# The WaZP galaxy cluster sample of the Dark Energy Survey Year 1

M. Aguena,1,2 C. Benoist,1,3 L. N. da Costa,1,4 R. L. C. Ogando,1,4 J.
Gschwend,1,4 H. B. Sampaio-Santos,1,4 M. Lima,1,5 M. A. G. Maia,1,4 S. Allam,6
S. Avila,7 D. Bacon,8 E. Bertin,10,9 S. Bhargava,11 D. Brooks,12 A. Carnero
Rosell,13,14 M. Carrasco Kind,15,16 J. Carretero,17 M. Costanzi,18,19 J. De
Vicente,20 S. Desai,21 H. T. Diehl,6 P. Doel,12 S. Everett,22 A. E.
Evrard,23,24 I. Ferrero,25 A. Ferté,26 B. Flaugher,6 P. Fosalba,27,28 J.
Frieman,29,6 J. García-Bellido,7 P. Giles,11 R. A. Gruendl,15,16 G.
Gutierrez,6 S. R. Hinton,30 D. L. Hollowood,22 K. Honscheid,31,32 D. J.
James,33 T. Jeltema,22 K. Kuehn,34,35 N. Kuropatkin,6 O. Lahav,12 P.
Melchior,36 R. Miquel,17,37 R. Morgan,38 A. Palmese,29,6 F. Paz-Chinchón,16,39
A. A. Plazas,36 A. K. Romer,11 E. Sanchez,20 B. Santiago,1,40 M. Schubnell,24
S. Serrano,27,28 I. Sevilla-Noarbe,20 M. Smith,41 M. Soares-Santos,24 E.
Suchyta,42 G. Tarle,24 C. To,43 D. L. Tucker,6 and R.D. Wilkinson,11
(Affiliations can be found after the references) E-mail<EMAIL_ADDRESS>
(Accepted 16 Febuary)

###### Abstract

We present a new (2+1)D galaxy cluster finder based on photometric redshifts
called Wavelet Z Photometric (WaZP) applied to DES first year (Y1A1) data. The
results are compared to clusters detected by the South Pole Telescope (SPT)
survey and the redMaPPer cluster finder, the latter based on the same
photometric data. WaZP searches for clusters in wavelet-based density maps of
galaxies selected in photometric redshift space without any assumption on the
cluster galaxy populations. The comparison to other cluster samples was
performed with a matching algorithm based on angular proximity and redshift
difference of the clusters. It led to the development of a new approach to
match two optical cluster samples, following an iterative approach to minimize
incorrect associations. The WaZP cluster finder applied to DES Y1A1 galaxy
survey (1,511.13 deg2 up to $m_{i}=23$ mag) led to the detection of 60,547
galaxy clusters with redshifts $0.05<z<0.9$ and richness $N_{gals}\geq 5$.
Considering the overlapping regions and redshift ranges between the DES Y1A1
and SPT cluster surveys, all SZ based SPT clusters are recovered by the WaZP
sample. The comparison between WaZP and redMaPPer cluster samples showed an
excellent overall agreement for clusters with richness $N_{gals}$ ($\lambda$
for redMaPPer) greater than 25 (20), with 95% recovery on both directions.
Based on the cluster cross-match we explore the relative fragmentation of the
two cluster samples and investigate the possible signatures of unmatched
clusters.

###### keywords:

Galaxies: clusters: general – Galaxies: distances and redshifts – Methods:
data analysis – Surveys

††pubyear: 2021††pagerange: The WaZP galaxy cluster sample of the Dark Energy
Survey Year 1–Affiliations

##  1 Introduction

The abundance and clustering properties of galaxy clusters have been shown to
be powerful probes to constrain cosmological models, provided that their
astrophysical properties are well characterized and linked to theoretical
predictions (e.g., Lima & Hu, 2005; Vikhlinin et al., 2009; Mantz et al.,
2010; Benson et al., 2013; Weinberg et al., 2013; Planck Collaboration et al.,
2016; DES Collaboration et al., 2020).

Galaxy clusters can be detected from X-ray observations (Kim et al. 2007;
Lloyd-Davies et al. 2011; Adami et al. 2018) and from the Sunyaev-Zel’dovich
(SZ) effect (Bleem et al., 2015), but on-going and future large photometric
surveys constitute a very promising approach to build large controlled galaxy
cluster samples for both cosmological and astrophysical studies. These include
the Kilo Degree Survey (KIDS, de Jong et al. 2013), the Dark Energy Survey
(DES, The Dark Energy Survey Collaboration Flaugher 2005), Pan-STARRS (Kaiser
et al., 2002), the Legacy Survey of Space and Time (LSST, LSST Science
Collaboration et al. 2009) and the European Space Agency Cosmic Vision mission
(Euclid, Laureijs et al. 2011).

However, detecting and characterizing clusters through their galaxy component
remains a nontrivial task, especially when considering lower mass or higher
redshift clusters. One has to distinguish between gravitationally bound groups
of galaxies and projection effects due to the underlying large scale
distribution of galaxies. Projection effects not only impact detection, but
also several fundamental properties of detected clusters, such as centering,
redshift, and mass proxy (e.g., cluster richness).

Many automated algorithms were developed in the last three decades to overcome
these difficulties. Automatic optical cluster finders can generally be
described as algorithms searching for cluster scale galaxy overdensities.
Galaxies are first filtered (or weighted) following prescriptions to increase
the detection contrast relative to background galaxies. The main techniques
used for searching galaxy overdensities include kernel smoothing (e.g.,
Shectman, 1985; Lumsden et al., 1992; Adami et al., 2010; Gladders & Yee,
2000), Friends-of-Friends (e.g., Botzler et al., 2004; Trevese et al., 2007;
Wen et al., 2012) or Voronoi tesselation (e.g., Ramella et al., 1999; Soares-
Santos et al., 2011). These techniques have been applied to galaxy catalogs
that are usually previously filtered in one or several dimensions (e.g.,
magnitudes, colors, or photometric redshifts). More sophisticated approaches
assume an underlying cluster model (e.g., density profile, luminosity
function, color content) and identify clusters in likelihood maps based on
matched filter techniques (e.g., Postman et al., 1996; Olsen et al., 1999;
Koester et al., 2007; Olsen et al., 2008; Rykoff et al., 2014, 2016;
Bellagamba et al., 2018).

A typical assumption of optical cluster finders is to consider the presence of
a red sequence of galaxies (Gladders & Yee, 2000; Koester et al., 2007; Hao et
al., 2010; Murphy et al., 2012). In low redshift clusters, the most luminous
galaxies define a tight sequence in the color-magnitude diagram, the so-called
"E/S0 ridge line", or "red sequence". Red sequence galaxies have very uniform
colors and are among the reddest galaxies at a given redshift. Because of the
strong 4000 Åbreak in their rest-frame spectra, their color is tightly
correlated with redshift and can be used to estimate cluster redshifts. This
feature has been observed in rich clusters up to $z\sim 1.7$ (e.g., Mei et
al., 2009; George et al., 2011; Wetzel et al., 2013; Strazzullo et al., 2019).
However, some galaxy clusters observed at high redshifts can display
appreciable star formation, even in cluster cores (e.g., Brodwin et al.,
2013), weakening the red sequence.

Cluster finders such as maxBCG (Koester et al., 2007), redMaPPer (Rykoff et
al., 2014, 2016) or RedGOLD (Licitra et al., 2016) rely on the red sequence
for cluster detection and redshift estimate. In the context of recent surveys,
cluster finders not based on the red sequence usually rely on photometric
redshifts. An alternative based on the knee of the cluster luminosity function
was also used in the context of surveys with a limited number of passbands
(e.g., Postman et al., 1996; Olsen et al., 1999).

Even if current automated optical cluster finders are all able to identify
rich clusters, evaluating their performances over broad ranges of masses and
redshifts and deriving the selection function of the resulting cluster samples
remain highly complex tasks. On the theoretical side, this requires the
development of ever more realistic simulated galaxy catalogs. On the
observational side, we need multiple surveys covering the same area at
different frequency domains to detect clusters through a variety of
signatures.

There is not a unique methodological framework to evaluate and compare the
performances of optical cluster finders. A variety of approaches have been
proposed, based either on mock galaxy catalogs (e.g., Euclid Collaboration et
al. (2019) and references therein), or on real data, or even on a mix of the
two approaches (e.g., Goto et al., 2002; Kim et al., 2002; Rykoff et al.,
2014; Costanzi, M. & Rozo, E. et al., 2019).

Within simulations assumptions, simulation-driven methods provide a truth
table useful for comparison, with clusters embedded in realistic large scale
structures. These methods also offer a direct link between galaxy clusters and
dark matter halos. However, they rely on sophisticated modeling that so far
does not fully reproduce all observed galaxy properties, specially at high
redshift (e.g., DeRose et al., 2019). In addition to this fundamental problem,
mock catalogs do not usually reproduce the variety and complexity of defects
occurring in observed images and introduced at the stage of source extraction
and classification. DES has recently started to deal with this using the
Balrog algorithm (Suchyta et al., 2016), which embeds simulations into real
data and should accompany future releases.

Addressing the cluster selection function based on real data is necessarily
limited by the absence of an absolute reference to confront the results of any
cluster finder. Nonetheless, useful information can be extracted from the
cross-match of a given optical cluster sample with detections based on
different tracers that do not suffer from the same projection effects (e.g.,
Saro et al., 2015) A better understanding of the galaxy cluster selection
function can also be improved from cross-matching samples from different
optical cluster finders. The resulting samples may differ not only due the
different adopted physical assumptions but also due to the details of cluster
finder implementation (Ascaso et al., 2017; Aguena & Lima, 2018), or even the
way the algorithms deal with specific features of real data (e.g. noise,
missing data, star/galaxy separation, etc.).

DES has produced galaxy cluster samples with the redMaPPer algorithm which
were published in Rykoff et al. (2016) and McClintock, T. & Varga, T. N. et
al. (2019), based on DES Science Verification and DES-Y1 data releases,
respectively. These samples led to several studies focusing on the mass-
richness relation (Melchior et al. 2015; Saro et al. 2015; Zhang et al. 2016;
Palmese et al. 2016; Saro et al. 2017; Melchior, P. & Gruen, D. et al. 2017;
Pereira et al. 2018; McClintock, T. & Varga, T. N. et al. 2019; Bleem et al.
2020; Pereira et al. 2020). Complementary analyses were also performed on
cluster luminosity function (Zhang et al., 2019b), baryon content (Chiu et
al., 2018), and cluster miscentering relative to X-ray detections (Zhang et
al., 2019a). These clusters were also used for the detection of voids (Pollina
et al., 2019). Most of these studies contribute to the work on cosmological
constraints using DES first year release redMaPPer clusters (DES Collaboration
et al., 2020).

In this paper, we present the Wavelet Z Photometric (WaZP) cluster finder and
apply it to DES-Y1 data. WaZP is an optical cluster finder designed to detect
clusters based mainly on the spatial clustering of galaxies using photometric
redshift information. The primary motivation for developing WaZP is to limit
assumptions on the properties of cluster galaxies such as the presence of a
red sequence, the shape of their luminosity function or radial profile,
assumptions that may impact cluster detection, in particular at high redshift
or at lower mass regime.

Here, the WaZP DES-Y1 sample is compared to cluster samples obtained from the
SPT survey based on the SZ effect and those obtained by the redMaPPer cluster
finder on the same DES-Y1 data set. The first comparison allows to test how
well the WaZP algorithm recovers the massive clusters detected by the SPT. The
second comparison, for which the two samples have similar cluster densities,
gives insights on the relative completnesses of the two optical cluster
samples, and on the derived properties of the common detections. Variations
may occur in the samples due to the different assumptions made in terms of
cluster modelling. They may also occur due to different uses of the underlying
galaxy dataset as the WaZP algorithm uses magnitude information from all bands
through photometric redshifts and $i$-band as a reference band, whereas
redMaPPer uses combinations of band pairs (colors) to select likely red
sequence galaxies and $z$-band as a reference. Considered survey coverage can
therefore be slightly different with one approach or the other. Depth
variability in all bands will also impact differently cluster detection with
each algorithm. While the comparisons performed here provide a valuable
heuristic approach to partly qualify the cluster samples, a complete
evaluation of the WaZP sample requires to address a quantitative assessment of
its purity, a work that will be presented in a companion paper based on mock
galaxy catalogs.

