1.11−0.52
5.61+1.96−1.68
0.74+1.33−0.48
1.51+1.17−0.56
1.97+2.09−1.06

2.40+0.85−0.57
1.05+0.44−0.32
0.90+0.36−0.32
0.98+0.52−0.42
3.27+1.12−1.00
1.27+0.60−0.39
1.19+0.59−0.37
3.87+1.07−0.99
0.50+0.48−0.31
1.15+0.64−0.40
1.49+1.11−0.80

HWL16a-038
HWL16a-039

0.463
0.310

3.25−2.68
0.95+2.51
−N/A

+N/A

1.10+0.62−0.53
0.90+0.70−0.41

0.89+0.46−0.42
0.69+0.40−0.31

3.33−2.79
1.52+3.43
−N/A

+N/A

1.06+0.61−0.53
0.98+0.59−0.40

0.86+0.46−0.42
0.77+0.38−0.32

HWL16a-041
HWL16a-045

0.194
0.260

1.69+2.47−1.11
0.55+1.30−0.52

1.24+0.74−0.44
0.70+0.51−0.35

0.97+0.44−0.33
0.54+0.32−0.27

1.74+2.82−1.20
0.52+1.32−N/A

1.19+0.65−0.43
0.68+0.49−0.34

0.93+0.42−0.32
0.53+0.32−0.27

HWL16a-046
HWL16a-047
HWL16a-050

0.135
0.472
0.481

4.88+8.25
−2.49
+N/A
9.26−4.79
0.50+0.79−0.46

1.31+0.50−0.37
2.50+0.74−0.63
1.30+1.60−1.13

1.05+0.34−0.29
2.01+0.55−0.49
0.46+0.67−0.35

4.41+7.59
−2.30
+N/A
10.26−5.51+0.73
0.44−N/A

1.29+0.52−0.37
2.53+0.73−0.63
1.53+1.63−1.35

1.03+0.36−0.29
2.04+0.55−0.50
0.48+0.71−0.37

HWL16a-051
HWL16a-052
HWL16a-053
HWL16a-056
HWL16a-057
HWL16a-058
HWL16a-059
HWL16a-060
HWL16a-064

0.254
0.252
0.167
0.470
0.450
0.248
0.561
0.469
0.144

1.67+1.23−0.78
0.76+1.83−0.68
1.66+2.45−1.12
1.72+1.70−0.96
1.50+1.70−0.93
2.42+2.84−1.28
1.76+3.61−1.33
0.96+0.91−0.59
0.32+0.69−N/A

4.40+1.73−1.43
0.92+0.60−0.39
1.59+1.07−0.49
1.92+1.47−0.90
1.71+1.93−0.77
1.43+0.80−0.50
0.96+0.87−0.67
4.29+2.06−1.72
3.03+2.19−2.23

2.66+0.80−0.73
0.71+0.37−0.29
1.25+0.58−0.37
1.28+0.72−0.52
1.23+0.84−0.51
1.11+0.47−0.37
0.72+0.61−0.52
2.25+0.95−0.89
0.99+0.95−0.41

1.43+1.16−0.72
0.67+1.63−0.63
1.68+2.72−1.18
1.71+1.71−0.95
1.28+1.57−0.84
2.23+2.69−1.21
2.24+4.66−1.60
0.97+0.96−0.61
0.26+0.71−N/A

4.28+1.89−1.54
0.89+0.59−0.40
1.53+0.86−0.48
1.75+1.48−0.81
1.63+2.08−0.74
1.37+0.81−0.49
1.05+0.85−0.68
4.09+2.10−1.80
1.10+3.58−0.45

2.50+0.85−0.77
0.69+0.37−0.30
1.20+0.53−0.36
1.20+0.72−0.50
1.19+0.84−0.50
1.06+0.48−0.37
0.80+0.61−0.53
2.14+0.98−0.91
0.88+0.77−0.36

HWL16a-070
HWL16a-071
HWL16a-076

0.645
0.534
0.296

0.82+0.83−0.60
1.10+1.65−0.82
4.09+4.52−1.93

4.10+2.57−1.95
1.43+1.24−0.65
1.61+0.69−0.37

2.07+1.10−0.99
1.05+0.59−0.46
1.29+0.44−0.29

0.87+0.85−0.60
1.08+1.58−0.79
4.10+4.95−1.98

3.94+2.51−1.92
1.47+1.34−0.66
1.58+0.63−0.36

2.01+1.10−0.99
1.07+0.62−0.46
1.27+0.40
−0.29

M500c
[1014 h−1 M⊙ ]1.51+0.49−0.36
0.86+0.49−0.42
0.72+0.32−0.29
1.78+0.81−0.75
1.60+0.66−0.43
0.86+0.41−0.29
0.81+0.40−0.29
1.77+0.72−0.69
0.76+0.48−0.38

Planck2018 cosmology
c200c
M200c
[1014 h−1 M⊙ ]4.74+4.74
1.85+0.77
−2.03
−0.49
2.76+7.11
1.08+0.66
−2.00
−0.53
+N/A
8.64−6.48
0.90+0.43−0.37
0.43+0.68
3.89+2.16
−0.41
−2.12
+2.74
2.29−1.30
1.89+1.25−0.56
2.46+3.11
1.15+0.62
−1.34
−0.39
2.02+2.46
1.05+0.67
−1.15
−0.39
+1.73
1.19+1.09
3.15
−0.64
−1.72
+1.84
0.75−N/A
0.93+0.79
−0.50

