# VST ATLAS Galaxy Cluster Catalogue I: cluster detection and mass calibration

Behzad Ansarinejad1,2, David Murphy3, Tom Shanks1, Nigel Metcalfe1
1Centre for Extragalactic Astronomy, Department of Physics, Durham University,
South Road, Durham DH1 3LE, UK
2School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
3Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Taking advantage of $\sim 4700$ deg2 optical coverage of the Southern sky
offered by the VST ATLAS survey, we construct a new catalogue of
photometrically selected galaxy groups and clusters using the orca cluster
detection algorithm. The catalogue contains $\sim 22,000$ detections with
$N_{200}>10$ and $\sim 9,000$ with $N_{200}>20$. We estimate the photometric
redshifts of the clusters using machine learning and find the redshift
distribution of the sample to extend to $z\sim 0.7$, peaking at $z\sim 0.25$.
We calibrate the ATLAS cluster mass-richness scaling relation using masses
from the MCXC, Planck, ACT DR5 and SDSS redMaPPer cluster samples. We estimate
the ATLAS sample to be $>95\%$ complete and $>85\%$ pure at $z<0.35$ and in
the $M_{\textup{200m}}$>$1\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ mass
range. At $z<0.35$, we also find the ATLAS sample to be more complete than
redMaPPer, recovering a $\sim 40\%$ higher fraction of Abell clusters. This
higher sample completeness places the amplitude of the $z<0.35$ ATLAS cluster
mass function closer to the predictions of a $\Lambda$CDM model with
parameters based on the Planck CMB analyses, compared to the mass functions of
the other cluster samples. However, strong tensions between the observed ATLAS
mass functions and models remain. We shall present a detailed cosmological
analysis of the ATLAS cluster mass functions in paper II. In the future,
optical counterparts to X-ray-detected eROSITA clusters can be identified
using the ATLAS sample. The catalogue is also well suited for auxiliary
spectroscopic target selection in 4MOST. The ATLAS cluster catalogue is
publicly available at
http://astro.dur.ac.uk/cosmology/vstatlas/cluster_catalogue/.

###### keywords:

cosmology: observations - large-scale structure of Universe - galaxies:
clusters: general - catalogues

††pubyear: 2022††pagerange: VST ATLAS Galaxy Cluster Catalogue I: cluster
detection and mass calibration–VST ATLAS Galaxy Cluster Catalogue I: cluster
detection and mass calibration

##  1 Introduction

In the context of hierarchical structure formation, primordial peaks in the
density field of the early Universe collapse and merge, leading to the
incremental formation of gravitationally bound halos of increasing mass (e.g.
Peebles 1980). Galaxy clusters are the largest of these gravitationally bound
structures in the Universe, occupying the extreme tail of the halo mass
function. As a result, the evolution of galaxy cluster abundance with mass and
redshift is extremely sensitive to variations in cosmological parameters.
Precise observations of large numbers of clusters over a wide range of masses
and redshifts can therefore provide powerful cosmological constraints (see
reviews by Allen et al. 2011; Weinberg et al. 2013). In the context of the
$\Lambda$CDM model, the study of galaxy clusters can place independent
constraints on $\Omega_{\textup{m}}$, the matter density parameter (Evrard
1997; Schuecker et al. 2003), $\Omega_{\Lambda}$, the dark energy density
parameter and $\omega$, the dark energy equation of state parameter (Morandi &
Sun 2016), as well as $\sigma_{8}$, the normalization of the matter power
spectrum on the scale of $8\ h^{-1}$Mpc (e.g. White et al. 1993). Furthermore,
extensions to $\Lambda$CDM such as massive neutrinos can be investigated via
the study of galaxy cluster number counts (Costanzi et al. 2013; Roncarelli et
al. 2015), while non-standard modified gravity models can be constrained via
the study of cluster properties (Brownstein 2009; Llinares & Mota 2013; Planck
Collaboration et al. 2016a; Bocquet et al. 2019). Galaxy clusters and groups
also provide useful tools in studying galaxy evolution in extreme environments
and their physical properties could aid our understanding of structure
formation, providing vital information on collapse of dark matter and
evolution of baryons in dark matter potentials (see reviews by Rosati et al.
2002; Voit 2005 and Kravtsov & Borgani 2012).

In the optical regime, the Sloan Digital Sky Survey (SDSS; York et al. 2000)
was one of the first wide-field surveys covering $\sim 10,000$ deg2 of the
sky, which formed the basis of the MAXBCG (Koester et al., 2007) and redMaPPer
(Rykoff et al., 2014) cluster catalogues. The VST ATLAS survey (Shanks et al.,
2015), which is the basis of this work, later provided coverage of $\sim 4700$
deg2 over the southern sky to similar depths as SDSS, but with superior
seeing. Other notable recent and upcoming photometric surveys include Dark
Energy Survey (DES; The Dark Energy Survey Collaboration 2005; Rykoff et al.
2016), Pan-STARRS (Kaiser et al. 2002; Ebeling et al. 2013), the Kilo-Degree
Survey (KiDS; de Jong et al. 2013; Radovich et al. 2017), the Hyper-Suprime
Camera (HSC; Aihara et al. 2018), and the Legacy Survey of Space and Time
(LSST; LSST Dark Energy Science Collaboration 2012), 4-metre Multi-Object
Spectroscopic Telescope (4MOST; de Jong et al. 2019) and Euclid (Laureijs et
al., 2011), providing enormous quantities of data upon which the search for
clusters of galaxies has and will be based.

In the absence of spectroscopic redshifts, galaxy redshifts can be estimated
photometrically based on techniques such as SED template fitting (e.g. Hyperz;
Bolzonella et al. 2000), using Bayesian probabilistic methods (e.g. BPZ;
Benítez 2011) or via machine learning approaches utilising artificial neural
networks or boosted decision trees (e.g. ANNz2; Sadeh et al. 2016). Due to
their large uncertainties, however, photometric redshifts alone are not
sufficient for accurate identification of galaxy clusters via 3D
reconstruction of the distribution of galaxies. A commonly adopted method of
detecting galaxy clusters in the optical regime is taking advantage of the
cluster red sequence (Baum 1959; Bower et al. 1992; Gladders & Yee 2000). The
red sequence refers to the tight correlation followed by the cluster galaxies
in the colour-magnitude space, this feature arises due to galaxy clusters
mostly containing early-type elliptical and lenticular galaxies which consist
of passively evolving stellar populations giving rise to strong metal
absorption lines at wavelengths blue-wards of $4000$Å. As a result, the
majority of cluster members appear red in colour and occupy a narrow ridge in
the colour-magnitude space, when viewed through broadband photometric filters
that straddle the $4000$Å break. In this way cluster galaxies can be isolated
from active, star-forming field galaxies. Algorithms utilising the red
sequence for cluster detection include Koester et al. (2007); Gladders et al.
(2007); Thanjavur et al. (2009); Hao et al. (2010); Rykoff et al. (2014) as
well as The Overdense Red-sequence Cluster Algorithm (orca; Murphy et al.
2012) used in this work.

Galaxy clusters have also been detected in X-ray by exploiting the radiation
due to thermal bremsstrahlung and line emission from the Inter Cluster Medium
(Cavaliere & Fusco-Femiano 1976; Allen et al. 2002). In this work, we compare
our optically detected cluster catalogue with the ‘MCXC: a meta-catalogue of
X-ray detected clusters of galaxies’ (Piffaretti et al. 2011) which contains
$\sim 1700$ X-ray detected clusters. In the near future, an ongoing all-sky
X-ray survey eROSITA (Merloni et al., 2012) with the aim of detecting $\sim
100,000$ galaxy clusters out to $z>1$ will significantly increase the cluster
sample size in the X-ray regime. The cluster catalogue introduced in this work
will provide a valuable resource which could be used to better characterise
the selection function of the eROSITA cluster in terms of sample completeness
and purity.

Inverse Compton scattering due to interaction with high energy electrons in
the ICM provides a boost in energy to the cosmic microwave background (CMB)
photons passing through clusters, making them detectable via a phenomenon
known as the Sunyaev-Zel’dovich (SZ) effect (Sunyaev & Zeldovich 1980). With
the benefit of a detection signature that is essentially redshift independent,
the SZ effect offers a new window on the cluster population providing a nearly
mass-limited sample at high redshifts, where cluster abundance can place
sensitive constraints on cosmological parameters (Carlstrom et al. 2002;
Planck Collaboration et al. 2014b). In recent years, several SZ surveys have
been undertaken with South Pole Telescope (SPT; Carlstrom et al. 2011) survey,
the Atacama Cosmology Telescope (ACT; Fowler et al. 2007), and Planck (Tauber
et al., 2010), providing the first SZ-selected cluster samples (Vanderlinde et
al. 2010; Menanteau et al. 2010; Planck Collaboration et al. 2011; Reichardt
et al. 2013; Hasselfield et al. 2013). In this work, we also draw a comparison
between our cluster catalogue and the second Planck catalogue of SZ sources
(henceforth Planck SZ; Planck Collaboration et al. 2016c), which contains 1653
detections, out of which 1203 are confirmed clusters based on identification
of counterparts in external datasets. Similarly, we compare the ATLAS cluster
catalogue with SZ detections from the ACT DR5 sample (Hilton et al., 2021),
which has a large overlap with part of ATLAS in the Southern Galactic Cap
(SGC). The ACT DR5 sample contains over 4000 cluster detections in a survey
area of $13,168$ deg2, probing lower cluster masses than the Planck SZ sample.

We then present a calibration of the ATLAS cluster mass-richness scaling
relation using cluster masses from the MCXC, ACT DR5, Planck and SDSS
redMaPPer samples. Our final aim in this work is to compare the observed mass
function of the ATLAS cluster catalogue to theoretical predictions of
$\Lambda$CDM based on the Tinker et al. (2008) model and other cluster
samples. We choose the Tinker et al. (2008) model due to its common use in
literature for cosmological analyses of various cluster samples over the past
decade. (see e.g. Vikhlinin et al. 2009; Allen et al. 2011; Bleem et al. 2015;
Planck Collaboration et al. 2016b; Bocquet et al. 2019; Abbott et al. 2020).
We perform this comparison in five redshift bins covering the range
$0.05<z<0.55$ in order to examine the evolution of the cluster mass functions.