The present paper is organized as follows. In Section 2, we describe the DES
Y1 data used in our analysis. In Section 3, we describe the main properties of
the WaZP cluster finder. In Section 4, we present our main results on DES-Y1
data. In Section 5 we matched the derived WaZP cluster samples to the SZ
sample and to the redMaPPer sample obtained from DES-Y1 data. Finally, in
Section 6 we analyze the properties of the catalog, and discuss the
differences between our catalog and the others compared in Section 5.

Throughout this work, we fix cosmological parameters from the Planck results
(Planck Collaboration et al., 2016) for a flat $\Lambda$CDM model with
$\Omega_{m}=0.308$ and $H_{0}=67.8$ km s-1 Mpc-1.

##  2 Data

The DES is an imaging survey covering 5,000 deg2 in 5 bands ($g,r,i,z,Y$ )
(e.g., Flaugher, 2005; Diehl et al., 2016; Dark Energy Survey Collaboration et
al., 2016). In this paper, we use the DES Year 1 data release, which has been
extensively studied by the DES collaboration (e.g., Troxel et al., 2018; Shipp
et al., 2018). The DES built the Dark Energy Camera (DECam, Flaugher et al.,
2015) with a field of view diameter of 2.2 deg covered by 520 Megapixels
distributed on a mosaic of 62 CCDs that are extra sensitive on the red part of
the electromagnetic spectrum, enhancing its capability of observing high
redshift galaxies. DECam is installed on CTIO 4-meter Blanco telescope prime
focus, and its observations follow a strategy that optimizes pointings based
on properties like weather and moon phase (Neilsen et al., 2019). The images
are reduced and calibrated by the DES Data Management (DESDM) team at the
National Center for Supercomputing Applications (NCSA). The DESDM pipeline
includes the reduction of single-exposure images, their co-addition into
deeper images, source extraction and calibration, all resulting in the
creation of the main scientific catalog (Morganson et al., 2018).

The DES Year 1 Annual Release (Y1A1, Abbott et al., 2018) co-added catalog
used in this analysis covers a total area of $\sim$1,520 deg2, split into two
main wide regions. One of them has an area of $\sim$140 deg2 overlapping the
SDSS Stripe 82 area (Aihara et al., 2011). The other part has an area of
$\sim$1,380 deg2 overlapping the South Pole Telescope footprint (SPT,
Carlstrom et al., 2011). In the following, we will refer to these two regions
as Y1-S82 and Y1-SPT, respectively. They were observed with three to four
exposures in each filter (Drlica-Wagner et al., 2018).

In addition to catalogs, we also use ancillary maps to track defects and
foreground objects (e.g., bright stars, very bright galaxies, and globular
clusters) all over DES footprint. The resulting coverage map is represented by
a detection fraction map, where pixels have values of area fraction from 0 to
1. We also use systematic maps to track observing conditions across the
footprint, such as number of exposures, seeing, and airmass (Leistedt et al.,
2016). These maps are combined to produce depth maps based on galaxy magnitude
limits, as described in Rykoff et al. (2015). All maps are recorded in Healpix
format (nside = 4096) (Górski et al., 2005).

The Y1A1 coadd catalog and maps produced by DES DM were transferred to the
Laboratório Interinstitucional de e-Astronomia
(LIneA)111http://www.linea.gov.br and ingested into the database associated to
the DES Science Portal (henceforth, the Portal) as described in Fausti Neto et
al. (2018). We used the Portal infrastructure to create a galaxy Value Added
Catalog (VAC) tailored for galaxy cluster search based on photometric
redshift. The creation of the galaxy VAC includes: computation of photometric
redshifts, star-galaxy classification, and pruning regions and objects to
produce a clean galaxy catalog with well controlled levels of completeness and
homogeneity. Along with the VAC, a final footprint map in Healpix format is
created, reflecting the selection and pruning applied to this VAC.

The computation of photometric redshifts relies on the machine-learning
algorithm DNF (Directional Neightbourhood Fitting, De Vicente et al. 2016),
operated in Euclidean Neighborhood Fitting (ENF) mode since tests using DES Y1
data have shown that ENF mode is considerably faster, while providing similar
results as in DNF mode. DNF uses as input observables SExtractor MAG_AUTO
magnitudes (Bertin & Arnouts, 1996). DNF was trained with a large sample of
spectroscopic redshifts extracted from a compilation of 29 public surveys
intercepting the DES footprint. Quality flags of all surveys are brought to a
common standard following OzDES approach (Yuan et al., 2015). As described in
Gschwend et al. (2018), sources with flags 0 and 1 have unknown redshift, flag
2 redshifts are not reliable, flag 3 redshift reliability is above 90%
confidence, and flag 4 is attributed to a trusted redshift (over 99%
confidence). The DES photometric catalog is matched to this spectroscopic
redshift sample with a 1.0 arcsec search radius and down to $m_{i}=23$ mag,
producing a catalog of 101,971 galaxies with a mean redshift of 0.63, and
covering the redshift range $z=0-1.1$. Although color and magnitude
distributions of this spectroscopic sample differ from the global photometric
set under study, we stress that it does cover the same color-magnitude ranges
with the exception of faint low redshift galaxies (typically magnitudes
fainter than 19 and redshifts below 0.15). The spectro-photometric catalog is
then randomly split into a training and a validation sets. Details about all
the steps carried out to compute photometric redshifts in the DES Science
Portal for Y1A1 data are described in Gschwend et al. (2018).

Star-galaxy classification follows a morphological prescription developed
within the DES consortium called MODEST, described by equations (3) and (4) of
Sevilla-Noarbe et al. (2018). It mainly depends on the SExtractor SPREAD_MODEL
(Desai et al., 2012; Bouy et al., 2013) and its error, assessing how extended
is the source to the local PSF. Note that this classification is based on the
DES $i$-band and extends to the faintest sources.

Based on bad regions maps (Drlica-Wagner et al., 2018), areas around bright
stars, bleed trails, bright foreground galaxies, or globular clusters (from
Harris, 2010) are removed from the footprint. Pixels with an effective
coverage $\leq$ 0.1 are discarded from our analysis. We also exclude regions
not covered simultaneously by $g,r,i,z$ bands with a minimum of 90s total
exposure time. Besides region-based filtering, we also discard individual
sources based on SExtractor FLAGS (only sources with FLAGS $\leq$ 3 are kept),
apply a magnitude cut ($m_{i}\geq 15$), and color cuts ($-2.0\leq
m_{g}-m_{r}\leq 10.0$; $-2.0\leq m_{r}-m_{i}\leq 4.0$; $-2.0\leq
m_{i}-m_{z}\leq 4.0$).

Depth maps, defined here as 10$\sigma$ limiting magnitude maps, were built
following the method described in Rykoff et al. 2015. These maps correspond to
the limit where the flux is at least 10 times its variance $\sigma$, computed
from the magnitude errors. This definition assures a galaxy completeness
larger than 90%. Figure 1 shows what survey area fraction is covered at a
given 10$\sigma$ $i$-band limiting magnitude (the $i$-band being the reference
band in this work). The whole survey is at least as deep as $m_{i}=22.27$ and
shallower than $m_{i}=23.25$ with half of the survey area reaching
$m_{i}=22.7$. In Figure 2 we compare galaxy number counts for Y1-S82 and
Y1-SPT with number counts from Arnouts et al. (2001) and Capak et al. (2007)
as compiled by Nigel Metcalfe222http://astro.dur.ac.uk/
nm/pubhtml/counts/counts.html. The surveys used for comparison are deeper than
our own, and number counts are comparable through a wide range of magnitudes
up to $m_{i}\sim 22.5$ mag, beyond which we observe an increasing deficit of
galaxies, consistent with the median depth of the survey.

Figure 1: The effective area coverage of DES-Y1 as a function of the limiting
magnitude in the $i$-band.  Figure 2: Galaxy density for Y1-S82 (thick blue
solid line) and Y1-SPT (thick green dashed line) compared to Arnouts et al.
2001 (A01), Capak et al. 2004 (C04) and Capak et al. 2007 (C07). Magnitude
limits of 22.5 mag and 23 mag are also noted as red vertical dotted and dashed
line, respectively.

Based on the survey depth shown above, the present analysis considers galaxies
down to a limiting magnitude $m_{i}=23$ (98% of the survey area). The
resulting galaxy VACs for Y1-S82 and Y1-SPT contain, respectively, 4,721,380
and 45,206,403 galaxies, in a total of 49,927,783 galaxies covering 1,511.13
deg2. Both regions have similar galaxy number densities and mean photometric
redshift (Table 1).

Figure 3 presents the projected galaxy distribution of Y1-S82 (top) and Y1-SPT
(bottom)333These were produced with skymapper by Peter Melchior. The
distributions are fairly uniform and galaxy densities comparable. Holes caused
by masking and dents on the footprint caused by unobserved regions can be seen
in both regions.

Normalized photometric redshift distributions are shown in Figure 4 for both
regioggns. Our magnitude cut leads to a mean $z_{phot}\sim 0.63$ in both cases
and very similar distributions. They both suffer from a counts drop around
$z\sim$0.4 due to the lack of the $u$-band. This can also be seen in the left
panel of Figure 5 where photometric redshifts are compared to spectroscopic
redshifts for a validation sample of 50,476 galaxies built during DNF
processing. There is an excellent correlation between spectroscopic and
photometric redshifts; however, galaxies around $z_{spec}\sim$0.3 show a very
large scatter, especially towards higher values of $z_{phot}$. This can also
be seen in the right panel of Figure 5, where we assessed the global quality
of the photometric redshifts by characterizing the average bias and standard
deviation of $(z_{ph}-z_{sp})/(1+z_{sp})$. These points will be examined in
detail in sections 4 and 6.

Figure 3: Spatial distribution of galaxies with $m_{i}\leq 23$ for Y1-S82 (top) and Y1-SPT (bottom). Color bar displays density of galaxies per pixel at NSIDE=1024 ($\approx 11$ arcmin2).  Region | Galaxies | Area | Density | Mean
---|---|---|---|---
|  | (deg2) | (Gal./arcmin2) | photo-z
Y1-S82 | 4,721,380 | 143.66 | 9.13 | 0.65
Y1-SPT | 45,206,403 | 1,387.47 | 9.05 | 0.63
Table 1: VACs properties for both Y1-S82 and Y1-SPT. Figure 4: Normalized
distribution of photometric redshifts for galaxies in Y1-S82 (dashed line) and
Y1-SPT (solid line).

Figure 5: The DNF photometric redshifts compared to their associated
spectroscopic redshifts for a validation sample of 50,476 galaxies within the
DES-Y1 footprint. The right panel shows the bias and scatter of photometric
redshifts relative to spectroscopic redshifts as a function of spectroscopic
redshift.

##  3 The WaZP cluster finder algorithm

The Wavelet Z-Photometric (WaZP) cluster finder is designed to detect galaxy
clusters from multi-wavelength optical imaging galaxy surveys. It searches for
projected galaxy overdensities in photometric redshift space without any
assumption on the red sequence. In a nutshell, WaZP first slices the galaxy
catalog in photometric redshift space, and then generates smooth wavelet-based
density maps for each slice where peaks are extracted (see Figure 6). These
overdensity peaks are then merged to create a unique list of clusters and
associated galaxy members. Hereafter, these various steps are described in
detail.

