11institutetext: University Observatory, Faculty of Physics, Ludwig-
Maximilians-Universität, Scheinerstr. 1, 81679, Munich, Germany
11email<EMAIL_ADDRESS>22institutetext: Max Planck Institute
for Extraterrestrial Physics, Giessenbachstrasse 1, 85748 Garching, Germany

# The ACT-DR5 MCMF Galaxy Cluster Catalog

M. Klein 11 J.J. Mohr 1122 C.T. Davies 11

(xxx; accepted xx)

Galaxy clusters are useful cosmological probes and interesting astrophysical
laboratories. As the cluster samples continue to grow in size, a deeper
understanding of the sample characteristics and improved control of
systematics becomes more crucial. In this analysis we create a new and larger
ACT-DR5-based thermal Sunyaev–Zel’dovich Effect (tSZE) selected galaxy cluster
catalog with improved control over sample purity and completeness. We employ
the red sequence based cluster redshift and confirmation tool MCMF together
with optical imaging data from the Legacy Survey DR-10 and infrared data from
the WISE satellite to systematically identify true clusters from a new cluster
candidate detection run on the ACT-DR5 dataset. The resulting ACT-DR5 MCMF
sample contains 6,237 clusters with a residual contamination of 10.7%. This is
an increase of 51% compared to the previous ACT-DR5 cluster catalog, making
this new catalog the largest tSZE-selected cluster catalog to date. The
$z_{\mathrm{phot}}$¿1 subsample contains 703 clusters, three times more than
in the previous ACT-DR5 catalog. Cross-matching the ACT-DR5 MCMF cluster
catalog with a deeper tSZE sample from SPTpol 500d allows us to confirm the
completeness and purity of the new ACT-DR5 MCMF sample. Cross-matching to the
two largest X-ray selected cluster samples, the all-sky RASS MCMF and the
western Galactic hemisphere survey eRASS1, confirms the sample purity of the
RASS MCMF sample and in the case of eROSITA eRASS1 reveals that 43% of the
matched clusters are designated in eRASS1 as X-ray point sources rather than
groups and clusters. Cross-correlating the ACT-DR5 MCMF cluster catalog with
ACT-DR6 lensing maps results in a 16.4 $\sigma$ detection of CMB lensing
around the clusters, corresponding to the strongest signal found so far for a
galaxy cluster sample. Repeating the measurement for the $z>1$ cluster
subsample yields a significance of 4.3 $\sigma$, which is the strongest CMB
lensing detection in a z¿1 cluster sample to date.

###### Key Words.:

surveys–galaxies: clusters: general–galaxies: clusters: intracluster
medium–X-rays: galaxies: clusters

##  1 Introduction

The thermal Sunyaev–Zel’dovich Effect (tSZE; Sunyaev & Zeldovich 1972) has
long been recognized as a means of creating nearly mass-limited samples of
galaxy clusters, independent of their redshift (Barbosa et al. 1998; Haiman et
al. 2001). The first tSZE-selected clusters were reported more than a decade
ago (Staniszewski et al. 2009), and in recent years, there has been
significant progress in utilizing tSZE-selected galaxy clusters together with
weak gravitational lensing datasets to study cosmic acceleration, to probe the
so-called S8 tension (e.g., Planck Collaboration et al. 2016; Bocquet et al.
2019; Planck Collaboration et al. 2020; Bocquet et al. 2024) and to explore
the promise of tSZE selected cluster samples for constraining modified gravity
and interacting dark matter models (e.g., Vogt et al. 2024; Mazoun et al.
2024).

The tSZE signature– as well as the X-ray emission from galaxy clusters– arises
from the hot plasma or intracluster medium (ICM) that compromises $\sim$15% of
the cluster mass. The tSZE signal depends on the electron pressure of the ICM
and therefore preferentially traces massive collapsed structures like the
centers of clusters. This is in contrast to selecting galaxy clusters through
overdensities of passive galaxies (Gladders & Yee 2000; Rykoff et al. 2014),
which is sensitive to collections of passive galaxies whether they lie within
a cluster halo or in the surrounding structures far outside the cluster. This
lack of contrast in galaxy-based cluster selection complicates the
cosmological modeling of such samples. Ways to account for this challenge have
been developed with mixed success (e.g., Costanzi et al. 2019; Murata et al.
2019; Abbott et al. 2020; Lesci et al. 2022), and this challenge remains as
one of the key issues for the success of future optically selected cluster
analyses using data from the Rubin LSST (Ivezić et al. 2019) and Euclid
surveys (Euclid Collaboration et al. 2024).

Of the two leading ICM-based cluster selection methods, the tSZE offers some
advantages. First, the tSZE signature of a cluster is strongly dependent on
cluster mass and approximately independent of cluster redshift, and so the
selected cluster sample has an effective mass selection that is only weakly
dependent on redshift for $z>0.2$. Second, the tSZE signature appears as a
shadow (negative signal) at frequencies below 200 GHz. Due to the uniqueness
of this signature, catalogs of tSZE selected cluster candidates are solely
contaminated by noise fluctuations in the mm-wave maps. This has the advantage
that the fraction of contaminants as a function of tSZE detection signal to
noise ratio (SNR) can be accurately predicted, and it falls rapidly with
increasing tSZE SNR. Combined with the redshift independence of the tSZE, this
has the benefit that no noise fluctuations can be incorrectly interpreted as
massive clusters.