The layout of this paper is as follows: In Section 2 we describe the VST ATLAS
galaxy input catalogue. Then in Section 3 we present a brief description of
the orca algorithm, as well as a description of our machine learning approach
to obtaining photometric redshifts for our clusters. We also present a
description of our cluster richness and mass estimation. In Section 4, we
present the ATLAS cluster catalogue and compare our results with external
multi-wavelength cluster samples, providing estimates of the completeness and
purity of the ATLAS cluster catalogue. This is followed by a comparison of the
observed cluster mass functions of the ATLAS, redMaPPer, Planck and ACT DR5
samples, versus theoretical predictions. Finally, we conclude by summarizing
the cosmological implications in Section 5. Unless otherwise specified, we
assume a Planck Collaboration et al. (2016a) $\Lambda$CDM cosmology with
$H_{0}=67.74$, $\Omega_{\Lambda}=0.6911$, $\Omega_{m}=0.3089$,
$\sigma_{8}=0.8159$, present all magnitudes in the AB system and define our
cluster masses as $M_{200\textup{m}}$ which is the mass enclosed in a halo
with a density $200\times$ the mean matter density of the Universe.

##  2 Dataset

The VST ATLAS (Shanks et al., 2015) is an European Southern Observatory (ESO)
public survey of the southern sky, designed to provide optical imaging in
$ugriz$ bands to similar depths as SDSS in the north. The data is taken using
the Very Large Telescope Survey Telescope (VST; Schipani et al. 2012) a 2.61-m
telescope with a $1\deg^{2}$ field of view located at the Paranal observatory.
The total survey area consists of $4711\deg^{2}$, with $2087\deg^{2}$ in the
Northern Galactic Cap (NGC) and $2624\deg^{2}$ in the Southern Galactic Cap
(SGC). The ATLAS coverage area is overlapped by the DES (The Dark Energy
Survey Collaboration, 2005), DESI (DESI Collaboration et al., 2016), KiDS (de
Jong et al., 2013) and eROSITA (Merloni et al., 2012) surveys. In the future,
the 4MOST (de Jong et al., 2019), Euclid (Laureijs et al., 2011) and LSST
(LSST Dark Energy Science Collaboration, 2012) surveys will also provide
multi-wavelength imaging and spectroscopic coverage of the VST ATLAS survey
area.

Based on the ATLAS DR4 data, we produce a band-merged $griz$ catalogue using a
pipeline developed with the ‘Starlink Tables Infrastructure Library Tool Set’
(STILTS; Taylor 2006) framework. During band-merging, we include all objects
with detections in a minimum of two bands. In this work, we utilise the VST
ATLAS aperture magnitudes provided by the Cambridge Astronomical Survey Unit
(CASU)111http://casu.ast.cam.ac.uk/surveys-projects/vst/technical/catalogue-
generation for obtaining galaxy colours while relying on the CASU Kron pseudo-
total magnitude for imposing magnitude limits on our galaxy samples. The ATLAS
Kron magnitudes are measured using circular apertures of $2\times$ the Kron
radius, with the definition of the Kron radius given by Bertin & Arnouts
(1996). The use of aperture magnitudes to measure galaxy colours was motivated
by the inspection of the red-sequence of rich, spectroscopically confirmed
Abell clusters, as well as comparison of galaxy colours between ATLAS Kron and
aperture magnitudes and SDSS model magnitudes in a $\sim 200$ deg2 overlap
area between the two surveys. These tests showed that aperture magnitudes have
a lower level scatter than Kron around the red-sequence and when compared to
SDSS. On the other hand comparison between ATLAS and SDSS shows that Kron
magnitudes provide a more reliable measure of the galaxies’ total magnitudes
compared to aperture magnitudes, which could miss a larger fraction of the
galaxy flux (see also Shanks et al. 2015). We denote aperture magnitudes
corresponding to the ATLAS Aperture flux 5 using the subscript ‘A5’. This
aperture has a radius of $2^{\prime\prime}$ and we apply the associated
aperture correction labelled as APCOR in the CASU catalogue. For $g,r,i$ and
$z$ bands, the mean values of APCOR5 are 0.12, 0.12, 0.11 and 0.12 mags.
Although these aperture corrections are derived for stars, they also provide a
first-order seeing correction for faint galaxies. Where Kron pseudo-total
magnitudes are used, we correct these to total magnitude for galaxies by
applying a $-0.15$ mag offset. This value is chosen based on an empirical
comparison of the ATLAS Kron and SDSS model magnitudes for galaxies in an
overlap area between the two surveys as shown by Shanks et al. (2015). We then
correct all magnitudes for Galactic dust extinction
$A_{\textup{x}}=C_{\textup{x}}E(B-V)$, with $x$ representing a filter
($griz$), taking the SDSS $C_{\textup{x}}$ values presented in Schneider et
al. (2007) (3.793, 2.751, 2.086, and 1.479 for $griz$ respectively) and using
the Planck $E(B-V)$ map (Planck Collaboration et al. 2014a).

In order to isolate galaxies from stars in the ATLAS data, we use the default
morphological classifications supplied in CASU catalogues, a description of
which can be found in González-Solares et al. (2008). Reflections from bright
stars in the VST ATLAS data could lead to the formation of circular halos
which in some cases can be misidentified as multiple extended sources and
misclassified as galaxies by the CASU source detection algorithm. To overcome
this, we mask circular regions around these bright stars based on cross-
matching the input catalogue to the Tycho-2 Catalogue of the 2.5 million
brightest stars (Høg et al., 2000). The Tycho bright stars are masked with
radii varying according to their $V$-band magnitudes. For this purpose we
choose the following radii based on visual inspection of stars with various
magnitudes: $V<8$: $340^{\prime\prime}$; $8<V<9$: $80^{\prime\prime}$;
$9<V<10$: $45^{\prime\prime}$; $10<V<11$: $30^{\prime\prime}$; $V>11$:
$20^{\prime\prime}$. However, depending on the position of the star on the CCD
chip, in some cases the halo can be off-centred from the stars. The remaining
stellar halos and other major remaining artefacts such as nearby galaxies or
satellite trails are manually removed by performing a visual inspection of the
data.

The input catalogue given to orca for cluster detection consists of objects
detected in two adjacent bands (i.e. $g-r$, $r-i$ or $i-z$ which are used for
cluster red-sequence detection). We also require objects to be classified as
galaxies in a minimum of two bands. Here, we only require objects to be
classified as galaxies in two bands as demanding a galaxy classification in
all bands was deemed too strict, resulting in a $\sim 10\%$ decrease in the
overall sample size. The slight increase in stellar contamination introduced
in this approach is unlikely to be a problem in our cluster detection, due to
orca’s reliance on the red sequence in selecting cluster members (see Section
3.1). We do not include the $u$ band due to its shallow depth and the fact
that the incompleteness of $u$-band observations at the time of conducting
this work would result in large gaps across the survey area.

As the VST ATLAS survey observations are conducted in $17$ deg2 blocks at
constant Declination, taking data in a single band at a time, nightly
variations in seeing, sky brightness and other observing conditions can result
in slight variations in survey depth across different concatenations. After
band merging, these slight fluctuations in object densities in some
concatenations could result in artificial inhomogeneities across the sky.
Consequently, we select magnitude limits of $g_{\rm Kron}<22.0$; $r_{\rm
Kron}<21.6$; $i_{\rm Kron}<21.1$; $z_{\rm Kron}<19.9$ as a compromise between
increasing the survey depth and increasing homogeneity between concatenations.
The final input catalogues contain $\sim 8,740,000$ galaxies in the SGC and
$\sim 7,825,000$ in the NGC of the survey.

##  3 Methodology

###  3.1 orca: The cluster detection algorithm

A detailed description of the ‘Overdense Red-sequence Cluster Algorithm’
(orca) which is used to create the ATLAS cluster catalogue can be found in
Murphy et al. (2012). The algorithm detects the red sequence in pairs of
adjacent bands or colours (in our case $g-r$, $r-i$ or $i-z$). In the first
stage, the algorithm applies a selection function to the input catalogue in
the form of narrow slices in the colour-magnitude space. This photometric
filtering separates galaxies within a specific redshift range from foreground
and background objects, broadly isolating the cluster galaxies via the $4000$Å
break. By detecting clusters in two colours concurrently, foreground and
background contamination can be significantly reduced, as galaxies follow
unique tracks in different colour-redshift spaces. During this stage, the red
sequence is isolated across a range of redshifts through modifications of the
colour slice in successive runs of the algorithm, systematically scanning the
entire photometric space.

Upon the application of photometric filtering, the algorithm estimates the
surface density of the remaining galaxies by calculating the Voronoi diagram
of their projected distribution on the sky. The Voronoi cells are then
separated into overdense and underdense cells based on a user-specified
probability threshold ($P_{\rm thresh}$), related to how likely they are to
belong to a random distribution (for more details, see Section 3.4 of Murphy
et al. 2012). Finally using the Friend-Of-Friends technique, the algorithm
connects adjacent overdense cells until the density of the whole system falls
below a user-defined critical density ($\sum_{\rm crit}$). At this stage, if
the system has at least $N_{\rm gals}$ linked galaxies (in this case we set
$N_{\rm gals}=5$), it is defined as a cluster.

We optimize the $P_{\rm thresh}$ and $\sum_{\rm crit}$ parameters for
performance on VST ATLAS based on multiple runs of the orca algorithm on a
$\sim 300$ deg2 area and assessing its performance based on recovering the
Abell, MCXC, redMaPPer and Planck SZ clusters in this region. We then set
$\sum_{\rm crit}=2.5\bar{\sum}$ (where $\bar{\sum}$ is the mean galaxy
density), and $P_{\rm thresh}=0.0125$, using the default for other adjustable
parameters of the algorithm as these do not have a major effect on improving
the results. As the adjustment to the colour slice is, by design, less than
the width of the red sequence, the same cluster can be identified multiple
times in successive runs of the algorithm.