Figure 6: A schematic view of the WaZP algorithm. Here are shown the wavelet
filtered galaxy density maps corresponding to three consecutive photometric
redshift slices. It shows how a cluster can propagate along several slices.

  1. 1.

Slicing in photometric redshifts. By photometric redshift slices, we mean here
the photometric redshift support over which individual galaxy redshift PDF’s
are integrated around a given redshift of interest. Therefore, at a given
considered redshift, galaxies are weighted by that quantity. These weights are
used to build density maps at different redshifts or estimate richnesses as
described in the next steps.

The adopted strategy to define photometric redshift slices is based on the
statistical comparison of the "best-estimate" discrete photometric redshifts
(taken here as the mean of the galaxy redshift PDF) and corresponding
spectroscopic redshifts if available. Based on available spectroscopic
samples, the mean bias and scatter of photometric redshifts ($z_{ph}$)
relative to spectroscopic redshifts ($z_{sp}$) were derived. Following the
standard way to evaluate the performance of photometric redshifts (e.g.,
Ilbert et al., 2006), we computed statistics of $(z_{ph}-z_{sp})/(1+z_{sp})$
as a function of both $z_{sp}$ and $z_{ph}$. The location and width of a slice
is then built in such a way that it includes 95% of the galaxies of a given
spectroscopic redshift. The separation between two slices corresponds to a
fourth of their width assuring a sufficient overlap to avoid missing clusters
being between two consecutive slices.

  2. 2.

Generation of galaxy number density maps. WaZP does not consider photometric
redshifts as discrete values. Instead, it operates with redshift PDF’s when
provided, or generates them from the errors provided by the chosen photometric
redshift algorithm. In each one of the slices defined above, galaxies are
weighted by the integral of their redshift PDF over that slice. The resulting
weighted RA-Dec distribution is then pixelized on a grid with a step of
physical size 1/16th of a Mpc. This image is finally filtered using the
wavelet task MR_FILTER from the multi-resolution package MR/1 (Starck et al.,
1998). This task incorporates a statistically rigorous treatment of the
Poisson noise, which allows us to keep significant structures in the desired
scale range. Here we select structures with scales in the range $0.5-2$ Mpc,
typical of cluster scales, and apply a 3$\sigma$ iterative multi-resolution
thresholding with a B-spline wavelet transform.

  3. 3.

Extraction of peaks. The smooth density maps obtained in the previous step are
segmented, and in each object domain, one or more peaks are extracted. In the
case of several peaks in one domain, depending on the distance between a peak
and the closest saddle point, the peak can be merged or preserve its identity.
Pixels of a domain are then distributed to peaks by proximity.

  4. 4.

Assessing peak significance The peak significance is chosen to be computed in
a radius of $R_{S}=300$ kpc, a radius that encloses typical cluster cores
(Adami et al., 1998). To perform background statistics, the survey is
pixelized with pixel areas equal to $\pi R_{S}^{2}$. Any pixel intersecting a
bad region or an edge is removed. Standard counts in cells are then applied to
estimate the mean density ($N_{bkg-global}$) and standard deviation
($\sigma_{bkg}$). The significance, defined as $SNR=(N-N_{bkg-
global})/\sigma_{bkg}$, where $N$ is the total density of galaxies in a
cylinder centered at the peak position, with a length that is the width of the
redshift slice and an angular radius $R_{S}$.

  5. 5.

Peak merging along the redshift direction. As $z_{ph}$ slices overlap, one can
expect clusters to be detected in several consecutive slices. To build the
final list of clusters, peaks of consecutive slices are associated, and only
the slice in which the system has maximum significance is kept. Note that two
clusters can be deblended along the line of sight if their distance in
redshift is larger than 2$\times 3\sigma_{dz/(1+z)}$ where $\sigma_{dz/(1+z)}$
denotes here the 68th percentile of the $dz/(1+z)=(z_{ph}-z_{sp})/(1+z_{sp})$
distribution.

  6. 6.

Centering and cluster redshift. The cluster center is defined as the location
of the density map peak. However, if the brightest cluster member is found
within the first neighbouring pixels, then this galaxy marks the center. This
leads to a maximum shift of 100 kpc from the peak location. Concerning the
redshift, an initial value is derived as the mode of the sum of the galaxy
redshift PDF’s within a 0.5 Mpc radius around the cluster center. This value
is refined iteratively based on the membership probabilities described below.

  7. 7.

Assignment of membership probabilities. Membership probabilities ($P_{mem}$)
are computed following the prescription given in Castignani & Benoist (2016).
In a nutshell, galaxies of the cluster field are piled up in a 3-dimensional
grid (cluster-centric distance, magnitude, photometric redshift) where
magnitudes and redshifts are included as probability distribution functions.
The same is done for local background galaxies in (magnitude, photometric
redshift) space. The local background galaxies are selected in a ring from 3
to 6 Mpc to the cluster center, whereas cluster field galaxies are selected
within a 3 Mpc disk. The membership probability is the combination of the
probability to be at the cluster redshift and the probability not to be a
background galaxy. The final membership probability at a given cluster-centric
distance, magnitude, and redshift is derived from the density ratio between
the cluster field and the background field. Note that, as in Castignani &
Benoist (2016), no parametric modelling is used for the radial density, nor
for the luminosity function.

  8. 8.

Richness and radius The cluster richness and radius are estimated jointly. The
richness is the sum of the membership probabilities within a radius that
corresponds to an overdensity of 200 times the mean galaxy background number
density (similar to Hansen et al. 2005). This is done considering galaxies,
both in the field and in the cluster, down to a given fraction of $L^{*}$
luminosity. Practically, galaxies brighter than ${\rm
m}^{\star}(z_{cluster})+\delta{\rm mag}$ are counted, where ${\rm m}^{\star}$
is the characteristic magnitude marking the knee of the luminosity function
and $\delta{\rm mag}$ is a fixed quantity, chosen here to be 1.5. The adopted
definition allows to produce "redshift independent richnesses", in the sense
that the same cluster seen at two different redshifts would have the same
richness. The evolution of the characteristic luminosity of the Luminosity
Function can be described by a passively evolving population formed in a
single burst (e.g., Lin et al., 2006). In this study, we derive ${\rm
m}^{\star}(z)$ from the passive evolution of a burst galaxy with a formation
redshift $z_{form}=3$ taken from the PEGASE2 library (burst_sc86_zo.sed, Fioc
& Rocca-Volmerange, 1997). It is calibrated using the value of ${\rm
K}^{\star}$($z=0.25$) derived by Lin et al. (2006) from an observed cluster
sample. The choice of $\delta{\rm mag}$ is critical as it sets the redshift
limit (${z}_{lim}$) of the final cluster sample through the relation ${\rm
m}^{\star}({z}_{cluster})+\delta{\rm mag}={\rm mag}_{lim}$, where ${\rm
mag}_{lim}$ is the survey apparent magnitude limit.

##  4 Application to DES-Y1 survey

###  4.1 Running WaZP cluster finder

As described in section 2, the DES-Y1 survey is split into two regions ("SPT"
and "S82"), for which two galaxy VACs are produced to feed the WaZP pipeline.
These catalogs are built based on the $i$-band, chosen here as a reference
band both for star-galaxy separation, and for defining apparent magnitude
cuts. Given this selection, cluster detection was performed with the same
setting independently from the position on the sky, assuming a sufficient
homogeneity over the whole survey. This is an approximation as we have seen
above that in some regions magnitude completeness limit can be lower by as
much as 1 mag.

The redshift limit of the constructed WaZP sample is constrained by the depth
of the survey reference band. It also depends on the adopted definition of the
richness estimate and in particular, the adopted magnitude limit used to count
galaxies entering the richness. We assume here that the same cluster, seen at
two different redshifts, would get the same richness by counting its galaxies
down to an apparent magnitude ${\rm m_{i}}^{\star}(z)+\delta_{mag}$, where
$\delta_{mag}$ is a fixed quantity and ${\rm m_{i}}^{\star}(z)$ is defined in
section 3. This can be achieved as long as this quantity remains lower than
the $m_{i}$-band depth of the survey. In the present case, richnesses are
chosen to be computed including galaxies down to ${\rm m_{i}}^{\star}+1.5$.

Based on the above considerations, given that some regions are not deeper than
$m_{i}=22.27$, there is an upper redshift limit, $z=0.60$, above which
richnesses start to become incomplete depending on the survey location. At the
limiting magnitude of the galaxy VAC, $m_{i}=23$, which corresponds to a
redshift limit $z=0.76$, richnesses are complete within only 2% of the survey
area. Based on the 10$\sigma$ $i$-band survey depth map (section 2), we have
derived a map indicating our local cluster $z_{max}$ at each position of the
survey, that is reported in the WaZP cluster catalog. Detection is performed
to slightly larger redshifts ($\sim 0.9$), but for clusters that would be
detected beyond their local $z_{max}$, their galaxy luminosity function is not
sampled homogeneously across redshifts and therefore richnesses for these
clusters would require some correcting factor. In this paper, we are not
introducing such a correction, and therefore richnesses are consistent over
the whole survey only up to $z=0.60$. As a lower redshift limit for cluster
detection, we adopted in this paper the value $z_{min}=0.05$.

Besides the considerations on redshift limits above, we also need to assess
the global quality of our photometric redshifts and determine the photometric
redshift slicing strategy on running WaZP. As can be seen in the right panel
of Figure 5, the average bias remains relatively modest and the scatter
roughly constant in the adopted redshift range. These properties should not
prevent cluster detection in general. Note, however, that this point will be
discussed in more details in section 6.

Operationally, the WaZP cluster finder runs on small sections of the sky. The
LIneA Science Portal manages data tiling, launches the code on each tile on
parallel cores and concatenates the final catalogs, both clusters and galaxy
members. The data tiling consists in dividing the survey in overlapping
rectangular tiles of typical area 20 deg2. The overlaps are set to assure a
tiling independent cluster detection for clusters with redshifts larger than
0.05 that would fall at the intersection of two tiles.

###  4.2 The WaZP cluster catalog

The WaZP pipeline was run on the DES Y1-S82 and Y1-SPT regions defined above.
This results in the detection of cluster candidates, not confirmed galaxy
clusters. However, these candidates will be referred to as "WaZP clusters"
throughout the paper for simplicity. For the combined sample, it led to the
detection of 60,547 clusters in the redshift range $z=0.05$ to $z=0.91$ with
richness $N_{gals}\geq 5$, corresponding to densities of 40.47 and 39.45
clusters/deg2 for Y1-S82 and Y1-SPT respectively . If we restrict to a sample
with more reliable redshifts and complete richenesses i. e. $0.1\leq z\leq
0.60$, we find 39,439 clusters, with a higher consistency in cluster densities
with 26.25 and 25.71 clusters/deg2 for the two DES regions. This result
supports strongly the high homogeneity over the sky of the galaxy VAC
construction, including photometric redshift computation, as well as the
subsequent cluster detection. This can also be seen in Figure 7 where the
projected distribution of detected clusters on the sky is shown. A description
of the WaZP cluster catalog is provided on LIneA’s
website444https://www.linea.gov.br/catalogs/wazp/.

Figure 7: Projected density of WaZP clusters for both Y1-S82(top) and
Y1-SPT(bottom) regions of DES first year data release. The green line is the
contour of the redMaPPer Y1 catalog and the blue line is the region of
detection for the SPT survey (Bleem et al. 2015).

The ranges of richnesses and redshifts covered by WaZP clusters are shown in
Figure 8. From the color coded SNR we can see that for a given richness, as
expected, the SNR decreases with redshift. This is mainly due to the
increasing scatter in the photometric redshifts leading to an increase of the
mean background density of galaxies. We see that above redshift $0.60$, the
number of rich clusters start to diminish rapidly, and above $0.76$ there is
only one cluster with richness greater than $100$.

Figure 8: Richness as a function of redshift for WaZP clusters color coded by
signal-to-noise ratio. The dashed line indicates the redshift (0.60) above
which cluster richnesses start to become incomplete and the dotted line marks
the redshift (0.76) above which all clusters have incomplete richnesses.