This is in contrast to X-ray selected cluster samples, where other physical
X-ray sources (AGN, Stars) exhibit similar X-ray fluxes and are significantly
more frequent than clusters. Due to the strong redshift dependence of the
X-ray signature of a cluster, even a low significance source can be mistakenly
interpreted to be a high mass cluster if it is at high redshift. Imposing a
selection on source extent to exclude point sources like AGN and stars helps
to increase the purity of the candidate list if the X-ray detector has
sufficient angular resolution, but it comes at the price of a significant
reduction of the number of selected clusters and a more complicated selection
function. Also, even with X-ray extent selection the sample still includes
significant contamination that requires modeling or cleaning (Klein et al.
2022; Chiu et al. 2023). The challenges associated with X-ray selection become
more important with larger cluster samples, as recently highlighted in the
eROSITA eRASS1 cosmological study (Ghirardini et al. 2024).

In this work, we are building upon our previous body of work creating large
ICM-selected cluster catalogs for the RASS, SPT, Planck and eROSITA eFEDS
surveys. (Klein et al. 2019a, 2022; Hernández-Lang et al. 2023; Klein et al.
2023a, b; Bleem et al. 2024). Motivated by the successful cluster-based
cosmological analyses (Chiu et al. 2023; Bocquet et al. 2024) of the two most
recent catalogs, we apply our Multi-Component Matched Filter (MCMF) cluster
confirmation technique (Klein et al. 2018) to tSZE data from the Atacama
Cosmology Telescope (ACT) data release 5 (ACT-DR5, Naess et al. 2020). This
dataset covers 13,211 deg2 of extra galactic sky and has been previously used
to create a cluster catalog of 4,195 tSZE-selected clusters (Hilton et al.
2021). Our main motivation in reanalysing this sample is to create a cluster
catalog with more homogeneous and complete optical confirmation so that we can
quantify the impact on the completeness and contamination of the final cluster
sample and estimate the level of contamination in the tSZE selected candidate
list. Thus, we expect to add value to the cluster catalog, enabling and
ideally simplifying cosmological studies employing this sample.

Throughout this paper we assume a flat $\Lambda$CDM model with
$\Omega_{\mathrm{m}}=0.3$ and $H_{0}=70$ km s${}^{-1}\,$Mpc-1 unless otherwise
noted.

Figure 1: Distribution of ACT cluster candidates in tSZE detection signal to
noise ratio (SNR) for different source detection thresholds. The threshold
SNR¿4 was used in the previous analysis (Hilton et al. 2021), while here we
adopt SNR¿3 for detection. Figure 2: Optical footprint of the LS-DR10 dataset
and distribution of ACT-DR5 sources (red points). White regions have
$g,r,z,w1,w2$-band coverage, while grey regions have $g,i,w1,w2$-band
coverage.

##  2 Datasets

###  2.1 The ACT-DR5 dataset

The dataset, footprint and detection algorithm used to create the list of tSZE
candidates is almost identical to those used to create the previously
published list of cluster candidates (Hilton et al. 2021). We therefore
restrict this section to a high-level overview and focus on the differences
between the Hilton et al. (2021) analysis and this work. For more details on
the tSZE datasets we refer the interested reader to Hilton et al. (2021) and
references therein.

Similar to Hilton et al. (2021) we make use of the fifth data release of ACT
data, which contains data obtained between 2008 and 2018. It includes data
obtained using the AdvACT camera (Henderson et al. 2016) starting in 2016. For
the tSZE source detection we use the 98 and 150 GHz channels, which have
approximate beam sizes of 2.2 and 1.4 arcmin FWHM. The coadded maps cover
$\approx 18,000$ deg2, but following Hilton et al. (2021) we exclude bad
regions such as regions of Galactic latitudes ($|b|<20$ deg), regions around
point sources and dusty regions. The remaining useful area is 13,211 deg2,
from which $\approx 12,000$ deg2 have noise levels of $<30\ \mu\mathrm{K}$.

For the source detection we use the NEMO111see
https://github.com/simonsobs/nemo/ pipeline (Hilton et al. 2021). The NEMO
algorithm includes a multi-frequency matched filter (Melin et al. 2006;
Williamson et al. 2011) applied to the 98 and 150 GHz maps that includes a
cluster based spatial and spectral filter along with the noise covariance
between maps. The spectral filter is based upon the cluster tSZE spectrum and
the spatial filter is based on the universal pressure profile (Arnaud et al.
2010) convolved with the beam appropriate for each frequency. To account for
different cluster sizes, 16 different matched filters are applied,
corresponding to cluster models with masses $\mathrm{M}_{\mathrm{500c}}$ of 1,
2, 4 and 8 $\times 10^{14}\ \mathrm{M}_{\odot}$ at four different redshifts
(0.2, 0.4, 0.8, 1.2).