###  3.2 ANNz2: Photometric redshift estimation

We make use of the publicly available
ANNz2222https://github.com/IftachSadeh/ANNZ (Sadeh et al., 2016) algorithm in
order to obtain photometric redshift estimates for the VST ATLAS cluster
members. For this purpose, we simultaneously make use of a combination of two
machine learning methods offered by the algorithm; Artificial Neural Networks
(ANNs) and Boosted Decision Trees (BDTs). After performing various tests, this
approach was shown to produce the best RMS scatter in comparison to using ANNs
or BDTs alone. Here the RMS error is given by:

$\sigma_{\Delta z/(1+z)}\equiv\sqrt{\frac{1}{n_{\rm gals}}\sum_{\rm
gals}\left(\frac{z_{\rm photo}-z_{\rm spec}}{1+z_{\rm spec}}\right)^{2}},$ (1)

where $n_{\rm gals}$ is the number of galaxies in the training set and $z_{\rm
photo}$ and $z_{\rm spec}$ are estimated photometric and measured
spectroscopic redshifts of these galaxies, respectively.

To estimate photometric redshifts using machine learning, we are required to
provide the algorithm with a training sample of galaxies with measured
redshifts, which overlap the VST ATLAS survey area. For this purpose, we make
use a total of 21,114 galaxies with redshifts obtained from various surveys as
detailed in Table 1. In cases where the same objects have redshifts provided
by more than one survey, we keep the redshift with the smaller uncertainty.
Figure 1 shows the redshift distribution of the galaxies included in the
training set used in our photometric redshift estimation.

Table 1: Details of the samples used in our photometric redshift training. Here the redshift coverage, the mean redshift of the samples, and the number of galaxies which corresponds to the number of objects with redshifts identified as ATLAS cluster members by the orca algorithm. Sample | Redshift coverage | $\bar{z}$ | Number of galaxies | Reference
---|---|---|---|---
2dFGRS | $0.00<z<0.30$ | 0.13 | 9,426 | Colless et al. (2003)
SDSS DR12 | $0.05<z<0.35$ | 0.15 | 5,260 | Alam et al. (2015)
GAMA G23 | $0.05<z<0.50$ | 0.21 | 3,008 | Liske et al. (2015)
Primus | $0.05<z<0.60$ | 0.24 | 177 | Cool et al. (2013)
2dFLenS | $0.05<z<0.75$ | 0.26 | 2,717 | Wolf et al. (2017)
BOSS LOWZ | $0.05<z<0.45$ | 0.26 | 375 | Dawson et al. (2013)
BOSS CMASS | $0.40<z<0.75$ | 0.52 | 141 | Dawson et al. (2013)

In order to improve our photometric redshift training, in addition to the
ATLAS $griz$ bands, we add the W1 and W2 magnitudes from the “unblurred coadds
of the WISE imaging” catalogue (unWISE; Schlafly et al. 2019)333We note that
despite the $6^{\prime\prime}$ WISE PSF, thanks to the improved modelling of
the blended sources in the unWISE catalogue, the addition of W1 and W2
information to our machine learning training still proved useful in improving
our photo-z RMS scatter.. Here we use a $2^{\prime\prime}$ radius to match
ATLAS and unWISE sources and we correct for galactic dust extinction in W1 and
W2 bands by subtracting $0.18\times E(B-V)$ and $0.16\times E(B-V)$ from the
W1 and W2 magnitudes respectively. These $E(B-V)$ values are taken from the
same Planck dust map (Planck Collaboration et al., 2014b) used to correct the
ATLAS magnitudes for galactic dust extinction and the $0.18$ and $0.16$
coefficients are taken from Yuan et al. (2013).

Figure 1: The redshift distribution of the training sample used to estimate
the photometric redshifts of the ATLAS galaxy cluster members by ANNz2.

Finally, we calculate the weighted mean photometric redshift of our clusters
using: $\bar{z}=(\sum z_{\rm i}/{\sigma_{\rm i}}^{2})/(\sum 1/{\sigma_{\rm
i}}^{2})$, where $z_{\rm i}$ and $\sigma_{\rm i}$ are the photometric redshift
and photometric redshift uncertainty of the $i$-th cluster member. The
uncertainty on the cluster redshift is then given by the standard error on the
mean, and we verify the uncertainty using the Jackknife technique.

###  3.3 Cluster catalogue post-processing

As discussed in 3.6 of Murphy et al. (2012), multiple detections of the same
cluster found in different colour-magnitude spaces are merged together by orca
based on five tests of ‘cluster similarity’. These criteria are based on the
similarity of the clusters’ red-sequence, the extent of spatial overlap and
the number of common galaxies between the two detections.

In this work, we utilise our photometric redshifts to further merge
overlapping cluster detections that are likely to belong to the same system
based on the following criteria:

  * •

Spatial overlap: A cluster centre lies within the cluster radius of a nearby
cluster. Here, the cluster centre is defined as the mean RA and Dec of its
cluster member and cluster radius is defined as the angular separation between
the furthest cluster member and the cluster centre. In addition, we require at
least one of the criteria below to be satisfied in order for overlapping
detections to be considered as part of the same system.

  * •

Photometric redshift overlap: The error weighted mean cluster photometric
redshift lies within one standard error of another spatially overlapping
cluster.

  * •

Red-sequence overlap: The mean colour of a cluster’s members (in either $g-r$,
$r-i$ or $i-z$) is within one standard error of another spatially overlapping
cluster’s mean colour.

Figure 2: The outcome of merging overlapping clusters based on spatial and
photometric redshift, or red-sequence overlap. Here, circles show the
corresponding cluster radii used to identify spatially overlapping clusters,
with the black circles marking clusters that are merged.

Requiring the combination of spatial overlap, with either photometric redshift
or red-sequence overlap results in the merging of $9\%$ of the clusters in the
catalogue. Figure 2 shows an example of merged overlapping clusters based on
the above criteria.

###  3.4 Cluster richness ($N_{200}$)

In order to provide an estimate of cluster richness, we first count the number
of cluster galaxies within a $1h^{-1}$ Mpc aperture of the cluster centre,
$N_{\textup{1Mpc}}$, which in the $i$-band, are fainter than the Brightest
Cluster Galaxy (BCG), and brighter than $0.4L_{*}$. Here $L_{*}$ is the
characteristic luminosity in the Schechter luminosity function, and following
Reyes et al. (2008), we take the $z=0.1$, $i$-band value of $L_{*}=2.08\times
10^{10}h^{-2}\textup{L}_{\odot}$. To calculate the K-corrections, we make use
of the ‘K-corrections calculator’ Python
algorithm444http://kcor.sai.msu.ru/getthecode/, which is based on the
procedure described in Chilingarian et al. (2010).

Once the values of $N_{\textup{1Mpc}}$ are determined, we calculate the
cluster radius $R_{200}$, defined as the radius within which the cluster
galaxy number density is $200\Omega_{m}^{-1}$ times the mean galaxy density of
the present Universe. We calculate $R_{200}$ (in units of $h^{-1}$Mpc) using
an empirical relation presented by Hansen et al. (2005):

$R_{200}=(0.142\pm 0.004)N_{\textup{1Mpc}}^{0.6\pm 0.01},$ (2)

based on their analysis of the SDSS maxBCG clusters. $R_{200}$ is in turn used
to obtain the final cluster richness $N_{200}$ which is calculated in the same
way as $N_{\textup{1Mpc}}$, but now using $R_{200}$ as the aperture within
which the cluster members are counted as opposed to the fixed aperture of 1
Mpc.

###  3.5 Scaling $N_{200}$ by $n(z)$

Due to the magnitude limits of our data, our ability to detect fainter cluster
members is reduced as a function of redshift, which could lead to an under-
estimation of our cluster $N_{200}$ values at higher redshifts. In order to
correct for the impact of the survey magnitude limits on our $N_{200}$ and
subsequent cluster mass ($M_{\textup{200m}}$) estimation, we up-weight our
$N_{200}$ values by a theoretical galaxy $n(z)$ curve which gives the relative
number of galaxies detectable as a function of redshift, given our $i<21.1$
magnitude limit. This theoretical $n(z)$ is obtained based on a luminosity
function which assumes a Schechter (1976) function, with the i-band, ‘red’
galaxy values of $\alpha=-0.46$ and the $z=0.1$ value of $M^{*}-5\log
h=-20.63$ taken from Table 3 of Loveday et al. (2012), $k$ and evolution
corrected (with a star-formation timescale of $\tau=2.5$ Gyr), using the
Stellar population synthesis model of Bruzual & Charlot (2003).

(a)

(b)

Figure 3: (a) orca cluster membership $N_{gal}$ as a function of photometric
redshift. We also highlight orca clusters with counterparts in Planck SZ,
MCXC, ACT DR5 and Abell cluster catalogues. The best fit curve (dashed pink
line), fitted to the normalized theoretical $n(z)$ described in Section 3.5 is
used to determine the redshift dependent $N_{200}$ scaling factor shown in
panel (b).

Figure 3(a) shows the cluster membership of our orca clusters (highlighting
orca clusters matched to Planck, ACT DR5, MCXC and Abell cluster samples) as a
function of redshift. Here, we are also plotting the theoretical $n(z)$
described above, normalized to match our maximum value of
$N_{\textup{gal}}=220$ at $z=0.1$. Also shown is a fitted curve to the
theoretical n(z) which is used to obtain the scaling factor, $C(\bar{z})$,
applied to our $N_{200}$ in the $z>0.1$ redshift range, where the number of
our cluster members begin to drop as a function of redshift. This scaling
factor takes the form of:

$C(\bar{z})=220/(571.7\bar{z}^{2}-819\bar{z}+292.8),$ (3)

i.e. the ratio of $N_{\textup{gal}}=220$ to our normalized theoretical $n(z)$
at each redshift and is shown in Figure 3(b) as a function of redshift.

(a)

(b)

Figure 4: (a) $n(z)$ scaled $N_{200}$ values as a function of photometric
redshift. A comparison of this plot with Figure 3(a) shows that this scaling
has compensated for our tendency to underestimate cluster richness values with
increasing redshift. For clarity, we do not show the error bars on the scaled
$N_{200}$ values in panel (a). However, in panel (b) we show the mean
uncertainty on the scaled $N_{200}$ values as a function of cluster
photometric redshift.