The redshift distribution of WaZP clusters is shown in Figure 9. The global
bell shape of the counts looks as expected except for a sharp concentration of
clusters at $z\sim 0.45$, similar to that observed on the galaxy photometric
redshift distribution of Figure 4. This peak becomes more prominent for lower
richness systems. In the next sections we investigate further the nature of
this peak.

Figure 9: The counts distribution of WaZP clusters detected in the combined
Y1-S82 and Y1-SPT DES regions as a function of cluster redshift for three
different richness cuts.

There are several ways to estimate the quality of the WaZP redshifts
($z_{\textsc{W}a\textsc{ZP}}$). They can be compared to known cluster
redshifts as shown in the next section, or, as done here, cluster members can
be cross-matched with available spectroscopic galaxy samples. The adopted
procedure here is to search for all galaxies with spectroscopic redshifts
within 0.5 Mpc around each detected cluster and likely to be cluster members.
We considered that a cluster could be associated with a spectroscopic redshift
if at least 5 galaxies were found within a range of $\pm 2000$ km/s. The
selected velocity window is the one that maximizes the number of spectroscopic
galaxies. The cluster spectroscopic redshift is then defined as the median of
the redshifts in that window. We also associated a redshift in the case the
central WaZP cluster galaxy has a spectroscopic redshift. This may lead to a
few outliers but increases statistics by a factor of 10. Based on public
spectroscopic surveys, 131 WaZP clusters covering the redshift range $z=0.05$
to $z=0.9$ could be associated a spectroscopic redshift with at least 5
concordant redshifts, and 1,859 clusters could be associated a spectroscopic
redshift based on their central galaxy. In Figure 10, the comparison with WaZP
redshifts is shown. Both spectroscopic redshift assignments led to the same
statistical differences with WaZP redshifts: an average bias of $\sim$0.014
and scatter of $\sim$0.026.

Figure 10: Comparison of cluster redshifts as derived by WaZP with estimated
spectroscopic redshifts. WaZP cluster members are cross matched to all
publicly available spectroscopic redshifts falling in the DES footprint. A
cluster spectroscopic redshift is derived each time at least 5 concordant
redshifts (within $\pm 2000$ km/s) are found within 0.5 Mpc to the cluster
center (blue points, 131 clusters), or if the WaZP central galaxy has a
spectroscopic redshift (red points, 1859 clusters).  Figure 11: Volume density
of WaZP clusters as a function of redshift in Y1-SPT (solid line) and Y1-S82
(dashed line) regions with different richness thresholds. The errorbars
correspond to Poisson noise.

Figure 11 shows the evolution of the volume density of clusters with redshift,
with Poisson errorbars. We can see that the density for both Y1 regions agree
with each other for both richness cuts.

##  5 Comparison to other cluster catalogs

In this section, we compare the WaZP Y1A1 clusters identified in the previous
section to those derived by other methods covering the same region. A first
comparison is made with clusters detected by the South Pole Telescope (SPT)
survey via the Sunyaev-Zel’dovich (SZ) effect (Bleem et al., 2015). A second
comparison is done with clusters detected by the redMaPPer optical cluster
finder, based on the same photometric data but using an algorithm searching
for overdensities of red sequence galaxies (Rykoff et al., 2014). These
comparisons are based on matching clusters from two different catalogs. This
is a complex operation that has led to a variety of proposed algorithms (e.g.,
Gerke et al., 2005; Knobel et al., 2009; Cucciati et al., 2010; Gerke et al.,
2012; Euclid Collaboration et al., 2019). We point out that adopting one
algorithm or another or using different configurations of the same may result
in different cluster associations. In particular, when a given cluster can
potentially be connected to several counterparts. As detailed below, we have
experimented with several of these approaches and finally adopted a hybrid
iterative procedure to optimally solve multiple matches. This appeared to be
the only way to avoid having systems left unmatched due to a wrong matching of
their obvious counterpart. Several such cases appeared in particular with
interacting clusters for which richness rankings were reversed. The resulting
pairing seems, from intensive visual inspection, optimal for addressing
statistically the different properties of the commonly detected clusters
(centering, redshift), and evaluating systems without any counterparts. In
particular, our matching procedure allowed us to decrease the number of
incorrect matches of rich clusters on each side significantly.

In this paper, we use a cylindrical matching where we require the angular
distance of cluster centers to be smaller than some defined length (be it
their respective radii or a fixed physical distance), and their redshift
separation to be constrained by the typical redshift errors from both samples.
In carrying out this comparison, some issues have to be considered. First, the
cluster radius definition for each cluster sample may be different. Second, we
must define the redshift window to be used. It should be large enough to take
into consideration the errors in photometric redshift assigned to clusters in
both samples. However, a large window may lead to an increased number of
multiple matches that need to be resolved as discussed in more detail below.
Finally, it is crucial to ensure that the projected area of overlap of the
samples is properly taken into account. To do that, footprints of the samples
are used to flag clusters falling outside the overlapping regions or near
their edges. This flag is useful when unmatched clusters are near the edges,
in which case they can be removed from the matching statistics.

As mentioned above, we compare the combined Y1-S82 and Y1-SPT clusters
identified by WaZP with those in the SZ and redMaPPer samples. In the first
case, we take the SZ sample as reference, treating the SZ clusters as true
representatives of the underlying mass distribution and test how well these
systems are recovered. In the redMaPPer case, we investigate the unmatched
cases in both directions to understand the specificities or the possible
limitations of each algorithm.

###  5.1 WaZP versus SPT clusters

The SZ sample (Bleem et al., 2015) covers an area of 2,500 deg2 (seen in
bottom panel if Figure 9), within which 516 clusters (out of 677 candidates)
were detected with signal-to-noise above 4.5. In Bleem et al. 2015, it is
stated that the catalog is highly complete for $M_{500^{c}}\geq 7\times
10^{14}M_{\odot}h^{-}1$ and $z\geq 0.25$. It is also mentioned that there were
a number of optical followups to confirm these clusters. Therefore, this
catalog will be utilized to validate the detectability of WaZP regarding
massive clusters. We only use the 331 SZ clusters that have information on
mass and redshift, and are located within the overlap with DES (external
envelope of the DES Y1-SPT region). We should stress that the redshifts
assigned to SZ clusters from Bocquet et al. (2019) are both spectroscopic
(106) and photometric (225).

We considered a one-way match, taking SZ clusters as reference and we looked
for WaZP clusters falling within SPT clusters radii. The adopted radius is
$R_{200}$, the radius where the average cluster overdensity is 200 times the
critical density, i.e., $\Delta=3M_{200}/4\pi
R_{200}^{3}\bar{\rho}_{crit}(z)=200$. It was computed from the available
values of $M_{200}$ and converted to an angular radius $\theta(R_{200})$ using
Planck Collaboration et al. (2016) cosmology. Here,
$\theta_{200}=R_{200}/D_{A}(z)$, where $D_{A}(z)$ is the comoving angular
diameter distance to the cluster redshift $z$. For a match to happen, the
redshift separation also has to fall within the interval defined by the
combined redshift errors. This interval was defined as $3\sigma$ where
$\sigma$ is the sum of redshift errors provided by the two catalogs. The
resulting matching cylinder is quite large in order to account for possible
centering offsets between the two wavelength domains and large redshift
discrepancies. The statistical comparison, a posteriori, of the differences in
centering and redshifts of the matched systems, allows us to evaluate the
adopted matching criteria.

Applying this method to the SZ and WaZP samples, we find that 292 SZ clusters
(out of 331) have at least one WaZP counterpart. Among these, 141 have only
one candidate for matching, while the rest have a multiplicity function as
shown in Figure 12. The large fraction of multiple matches is not surprising
considering the very different selection functions of the two samples, their
relative densities and the adopted matching criteria. The multiple WaZP
matches were resolved by choosing the richest associated counterpart.

Figure 12: Number of WaZP clusters found to match individual SZ clusters.

Out of the 39 unmatched SZ clusters, 12 are located near the WaZP footprint
edges and 27 have redshifts beyond $z=0.76$, where WaZP cluster finder reaches
its expected limit of completeness for DES-Y1 (9 of those have $z>1.1$, and
are completely beyond WaZP reach). There was one unmatched SZ cluster (SPT-
CLJ2218-5532) with $z=0.77$, just above the redshift limit for WaZP clusters
with complete richnesses. However, the local $z_{max}$ at this cluster
position is $0.71$ and, upon visual inspection, we found no clear visible
optical counterpart.

In Figure 13, we show the characteristics of the matching for three mass bins,
both in terms of angular separation (left panel) and redshift separation
(right panel). The average distance of WaZP-SZ centers is
$\langle\Delta\theta\rangle=0.16\,R_{200}$, with 80% of clusters within
$0.2\,R_{200}$ and 95% within $0.8\,R_{200}$. If we consider the different
mass bins in the figure, there is a small systematic improvement on
$\langle\Delta\theta\rangle$ for higher masses ($0.177$, $0.178$ and $0.134$
respectively), even though there is only $\approx 100$ clusters per mass bin.
We also see that there is a reasonable agreement in redshift, with 79% of
matches within the average redshift uncertainties of the clusters $\langle
z^{err}\rangle=\sqrt{\langle
z^{err}_{\textsc{W}a\textsc{ZP}}\rangle^{2}+\langle z^{err}_{SZ}\rangle^{2}}$
(gray shaded region). There is also a slight improvement on redshift scatter
and bias was we look at higher mass bins.

We also compare the photometric estimates of WaZP redshifts with those
assigned to SZ clusters. As 93 of the matched SZ clusters have been assigned a
spectroscopic redshift (Bocquet et al., 2019), we can assess the accuracy of
the estimated WaZP redshifts. Figure 14 shows the distribution of redshift
separations (left) and relation of redshifts (right), splitting the SZ cluster
redshifts into spectroscopic and photometric subsamples. As can be seen, WaZP
redshifts show a good agreement with SZ spectroscopic redshifts and some
residual relative bias when compared with SZ photometric redshifts. A
quantitative description is provided in table 2, which gives in column (1) the
type of sample; in column (2) the number of clusters and in columns (3) and
(4) the bias and the scatter, defined as the mean and standard deviation of
$(z_{\textsc{W}a\textsc{ZP}}-z_{\textsc{SZ}})/(1+z_{\textsc{SZ}})$, and column
(5) is the combined redshift errors $\langle z^{err}\rangle/(1+z_{SZ})$. When
compared to photometric redshifts, we measure a relative bias of $\sim 0.015$
and a scatter $\sim 0.029$, a value similar to the combined redshift error
0.030. However, when comparing to SZ clusters with spectroscopic redshifts,
WaZP redshifts are almost unbiased (bias $=0.007$), and show a significantly
lower scatter ($\sigma=0.017$, also compatible with the combined errors
0.015), corresponding to roughly half the average galaxy photometric redshift
scatter. It is interesting to note that in all samples, the scatter was very
close to the redshifts uncertainties, even though the matching conditions only
imposed a redshift difference of three times the errors. From the right panel
of Figure 14, it can also be seen that the moderate average bias is actually
mainly due to low redshift clusters ($z\lesssim 0.4$).

Figure 13:  Distribution of angular distances (left panel) and redshift
separations (right panel) of WaZP-SZ matched clusters binned by
$\log(M_{200}[M_{\odot}h^{-1}])$. The gray shaded region on the right plot is
the combined average photometric redshifts uncertainties of the matched
clusters from both catalogs divided by $1+z_{SZ}$.

Figure 14:  Distribution of redshift separations (left) and relation between the redshifts (right) of WaZP-SZ matched clusters split into spectroscopic and photometric redshift samples. Sample | N | bias | scatter | errors
---|---|---|---|---
All redshifts | 292 | 0.012 | 0.026 | 0.025
Phot z only | 200 | 0.015 | 0.029 | 0.030
Spec z only | 92 | 0.007 | 0.017 | 0.015
Table 2: Bias, scatter and combined redshift errors of WaZP-SZ matched
clusters for different subsamples as described in the text.