The version of NEMO used to construct our cluster candidate list differs
slightly from that used in the original 2021 analysis (Hilton et al. 2021). We
first run the NEMO code with the same settings used in Hilton et al. (2021),
which yields 8,727 candidates, which differs by only 1.7% from that reported
in Hilton et al. (2021) (8,878). This gives us confidence that any changes in
the pipeline have only a minor impact on the candidate list. Examining the
distribution of candidates in the reported signal to noise ratio (SNR) (see
Fig. 1), we find that the distribution drops at SNR¡4.3, well above the
nominal detection threshold of SNR=4. Because the expected number of noise
fluctuations as well as that of real sources should both continuously grow
towards lower SNR values, this behaviour suggests an enhanced incompleteness
in the candidate list when approaching the specified detection threshold. To
avoid this effect, we run the source detection with lower detection
thresholds. We find that the source list is stable for sources with SNR that
lie $\sim$0.5 above the detection threshold. To guarantee a truly SNR selected
sample, we define our candidate list as all sources above SNR¿4 from a source
detection run that uses a threshold of SNR=3. In contrast to the original
threshold this setting provides a list of 12,671 candidates, 45% more than are
found when running with a threshold of SNR=4.

###  2.2 Legacy Survey (LS)

Similar to our previous work on ROSAT X-ray selected galaxy clusters (Klein et
al. 2023a), we use the most recent data release (DR10) from the DESI Legacy
Imaging Surveys (LS; Dey et al. 2019). Up to data release 8, LS was a
combination of four imaging surveys, the 9,000 $\text{deg}^{2}$ $grz$-band
DECam (Flaugher et al. 2015) based DECaLS survey, the 5,000 $\text{deg}^{2}$
BASS and MzLS surveys providing photometry in $g,r$ and $z$-band,
respectively, and the WISE and NEOWISE surveys in the mid-IR at 3.4 $\mu$m and
4.6 $\mu$m. LS-DR10 includes all imaging data from the Dark Energy Survey as
well as from various other survey programs such as BLISS and the DeROSITAS
surveys, which resulted in a significant increase of the total survey area
compared to DR9. Compared to previous data releases DR10 also includes DECam
imaging data from the $i$-band, increasing the amount of color information of
sources over significant parts of the survey with declination below 30 deg. In
Fig. 2 we show the LS-DR10 footprint along with the distribution of ACT-DR5
sources in red.

The imaging depth depends on sky position. The 5,000 deg2 BASS and MzLS
surveys, covering the sky at Declination ¿30 deg (Legacy Survey North),
exhibits a typical 5$\sigma$ point source depth of $\sim$24.3, $\sim$23.8 and
$\sim$23.4 mag in the $grz$ bands respectively. The DECam based surveys
typically show a double peaked imaging depth distribution 222See
https://www.legacysurvey.org/dr10/description/. The deeper peak is associated
with the DES survey with 5$\sigma$ point source depths of $\sim$25.3,
$\sim$25.0 and $\sim$23.9 mag, and the shallower peak is mostly associated
with the DECaLS survey with depths of $\sim$24.8, $\sim$24.2 and $\sim$23.3
mag ($grz$).

The ACT-DR5 footprint ranges in Declination from -60 to +20 and therefore lies
nominally in the southern part of the LS. While recognizing the fact that
additional Legacy Survey North data are present within the footprint, we do
not use those data, because they do not expand the coverage of the ACT-DR5
footprint.

###  2.3 WISE

To expand the redshift reach of our cluster search we follow our previous work
on SPT tSZE-selected clusters (Klein et al. 2023b; Bleem et al. 2024) and use
WISE to identify high redshift clusters. The WISE satellite is a mid-infrared
telescope with a main mirror of 40 cm observing in four bands at 3.4 µm, 4.6
µm, 12 µm and 22 µm (referred to as $w1$, $w2$, $w3$, $w4$, respectively). The
WISE observation campaign can be divided into three phases. During the main
phase there was sufficient coolant to observe the entire sky 1.5 times in all
four bands. Immediately following the main campaign was a second phase known
as NEOWISE, which completed the second full-sky observations in the w1 and w2
bands, where coolant was not required. A third phase of WISE observations
(NEOWISE-R) began in September 2013 after WISE was reactivated following more
than two years of inactivity. Since then, WISE has been conducting a full-sky
survey approximately every six months.

In our study, we utilize the most recent version of the unWISE catalog
(Meisner et al. 2019) that is based on the nine-year full-depth unWISE coadds
(Meisner et al. 2022). This version of the catalog incorporates all WISE data
up until December 13 2021. It was constructed using the unblurred coadds of
WISE imaging data (unWISE Lang 2014; Meisner et al. 2022) and incorporates
enhanced source detection and deblending modeling for densely populated areas.