In order to apply the $C(\bar{z})$ scaling to our $N_{200}$ values, we first
scale our $N_{\textup{1Mpc}}$ values which are used to estimate the value of
$R_{200}$ for each cluster, thus ensuring our $R_{200}$ values are not
underestimated at higher redshifts. We then count the number of cluster
galaxies within $R_{200}$ and scale this number by the $C(\bar{z})$ to obtain
the $N_{200}$ values plotted in Figure 4(a). In this manner, when estimating
the richness of our clusters, we account for the increased likelihood of
cluster galaxies remaining undetected with increasing redshift. The
uncertainty on the scaled $N_{200}$ is then given by propagation of the
$\sqrt{n}$ error on $N_{200}$ and the uncertainty on $C(\bar{z})$:

$\sigma_{N_{200}}=2200\times\\\
\sqrt{\frac{4N_{200}^{2}(5717\bar{z}-4095)^{2}\sigma_{\bar{z}}^{2}+N_{200}(5717\bar{z}^{2}-8190\bar{z}+2928)^{2}}{(5717\bar{z}^{2}-8190\bar{z}+2928)^{4}}}.$
(4)

Note that in the above equation the $N_{200}$ values are not scaled by
$C(\bar{z})$. For the remainder of this work, however, unless otherwise
specified, our use of $N_{200}$ refers to these $n(z)$ scaled $N_{200}$
values.

We note that our cluster mass estimates and the resulting mass functions as
presented in Sections 4.5 and 4.6 are not very sensitive to the chosen value
of normalisation and exact parameters of equation 3. This is because any
systematic redshift-dependent offsets introduced during the $n(z)$ scaling is
removed when we calibrate our cluster mass-richness scaling relation to
cluster masses from external samples in the next Section.

###  3.6 ATLAS cluster mass-richness scaling relation

Using our $N_{200}$ estimates from Section 3.4, along with the mean
photometric redshifts of our clusters, $\bar{z}$, we provide cluster masses
$M_{200\textup{m}}$ (in units of $10^{14}h^{-1}\textup{M}_{\odot}$) based on
the following scaling relation:

$M_{200\textup{m}}=\left(\frac{N_{200}}{20}\right)^{1.1}\times 3\bar{z}\
^{0.9}+1.$ (5)

The form and parameters of this relation are chosen based on comparison555The
ATLAS mass-richness scaling relation is chosen to minimise the offset and
scatter in Figure 6(a). of the resulting ATLAS cluster mass estimates to
masses of $626$ clusters from external samples. These include $218$ ACT DR5
and $95$ Planck SZ clusters, $185$ SDSS redMaPPer clusters and $118$ MCXC
X-ray clusters. The error on our cluster masses are then approximated using:

$\sigma_{M_{200\textup{m}}}=\sqrt{0.015\sigma_{N_{200}}^{2}N_{200}^{0.2}\bar{z}^{1.8}+\frac{0.01N_{200}^{2.2}\sigma_{\bar{z}}^{2}}{\bar{z}^{0.2}}},$
(6)

where $\sigma_{\bar{z}}$ and $\sigma_{N_{200}}$ are the uncertainties on
$\bar{z}$ and $N_{200}$ respectively.

(a)

(b)

Figure 5: Redshift (a) and mass (b) distributions of the clusters from the ACT
DR5, redMaPPer, MCXC and Planck SZ cluster catalogues, used to calibrate the
ATLAS cluster masses.

Figure 5 shows the redshift and mass distribution of the clusters from these
four samples. The combination of these samples provide a reasonable number of
counterparts to ATLAS clusters across the redshift range of $0.05<z<0.55$ and
the mass range of $10^{14}-10^{15}$ M$\odot$. To select the cluster
counterparts, we use the orca cluster radius ($R_{\rm ORCA}$) as our matching
radius for the ACT DR5, redMaPPer, MCXC and Abell cluster catalogues, while
using the Planck position uncertainty on the SZ cluster centre as the matching
radius. Here, we calibrate the parameters in our cluster mass-richness scaling
relation (Equation 5) using all available cluster masses from these four
samples simultaneously, to avoid biasing our masses towards the range of
masses covered by the individual samples.

We obtain the masses of the redMaPPer clusters using the weak lensing
calibrated mass-richness scaling relation of Simet et al. (2016), with
$M_{\textup{200m}}=10^{14.344}(\lambda/40)^{1.33}$. Here, $M_{\textup{200m}}$
is the $M_{200}$ cluster mass with respect to the mean density of the universe
(in units of $10^{14}h^{-1}$$\textup{M}_{\odot}$) and $\lambda$ is the
richness of redMaPPer clusters as defined by Rykoff et al. (2014). Similarly,
we use the $M_{\textup{200m}}$ cluster masses from the ACT DR5 catalogue. As
the MCXC cluster masses are defined as $M_{500}$, we multiply these by a
factor of 1.5 in order to remove the mean offset found between the MCXC
$M_{500}$ masses and the $M_{\textup{200m}}$ masses of their redMaPPer and ACT
DR5 counterparts. We correct for the mean offset between the Planck SZ mass
proxy values, $M_{\textup{SZ}}$, and the $M_{\textup{200m}}$ masses of their
redMaPPer and ACT DR5 counterparts, by multiplying the Planck masses by a
factor of 1.3. Once the Planck and MCXC masses are scaled to roughly
correspond to $M_{\textup{200m}}$ masses, we use the masses from the four
samples to obtain the ATLAS mass-richness scaling relation given by Equation
5. In the case of the ACT DR5 and Planck SZ samples, we limit the sources to
those with a SNR>5.

(a)

(b)

Figure 6: (a) The ratio of our orca cluster masses to masses of their cluster
counterparts from external samples. The dotted lines show regions where our
masses are in agreement with external masses within $\pm 50\%$. (b) Histogram
showing the ratio of the orca cluster masses to external cluster masses shown
in (a). Here, we mark the percentage of clusters with mass ratios below, above
and within the $\pm 50\%$ region.

The ratio of our orca cluster masses to masses based on their counterparts
from these four samples are shown in Figure 6(a). Our cluster masses are
calibrated on the basis of minimizing the mean offset and scatter in this plot
as well as the mass ratio histogram shown in Figure 6(b). The level of scatter
found in the ratio of our richness-based cluster mass estimates to masses from
external X-ray and SZ samples is comparable to that found when we compare the
redMaPPer richness-based cluster masses to masses from these external samples.

The photometric redshift term in our ATLAS mass-richness scaling relation
(Equation 5) is added to remove a redshift dependent offset found between our
$M_{\textup{200m}}$ masses and the masses from the four external catalogues.
This redshift dependence may be partially due to the presence of a slight
systematic offset towards lower redshifts in our photometric redshift
estimates in the $z>0.35$ redshift range, which is likely caused by the
relatively lower number of galaxies in our spectroscopic training set in this
redshift range (see Figures 1, 8 and the discussion in Section 4.2). The
presence of such systematic offset towards lower redshift will, in turn,
result in a lower $n(z)$ scaling factor being applied to the high redshift
cluster richness values, biasing our estimates of $N_{\textup{200}}$ and
$M_{\textup{200m}}$ low.

In addition, although no evidence has been found to suggest cluster mass-to-
light ratios ($M/L$) increase as a function of redshift (e.g. Lin et al. 2006;
Muzzin et al. 2007; Soucail et al. 2015), the increase of cluster $M/L$ as a
function of cluster mass has been well established. Over a wide range of
masses, various studies have found a ratio of $M/L\propto M^{\alpha}$, with
$\alpha$ ranging from $0.2-0.4$ (Girardi et al. 2002; Lin et al. 2003; Rines
et al. 2004; Popesso et al. 2007). Given that with increasing redshift (and in
the $z>0.3$ range in particular) due to the magnitude limits of our sample we
are increasingly likely to miss smaller, lower mass clusters, with the same
being true (to varying degrees) for the external cluster samples used in our
mass calibrations666This is due to the flux limit of the redMaPPer and MCXC
samples, and for Planck and ACT DR5, due to the SZ lower-limit of cluster mass
observability., the increased fraction of higher mass clusters at higher
redshifts could mean that our $N-M$ scaling relation which provided a good
estimate of total cluster masses at lower redshifts, is under-estimating the
mass of our high redshift population which tend to have higher $M/L$ ratios
due to their higher mass.

##  4 Results and Discussion

###  4.1 The VST ATLAS cluster catalogue

Table 2 provides a description of the ATLAS catalogue columns, detailing the
various properties of clusters including redshift, richness, cluster radius
and cluster mass. For each catalogue column, we provide the column title given
in the catalogue, the corresponding symbol as used in the text of this work,
followed by the units and a description of the values in each column. Table 3
provides a description of the columns in the ATLAS cluster members catalogues
providing unique identifiers, coordinates and photometric redshifts of each
cluster galaxy.