###  5.2 WaZP versus redMaPPer

In this paper we also match WaZP clusters with those detected by the redMaPPer
algorithm. Here we use the redMaPPer volume limited cluster catalog of DES-Y1
presented in McClintock, T. & Varga, T. N. et al. (2019) which consists of
83,238 clusters with richness $\lambda\geq$5 found in the Y1-SPT and Y1-S82
regions (seen in Figure 9) over the redshift range 0.1-0.95. This volume
limited catalog considers clusters for which the local $z$-band depth assures
a complete galaxy catalog down to the adopted magnitude limit of the richness
definition. The variable $z$-band depth translates into a variable redshift
limit ($z_{max}$) map that characterizes the cluster sample. When evaluating
the recovery rates of clusters at high redshifts, we also consider the full
redMaPPer cluster catalog over the same region, defined by a constant
$z_{max}=$ 0.95 and a richness threshold of 20.

####  5.2.1 Differences in the detection algorithms

redMaPPer is an optical cluster finder based on the detection of spatial
overdensities of red sequence galaxies (Rykoff et al., 2016). Although WaZP
does not make any assumption relative to the cluster galaxy population when
searching for galaxy overdensities, we do expect these two algorithms to yield
similar samples up to redshifts $\sim$0.7, at least when considering the
richest systems. However, a number of differences can be expected in the
cluster characterization for several reasons. First, as it was stressed above,
galaxies used for searching for overdensities are not selected in the same
way. redMaPPer selects them based on colors whereas WaZP selects them based on
redshifts. Second, the two algorithms differ in defining cluster centers. In
the case of redMaPPer centers are associated to a bright galaxy with some
probability of being a central galaxy, whereas WaZP defines the center as a
centroid. Note however, that, as described above, WaZP moves the center to the
brightest cluster member position if its distance is less than 100 kpc, which
happens here for 68% of the WaZP clusters. Third, redMaPPer redshifts are
assigned based on an empirical modelling of red sequence colors, whereas WaZP
assigns redshifts based on a concentration in photometric redshift space
including all galaxy types at the cluster location. Finally, we also expect
differences on how each cluster finder performs in terms of deblending, or in
terms of fragmentation and over-merging.

The above effects make the matching between the two samples non-trivial, since
the key elements to perform a proximity matching, like centering and redshift,
can have distinct behavior, that may not lead to a unique solution. Despite
its complexity, it should be able to provide us with a measure of the
statistical consistency of the two catalogs. It should also help us infer a
lower limit for centering uncertainty, as both cluster finders have optical
centering estimations. Finally, by carefully dealing with footprint coverage
and edge effects, it should allow us to identify a number of missing systems
and provide feedback on the respective selection functions, on possible ways
to improve cluster detection algorithms and improve aspects of the
construction of the underlying galaxy catalog.

####  5.2.2 Matching procedure

In contrast to what was done in the comparison between WaZP and SPT clusters,
here, each cluster sample is considered as a reference to the other.
Therefore, not only we consider a one-way match using redMaPPer as a reference
(redMaPPer-matched), but we also analyse the case where WaZP (WaZP-matched) is
the reference catalog. These one-way matches are used to investigate the
fraction of missed detections. In addition to the one-way matches, in order to
estimate differences in cluster properties (e.g., centering, redshift,
richnesses), we also carry out a two-way (unique) match, for which it is
required that both one-way matches point to the same cluster.

We recall that the cluster matching is performed within a specified redshift
window. Following what is done in section 5.1 when matching with SZ clusters,
this window was first defined considering the sum of the redshift errors
provided for each cluster in the different samples. However, while visually
inspecting a sample of unmatched systems, it was noticed that for relatively
low redshifts ($z\lesssim 0.4$), obvious pairs (i.e., sharing exactly the same
center without any other overdensity on the line of sight) were not associated
due to large redshift discrepancies. This fact is not surprising due to the
systematic errors in photometric redshifts occurring at low redshifts. In
order to take this effect into account, we used an empirical approach to
define the redshift window for matching. We first matched systems by angular
center proximity only, without a redshift window, but imposing the center
angular positions to be closer than 0.05 Mpc, computed at the largest redshift
of the cluster pair. The clusters were ranked by richness, and when multiple
candidates were found (about 10% of the time for WaZP clusters and 24% for
redMaPPer), the richest candidate was selected. As judged by eye inspection,
with this criterion, most matches refer to the same system.

Figure 15 presents the resulting relation between redshifts for WaZP and
redMaPPer matched clusters. As can be seen, this relation shows some
deviations from a linear relation, in particular at WaZP redshift $\sim 0.4$,
where redshifts are distributed from 0.2 to 0.5, so a large scatter. This
issue is discussed in more details in the following section. Based on this
plot, we defined a new redshift window to carry out the matching, which
corresponds to the union of the 99 percentile of the redshift differences
using redMaPPer as reference (as it is covering a smaller redshift baseline)
and a $3\sigma(1+z)$ scatter.

Figure 15: Redshift relation for WaZP-redMaPPer matched clusters within
$50$kpc. Yellow lines are the $99\%$ percentiles, the green lines are the
$3\sigma$ levels, and the orange dashed lines are the union of the two, which
will be used for the last step of matching.

Matching the catalogs by considering the resulting large redshift window
combined to larger angular radii than in figure 15 unavoidably leads to a
large fraction of multiple associations. Resolving these multiples by
selecting the richest available system on both sides resulted in many false
matches. An emblematic case that appeared several times in our visual
inspections, is the case of interacting clusters of similar richnesses. Both
cluster finders would detect the two components but not necessarily with the
same richness ranking. In that case the matching could lead to one mis-match
and one unmatched cluster, or more mis-matches due to a cascade effect.
However, in the case of absence of an interacting system or if neighbouring
systems are much poorer, the richness ranking is more adequate.

We found that for maximizing the number of correct associations, the best
option is to go beyond a single matching rule. Therefore, we decided to
perform the matching following a several steps process where the most
unambiguous pairs are matched first and then proceed to the rest of the list.
The steps are detailed below.

It was determined empirically that a four-step process is optimal, where, at
each step, the matching would only be performed on clusters not previously
matched. In the first step, we do not consider the clusters’ redshift, and
match all clusters that have the exact same centering (which happens when the
two cluster finders are centered on the same galaxy). By construction, each
one way match finds the same corresponding pairs, therefore all cluster pairs
found here will also be a match in the two way matching. This led to a total
of 15,534 matched clusters. In the second step, remaining clusters are matched
within an angular distance of $300$ kpc (computed at the lowest redshift of
the pair) from each other, and a redshift difference less than $1\sigma_{z}$
(computed from Figure 15). When more than one candidate is found, the richest
one is considered to be the correct match. This step also results in having
the same number of matched clusters for both one way matches, adding an extra
6,431 matched clusters in each catalog. The third step expands on the second
one with a $3\sigma_{z}$ window, leading to additional 3,783 matches in each
catalog. In the last step, we match the remaining clusters with the empirical
redshift window shown in Figure 15 and use the radius provided by each cluster
finder as a parameter for angular distance. Here, the matching is not
symmetric and results in another 4,915 and 3,027 matched clusters for
redMaPPer and WaZP respectively. Hence, we obtained a total of 30,663 one-way
matches for redMaPPer and and 28,775 for WaZP. We note that, in this four-step
matching, if we do not remove matched clusters at each step and allow for
multiple matches, we obtain 32,467 redMaPPer matched clusters and 33,498 WaZP
matched clusters.

Finally, two-way matches are obtained when the two one-way matches point to
each other. This results in 28,621 redMaPPer-WaZP cross-matched clusters.

####  5.2.3 Comparison of the matched clusters

We start by comparing the individual properties of the matched clusters (i. e.
centering, redshift and richness). Cross-matched clusters (28,621 pairs) will
be used in this evaluation, as a reliable one-to-one correspondence between
clusters is required. Figure 16 shows the distribution of angular separation
(left panel) and the redshift difference (right panel) of two-way matched
clusters. As it can be seen, all matched clusters are well within the mean
radius of the clusters ($\sim 600$kpc), that was used in the last step of the
matching. In 56% of the cases, clusters have the exact same center. Those
occur when WaZP defines the same BCG as redMaPPer to be its center. The
average distance of central position for the clusters that do not share the
same center is of 86kpc, with 86% of matched clusters within 100kpc of each
other and over 99% within 300kpc. The angular separation only shows a very
weak tail beyond the typical cluster core radius. In addition, we note that
the centering statistics do not seem to depend significantly on the richness.

Turning to the distribution of the redshift separations, we find that for the
vast majority of pairs ($>99\%$), the redshift separation is well within one
third of the redshift window used in the last step of the matching, which has
an average size of $\sim\pm 0.15(1+z)$. Additionally, we find over 75% of
pairs with redshift separation within the combined uncertainty of the cluster
redshifts $\langle z^{err}\rangle=\sqrt{\langle
z^{err}_{WaZP}\rangle^{2}+\langle z^{err}_{RM}\rangle^{2}}=0.028$ (gray shaded
region). We note, however, a small average redshift bias of $0.010$ (Table 3)
exists between WaZP and redMaPPer , with redshifts derived by WaZP being on
average slightly larger than redMaPPer redshifts. In Figure 17, redMaPPer and
WaZP redshifts are compared, showing that the bias is mainly due to a
significant fraction of $z\sim 0.3$ clusters that were pushed to $z_{wazp}\sim
0.4$, an effect that is discussed in the next section. Additionally, we see on
the right panel that this effect occurs mainly on poor ($\lambda<5$) clusters.
Overall, these results show that most matches are well within the ranges
adopted in the matching procedure. It strongly supports the idea that we are
detecting on average the same systems. We also note that the scatter is very
similar to the combined redshift uncertainties in all richness limited samples
(Table 3), and the biases are well within these values.

$\lambda$ bin | # clusters | $\frac{z_{\textsc{W}a\textsc{ZP}}-z_{RM}}{1+z_{RM}}$ | $\frac{\sigma_{z}}{1+z_{RM}}$ | $\frac{\sqrt{(z_{RM}^{err})^{2}+(z_{\textsc{W}a\textsc{ZP}}^{err})^{2}}}{1+z_{RM}}$
---|---|---|---|---
All | 28,621 | 0.010 | 0.027 | 0.028
5-12 | 15,099 | 0.010 | 0.029 | 0.031
12-20 | 7,531 | 0.010 | 0.025 | 0.027
20-30 | 3,440 | 0.011 | 0.023 | 0.024
30-50 | 1,918 | 0.011 | 0.021 | 0.021
50-234 | 632 | 0.014 | 0.019 | 0.018
Table 3: Bias, scatter and uncertainty of WaZP-redMaPPer matched clusters.

Figure 16:  Distribution of distances for WaZP-redMaPPer matched clusters
binned by $\lambda$. On the left is the angular separation between cluster
centers (converted into physical units at the mean redshift of the cluster
pair). The first bin on the left plot represents all cluster pairs that have
the exact same position. On the right is redshift differences and the gray
shaded region is the average combined redshift uncertainties divided by
$1+z_{RM}$.

Figure 17:  Redshift comparison of WaZP and redMaPPer matched clusters by the
four-step match, with the colors corresponding to a count map. The left panel
is the matching to the redMaPPer volume-limited catalog with $\lambda\geq 7$.
The right panel is the matching to the redMaPPer full catalog $\lambda\geq
20$, showing many additional matches at redshifts above $\sim$0.7. The
matching of both catalogs was done as described in section 5.2.2, with the
yellow dashed lines being the windows defined in Figure 15. The top and right
blue lines are the bias of the redshift relations defined by
$(z_{RM}-z_{WaZP})/(1+z_{WaZP})$ and $(z_{WaZP}-z_{RM})/(1+z_{RM})$
respectively. The green shaded regions in those panels are the average
combined uncertainties the clusters redshifts $\langle
z^{err}\rangle=\sqrt{\langle z^{err}_{WaZP}\rangle^{2}+\langle
z^{err}_{RM}\rangle^{2}}$. Comparing both panels, we can see that most
outliers present at $z_{WaZP}\sim 0.4$ are low richness clusters.