##  3 Confirming ACT-DR5 clusters

To confirm clusters of galaxies in the ACT-DR5 based candidate list, we follow
a similar approach to our previous work on SPT tSZE-selected clusters (Klein
et al. 2023b; Bleem et al. 2024) by running two instances of the MCMF cluster
confirmation tool (see details in Klein et al. 2018, 2019a). The first
instance uses optically selected galaxies from LS-DR10 and is the version used
in our work on RASS X-ray-selected clusters (Klein et al. 2023a). The
availability of WISE photometry in LS-DR10– as with our work on SPT tSZE-
selected clusters using DES– allows us to expand the redshift reach of this
instance of MCMF out to z=1.5. The second instance of MCMF, which we call the
high-z extension, is based on IR selected galaxies from the unWISE catalog.
Compared to LS-DR10 the IR selection allows one to identify massive galaxies
to higher redshift, which enhances the capability to confirm clusters at the
high redshift end of the sample. Given our experience with SPT tSZE-selected
clusters we expect that the vast majority of ACT-DR5 clusters will be
confirmed with our main instance of MCMF using LS-DR10, while only $\sim
5-10$% of the sample will be confirmed by our high-z extension. Because in
this work we closely follow our previous work on confirmation of ICM selected
clusters, we will only give a brief overview of MCMF in the following
subsections and refer the interested reader to the additional details provided
in our previous work (Klein et al. 2023a, b).

###  3.1 MCMF

The MCMF algorithm was created to validate and describe cluster candidates
identified in large X-ray or tSZE surveys. MCMF has proven successful in
analyzing SPT tSZE-selected sources within the DES footprint (Klein et al.
2023b; Bleem et al. 2015). Furthermore, it has been utilized in conjunction
with the LS-DR10 dataset to establish the optically-confirmed X-ray cluster
catalog covering the entire sky (RASS-MCMF; Klein et al. 2023a). Additionally,
it played a crucial role in the optical follow-up of the initial eROSITA-based
galaxy cluster catalog in the eFEDS early mission test field (Klein et al.
2022). Moreover, it has been employed in the examination of new S/N¿3 Planck
tSZE-selected catalogs across the DES region (MADPSZ Hernández-Lang et al.
2023). Throughout these various applications, enhancements and expansions have
been made to the original method. The newly generated MCMF-based cluster
catalogs have significantly increased the number of clusters that were
previously identified in the same X-ray or tSZE datasets and subsequently
analyzed through individual imaging and spectroscopy. In addition to extending
the samples, the MCMF method allows one to control the level of contamination
in the new cluster samples.

The MCMF algorithm incorporates a red sequence technique (Gladders & Yee 2000;
Rykoff et al. 2014) that utilizes redshift and magnitude dependent color
filters, such as $g-r$, $r-z$, $z-w1$. It also employs radial weighting using
a projected NFW profile centered at the tSZE or X-ray selected candidate
position, as well as a specific magnitude range depending on redshift. It also
incorporates a tSZE- or X-ray-based mass proxy to estimate $R_{500}$ for a
given redshift z. For each cluster candidate, the color and radially weighted,
background-subtracted richness $\lambda(z)$ within R500 is calculated as a
function of the candidate redshift, which is initially unknown. The peaks in
$\lambda(z)$ are then identified and modeled with so-called “peak profiles”,
which is described below. Examples of peak profiles and their best fit to
$\lambda(z)$ profiles of clusters are presented in previous MCMF analyses
(e.g. Figs. 4 & A2, Klein et al. 2019a). These peaks with associated
richnesses and redshifts are then collected and processed further as described
in the following subsections. Note that the peak profile models are built
using renormalised stacks of individual $\lambda(z)$ profiles from real
clusters with spectroscopic redshift measurements (spec-z’s). This approach
ensures that no additional correction between MCMF and spec-z redshifts are
needed.

To confirm cluster candidates, we assess the probability that a given optical
counterpart is a coincidental overlap rather than a genuine counterpart to the
ICM-selected cluster candidate. This assessment requires knowledge of the
typical distribution of contaminants based on their richness and redshift
within the survey region. To obtain this information we supplement the
analysis that we do on the candidates positions with an analysis of random
positions within the tSZE or X-ray survey footprint. In the ACT-DR5 analysis,
these random positions have a similar density and SNR distribution as the
candidate clusters but exclude regions with real ACT candidates. The richness
distribution of these random positions provides an estimate of the non-tSZE
detected structures (noise, projections, undetected clusters) within the
candidate list.

To be able to control the contamination of the final ACT-DR5 MCMF cluster
sample, we calculate a quantity $f_{\mathrm{cont}}$, where ”cont” is short for
contamination. High values of $f_{\mathrm{cont}}$ indicate a higher
probability that the candidate in question is a chance superposition rather
than a real cluster. $f_{\mathrm{cont}}$ is calculated using the richness
distributions along the random lines-of-sight $f_{\mathrm{rand}}(\lambda,z)$
and the richness distributions $f_{\mathrm{obs}}(\lambda,z)$ towards the
candidates at each redshift $z$.

That is, for each candidate $i$ we calculate the number of random lines-of-
sight within a redshift bin with richness $\lambda\geq\lambda_{i}$ and divide
by the number of tSZE candidates within the same redshift bin with
$\lambda\geq\lambda_{i}$. This ratio is then re-scaled according to the total
number of SPT candidates and random lines-of-sight.