Table 2: A description of the columns of the ATLAS cluster catalogue. For each catalogue column, the symbols column of this Table shows the corresponding symbol used in the text and figures of this paper. Column | Symbol [units] |  Description
---|---|---
ClusterID | — |  A unique cluster identification number assigned to each cluster by the orca cluster detection algorithm.
RA | [Degrees J2000] |  Right Ascension of the cluster centre (mean RA of cluster members).
DEC | [Degrees J2000] |  Declination of the cluster centre (mean Dec of cluster members).
Photoz | $\bar{z}$ |  Error-weighted mean photometric redshift of the cluster.
Photoz_err | $\sigma_{\bar{z}}$ |  Standard Error on the mean cluster photometric redshift.
R_ORCA |  $R_{\rm ORCA}$ [arcmin] |  The projected radius of the orca cluster on the sky, defined as the angular separation between the cluster centre and the furthest cluster galaxy.
ORCA_Ngal | $N_{gal}$ |  The number of cluster galaxies detected by orca.
N_1Mpc | $N_{1Mpc}$ |  The number of cluster galaxies detected by orca within a radius of $1h^{-1}$Mpc from the centre of the cluster. This number is scaled by the theoretical $n(z)$ following Equation 3.
R_200 |  $R_{200}$ [arcmin] |  The radius from the cluster centre within which the density is $200\times$ the mean density of the Universe (given by Equation 2).
N_200 | $N_{200}$ |  The number of cluster galaxies within a radius of $R_{200}$ from the centre of the cluster. This number is scaled by the theoretical $n(z)$ following Equation 3.
N_200_err | $\sigma_{N_{200}}$ |  The error on $N_{200}$ given by Equation 4.
M_200m |  $M_{200\textup{m}}$ [$10^{14}h^{-1}\textup{M}_{\odot}$] |  The cluster mass enclosed within a radius of $R_{200}$ from the centre of the cluster (given by Equation 5). The cluster masses in our catalogue are measured with respect to the mean density of the Universe (often represented using the symbol $M_{200m}$ in literature).
M_200m_err |  $\sigma_{M_{200\textup{m}}}$ [$10^{14}h^{-1}\textup{M}_{\odot}$] |  The uncertainty on the $M_{200\textup{m}}$ cluster mass given by Equation 6.
Table 3: A description of the columns of the ATLAS cluster members catalogue. For each catalogue column, the symbols column of this Table shows the corresponding symbol used in the text and figures of this paper. Column | Symbol [units] |  Description
---|---|---
ClusterID | — |  A unique cluster identification number assigned to each cluster by the orca cluster detection algorithm.
ObjID | — |  A unique object identification number assigned to each cluster galaxy by the orca cluster detection algorithm.
RA | [Degrees J2000] |  Right Ascension of the cluster member.
DEC | [Degrees J2000] |  Declination of the cluster member.
g_mag | $g_{\textup{A5}}$ |  VST ATLAS $g$-band Aperture 5 (aperture radius of $2^{\prime\prime}$) magnitude in the AB system.
r_mag | $r_{\textup{A5}}$ |  VST ATLAS $r$-band Aperture 5 (aperture radius of $2^{\prime\prime}$) magnitude in the AB system.
i_mag | $i_{\textup{A5}}$ |  VST ATLAS $i$-band Aperture 5 (aperture radius of $2^{\prime\prime}$) magnitude in the AB system.
z_mag | $z_{\textup{A5}}$ |  VST ATLAS $z$-band Aperture 5 (aperture radius of $2^{\prime\prime}$) magnitude in the AB system.
W1_mag | W1 |  unWISE W1 magnitude in the AB system.
W2_mag | W2 |  unWISE W2 magnitude in the AB system.
Photoz | $z$ |  Cluster galaxy photometric redshift as determined by ANNz2 machine learning algorithm (see Section 3.2).
Photoz_err | $\sigma(z)$ |  Error on cluster galaxy photometric redshift as determined by ANNz2.

###  4.2 Photometric redshifts

(a)

(b)

(c)

(d)

Figure 7: (a) Photometric vs. spectroscopic redshift of the orca cluster
members. The photometric redshifts are calculated following the procedure
described in Section 3.2. Outliers (red) are defined as $(|z_{\rm
photo}-z_{\rm spec}|/(1+z_{\rm spec}))>0.15$. (b) The distribution of the
(error-weighted) mean photometric redshift of the VST ATLAS clusters, with the
standard error on the cluster photometric redshift shown in panel (c). (d) A
comparison of the Jackknife and standard error estimates of the cluster
photometric redshift uncertainties shows a good agreement between the two
estimates. Here the error bars represent the error in the error given by
$1/\sqrt{2N-2}$ where $N$ is the number of clusters in each redshift bin.
Figure 8: A comparison of our ATLAS cluster photometric redshifts to
spectroscopic redshifts from the MCXC, SDSS redMaPPer, Planck and ACT DR5 SZ
cluster samples. The dotted lines mark the $\pm 0.1$ ($\sim 3\sigma$) error
region.

Following the procedure described in Section 3.2, we use ANNz2 to obtain
photo-z estimates for our cluster galaxies with an RMS scatter of $\sim 0.024$
(see Figure 7(a)), upon removing the outliers given by $((z_{\rm photo}-z_{\rm
spec})/(1+z_{\rm spec}))>0.15$. Inclusion of outliers results in an RMS of
$\sim 0.026$. The error-weighted mean photometric redshift of the full ATLAS
sample containing $\sim 40,000$ orca detections with $N_{200}\geq 5$, as well
as $\sim 22,000$ clusters with $N_{200}>10$ and $9,000$ clusters with
$N_{200}>20$ are shown in Figure 7(b). Here, the peaks of the distributions
lie at $z\sim 0.25$ and clusters are detected up to $z\sim 0.7$, (or $z\sim
0.6$ for clusters with $N_{200}>20$).

Figure 7(c) shows the distribution of the Standard error on the mean cluster
redshift peaking at $\sim 0.02$ for the full sample, and $\sim 0.01$ for
clusters with $N_{200}>20$. Finally, for the full sample, we show the
uncertainty on the mean cluster photometric redshifts as a function of
redshift in Figure 7(d), finding a good agreement between the Jackknife and
standard error estimates of the uncertainty.

In Figure 8 we compare the photometric redshift of our ATLAS clusters with
spectroscopic redshifts from their cluster counterparts from MCXC, SDSS
redMaPPer, Planck and ACT DR5 samples. Here the $>3\sigma$ outliers (based on
mean cluster photo-z error of $\sim 0.03$) are marked by the dotted lines and
constitute $\sim 6\%$ of the sample. While we find a general agreement between
the ATLAS photometric redshifts and the spectroscopic redshifts from external
samples, we also note hints of systematics at $z>0.3$ where our photometric
redshifts appear more likely to be under-estimated.

###  4.3 Mass and redshift completeness

(a) Relative redshift completeness

(b) Mean redshift completeness

Figure 9: (a) The redshift completeness of the ATLAS cluster sample, based on
the fraction of the recovered SDSS redMaPPer, ACT DR5, Planck, MCXC and Abell
clusters overlapping the ATLAS coverage area (see text for selection and
matching criteria). The error bars are given by the propagation of $\sqrt{n}$
error estimates and for clarity, the data points corresponding to different
datasets have been slightly shifted along the x-axis. (b) The mean redshift
completeness of the ATLAS clusters sample based on the comparison to external
clusters in panel (a). Here, the error bars are given by the standard error on
the mean.

Figure 9(a) shows the fraction of clusters from the SDSS redMaPPer, ACT DR5,
Planck, MCXC and Abell catalogues overlapping the ATLAS coverage area, that
are detected in the ATLAS cluster catalogue. This provides a measure of the
completeness of the ATLAS cluster sample as a function of redshift. In the
case of the ACT DR5 and Planck samples, we limit the match to clusters with SZ
detections with $SNR>5$. In all cases, the cluster samples are limited to
clusters with masses greater than $1\times 10^{14}h^{-1}$$\textup{M}_{\odot}$,
roughly corresponding to the lower mass limit of the ATLAS clusters. Figure
9(b) shows the mean completeness of the ATLAS cluster samples as a function of
redshift, where the sample is $>95\%$ complete in the range $z<0.3$ and
$>80\%$ complete up to $z=0.4$. We note that the sharp fall in our
completeness comparison to Planck at $z=0.5$ is due to small number statistics
of the Planck sample at this redshift range where we detect 3/7 Planck SZ
clusters overlapping the ATLAS coverage area.

(a) Relative cluster mass completeness

(b) Mean cluster mass completeness

Figure 10: (a) The fraction of SDSS redMaPPer, ACT DR5, Planck and MCXC
clusters overlapping the ATLAS coverage area which are detected in the ATLAS
cluster catalogue. This provides an estimate of the completeness of the ATLAS
cluster sample as a function of mass. The error bars shown in this panel are
given by the propagation of $\sqrt{n}$ error estimates and for clarity, the
data points corresponding to different datasets have been slightly shifted
along the x-axis. (b) The mean completeness of the ATLAS sample as a function
of cluster mass, based on comparison to the samples shown in panel (a). Here,
the error bars are given by the standard error on the mean.

Figure 10(a) shows the mass completeness of the ATLAS cluster catalogue by
assessing the fraction of SDSS redMaPPer, ACT DR5, MCXC and Planck clusters
detected in the ATLAS sample as a function of cluster mass. Here, all samples
are limited to the redshift range $0.1<z<0.3$ in order to ensure the mass
completeness is not impacted by our reduced completeness at higher redshifts.
Figure 10(b) shows the mean cluster mass completeness of the ATLAS sample,
with the sample being $>95\%$ complete across the full $1\times
10^{14}-1.5\times 10^{15}h^{-1}$$\textup{M}_{\odot}$ mass range, with a near
full recovery rate of external clusters for masses $>5\times
10^{14}h^{-1}$$\textup{M}_{\odot}$.

###  4.4 Comparison to redMaPPer

Figure 11 shows the number of ATLAS clusters as a function of cluster richness
($N_{200}$). A similar histogram showing the richness ($\lambda$) of the SDSS
redMaPPer sample is also added for comparison. Although redMaPPer clusters
with ($\lambda<20$) are not available with the public release of the
catalogue, it can be seen that both samples follow similar cluster richness
distributions with thousands of clusters in bins of richness smaller than
$\sim 40$, hundreds in richness bins between $\sim 40$ and 80, and tens of
clusters in bins of richness ranging from $\sim 80-140$. The ATLAS sample
however contains a slightly larger number of richer clusters ($\sim 10\%$),
which is likely due to differences in cluster detection algorithms and the
definitions of cluster richness between the two samples.

Figure 11: ATLAS cluster richness $N_{200}$ distribution in comparison to SDSS
redMaPPer cluster richness $\lambda$. Here both samples are limited to the
redshift range $0.1<z<0.4$, and the redMaPPer histogram is scaled down by a
factor of 2.2 to account for the differences between survey areas.

For the remainder of this section, we limit the ATLAS and redMaPPer samples to
their mutual survey overlap area of $\sim 200$ deg2, which constitutes $\sim
4\%$ and $\sim 2\%$ of the total ATLAS and SDSS survey areas respectively. As
the overlap area is situated at the edge of both surveys, we remove any
cluster that lies within $5^{\prime}$ of the survey boundaries. We highlight
that given the limited overlap area, one should keep in mind that the
comparisons in this section may not be representative of the complete cluster
samples. In addition, unless otherwise specified, we limit the cluster samples
to the $M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ mass
range in the comparisons performed in this section.

In Figure 12, we show a comparison of the ATLAS, redMaPPer, Abell and ACT DR5
cluster catalogues in the ATLAS/SDSS survey overlap areas. For reasons that we
will explore in more detail later, in the redshift range $0.05<z<0.35$ ATLAS
appears to perform better than redMaPPer in recovering Abell and ACT DR5
clusters777We note that the Abell sample is not as complete as the ATLAS and
redMaPPer samples as the latter surveys have the advantage of detecting
clusters in multiple colours, (with both catalogues detecting $\sim 3-4$ times
as many clusters as the Abell sample across their full survey footprints in
the $z<0.3$ redshift range). However, Abell is still a useful sample for
comparison to ATLAS and redMaPPer, as clusters detected in both ATLAS and
Abell samples are likely to be genuine rich clusters which should be detected
by redMaPPer.. In the $0.35<z<0.55$ redshift range, however, redMaPPer appears
to recover a larger fraction of ACT DR5 clusters than ATLAS. We also note that
at higher redshifts there appears to be a larger number of redMaPPer clusters
with no detections in ATLAS or ACT DR5 catalogues compared to lower redshift
ATLAS clusters with no detections in the other catalogues.