We now compare the values of richnesses as derived from the two algorithms. We
do not expect them to be equal in average as they are derived with different
definitions. redMaPPer richness ($\lambda$) considers red sequence galaxies
down to 0.2L∗ in the $z$-band, whereas WaZP richness ($N_{gals}$) considers
all galaxies down to 0.25$L^{*}$ in the $i$-band. Moreover these quantities
are not necessarily computed in the same angular radius. Despite these
differences, we expect some correlation between these two richness estimates.
To perform this comparison we restrict to two-way matched clusters with same
centers and redshift offsets $\leq 0.02(1+z)$. We also considered clusters in
the redshift range 0.1 - 0.6 in order to assure richnesses to be complete for
both cluster finders. Figure 18 shows a strong correlation between the
richnesses of the two cluster finders, with WaZP richnesses on average
systematically larger than redMaPPer ones. To quantify the effect, we
performed a linear fit in $log$ space. To do so, cluster richnesses were first
sliced in $\lambda$ and for each slice the mode of the smoothed distribution
of $N_{gals}$ computed. This procedure minimizes the effect from Malmquist
bias when constraining the relation between richnesses. The fit of the
resulting $(N_{gals},\lambda)$ pairs led to the relation:

$\log\lambda=(0.92\pm 0.09)\log N_{gals}+(0.01\pm 0.11).$ (1)

An independent comparison of the two estimated richnesses is based on the
cluster density given a richness threshold. This is shown as the blue dashed
line in Figure 18. Each point of this line provides the threshold in
$N_{gals}$ and in $\lambda$ richnesses to obtain the same density in the two
cluster samples, and is independent of any matching. It is remarkable that
this measurement is very close to the mean relation between the two richness
estimators. Another way to look at this is to compare directly the densities
of the two cluster samples. In Figure 19, we compare WaZP and redMaPPer
cluster densities considering clusters with redshifts in the range 0.1 - 0.6
and with richnesses above a given threshold, where the threshold in $N_{gals}$
and $\lambda$ are related following Eq. 1 (e.g. $\lambda\geq 20$ is equivalent
to $N_{gals}\geq 25$). Cluster densities are very similar over a very wide
range of richnesses. This result supports the idea that, on average, the
ranking of the two cluster samples by their richness is similar.

Figure 18:  Comparison of richnesses computed by WaZP and redMaPPer for 13,664
two-way matched clusters with redshifts in the range 0.1-0.6, with same
centering and redshift offsets $\leq 0.02(1+z)$. The orange line is a power-
law fit of the richness relation. The dashed blue line represents the
thresholds to be applied to the two richnesses to obtain the same cluster
densities in the two cluster samples independently from any matching.  Figure
19:  Density of WaZP and redMaPPer catalogs as a function of $N_{gals}$
threshold. The thresholds for redMaPPer were computed by converting $\lambda$
to $N_{gals}$ using Eq. 1, with the shaded regions being the uncertainties
propagated and errobars from Poisson noise. The top panel shows the ratio
between the densities.

Figure 20: Fraction of clusters detected by both redMaPPer and WaZP binned by
richness as a function of redshift. The dotted line is the minimum redshift of
the redMaPPer catalog, the gray shaded area is the region where redMaPPer
footprint decreases due its local $z_{max}$ and the dashed line is the mean of
$z_{max}$. The solid lines are the recovery rates for the matching with
redMaPPer volume limited catalog, and the light shaded lines correspond to
matching with redMaPPer volume limited plus redMaPPer full ($\lambda\geq 20$)
catalog. On the left redMaPPer is used as the reference catalog (binning by
$\lambda$) and on the right WaZP is the reference catalog (binning by
$N_{gals}$).

####  5.2.4 Statistics of unmatched clusters

We now evaluate the recovery rates between catalogs. Our main goal in this
section is to check if each cluster finder could have missed a detection,
therefore we take a very conservative approach to label clusters as unmatched.
In principle, these rates could be computed considering a two-way matching.
However, in that case, if a cluster of the first sample appears to be
fragmented in the second one, the extra cluster will be counted as not
recovered. Here, we wish to separate the absence of a counterpart from
fragmentation, which should be treated separately. Therefore, we defined the
recovery rate as the fraction of clusters having one or more counterparts in
the matched cluster sample. Obviously, the absence of a match does not exclude
completely the existence of a counterpart. Some systems could suffer from a
strong mis-centering, larger than tolerated by the matching criteria. Other
systems could also suffer from edge effects that may occur at the periphery of
the survey or close to a masked region within the survey. To minimize the
latter, we do not consider unmatched clusters outside the intersection of the
two cluster sample footprints (constructed as nside $=4096$ Healpix maps) or
unmatched cluster located in edge pixels. In addition, as redMaPPer removes
clusters with over 20% of their area masked, an equivalent consideration had
to be made when looking for WaZP counter-parts. Hence, unmatched clusters
whose cover fraction was less than 80% on the other catalog’s detection
fraction footprint were also discarded from the analysis. These cover
fractions were computed using the same weighted methodology as (Rykoff et al.,
2014) considering the other catalog footprint. Although this is not a major
contribution for the values of the recovery fraction of WaZP clusters,
ignoring this effect leads to lower recovery close to the footprint edges and
holes. It is also important to note that, because both catalogs have a
footprint with a variation on $z_{max}$ at different locations, the
computation of whether the cluster is inside the footprint or in a edge pixel
and its cover fraction depends on the cluster position and redshift.

These cuts, based on the footprints and cover fraction, removed 16,283 and
18,737 redMaPPer and WaZP clusters without counterpart, respectively, from our
recovery rate analysis. These clusters certainly contain information regarding
each cluster finder selection function and limitations, however the study of
these objects require a different analysis on a object-by-object case and will
be done in a future work.

With these considerations, for one-way matching including the possibility of
multiple associations, the recovery rate analysis is based on the total of
66,955 redMaPPer and 43,980 WaZP clusters with 32,467 and 33,498 matched
respectively.

The recovery rate for each catalog as a function of redshift in different
richness bins is shown in Figure 20. The left panel shows the fraction of
redMaPPer clusters recovered by WaZP and the right panel the other way around.
The gray shaded area is the redshift region where the redMaPPer (left panel)
and WaZP (right panel) footprints decreases in size, with the dashed line
being the median value of this footprint redshift limitation (i. e. where the
area drops to 50% of the total footprint). The different shades correspond to
the 95% and 100% percentiles of the $z_{max}$ distribution. We binned
redMaPPer clusters into 5 samples, 2 bins for lower richness ($\lambda<20$)
clusters, that were not used for cosmological constraints (DES Collaboration
et al., 2020), and 3 sample with higher richness. WaZP clusters were binned on
the corresponding richness using our fit on Eq. 1. One can see that redMaPPer
clusters with $\lambda\geq 20$ are mostly ($\geq 90\%$) recovered up to the
WaZP redshift limit. Similarly, at the same level of richness ($N_{gals}\geq
25$), we find that WaZP clusters are also recovered at more than $90\%$ up to
the redMaPPer redshift limit of $\sim 0.7$. We also note rapid increase in the
recovery rate of clusters with richness in both cases. Considering the
$0.1<z<0.7$ range, the overall recovery rate of redMaPPer clusters is 93.3%,
98.4%, 99.7% for $\lambda\geq 20$ bins ($20-30$, $30-50$ and $50-234$
respectively). For WaZP clusters, similarly, we have 95.4%, 97.9% and 99.7%
for $N_{gals}\geq 25$ bins ($25-39$, $39-68$ and $68-288$ respectively). Major
differences occur when considering clusters less rich than $\lambda\sim 20$
($N_{gals}\sim 25$). It is remarkable that even in the $\lambda$ range 5 to 20
($N_{gals}$ 5.6 to 25), clusters are still recovered at rates between 50 and
60%, depending on the redshift. The dotted vertical line corresponds to the
minimum redshift of redMaPPer clusters ($z=0.1$), hence the low recovery rate
for WaZP clusters in the first redshift bin.

We note that the recovery rates reach 100% at high redshifts in both panels of
Figure 20. This is an effect of the footprints variable redshift limit leading
to smaller effective area as redshift increases, with only $3\%$ ($2\%$) of
the redMaPPer (WaZP) area remaining at $z=0.7$ ($0.76$). Consequently, at
those redshifts, all unmatched clusters are removed from the analysis,
resulting in an artificially perfect recovery. To obtain a more relevant WaZP
recovery rate at high redshifts, we combined redMaPPer volume limited catalog
($\lambda\geq 5$) with the redMaPPer full catalog (uniform $z_{max}$ of $0.97$
and $\lambda\geq 20$) and matched it to the WaZP catalog using the procedure
described above. These results are represented by the shaded lines in Figure
20. We see now that the redMaPPer recovery rate for $\lambda\geq 20$ clusters
extends to higher redshifts. Comparing the WaZP recovery fraction to the
matching with redMaPPer volume limited only, we have a general decrease at
high redshifts. The lower recovery rate of $N_{gals}\leq 25$ clusters is
directly correlated with poorer clusters ($\lambda\leq 20$) missing in the
redMaPPer full catalog. Richer clusters ($N_{gals}\geq 25$) are affected by
the scatter down of the richness relation between both cluster finders.

We conclude from this analysis that, statistically, rich systems are found by
both cluster finders, independent of their redshifts, with very few individual
differences that are investigated in the next section.

##  6 Discussion

We have shown that all SZ clusters with redshifts $\leq$ 0.76 intersecting our
footprint are recovered by WaZP cluster finder applied on DES-Y1 data. We have
also shown that more than 90% of clusters with richnesses above $\sim$20
detected by redMaPPer (or 25 by WaZP) are matched to those detected by WaZP
(or by redMaPPer). In this section we compare some properties of the the two
optical cluster samples. In particular, we explore differences such as
redshift discrepancies, overmerging / fragmentation, and the reasons for
unmatched systems on both sides.

###  6.1 Redshift discrepancies between WaZP and redMaPPer

Whereas the comparison of WaZP redshifts with spectroscopic redshifts assigned
to SZ clusters (see table 2) showed moderate bias and small scatter, the
comparison with redMaPPer clusters revealed stronger discrepancies. This is
related to the fact that redMaPPer (and WaZP) clusters are on average much
less massive than SPT clusters. This is confirmed if we restrict the cross
match between WaZP and redMaPPer to richer clusters. In that case, as shown in
the right panel of Figure 17, a reduced redshift bias and scatter is observed
between the two samples.

The fraction of redshift outliers (defined by a redshift difference $\geq
3\sigma_{z}(1+z)$) is less than 5% of our clusters, with 78% (82%) of them
having $N_{gals}\leq 25$ ($\lambda\leq 20$). These redshift outliers are
mainly produced at $z_{\textsc{W}a\textsc{ZP}}\sim 0.4$. This also reflects on
WaZP cluster number counts (Figure 9), that showed a peak at redshift
$\sim$0.4 that becomes more prominent when considering the poorest clusters.
From the same cluster number counts, a deficit of clusters at $z\sim 0.3$ can
also be noticed. These points support the idea that in the redshift range 0.15
- 0.35, WaZP detects on average the same clusters as redMaPPer but shifts a
fraction of these to $z\sim 0.4$.