$f_{\mathrm{cont}}(\lambda_{i},z_{i})=\frac{\int_{\lambda_{i}}^{\infty}f_{\mathrm{rand}}(\lambda,z_{i})d\lambda}{\int_{\lambda_{i}}^{\infty}f_{\mathrm{obs}}(\lambda,z_{i})d\lambda},$
(1)

This $f_{\mathrm{cont}}$ parameter is calculated for each richness peak
associated with a candidate. The peak showing the lowest value of
$f_{\mathrm{cont}}$ is assigned as the best optical counterpart for the SPT-SZ
candidate, because it is the most likely to be a real cluster.

The cluster sample itself can then be defined as those candidates showing an
$f_{\mathrm{cont}}$ below a certain threshold value
$f_{\mathrm{cont}}^{\mathrm{max}}$. The threshold value corresponds to the
fraction of the contamination in the initial candidate list that makes it into
the final cluster catalog at each redshift $z$. The contamination of the
resulting final cluster catalog would then be $f_{\mathrm{tSZE-cont}}\times
f_{\mathrm{cont}}^{\mathrm{max}}$, where the confirmed catalog contains all
candidates with $f_{\mathrm{cont}}(\lambda_{i},z_{i})\leq
f_{\mathrm{cont}}^{\mathrm{max}}$ and the initial contamination of the
candidate catalog is $f_{\mathrm{tSZE-cont}}$. As an example, if the input
catalog is known to be 50% pure and an $f_{\mathrm{cont}}$ threshold value
$f_{\mathrm{cont}}^{\mathrm{max}}=0.2$ is employed, then the contamination
fraction of the confirmed cluster catalog would be $0.5\times 0.2=0.1$ or 10%.

####  3.1.1 MCMF on Legacy Survey DR10

The version of MCMF applied here is– aside from the different mass proxy–
largely the same as the version applied to X-ray cluster candidates from the
2RXS catalog (Boller et al. 2016) to create the RASS-MCMF catalog (Klein et
al. 2023a). The most significant change between the MCMF version used to
create RASS-MCMF and this work is that we exclude the $w1-w2$ color in our
baseline approach, and we recalibrate the scatter around the red sequence
model in the $z-w1$ color. This results in a better behaved redshift evolution
of the richness-mass relation. These changes imply that the there are
systematic differences between our ACT-DR5 MCMF and RASS-MCMF richnesses for
clusters of similar redshift and mass. We therefore provide a table to
transform RASS-MCMF richnesses to our new baseline richnesses in the appendix.
We provide a more detailed comparison between RASS-MCMF and our new ACT-DR5
catalog below in Section 5.2.

In addition, following the approach we used in our work on SPT tSZE selected
candidates (Klein et al. 2023b; Bleem et al. 2024), we improve the estimate of
the richness distributions along random lines-of-sight with a second
iteration. The richness distribution along random lines-of-sight
$f_{\mathrm{rand}}(\lambda,z)$ is supposed to resemble the expected richness
distribution of contaminants as a function of redshift. Given the correlation
between the tSZE observable SNR (often referred to as $\hat{\zeta}$ or $\xi$
in our previous SPT analyses), cluster mass, and likelihood of a source being
a real cluster, the initial choice of using the identical SNR distribution as
the full candidate list causes the estimate of $f_{\mathrm{cont}}$ to be
mildly biased high. To avoid this bias we use the SNR distribution of rejected
candidate systems ($f_{\mathrm{cont}}$¿0.3) as a proxy of the distribution of
contaminants. We select a subsample of random lines of sight that follows this
tSZE observable distribution and use that to remeasure $f_{\mathrm{cont}}$ for
all candidates.

####  3.1.2 MCMF on unWISE

In addition to passive galaxies becoming fainter as redshift and distance
increase, the spectral range covered by the DECam bands no longer encompasses
the 4000,Å break at redshifts $z\gtrsim 1$. Consequently, photometric redshift
estimation becomes increasingly uncertain at these redshifts. For confirming
high-redshift clusters and improving photometric redshift estimates,
transitioning to redder bands like the near- or mid-IR range covered by the
Spitzer or WISE observatories proves advantageous. Previous studies have
utilized data from both observatories for high-redshift cluster searches
(Muzzin et al. 2009; Gonzalez et al. 2019) and cluster validation (Bleem et
al. 2015, 2020, 2024; Klein et al. 2023b). In this work we follow closely the
approach presented in Klein et al. (2023b).

In our current analysis, we primarily rely on a version of the unWISE catalog
(Schlafly et al. 2019), which incorporates eight years of $w1$ and $w2$ band
WISE imaging data from the NEOWISE-R phase. WISE data cover the entire sky,
enabling the matching of unWISE and LS-DR10 catalogs for optical+IR photometry
of all WISE sources. Optical to WISE galaxy colors (e.g., $z$-$w1$) exhibit
strong redshift dependence, making them suitable for obtaining high-quality
cluster photometric redshift estimates.

However, a limitation of WISE $w1$ is its large PSF ($\sim 6"$), which becomes
problematic in regions with high source density like the cores of galaxy
clusters. In such dense areas, distinguishing individual sources and
deblending fluxes pose challenges. Here, the improved modeling of crowded
regions in unWISE compared to the previous ALLWISE catalog becomes important.