M500c
[1014 h−1 M⊙ ]1.46+0.50−0.37
0.85+0.47−0.41
0.73+0.33−0.30
1.55+0.96−0.68
1.49+0.66−0.42
0.90+0.38−0.29
0.82+0.40−0.30
1.70+0.81−0.71
0.70+0.48
−0.37

30

Publications of the Astronomical Society of Japan, (2018), Vol. 00, No. 0

Table 5. (Continued)

ID

zcl

HWL16a-077
HWL16a-080
HWL16a-081
HWL16a-084

0.294
0.312
0.283
0.549

WMAP9 cosmology
c200c
M200c
[1014 h−1 M⊙ ]3.84+5.92
1.25+0.53
−2.02
−0.40
+2.74
3.04−1.39
2.32+1.43−0.69
N/A
1.19+0.37−0.32
0.33+1.02
1.15+2.41
−0.70
−N/A

HWL16a-088
HWL16a-090
HWL16a-091
HWL16a-093
HWL16a-094
HWL16a-095
HWL16a-097
HWL16a-098

0.528
0.295
0.523
0.286
0.592
0.304
0.220
0.395

2.35+3.01−1.44
1.98+2.76−1.18
1.51+1.54−0.86
2.13+1.70−0.95
0.95+0.78−0.57
3.97+2.49−1.42
2.30+4.30−1.40
0.38+1.03−N/A

1.57+1.05−0.61
1.25+0.72−0.43
3.31+2.78−1.43
1.82+1.17−0.80
10.90+5.99−4.25
3.07+1.30−1.03
1.00+0.52−0.38
1.27+1.89−0.53

1.22+0.64−0.47
0.98+0.42−0.33
2.23+1.16−0.72
1.20+0.61−0.42
5.59+2.00−1.76
2.26+0.76−0.65
0.79+0.36−0.30
0.94+0.56−0.37

2.42+3.07−1.43
1.90+2.76−1.17
1.47+1.44−0.82
2.12+1.75−0.97
0.85+0.78−0.56
3.81+2.41−1.38
1.89+3.24−1.17
0.55+1.31−N/A

1.54+1.00−0.60
1.25+0.71−0.44
3.57+2.66−1.70
1.69+1.32−0.71
10.51+5.91−4.19
3.07+1.37−1.08
0.95+0.54−0.39
1.24+1.30−0.49

1.19+0.63−0.46
0.98+0.42−0.33
2.23+1.18−0.75
1.18+0.64−0.41
5.26+1.95−1.71
2.24+0.81−0.67
0.75+0.37−0.30
0.95+0.51−0.36

HWL16a-0101
HWL16a-0102
HWL16a-0103
HWL16a-0104
HWL16a-0107
HWL16a-0110
HWL16a-0112
HWL16a-0114
HWL16a-0115
HWL16a-0117

0.152
0.287
0.260
0.350
0.308
0.323
0.154
0.402
0.281
0.338

7.60+8.72−3.20
1.20+1.14−0.71
5.71+11.68−3.02
0.71+0.74−0.51
1.99+1.75−0.97
1.76+1.36−0.84
6.34+7.26−2.75
2.80+5.43−1.90
6.52+12.45−3.29
3.06+2.65−1.36

2.22+0.67−0.48
1.76+1.18−1.04
1.15+0.45−0.37
3.67+1.79−1.41
1.55+1.28−0.65
2.52+1.35−1.20
2.12+0.71−0.49
0.99+0.53−0.40
1.68+0.55−0.44
2.48+1.67−0.76

1.77+0.47−0.37
0.91+0.62−0.44
0.92+0.33−0.29
1.75+0.73−0.65
1.09+0.61−0.39
1.51+0.74−0.63
1.69+0.48−0.38
0.78+0.37−0.31
1.35+0.39−0.35
1.92+0.87−0.54

6.87+7.78−2.91
1.10+1.12−0.68
5.21+9.56−2.71
0.75+0.76−0.52
1.90+1.70−0.95
1.55+1.24−0.77
6.10+6.96−2.65
2.23+4.37−1.60
6.99+15.38−3.62
2.79+2.43−1.27

2.19+0.72−0.50
1.76+1.31−1.11
1.22+0.48−0.39
3.49+1.83−1.43
1.50+1.39−0.63
2.70+1.49−1.33
2.17+0.75−0.52
0.93+0.53−0.41
1.71+0.56−0.44
2.49+1.84−0.80

1.75+0.49−0.39
0.86+0.68−0.41
0.97+0.34−0.30
1.69+0.75−0.67
1.08+0.63−0.40
1.57+0.78−0.69
1.72+0.51−0.39
0.72+0.37−0.31
1.38+0.39−0.35
1.91+0.93
−0.55

M500c
[1014 h−1 M⊙ ]0.99+0.36−0.31
1.80+0.74−0.49
0.97+0.28−0.26
0.78+0.75−0.45

Planck2018 cosmology
c200c
M200c
[1014 h−1 M⊙ ]3.80+5.93
1.28+0.55
−2.01
−0.40
+2.92
2.99−1.41
2.23+1.35−0.63
N/A
1.20+0.36−0.33
0.31+1.04
1.05+2.24
−0.64
−N/A

M500c
[1014 h−1 M⊙ ]1.02+0.37−0.32
1.75+0.72−0.46
0.98+0.28−0.27
0.73+0.71−0.43