From global statistics of photometric redshifts, only moderate bias is
measured (see right panel of Figure 5). However, this global bias includes
galaxies of all magnitudes down to $m_{i}^{*}+1.5$. In order to understand the
shift in redshift of a fraction of WaZP clusters, one needs to investigate the
photometric redshift bias at least as a function of both redshift and
magnitude. This is what is shown in Figure 21. We binned our spectroscopic
sample in $i$-band magnitude and spectroscopic redshift and computed, for each
bin with at least 100 galaxies, the median and standard deviation of
$(z_{phot}-z_{spec})/(1+z_{spec})$. The amplitude of the photometric redshift
bias is shown as a color code. In most regions of this diagram, the bias is
moderate, consistent with the global bias. However, for redshifts between 0.15
and 0.35 and $i$-band magnitudes fainter than $\sim 20.$, we find a strong
bias that reaches values of 0.1-0.2.

The origin of this strong bias is two-folded. It is first due to the lack of
$u$-band and to the transition of the 4000Å break between $g$ and $r$ band at
redshift $z\sim 0.3-0.4$. Second, it is due to the lack of faint ($m_{i}\geq
20$) red galaxies at redshifts below $\sim 0.35$ in our spectroscopic training
sample. We stress that for these faint, low-redshift and red galaxies, the
bias may be even larger as it cannot be estimated properly. The consequence is
that their photometric redshift is overestimated, pushed to redshifts where
the training set samples better the same (magnitude, colors) space. This
effect has already been stressed in several other studies (e.g., Figure 25 of
Rykoff et al., 2016).

This statement can actually be tested by comparing how a DES redshift biased
WaZP cluster is detected in the Sloan Digital Sky Survey, that is covered in
$u,g,r,i,z$ bands in the overlapping Stripe 82 region. As an example, we
selected from our DES-Y1-S82 run, one cluster of richness above 60 with a
redshift 0.43, whereas the same cluster is detected by redMaPPer at a redshift
of 0.28. WaZP was run on a small section of SDSS-S82 around that cluster with
the same settings but based on SDSS DR-12 photometric redshifts from Beck et
al. (2016). Based on these redshifts, WaZP recovers a much lower redshift
($z=0.28$) for the cluster, which is consistent with redMaPPer and with the
available BCG spectroscopic redshift. In Figure 22 we show the redshifts and
magnitudes of galaxies classified as cluster members for the two detections,
based on DES and on SDSS. One can first notice a large overlap between the
members. Then, one can clearly see that these common members are
systematically shifted to larger redshifts within the DES. As this effect is
stronger for fainter objects, one can also notice for instance that the BCG,
at a magnitude of 16.5, has an unbiased redshift. The consequence is that the
BCG was not considered as a member in the DES based membership. To conclude,
WaZP based on DNF-DES photometric redshifts seems able to recover clusters at
$z\sim 0.3$, but redshift, membership and therefore richness may be severely
affected.

Figure 21: Bias of the galaxy DNF photometric redshifts relative to
spectroscopic redshifts as a function of spectroscopic redshift and magnitude.
Computation is based here on the SPT region. We require a minimum of 100
spectroscopic redshifts at each redshift - magnitude position. Pixels not
satisfying this condition remain white on these maps.

New approaches are currently being investigated to correct for the impact on
cluster detection and characterization of the photometric redshift bias
effect.

Figure 22:  This figure illustrates how the bias in photometric redshifts
shown statistically in right panel of Figure 5 may affect cluster members at
$z\sim 0.3$. It displays the members of a galaxy cluster detected in Y1-S82 by
WaZP at redshift $z=0.43$ (black stars). This cluster is also found by
redMaPPer but with a redshift $z=0.28$. When using SDSS photometry WaZP
recovers the cluster at the same redshift as redMaPPer (blue dashed line), and
associated members are shown in blue. These SDSS based members have a
different photometric redshift when computed with DNF based on DES photometry
(i.e., without the $u$-band). The latter are shown in red, and are linked to a
blue point when it is the same galaxy. The common red and black symbols show
that the two detections have many members in common. However, the BCG (at
magnitude 16.5), for instance, was missed as its redshift is not biased. As
the DNF+DES-Y1 bias is strongly magnitude dependent, it pulls members apart in
two different redshift bins.

###  6.2 Relative fragmentation

In the previous section, the relative completeness of the two optical cluster
finders presented in figure 20 is meant to highlight the fraction of clusters
without any counterpart. For those clusters tagged as matched, this matching
does not assure a one-to-one correspondence for the matched clusters, but only
that a cluster from one sample has at least one counterpart in the matched
sample. Here, we examine clusters from one sample that are matched to more
than one cluster in the opposite sample. In the case of a two-way match, the
richest counterparts are selected letting the additional possibilities
unmatched. The extra component(s) involved in the one-way match only could be
interpreted as a cluster sub-structures in the other sample, or as a missed
cluster, depending on the adopted definition of each cluster finder.

In terms of one-way matching, from figure 20, we found that 96% of WaZP
clusters in the redshift range 0.1 - 0.6 with $N_{gals}\geq 25$ have a
redMaPPer counterpart, and conversely, 94% of redMaPPer clusters with
$\lambda\geq 20$ have a WaZP counterpart in the same redshift range. If we now
consider two-way matches, only 87% of WaZP clusters have a redMaPPer
counterpart, whereas 91% of redMaPPer clusters are two-way matched, about the
same fraction as for the one-way matching. The larger decrease of matches for
WaZP clusters when going from one to two-way matching suggests that they are
in average relatively more fragmented (or redMaPPer clusters relatively more
merged). This is what we investigate below.

The apparent larger fragmentation of WaZP clusters could be due to the
presence of very low richness clusters in the periphery of richer ones. To
test this, we evaluated the relative fragmentation rate of the two cluster
finders considering different richness cuts in the associated systems. The
relative fragmentation rate is estimated as the fraction of matches that have
more than one counterpart. If we start from WaZP clusters with $N_{gals}\geq
25$, the relative fragmentation rate is 22%, 6% and 2% when considering
counterparts with, respectively, $\lambda\geq 5,10,20$. Conversely, starting
from redMaPPer clusters with $\lambda\geq 20$, the relative fragmentation rate
is 30%, 24% and 14% when considering counterparts with, respectively,
$N_{gals}\geq 6,12,25$. The ratio (WaZP to redMaPPer) of the fragmentation
rates increases strongly when considering richer multiple counterparts. We can
conclude from this that WaZP tends to find pairs of relatively rich clusters
more frequently than redMaPPer.

Should multiple systems be seen as one or several clusters is a matter of
cluster definition for each cluster finder. They may also suggest a wrong
tuning of the detection algorithm leading to undesirable fragmentation within
a clearly unique cluster. To address this point, we visually inspected the 50
richest WaZP cluster pairs and found that the vast majority do correspond to
clear separate groups. In very few cases only WaZP detected two peaks clearly
within the same cluster. In figure 23 we show two relatively rich cases at
redshifts 0.38 and 0.68. In both cases, the redMaPPer cluster radii are only
slightly larger than the distance between the two WaZP clusters, which assured
the one way matching of both WaZP systems. It is likely here that the galaxies
from the extra WaZP clusters were percolated to the most likely redMaPPer
cluster reducing their weight as members of a secondary cluster and eventually
leading to only one detection (Rykoff et al. (2014), section 9). However, let
us stress that the detection algorithms may be tuned to find different
overdensities, leading to different samples with their own selection function.

Figure 23: Examples of two clusters that are one-way matched but not two-way
matched. In these examples, WaZP detects two clusters (yellow circles),
whereas redMaPPer detects one (red circle), with a cluster radius reaching the
second component. In the left panel (RA=34.312, Dec=-52.760), WaZP detected
two clusters at $z=0.39/0.38$ separated by 1 Mpc. The top right detection is a
redMaPPer cluster with $z=0.34$ ($\lambda=119$ and $N_{gals}=118$). The bottom
left detection ($N_{gals}=81$) also matches a SPT detection (cyan circle). In
the right panel (RA=28.125, Dec=-40.944), WaZP detected two clusters at
$z=0.67/0.69$ separated by 1 Mpc. The bottom left detection is a redMaPPer
cluster with $z=0.68$ ($\lambda=32$ and $N_{gals}=47$). The top right
detection has a richness $N_{gals}=66$.

###  6.3 Unmatched systems

Let us now turn to the WaZP or redMaPPer clusters for which no counterpart was
found using our one-way matching procedure. Our goal here is to provide some
insight on the reasons why some systems, or types of systems would not be
detected by one algorithm or the other. We should first stress that our
matching procedure is designed in such a way that we have strongly limited the
number of unmatched clusters that could be due e.g. to edge effects, variable
depths of the used reference bands or differences in estimated redshifts.

Treating edge effects properly appeared to be a critical issue as it concerns
a significant number of detections due to the complex geometry of the masked
regions. Moreover each cluster sample was not built using exactly the same
footprint, in particular due to the different reference band used. We
considered regions covered by the two footprints, and also followed
redMaPPer’s prescription and discarded clusters that would intersect empty
regions of the galaxy catalogue by more than 20% within a 1 Mpc radius. Note
that this area fraction is actually weighted by a projected NFW profile as
described in Rykoff et al. (2012). Concerning the adopted tolerance in
redshift difference, as shown above, we have carried out an empirical
approach, precisely to avoid unmatched systems that would be detected on both
sides but with a too large redshift discrepancy. This case may still happen in
our matched catalog, but with a lower occurrence.

In order to qualify the unmatched clusters, we have carried out a visual
inspection of the 60 richest ones (for each cluster finder) in the redshift
range 0.1 to 0.65. These systems have richnesses $N_{gals}\geq 30$ and
$\lambda\geq 25$.

Without trying to derive precise statistics from this inspection, unmatched
systems clearly enter two categories common to the two cluster finders. The
first one, corresponding to one third of the inspected systems, is made of
clear concentrated overdensities of red galaxies (two examples are shown in
Figure 24, one detected by WaZP and the second by redMaPPer). For these
systems, possible edge or depth effects were checked and discarded. Those not
found by redMaPPer have redshifts ranging uniformly from 0.3 to 0.6, whereas
those not found by WaZP are concentrated in two redshift bins, around
0.25-0.35 (possibly due to the photometric redshift bias described above) and
the second around 0.5-0.6.

A second category covering more than half of the inspected clusters is
composed of much looser systems, without any obvious central concentration,
sometimes possibly fragments of larger scale filamentary structures, and in
some few cases no apparent cluster at all. These loose systems may appear as
poorer clusters even though they are selected among the richest unmatched,
typically $\lambda$ (or $N_{gals}$) $\sim 30-35$. One typical example is shown
in Figure 25 where we compare the case of two redMaPPer clusters at the same
redshift ($z\sim 0.6$) and with similar richnesses ($\lambda\sim 30$). The
concentrated system is well recovered by WaZP at the same redshift and with
similar richness, whereas no counterpart was found for the looser one. Similar
opposite situations occur when considering WaZP clusters as a reference.

What seems to be common to these loose unmatched clusters is that they are
often characterized, at a given richness, by a lower SNR (in the case of WaZP)
and a lower likelihood (in the case of redMaPPer). In order to verify this
observation statistically, we have compared the WaZP SNR and redMaPPer
likelihood of the matched and unmatched clusters. To do this, as these two
quantities depend in average on both redshift and richness, we have computed
the median and 68 percentile of the SNR and likelihood in bins of redshift and
richness. We can then compare how each matched and unmatched cluster deviates
relative to its local (in redshift-richness space) median SNR or likelihood.
The result, considering all clusters in the redshift range 0.1-0.6 and with
$N_{gals}\geq 25$ and $\lambda\geq 20$, is shown in Figure 26. Clearly,
unmatched clusters have in average lower SNR or lower likelihood than the
average, suggesting that these quantities should be considered in the cluster
selection function.

Figure 24: Examples of a WaZP cluster (left panel, yellow circle) and a
redMaPPer cluster (right panel, red circle) both tagged as unmatched. The
redshifts and richnesses of these two clusters are $z=0.44$ / $N_{gals}=56.$
(left panel) and $z=0.38$ / $\lambda=32.$ (right panel).