Finally, we accommodate masks or missing data in different surveys by deriving
separate richness estimates for cluster regions with different coverage (LS-
DR10 + $w1$ only, WISE $w1$ only, WISE $w1$, $w2$ & LS-DR10) and summing them
for the final cluster richness estimate. With this approach, we need only
track the masked area in $w1$ imaging. The total richness in the high-redshift
code is therefore given as

$\displaystyle\lambda_{\mathrm{HZ}}(z)=$
$\displaystyle\lambda_{\mathrm{LSDR10+w1}}(z)+\lambda_{\mathrm{LSDR10+w1+w2}}(z)+\lambda_{\mathrm{w1+w2}}(z)+\lambda_{\mathrm{w1}}(z),$

where the individual richnesses are defined in the same manner as the standard
MCMF richness (see Klein et al. 2018, 2019a), with the color-weights depending
on the availability of the bands ($i,z,w1,w2$ for
$\lambda_{\mathrm{LSDR10+w1+w2}}$, $i,z,w1$ for
$\lambda_{\mathrm{LSDR10+w1}}$, $w1$, $w2$ for $\lambda_{\mathrm{w1+w2}}$ and
no color weight for $\lambda_{\mathrm{w1}}$).

The high-$z$ cluster confirmation code has been applied to all candidates over
the redshift range $0.63<z<2.0$. Similar to the optically-based MCMF code,
runs are performed along random lines-of-sight, and $f_{\mathrm{cont}}$ is
calculated for potential counterparts to each tSZE candidate. Clusters with
spectroscopic redshifts available for a subset of the sample are utilized to
calibrate the WISE-based measurements. Moreover, the overlap in redshift
between the optically-based MCMF and the high-z WISE-based run of $0.63<z<1.5$
enables the comparison of richnesses.

####  3.1.3 Spectroscopic redshifts

We follow the same approach as adopted in Klein et al. (2023a) and estimate
spectroscopic redshifts for the best optical counterpart identified using MCMF
by employing public spectroscopic galaxy redshifts. For this analysis we use
the same collection of spectroscopic redshifts as was used in Klein et al.
(2023a) that mainly includes redshifts from the SDSS DR17 (Abdurro’uf et al.
2022), 2dFGRS (Colless et al. 2001), 6dFGS (Jones et al. 2009), 2MRS (Huchra
et al. 2012) and GLADE+ (Dálya et al. 2022). Additionally, we match to known
clusters with spectroscopic redshifts. We first match the spectroscopic
catalog with the BCG positions using a maximum positional offset of 2 arcsec.
We then search for all spectroscopic galaxies within 2 Mpc around the ACT
position and
$|z_{\mathrm{cluster}}-z_{\mathrm{spec}}|<0.025(1+z_{\mathrm{cluster}})$. From
this list of potential spectroscopic member galaxies, we derive the median
redshift and finally derive the cluster redshift using all galaxies within
$\left|\delta z\right|<0.015$ from the median redshift. If a BCG spectroscopic
redshift exists, we instead select galaxies within
$\left|\delta\right|z<0.015$ from the BCG redshift. In our final cluster
catalog, we only list spectroscopic redshifts that have been derived from at
least two members, that has a BCG redshift or that has matches to previously
known clusters with spectroscopic redshifts.

Figure 3: Measured distribution of ACT-DR5 candidates with SNR¿4.5 as a
function of $\log_{10}(\lambda/M_{500})$. At low values the distribution is
dominated by contamination caused by noise fluctuations in the mm-wave maps,
which are well described by the contamination model (blue), while the high
values are dominated by true galaxy clusters. The joint contamination plus
cluster distribution is shown in green and blue.

###  3.2 Contamination in the ACT-DR5 candidate list

Figure 4: Measured cumulative number of false detections from noise
fluctuations in the mm-wave maps as a function of the tSZE SNR threshold. The
blue line shows the best fit Gaussian noise model. The two parameter fit
describes the data over full regime explored, corresponding to almost four
orders of magnitude.

Systematic optical and IR follow-up of ACT-DR5 candidates with MCMF not only
allows one to confirm clusters, differentiating between likely clusters and
noise fluctuations, but it also allows one to statistically estimate the total
number of clusters and noise fluctuations in the sample (Klein et al. 2022;
Hernández-Lang et al. 2023; Klein et al. 2023a, b). One does this by using the
fact that the richness and redshift distributions along random lines of sight
are representative of the distribution of noise fluctuations in the candidate
list. The accuracy of the estimate improves when there is a region in the
parameter space that is dominated by noise fluctuations. In cases where the
ICM-based mass proxies are noisy or not reliable it is most robust to use
solely the richness distribution of the sample selected along random lines of
sight and to employ the sources with low richness to estimate the total number
of contaminants (Hernández-Lang et al. 2023; Chiu et al. 2023). For cases with
meaningful mass proxies, such as in this work or in our previous work using
SPT-SZ or eROSITA eFEDS extent-selected clusters (Klein et al. 2022, 2023b),
one can model the distribution in $\log_{10}(\lambda/M_{500})$ space, where
$M_{500}$ is the ICM-based mass estimate derived from a calibrated observable-
mass relation. Given that the mass trend of the richness-mass relation
$\lambda_{M500}\sim 1$, moving to $\log_{10}(\lambda/M_{500})$ space causes
the cluster population to be projected onto one location (value) with some
scatter around that value due to scatter in the richness-mass relation. In
this space the clusters and contaminants tend to be better separated. In
addition, this space is a well suited for building a more sophisticated model
of the full population that includes a contamination component calibrated
using systems identified along random lines of sight, and a cluster model that
we approximate using a log-normal distribution.