Figure 12: (a) Comparison of ATLAS, redMaPPer and Abell clusters limited to
the redshift range $0.05<z<0.35$ in the overlapping areas of ATLAS and SDSS.
(b) Same as (a) but now ATLAS and redMaPPer are compared to ACT DR5 clusters
with all samples limited to the redshift range $0.35<z<0.55$. In all cases,
ATLAS, redMaPPer and ACT clusters are limited to the
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ mass range. We
also note that the number density of Abell clusters in the ATLAS-SDSS overlap
area in the NGC is $\sim 50\%$ lower compared to the SGC.

Table 4 shows a comparison of the sky density of the ATLAS cluster catalogue
to the SDSS redMaPPer catalogue in five bins of redshift ranging from
$z=0.05-0.55$. Similar to what we saw in Figure 12, one can see here that in
the redshift range $z<0.35$ the ATLAS catalogue has a $\sim 3\times$ higher
cluster sky density compared to redMaPPer, while the two samples become more
comparable at $z>0.35$. A similar pattern can be seen in Figure 13(b) where we
compare the photometric redshift distributions of ATLAS and redMaPPer samples
(without limiting the two samples to the survey overlap areas).

Table 4: Number and sky density (number of clusters per square degree) of ATLAS and SDSS redMaPPer cluster samples in different redshift bins. Here, the samples are limited to the overlap area of the two surveys and we apply a mass cut of $M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ to both samples. | Number of clusters | Sky density
---|---|---
Redshift | ATLAS | redMaPPer | ATLAS | redMaPPer
$0.05<z<0.15$ | 7 | 2 | 0.03 | 0.01
$0.15<z<0.25$ | 21 | 7 | 0.10 | 0.03
$0.25<z<0.35$ | 47 | 14 | 0.22 | 0.07
$0.35<z<0.45$ | 56 | 52 | 0.26 | 0.24
$0.45<z<0.55$ | 90 | 72 | 0.42 | 0.34

To further examine this, we calculate the detection rate of Abell clusters
(which are classified as clusters with 30 or more members), by the redMaPPer
catalogue across the full SDSS survey footprint. We find that redMaPPer only
recovers $\sim 60\%$ of the $0.05<z<0.3$ Abell clusters, compared to a $\sim
85\%$ recovery rate in the ATLAS sample with a mass cut of
$M_{\textup{200m}}$>$0.9\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ (which
approximately corresponds to the lower mass limit of the redMaPPer sample
given its $\lambda>20$ richness cut). Consequently, the higher number of ATLAS
detections in this redshift range relative to redMaPPer is likely to be
predominantly due to the incompleteness of the redMaPPer sample at lower
redshifts.

(a)

(b)

Figure 13: (a) A comparison of the photometric redshift distribution of full
ATLAS and SDSS redMaPPer cluster catalogues. Here, both samples are restricted
to clusters with $M_{\textup{200m}}$>$3\times
10^{14}h^{-1}$$\textup{M}_{\odot}$. The redMaPPer histogram is scaled down by
a factor of 2.2 to account for the difference between survey areas. (b) The
number density of ATLAS and redMaPPer clusters per Gpc3.

We now explore the completeness and purity of the ATLAS sample as a function
of redshift, based on direct comparison to redMaPPer in the
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ mass range. In
column (2) of Table 5, for each redshift bin, we calculate the fraction of
SDSS redMaPPer clusters (within the ATLAS coverage area) that are also
detected by orca using the ATLAS data888Similar to Section 4.3, we use the
orca radius $R_{\rm ORCA}$ as our matching radius here and throughout the rest
of this Section when comparing the two catalogues.. Here, our aim is to simply
compute the likelihood that a SDSS redMaPPer cluster detection is also
identified by orca in the ATLAS data. As such, when performing the matching,
we only impose the mass and redshift cuts on the redMaPPer catalogue to allow
for matches to be made in cases where the two catalogues assign the same
cluster to different mass and redshift bins. These selections in effect ensure
that our completeness estimates are not biased due to differences in
photometric redshift and richness estimates between the two samples.

With these caveats in mind, we convert the recovery fractions shown in column
(2) to percentages giving an estimate of the completeness of the ATLAS
catalogue as a function of redshift (under the assumption that the redMaPPer
sample is $100\%$ pure). Based on this comparison, we find that the ATLAS
sample is $100\%$ complete up to $z<0.25$, with a slight reduction in
completeness to $93\%$ between $0.25<z<0.35$ and a further reduction down to
$63\%$ and $46\%$ completeness at $0.35<z<0.45$ and $0.45<z<0.55$
respectively. Although these values are presented here to show a full
comparison of the two samples, we note that the mean completeness shown in
Figure 9(b) is likely to be a more reliable estimate of the completeness of
the ATLAS cluster sample than estimates based on comparison to any one
external sample alone.

Table 5: Various comparisons of the intersections between the ATLAS and SDSS redMaPPer cluster catalogues. (columns 2 & 3) Fraction & percentage of SDSS redMaPPer clusters with detections in the ATLAS cluster catalogue. (4) The fraction of ATLAS clusters with counterpart detections in the redMaPPer catalogue. (5) Among the ATLAS clusters not matched to redMaPPer in column (4), how many clusters we constitute as genuine and rich detection clusters based on Visual Inspection (VI) of their photo-z $n(z)$. (6) The fraction of ATLAS detections that are either detected by redMaPPer or appear as likely detections based on the VI of the cluster $n(z)$s. (7) An estimate of the completeness of the redMaPPer based on the ratio of redMaPPer cluster detections to the number of “pure” ATLAS detections (from column 6). We refer the reader to the discussion in the text for full details of the sample selections implemented prior to comparing the catalogues. Redshift | AT/RM | AT Comp. | RM/AT | VI | AT Purity | RM Comp.
---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7)
$0.05<z<0.15$ | 2/2 | 100% | 5/7 | +2/7 | 7/7 (100%) | 5/7 (71%)
$0.15<z<0.25$ | 7/7 | 100% | 16/21 | +6/21 | 21/21 (100%) | 16/21 (76%)
$0.25<z<0.35$ | 13/14 | 93% | 29/47 | +12/47 | 41/47 (87%) | 29/41 (70%)
$0.35<z<0.45$ | 35/52 | 63% | 30/56 | +10/56 | 40/56 (71%) | 30/40 (75%)
$0.45<z<0.55$ | 33/72 | 46% | 23/90 | +27/90 | 50/90 (56%) | 23/50 (46%)

In columns (4, 5 & 6) of Table 5, we estimate the purity of the ATLAS cluster
catalogue based on comparison to the cluster detections by redMaPPer in the
survey overlap region. As we are interested in assessing our sample purity as
a function of redshift, we limit the ATLAS sample to photometric redshift bins
shown in column (1) and look for confirmation that the ATLAS detection is also
identified by redMaPPer. The fraction of ATLAS detections confirmed by
redMaPPer in this way at each redshift bin is shown in column (4).

We note, however, that without access to the full redMaPPer catalogue we have
no way of checking whether our clusters with no successful matches to
redMaPPer may have been detected and classified by redMaPPer as clusters with
$\lambda<20$, or indeed, remain undetected by redMaPPer. To overcome this
limitation, in the next step we visually inspect the photometric redshifts of
the clusters that were detected as $M_{\textup{200m}}$>$3\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ systems in ATLAS, but did not have a
successful match in redMaPPer. We also count our cluster detections as “pure”
(i.e. non-spurious detections) if $>50\%$ of the cluster members appear
concentrated in a histogram with a width of $\bar{z}\pm 0.025$ (corresponding
to the ATLAS RMS error on cluster galaxy photometric redshifts). The choice of
this criterion was motivated based on visual inspection of the photometric
redshift histograms of ATLAS clusters with successful matches to redMaPPer,
where almost all clusters detected in both catalogues had ATLAS or SDSS
photometric redshift histograms which were centred around a redshift with more
than half of cluster members lying within $\bar{z}\pm 0.025$ of the histogram
peak. Figure 14 shows colour images and photometric redshift histograms of two
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ ATLAS clusters
with no SDSS redMaPPer detections, which were confirmed as pure following the
procedure described above.

(a)

(b)

(c)

(d)

Figure 14: (a & b) PanSTARRS colour images of two $N_{200}>20$ ATLAS clusters
with no redMaPPer detections in SDSS. Here the cluster members identified in
ORCA are shown by the red squares. (c & d) The photometric redshift histograms
for clusters in panels (a & b) respectively.

The fraction of additional clusters confirmed as genuine cluster detections
based on visual inspection is presented in column (5), with column (6) showing
the sum of the fractions in columns (4 & 5) with the percentages in this
column providing an estimate of the purity of the ATLAS cluster detections.
Based on this estimate we find our $M_{\textup{200m}}$>$3\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ clusters to be $100\%$ pure at $z<0.25$,
and $87\%$ pure in the redshift range $0.25<z<0.35$. The purity of the sample
then falls to $71\%$ and $56\%$ in our final two redshift bins. This reduction
of sample purity with increasing redshift could be partially due to the fact
that as we reach the magnitude limits of the ATLAS and SDSS surveys the
likelihood of real but faint cluster members not being detected by redMaPPer
increases, bringing clusters with richness greater than 20, below their
$\lambda>20$ limit. Similarly, the increase in the uncertainty on our
estimated photometric redshifts with increasing redshift, reduces the number
of clusters classified as pure on the basis of their photo-z histograms in
column (5). On the other hand, it is also possible that at higher redshifts
redMaPPer performs better than ORCA in overcoming projection effects which
could artificially increase the richness of our clusters, or result in false
cluster detections. We refer the reader to Section 6.4.3 of Murphy et al.
(2012), for a different analysis of the purity of orca cluster detections as a
function of cluster mass and redshift, based on comparison to SDSS-like
simulated mock cluster catalogues. Based on this comparison, the purity of
orca detections at the median redshift of the survey was shown to be $>70\%$
across all cluster masses.