Figure 25: Example of a redMaPPer cluster (red circle) not found by WaZP (left
panel) compared to a redMaPPer cluster with same richness and redshift that is
matched to WaZP (right panel, yellow circle). The two redMaPPer clusters have
a richness $\lambda=30$, and a redshift $z=0.58-0.59$. The matched WaZP
cluster was found at $z=0.6$ with a richness $N_{gals}=37$.

Figure 26: Comparison of the distributions of redMaPPer cluster likelihoods
relative to the median likelihood at the clusters redshift and richness (left
panel) and WaZP cluster SNRs relative to the median SNR at the clusters
redshift and richness (right panel) for matched (blue) and unmatched (red)
clusters. For both samples we have considered here clusters with richnesses
$\lambda\geq 20$ and $N_{gals}\geq 25$ and redshifts in the range 0.1 - 0.6.
Note that the distributions are normalized; in both cases, the unmatched
clusters are $\sim 20$ times less numerous than the matched ones.

##  7 Conclusions

In this paper we present WaZP a new (2+1)D cluster finder based on photometric
redshifts. It is applied to DES-Y1A1 data and resulting samples are compared
to those derived from the SPT survey based on the SZ effect, and from the
redMaPPer cluster finder applied to the same photometric data.

Our conclusions can be listed as follows.

  * •

A galaxy Value Added Catalog was derived from the DES Y1A1 survey. It is
controlled by the $i$-band that is used for star-galaxy separation and for
determining the local depth. Depending on the location, complete galaxy
samples can be built down to $m_{i}=22.25-23$, with a median (corresponding to
half of the survey area) completeness magnitude limit of $m_{i}=22.7$.

  * •

The WaZP cluster finder was applied to DES Y1A1 survey led to the detection of
60,547 clusters over 1,511.13 deg2, with redshifts ranging from 0.05 to 0.9
and richness greater than $5$. Due to the $i$-band limiting magnitude of the
survey and the adopted limiting magnitude for estimating richnesses, complete
richnesses are derived for clusters in the redshift range 0.05-0.60. Clusters
detected at larger redshifts get increasingly incomplete richnesses depending
on the local depth.

  * •

Considering the SPT cluster sample intersecting the DES Y1A1 footprint in the
redshift range 0.05 - 0.76, WaZP was shown to recover all 293 SZ clusters.
Comparing redshifts of both cluster finders, we found a bias of $0.012$ and a
scatter of the same order of redshift uncertainties ($\sim 0.025$). When we
restrict to SZ clusters with an assigned spectroscopic redshift, all these
quantities are lowered by 40%.

  * •

Cross-matching WaZP and redMaPPer catalog led to the development of an
iterative matching algorithm to minimize incorrect associations. Special care
was taken to deal with edge effects and depth variations. It resulted in
matching 28,621 clusters in the two-way criteria with richnesses $N_{gals}$
(and $\lambda$) down to 5. Considering one-way matching for clusters richer
than $N_{gals}=25$ ($\lambda=20$) with $0.1<z<0.6$, we showed that WaZP
recovered 96% redMaPPer clusters, and, symmetrically, redMaPPer recovered 94%
WaZP clusters.

  * •

The centering offset between WaZP and redMaPPer is less than 200 kpc in most
cases (97%), which is much less than the matching criteria used.

  * •

Comparison of the estimated redshifts from redMaPPer and WaZP shows an overall
good agreement. However, a fraction of WaZP clusters suffer from a redshift
bias, reflecting the underlying galaxy photometric redshift bias. Note that
this effect does not seem to prevent detection in general.

  * •

Despite different definitions, the comparison of WaZP and redMaPPer richnesses
shows a strong correlation. The scatter of this relation will be analyzed in
details on a separate paper in a future work. We also computed a richness
relation based on thresholds that provided the same densities of clusters.
This relation is remarkably close to the fit from matched clusters, supporting
the idea that, on average, the ranking of the two cluster samples by their
richness is similar.

  * •

The study of the relative fragmentation of WaZP and redMaPPer clusters showed
that WaZP tends to find pairs of relatively rich clusters more frequently than
redMaPPer.

  * •

The visual inspection of 60 unmatched clusters richer than $N_{gals}\geq 25$
(for WaZP) and $\lambda\geq 20$ (for redMaPPer), showed that (for both
samples) 1/3 of these are clear concentrations of red galaxies, which, in
itself could help to improve the completeness of both samples. A second
category representing more than half of the inspected unmatched systems is
composed of much looser systems or more filamentary structures. This dominant
category of unmatched clusters has been shown statistically to be
characterized by lower SNRs (for WaZP) and lower likelihoods (for redMaPPer)
than for the matched clusters.

One of the main aspects of this work is to address the relative completenesses
of two optical cluster finders based on different methods and applied to the
same survey. As shown in this paper, the comparison of the resulting samples
in itself is not a trivial task free of assumptions. However, it is a useful
guide to detect features of cluster finding algorithms when applied to large
surveys with all their complexity (missing data, mis-classified sources, depth
variation, etc.). In that sense, such comparisons are complementary to those
performed with mock catalogs, that can be used, for instance, to address the
sample purity in addition to completeness.

A limitation of the present study is the quality of the photometric redshifts
partly hampered by the poor representation of faint red galaxies in our
spectroscopic training set, in particular at low redshifts ($z\leq 0.35$). In
a companion paper we are exploring the impact of using a much larger
spectroscopic training set, and the use of several photometric redshift codes.
We will also study the impact of the star-galaxy classification algorithm.
Finally, we will compare WaZP cluster samples based on $i$-band and $z$-band
reference magnitudes with a special focus on high redshift cluster detection.

## Acknowledgments

M.A. was supported by FAPESP (2013/26612-2) and CNPq (380101/2017-3,
165049/2017-0, 381135/2018-7). C.B. acknowledges the support obtained from the
Bonus-Qualité-Recherche 2013 & 2017 Programs of the J.-L. Lagrange Laboratory
(UMR 7293).

C.B. thanks J.-L. Starck for his help in using and configuring the MR/1 multi-
resolution to optimally build galaxy density maps. C.B. also thanks S.
Maurogordato, A. Cappi and E. Slezak for many stimulating discussions during
the development of the WaZP cluster finder algorithm.

Funding for the DES Projects has been provided by the U.S. Department of
Energy, the U.S. National Science Foundation, the Ministry of Science and
Education of Spain, the Science and Technology Facilities Council of the
United Kingdom, the Higher Education Funding Council for England, the National
Center for Supercomputing Applications at the University of Illinois at
Urbana-Champaign, the Kavli Institute of Cosmological Physics at the
University of Chicago, the Center for Cosmology and Astro-Particle Physics at
the Ohio State University, the Mitchell Institute for Fundamental Physics and
Astronomy at Texas A&M University, Financiadora de Estudos e Projetos,
Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro,
Conselho Nacional de Desenvolvimento Científico e Tecnológico and the
Ministério da Ciência, Tecnologia e Inovação, the Deutsche
Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy
Survey.

The Collaborating Institutions are Argonne National Laboratory, the University
of California at Santa Cruz, the University of Cambridge, Centro de
Investigaciones Energéticas, Medioambientales y Tecnológicas-Madrid, the
University of Chicago, University College London, the DES-Brazil Consortium,
the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH)
Zürich, Fermi National Accelerator Laboratory, the University of Illinois at
Urbana-Champaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the
Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the associated Excellence
Cluster Universe, the University of Michigan, NFS’s NOIRLab, the University of
Nottingham, The Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, Texas A&M University, and the OzDES
Membership Consortium.

Based in part on observations at Cerro Tololo Inter-American Observatory at
NSF’s NOIRLab (NOIRLab Prop. ID 2012B-0001; PI: J. Frieman), which is managed
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation.

The DES data management system is supported by the National Science Foundation
under Grant Numbers AST-1138766 and AST-1536171. The DES participants from
Spanish institutions are partially supported by MICINN under grants
ESP2017-89838, PGC2018-094773, PGC2018-102021, SEV-2016-0588, SEV-2016-0597,
and MDM-2015-0509, some of which include ERDF funds from the European Union.
IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.
Research leading to these results has received funding from the European
Research Council under the European Union’s Seventh Framework Program
(FP7/2007-2013) including ERC grant agreements 240672, 291329, and 306478. We
acknowledge support from the Brazilian Instituto Nacional de Ciência e
Tecnologia (INCT) do e-Universo (CNPq grant 465376/2014-2).

This manuscript has been authored by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics.

## Data Availability

The WaZP DES-Y1A1 cluster products with a full description of the catalogs and
columns can be found are available in its online supplementary material
555https://www.linea.gov.br/catalogs/wazp/.

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## Affiliations

1\. Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
2\. Laboratoire d’Annecy De Physique Des Particules (LAPP), 9 Chemin de
Bellevue, 74940 Annecy, France
3\. Université Côte d’Azur, OCA, CNRS, Lagrange, UMR 7293, CS 34229, 06304,
Nice Cedex 4, France
4\. Observatório Nacional, Rua Gal. José Cristino 77, Rio de Janeiro, RJ -
20921-400, Brazil
5\. Departamento de Física Matemática, Instituto de Física, Universidade de
São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
6\. Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510,
USA
7\. Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid,
28049 Madrid, Spain
8\. Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, UK
9\. CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
0\. Sorbonne Universités, UPMC Univ Paris 06, UMR 7095, Institut
d’Astrophysique de Paris, F-75014, Paris, France
11\. Department of Physics and Astronomy, Pevensey Building, University of
Sussex, Brighton, BN1 9QH, UK
12\. Department of Physics & Astronomy, University College London, Gower
Street, London, WC1E 6BT, UK
13\. Instituto de Astrofisica de Canarias, E-38205 La Laguna, Tenerife, Spain
14\. Universidad de La Laguna, Dpto. Astrofísica, E-38206 La Laguna, Tenerife,
Spain
15\. Department of Astronomy, University of Illinois at Urbana-Champaign, 1002
W. Green Street, Urbana, IL 61801, USA
16\. National Center for Supercomputing Applications, 1205 West Clark St.,
Urbana, IL 61801, USA
17\. Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
18\. INAF-Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
19\. Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
20\. Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
21\. Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
22\. Santa Cruz Institute for Particle Physics, Santa Cruz, CA 95064, USA
23\. Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
24\. Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
25\. Institute of Theoretical Astrophysics, University of Oslo. P.O. Box 1029
Blindern, NO-0315 Oslo, Norway
26\. Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Dr., Pasadena, CA 91109, USA
27\. Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
28\. Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can
Magrans, s/n, 08193 Barcelona, Spain
29\. Kavli Institute for Cosmological Physics, University of Chicago, Chicago,
IL 60637, USA
30\. School of Mathematics and Physics, University of Queensland, Brisbane,
QLD 4072, Australia
31\. Center for Cosmology and Astro-Particle Physics, The Ohio State
University, Columbus, OH 43210, USA
32\. Department of Physics, The Ohio State University, Columbus, OH 43210, USA
33\. Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
34\. Australian Astronomical Optics, Macquarie University, North Ryde, NSW
2113, Australia
35\. Lowell Observatory, 1400 Mars Hill Rd, Flagstaff, AZ 86001, USA
36\. Department of Astrophysical Sciences, Princeton University, Peyton Hall,
Princeton, NJ 08544, USA
37\. Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona,
Spain
38\. Physics Department, 2320 Chamberlin Hall, University of Wisconsin-
Madison, 1150 University Avenue Madison, WI 53706-1390
39\. Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
40\. Instituto de Física, UFRGS, Caixa Postal 15051, Porto Alegre, RS -
91501-970, Brazil
41\. School of Physics and Astronomy, University of Southampton, Southampton,
SO17 1BJ, UK
42\. Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37831
43\. Department of Physics, Stanford University, 382 Via Pueblo Mall,
Stanford, CA 94305, USA