In this work we follow closely our analysis of SPT tSZE-selected cluster
candidates and fit a population model described by four parameters: three
parameters for the log-normal distribution describing the true clusters and
one for the relative normalisation of contaminant distribution. In Fig 3 we
show an example for the ACT-DR5 candidates with SNR¿4.5. Using this method, we
measure the number of contaminants as a function of SNR threshold SNRmin.
Given that the cluster candidate search was performed in the 90 and 150 Ghz
bands where the tSZE signature appears as a negative signal, we expect the
only true or physical sources to be clusters, while the contamination is
solely caused by noise in the mm-wave maps. Given the noise characteristics,
the number of noise fluctuations as a function of SNRmin should be well
described by a Gaussian noise model with only two free parameters: the
standard deviation $\sigma$and the amplitude $A$, with a mean set to 0. The
number of false detections then is

$N_{\mathrm{false}}=A\int_{\mathrm{SNR}_{\mathrm{min}}}^{\infty}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{x^{2}}{2\sigma^{2}}}dx.$
(2)

In this case the standard deviation roughly corresponds to the noise of the
map used for source detection. Although the analysis is done in terms of SNR,
we do expect some mild offset from the expected value from unity, even in the
case of a correct estimate of the noise properties of the map. This bias is
caused by the fact that the detection pipeline tries to maximize the SNR at a
given position by adopting different filter sizes (see Sec.2.1). The amplitude
of the Gaussian model is given by the effective number of independent
detection cells. It therefore scales with survey area and the size of the
matched filter used for the detection. As clusters are only barely resolved in
ACT one can expect that the size of the filter (or detection cell) is close to
the beam size of ACT, our expectation value for the amplitude therefore is
close to the number of beams that fit into the ACT footprint.

In Fig. 4 we show the number of false detections as a function of SNRmin
together with the best fit Gaussian noise model. As is clear, the model
describes the data remarkably well over almost four orders of magnitude in
dynamical range. The best fit amplitude of the Gaussian model is $A=(28.1\pm
1.4)\times 10^{6}$. Using the size of the footprint this cell area corresponds
to 1.7 arcmin2, which is, as expected, close but mildly larger than the area
covered by the beam at 150 GHz of $\sim 1.5$ arcmin2. The measured standard
deviation $\sigma=1.146\pm 0.004$ is, as expected, close but significantly
larger than the naive assumption of $\sigma=1$.

Using our best fit model and the number of candidates, we estimate the
fraction of contaminants in the candidate list to be $53.7\pm 0.5$%. We note
that this result is robust against the particular choice of method to model
the population of noise fluctuations in the candidate list. Repeating the same
exercise using the method applied in the MCMF analysis of the Planck cluster
candidate list (Hernández-Lang et al. 2023), where the method uses richness
and constrains the contamination in the contamination dominated regime
($\log_{10}(\lambda/M_{500})<0.7$), yields contamination fractions within 0.5%
of those we estimate with our baseline analysis.

With the model describing the number of contaminants and the number of
observed candidates we can measure the purity
($1-N_{\mathrm{false}}/N_{\mathrm{candidates}}$) as a function of SNR
threshold SNRmin, as shown in Fig.5. Assuming that the Gaussian noise model
holds also at lower SNRmin¡4.0, we show the expected purity of the candidate
list down to lower SNRmin (dashed line), which yields a purity of only 25% at
SNR${}_{\mathrm{min}}=3.5$. In the SNR range between 3.5 and 4.0 we would
expect $\sim 4,800$ additional clusters, but at the same time adding $\sim
25,000$ noise fluctuations. Lowering the threshold therefore significantly
increases the amount of contamination, which would require stricter optical
cleaning with MCMF. The impact of optical cleaning on the over-all cluster
sample would become more important, and uncertainties in that cleaning would
begin to play an important role in the selection function.

Figure 5: Measured purity of the tSZE selected candidate list as function of
SNR threshold. The ACT-DR5 candidate list selected with SNR¿4.0 has a purity
of 53.5($\pm$0.5)%.

##  4 The ACT-DR5 MCMF cluster catalog

In this section we define the cluster sample considering completeness and
purity and report the basic properties of the sample. With the estimate of the
purity of the initial list of ACT-DR5 candidates, the total number of true
clusters is known to be $5872\pm 99$, 97% of which have good optical and NIR
data for confirmation. The MCMF-based $f_{\mathrm{cont}}$ estimator together
with the initial level of contamination provides an estimate of the
contamination fraction of the final $f_{\mathrm{cont}}$-cleaned sample. When
this information from the tSZE selection and optical cleaning is combined, it
allows one to trace the completeness and purity of the
$f_{\mathrm{cont}}$-cleaned sample and provides guidance for the optimal
sample selection.