Column (7) of Table 5 shows an estimate of the completeness of the redMaPPer
sample, which is defined as the fraction of “pure” ATLAS detections co-
detected by redMaPPer999We refer the reader to Section 11 of Rykoff et al.
(2014) for a more systematic analysis of the completeness of the redMaPPer
sample.. We find the completeness of redMaPPer to be in the $\sim 70-75\%$
range at $z<0.45$, before falling down to $\sim 50\%$ at $0.45<z<0.55$. At
this point we remind the reader that although we were able to estimate the
completeness of the ATLAS sample in column (3) by including successful matches
between ATLAS and redMaPPer in cases where a $\lambda>20$ redMaPPer cluster
had an ATLAS detection with $N_{200}<20$, we are unable to do the reverse in
column 6. Here we recall that, while it is possible that some of our ATLAS
detections may have redMaPPer counterparts with $\lambda<20$ which would
increase the redMaPPer completeness estimate; we have no way of verifying
this. Indeed, this could explain the apparent fall in the completeness of the
redMaPPer sample in the highest redshift bin where it is more likely for
cluster richness to be under-estimated due to the increased likelihood of
cluster galaxies falling out of the survey magnitude limits. Nonetheless, the
information in column (7) provides a useful estimate of the percentage of low-
redshift ATLAS clusters with $M_{\textup{200m}}$>$3\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ that are missed by redMaPPer, (possibly due
to differences in our definitions of cluster richness and what constitutes a
“pure” cluster detection).

To summarise, comparison of the ATLAS cluster catalogue to redMaPPer shows
that at $z<0.35$ the ATLAS sample is highly complete, and the majority of our
clusters appear to be genuine detections. At $z<0.35$ we also find that a
number of $M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$
ATLAS clusters (which were visually confirmed as rich clusters) are not
detected as $\lambda>20$ clusters by redMaPPer. Similarly, we found that ATLAS
generally performs better than redMaPPer at recovering $z<0.35$ Abell and ACT
DR5 clusters, with ATLAS recovering $\sim 85\%$ of $0.05<z<0.3$ Abell clusters
while redMaPPer only recovers $\sim 60\%$ of Abell clusters. At $z>0.35$ while
ATLAS appears to be less complete ($65\%$) compared to redMaPPer, Figure 13(b)
suggests that the additional redMaPPer cluster detections at higher redshifts
tend to be clusters of lower mass with a good agreement being found between
the $n(z)$ of the two samples for clusters with masses
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$.

###  4.5 ATLAS cluster masses

Figure 15 shows a comparison of the ATLAS cluster masses, obtained using
Equation 5, to SZ cluster masses of the full ACT DR5 and Planck samples. As
seen in Figure 15(a) the ATLAS cluster catalogue provides a complementary
sample to the Planck catalogue, by detecting clusters in a lower mass range
than possible with Planck. On the other hand, SZ cluster surveys offer the
ability to detect all clusters above a certain mass threshold with little
dependence on redshift, whereas optical surveys are limited in their ability
to detect clusters at higher redshifts by the magnitude limit of the survey.
As a result, the two approaches to cluster detection are highly complementary
in maximizing the mass and redshift completeness of cluster samples.

(a)

(b)

(c)

(d)

Figure 15: Comparison of the ATLAS $M_{\textup{200m}}$ cluster masses to
Planck SZ masses $M_{\textup{SZ}}$ (panel a); ACT DR5 $M_{\textup{200m}}$ SZ
masses (panel b); MCXC $M_{500}$ cluster masses (panel c), and SDSS redMaPPer
$M_{200m}$ cluster masses (panel d).

As seen in Figure 15(b), the ATLAS cluster sample also complements the ACT DR5
SZ sample in terms of detection of lower mass clusters in the redshift range
$z<0.5$. We note however, that due to it’s higher angular resolution and
superior flux sensitivity, ACT DR5 is able to detect lower mass SZ clusters
than Planck. The larger Planck beam size, however, makes it more sensitive to
clusters at $z<0.05$ when compared to ACT DR5 (due to the larger projected
area of these low redshift clusters on the sky).

Figure 15(c) shows a comparison of our ATLAS cluster masses, with X-ray
cluster masses ($M_{500}$) from the MCXC sample. While most X-ray surveys
included in the MCXC sample tend to be more sensitive to low-mass, low-
redshift clusters in comparison to optical cluster catalogues, the
completeness of these X-ray flux-limited cluster samples is reduced with
redshift, with optical and SZ surveys providing more complete cluster samples
at higher redshifts. Nevertheless, X-ray cluster samples play an important
role in calibrating optical and SZ cluster masses and in determining the
cluster gas fractions. The next generation of deeper X-ray cluster samples
such as eROSITA will significantly improve on the completeness and statistics
of current X-ray cluster samples by detecting $\sim 100,000$ galaxy clusters
with masses $>5\times 10^{13}h^{-1}$$\textup{M}_{\odot}$ (Pillepich et al.,
2012), a 2 orders of magnitude improvement over the MCXC cluster sample size.

Finally, we compare the ATLAS cluster masses to cluster masses from the SDSS
redMaPPer cluster catalogue in Figure 15(d). It can be seen that the two
samples cover a similar range of cluster masses and redshifts, which is
expected, given that both samples detect clusters using optical $griz$ bands
with a similar depth. However, as the ATLAS catalogue covers areas of the
southern sky, not covered by the SDSS redMaPPer catalogue, the two catalogues
are highly complementary. For reasons which we discussed in Section 4.4, the
ATLAS sample also contains a larger number of cluster detections in the
redshift range $0.05<z<0.3$, while the redMaPPer catalogue contains more
cluster detections in the $0.3<z<0.55$ range, making this another aspect in
which the two catalogues are complementary.

###  4.6 Comparison of cluster mass functions

(a)

(b)

(c)

(d)

(e)

Figure 16: A comparison of ATLAS, redMaPPer, Planck and ACT DR5 cluster mass
functions to the theoretical predictions of the Tinker et al. (2008)
$\Lambda$CDM models assuming a Planck Collaboration et al. (2016a) cosmology.
Here, the error bars on the observed mass functions are simply estimated as
$\sqrt{n}$.

We now compare the observed cluster mass functions of ATLAS, redMaPPer, Planck
and ACT DR5 samples to the theoretical predictions of the Tinker et al. (2008)
model. The mass function models are generated for spherical overdensities
defined as $M_{200m}$, assuming the Planck 2015 cosmology (Planck
Collaboration et al. 2016a; Table 4, column 6). We compare the observed mass
functions to the models in five redshift bins, in order to examine the
evolution of the cluster mass function in the redshift range $0.05<z<0.55$.

In Figure 16 we compare the redshift evolution of observed optical and SZ
cluster mass functions to the predictions of $\Lambda$CDM. For the purpose of
visual comparison of the cluster mass functions, the error bars shown in this
plot are simply estimated by $\sqrt{n}$, where $n$ is the number of clusters
in each mass bin. However, in Paper II, we shall perform a more detailed
analysis of the mass function uncertainties prior to using the mass functions
for cosmological parameter extraction. The fall in the amplitude of the
cluster mass functions at lower masses seen in this figure is due to the
reduced completeness of the samples as they approach their lower limit of
cluster mass detection. As expected given the selection functions of ACT DR5
and Planck, the optical samples probe a lower range of cluster masses than the
two SZ samples.

As seen in Figure 16 in the $0.05<z<0.35$ redshift range, we find the ATLAS
cluster mass functions to have a higher amplitude relative to redMaPPer,
placing them in better agreement with the $\Lambda$CDM model predictions.
Despite this closer agreement however, strong and unexplained tensions between
the observed ATLAS mass functions and the model predictions are present,
particularly at higher masses. The lower amplitude of redMaPPer could be in
part due to an underestimation of redMaPPer cluster masses at lower redshifts.
This is in line with the discussion presented in Section VI.B of Abbott et al.
(2020), where the lower than expected $\Omega_{\textup{m}}$ obtained from
cluster counts of the DES redMaPPer sample is attributed to a possible
underestimation of the weak lensing estimated cluster masses for $\lambda<30$
redMaPPer clusters.

However, in Section 4.4 we also found a higher density (per unit volume) of
ATLAS clusters at $z<0.35$ compared to redMaPPer. Similarly, our comparison of
ATLAS and redMaPPer samples to the Abell cluster catalogue at lower redshifts
showed that for clusters with $N_{200}>20$, ATLAS recovers $\sim 85\%$ of
Abell clusters, while redMaPPer recovers Abell clusters at a lower rate of
$\sim 60\%$. This could be taken as an indication that at lower redshifts
redMaPPer is likely to miss a larger fraction of genuine (and relatively rich)
clusters compared to ATLAS. Consequently, redMaPPer’s lower completeness at
$z<0.35$ could also be an important contributing factor towards the lower
amplitudes of its mass function relative to $\Lambda$CDM predictions.

As one can see in Figure 16, although the ACT DR5 survey probes a lower range
of cluster masses than Planck101010Note that in the $z=0.1$ bin (Figure 16(a))
the fact that Planck probes lower redshifts than ACT DR5, offsets ACT DR5’s
ability to probe clusters of lower masses, resulting in both samples having a
similar completeness in the $M_{\textup{200m}}$¡$3\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ mass range., at all redshifts we find a
good agreement between the mass functions of the two SZ samples in the range
of masses where both surveys are complete. With the exception of the $z=0.5$
redshift bin (Figure 16(e)), the Planck and ACT DR5 mass functions also appear
to be in general agreement with redMaPPer, placing them below the ATLAS
cluster mass functions and the predictions of $\Lambda$CDM.

The lower amplitude of the SZ cluster mass functions compared to $\Lambda$CDM
is likely to be due to systematics in the SZ flux-cluster mass scaling
relation resulting in an under-estimation of cluster masses of these samples.
Indeed, some SZ mass calibration techniques place the Planck cluster counts in
closer agreement to the prediction of the $\Lambda$CDM model. However, the
incompleteness of SZ samples could be another explanation for the lower
amplitude of their mass functions relative to the models. To investigate this
possibility, we compare the ACT DR5111111We note that although Planck SZ
detections with no measured redshifts are included in the catalogue used in
this comparison, at the time of this writing, we only had access to ACT DR5 SZ
detections with cluster counterparts identified based on their photometric or
spectroscopic redshifts. and Planck samples to the Abell clusters with
richness classes $>2$ and $>3$, (which limit the catalogue to clusters with
greater than 80 and 130 members respectively). A summary of our results is
presented in Table 6, where we find the SZ samples to recover $\sim 30\%$ of
Abell clusters with a richness class>2 and $\sim 50\%$ of Abell clusters with
richness>3\. Figure 17 shows examples of four $z\sim 0.2$ ATLAS clusters with
mass estimates of $M_{\textup{200m}}$>$4\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ placing them above the Planck lower limit
on mass observability. In the future, spectroscopic follow up of SZ clusters
guided by optical cluster catalogues could improve the completeness of current
SZ samples. Similarly, a more detailed study of potential issues which could
lead to the lack of detection of rich clusters in SZ samples based on
comparison to optical catalogues such as ATLAS and redMaPPer would be an
interesting topic for future works.

Table 6: Fraction / percentage of rich Abell clusters overlapping the Planck and ACT DR5 surveys recovered by each SZ survey. ACO2 and ACO3 denote Abell clusters with richness class>2 and 3 respectively. Redshift | ACT DR5/ACO2 | PSZ/ACO2 | ACT DR5/ACO3 | PSZ/ACO3
---|---|---|---|---
$0.05<z<0.15$ | 16/73 ($22\%$) | 50/170 ($29\%$) | 4/9 ($44\%$) | 13/29 ($65\%$)
$0.15<z<0.25$ | 34/105 ($32\%$) | 105/300 ($35\%$) | 13/28 ($46\%$) | 36/64 ($56\%$)
Figure 17: DES, DECaLS or PanSTARRS images of four $z\sim 0.2$,
$M_{\textup{200m}}$>$4\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ ATLAS clusters
which are also detected in the Abell sample, with no detections in the Planck
SZ catalogue.

Although the lower amplitude of SZ cluster mass functions relative to the
$\Lambda$CDM predictions could be due to the issues described above, we note
that a value of $\sigma_{8}\approx 0.7$ (with $\Omega_{\textup{m}}=0.3$),
could also produce such observations. Furthermore, suppression of observed
cluster mass functions compared to $\Lambda$CDM predictions at higher masses
(similar to those seen in the ATLAS mass functions) could originate from
primordial non-Gaussianities (Dalal et al. 2008; Verde 2010) or signal the
presence of massive neutrinos (e.g. Costanzi et al. 2013; Castorina et al.
2014; Biswas et al. 2019).

While this comparison provides a preliminary indication of the divergence of
observations to predictions of the model, a more comprehensive analysis of the
systematic and statistical uncertainties on the ATLAS cluster mass function is
needed before the statistical significance of the divergence between the
observations and the predictions can be quantified. We leave this, as well as
comparison of the observations to different mass function models, and
obtaining constraints on various cosmological parameters from the ATLAS
cluster mass functions to Paper II in this series.

##  5 Conclusions

In this work, we have presented a new catalogue of photometrically detected
galaxy groups and clusters, using the orca cluster detection algorithm in
combination with the $griz$ bands of the VST ATLAS survey, covering $\sim
4700$ deg2 of the Southern sky. The catalogue contains $\sim 22,000$
detections with richness $N_{200}>10$ and $\sim 9,000$ clusters with
$N_{200}>20$. Using the ANNz2 machine learning algorithm we obtain photometric
redshift estimates with an RMS of $\sim 0.025$ for our cluster galaxies and a
mean redshift uncertainty of $\sim 0.01$ for our clusters with richness
$N_{200}>20$. The photometric redshift of our sample peaks at $z\sim 0.25$,
extending up to $z=0.7$.

We described our calculation of cluster richness ($N_{200}$) which we use as a
proxy for cluster mass ($M_{\textup{200m}}$). To this end, we calibrated the
mass-richness scaling relation of the ATLAS sample to cluster masses from the
SDSS redMaPPer, MCXC, Planck and ACT DR5 samples. We found the ATLAS sample to
be $>95\%$ complete at $z<0.35$, $>80\%$ complete up to $z=0.45$, and $\sim
60\%$ complete in the $0.45<z<0.65$ redshift range. In terms of cluster mass,
we found the sample to be greater than $\sim 95\%$ complete for all cluster
masses, with near full completeness in the $>5\times
10^{14}h^{-1}$$\textup{M}_{\odot}$ mass range. Based on a comparison to the
SDSS redMaPPer cluster detections as well as visual inspection of our
clusters, we estimate the purity of our cluster detections with
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$ to be $100\%$
in the range $z<0.25$, $87\%$ at $0.25<z<0.35$, $71\%$ at $0.35<z<0.45$ and
$56\%$ at $0.45<z<0.55$.

Our comparison to the redMaPPer catalogue showed that in the $0.05<z<0.35$
redshift range, the ATLAS sample contains a larger number of cluster
detections and recovers an $\sim 40\%$ higher fraction of Abell clusters
compared to redMaPPer. At higher redshifts ($0.35<z<0.55$) the redMaPPer
catalogue appears to perform better than ATLAS at recovering ACT DR5 clusters,
but also detects a large number of clusters not found in the ACT DR5 sample
which could be lower mass groups which are misclassified as $\lambda>20$
clusters. At $z>0.35$ we also find a good agreement between the redshift
distributions of the ATLAS and redMaPPer samples above a mass limit of
$M_{\textup{200m}}$>$3\times 10^{14}h^{-1}$$\textup{M}_{\odot}$.

We then compared the cluster mass functions of ATLAS, redMaPPer, Planck and
ACT DR5 samples to the theoretical predictions of Tinker et al. (2008) models
(assuming a Planck Collaboration et al. 2016a $\Lambda$CDM cosmology), in the
$0.05<z<0.55$ redshift range. At $z<0.35$, ATLAS cluster mass functions have a
higher amplitude compared to those of the SZ samples and the redMaPPer mass
functions. This places the ATLAS measurements in better agreement with
$\Lambda$CDM predictions with $\sigma_{8}\approx 0.82\pm 0.01$ (based on the
CMB analysis of Planck Collaboration et al. 2016a), rather than some of the
previous constraints from the Planck SZ cluster counts, which depending on the
SZ mass calibrations, can give $\sigma_{8}$ measurements as low as $0.71\pm
0.03$ (Planck Collaboration et al., 2016b). Despite this closer agreement,
however, at higher masses we found the observed ATLAS mass functions to have a
significantly lower amplitude than the $\Lambda$CDM model and the cause of
this discrepancy is currently unknown.

Based on our earlier findings, we suggest that the incompleteness of the SDSS
redMaPPer sample at lower redshifts could be a contributing factor to the
lower amplitude of its mass functions relative to the predictions of
$\Lambda$CDM. For the SZ samples, while mass calibration systematics are
likely to be the dominant contributing factor to their lower mass function
amplitudes relative to $\Lambda$CDM predictions, we show that sample
incompleteness is also likely to have a non-negligible contribution and will
need to be carefully characterized and accounted for. Future follow-up
spectroscopic observations of SZ clusters guided by optical cluster catalogues
such as ATLAS and redMaPPer could improve the completeness of SZ samples,
while detailed studies of rich optical clusters which remain undetected in SZ
samples could help with the identification of any unknown issues impacting the
completeness of SZ catalogues.

In paper II of this series, we shall perform a detailed analysis of the
systematic and statistical uncertainties of the ATLAS cluster mass functions.
This will in turn enable us to constrain various cosmological parameters
including, $\sigma_{8}$, $\Omega_{m}$, $\omega$ and $\sum m_{\nu}$ and compare
these to constraints from other cluster samples and different cosmological
probes.

## Acknowledgements

BA acknowledges support from the Australian Research Council’s Discovery
Projects scheme (DP200101068). This work used the DiRAC@Durham facility
managed by the Institute for Computational Cosmology on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
funding via STFC capital grants ST/K00042X/1, ST/P002293/1, ST/R002371/1 and
ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC
is part of the National e-Infrastructure. For the purpose of open access, the
author has applied a Creative Commons Attribution (CC BY) licence to any
Author Accepted Manuscript version arising.

The mass function models in Fig. 16 are generated using the ‘Halo mass
function’ module of the COLOSSUS python package described by Diemer (2018).
This research has made extensive use of Python 2 & 3 (Van Rossum & Drake Jr
1995; Van Rossum & Drake 2009), IPython (Perez & Granger, 2007), Matplotlib
(Hunter, 2007), SciPy (Virtanen et al., 2020), NumPy (van der Walt et al.,
2011), pandas (Wes McKinney, 2010), AstroPy (Price-Whelan et al., 2018), as
well as TOPCAT & STILTS121212http://www.star.bris.ac.uk/
mbt/topcat/sun253/index.html packages (Taylor, 2005). This research has made
use of the NASA/IPAC Extragalactic Database, which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology, as well as the “Aladin sky atlas” developed at CDS, Strasbourg
Observatory, France.

This publication makes use of data products from the Wide-field Infrared
Survey Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration.Funding for the
Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan
Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org. The Pan-STARRS1 Surveys (PS1) and the PS1 public
science archive have been made possible through contributions by the Institute
for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the
Max-Planck Society and its participating institutes, the Max Planck Institute
for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial
Physics, Garching, The Johns Hopkins University, Durham University, the
University of Edinburgh, the Queen’s University Belfast, the Harvard-
Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global
Telescope Network Incorporated, the National Central University of Taiwan, the
Space Telescope Science Institute, the National Aeronautics and Space
Administration under Grant No. NNX08AR22G issued through the Planetary Science
Division of the NASA Science Mission Directorate, the National Science
Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand
University (ELTE), the Los Alamos National Laboratory, and the Gordon and
Betty Moore Foundation. This publication makes use of data products from the
Wide-field Infrared Survey Explorer, which is a joint project of the
University of California, Los Angeles, and the Jet Propulsion
Laboratory/California Institute of Technology, funded by the National
Aeronautics and Space Administration.

## Data Availability

The cluster catalogues described in Tables 2 and 3 are publicly available at
http://astro.dur.ac.uk/cosmology/vstatlas/cluster_catalogue/. The input galaxy
catalogues used for cluster detection are also available from the primary
author upon request.

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