###  4.1 Definition of the ACT-DR5 MCMF cluster catalog

Figure 6: Left: Completeness with respect to the initial tSZE selection versus
purity after optical cleaning with $f_{\mathrm{cont}}$
($f_{\mathrm{cont}}^{\mathrm{max}}$ increasing from left to right) for
candidate lists imposing SNR¿4 (black) and SNR¿4.5 (blue). Right: Distribution
of ACT-DR5 candidates in richness vs. SNR, color coded according to the
$f_{\mathrm{cont}}$ ranges.

We follow the approach used in the creation of the SPT-SZ MCMF selected
cluster sample (Klein et al. 2023b) and explore the contamination fraction and
completeness as a function of the $f_{\mathrm{cont}}^{\mathrm{max}}$ selection
threshold. If a cluster sample is created by imposing an upper limit in
$f_{\mathrm{cont}}^{\mathrm{max}}$, then the corresponding contamination
fraction of the sample is the product of $f_{\mathrm{cont}}^{\mathrm{max}}$
and the contamination fraction of the initial candidate list (i.e., $53.7\pm
0.5$% for the SNR¿4 ACT-DR6 sample). In the left plot of Fig. 6 we show the
completeness with respect to the initial tSZE selection and the purity
achieved after imposing a $f_{\mathrm{cont}}^{\mathrm{max}}$ selection that
decreases from left to right along each of the lines. On the right side of
Fig. 6 we show the distribution of candidates in richness versus ACT-DR5 SNR.
There is a clear correlation between richness $\lambda_{\mathrm{MCMF}}$ and
tSZE SNR, even in the presence of variations in ACT-DR5 survey depth over the
sky. The number of low-richness systems, which are likely noise fluctuations,
significantly rises towards lower SNR, while the typical richness of likely
true clusters decreases towards lower SNR. The boundary between the noise
fluctuation dominated region and the real cluster dominated region is around
$\lambda_{\mathrm{MCMF}}=20$.

One rough rule of thumb for defining a cluster sample where the residual
contamination is unimportant for most analyses is to ensure that the residual
contamination is smaller than the expected Poisson noise, which for a sample
of $\sim 5,900$ clusters is about 1.3%. For some analyses it is important to
avoid a situation where a significant number of real clusters are lost due to
the optical cleaning. In terms of completeness this threshold is reached by
imposing a cut of $f_{\mathrm{cont}}^{\mathrm{max}}=0.2$, which corresponds to
a purity of $\sim 90$% (i.e., contamination fraction is 0.2$times$0.53 in the
case of the SNR¿4 sample). This selection yields a sample with significant
contamination but the impact of the optical confirmation is negligible in
terms of removing real clusters. To obtain samples with negligible levels of
contamination one either has to accept $\sim 16$% incompleteness introduced by
the more aggressive cleaning implied by a dropping threshold in
$f_{\mathrm{cont}}^{\mathrm{max}}$, or one has to create the cluster sample
from a candidate list with a higher SNR threshold to increase the initial
purity of the tSZE selected candidate list. As an example, imposing
SNR${}_{\mathrm{min}}=4.5$ increases the initial purity to 76% allowing for a
cluster sample with 98% purity and completeness, which is close to the level
of the Poisson noise for a cluster sample of this size.

Because all cleaner subsamples can be constructed from the
$f_{\mathrm{cont}}^{\mathrm{max}}=0.2$ and SNR${}_{\mathrm{min}}>4.0$ selected
sample, we use these two selection thresholds in defining our ACT-DR5 MCMF
cluster catalog. In Table 1 we show sample selection parameters and key
numbers for the ACT-DR5 MCMF sample along with two example subsamples: the
aforementioned sample with SNR${}_{\mathrm{min}}=4.5$ and 98% purity and
completeness (SNR-4.5) and the Purity-95 sample that has increased purity and
decreased completeness with respect to the full ACT-DR5 MCMF cluster sample.
Following A&A policies the catalog will be made available on CDS after
acceptance of the paper. Earlier access to the catalog can be granted upon
reasonable request. The description of the catalog entries can be found in
Appendix B.

Table 1: Properties of the ACT-DR5 MCMF cluster catalog along with two
subsamples. The table shows sample name, the optical selection criterion
$f_{\mathrm{cont}}^{\mathrm{max}}$, the tSZE selection criterion SNRmin, the
expected final sample purity, the completeness with respect to the tSZE
selection and the number of selected clusters $N_{\mathrm{cl}}$.

|  |  | Purity | Comp. |
---|---|---|---|---|---
Cluster Sample | $f_{\mathrm{cont}}^{\mathrm{max}}$ | SNRmin | $[\%]$ | $[\%]$ | $N_{\mathrm{cl}}$
ACT-DR5 MCMF | 0.2 | 4.0 | 89.7 | 97.8 | 6237
Purity-95 | 0.1 | 4.0 | 94.6 | 93.1 | 5604
SNR-4.5 | 0.1 | 4.5 | 97.6 | 98.2 | 3747

###  4.2 Properties of the ACT-DR5 MCMF cluster catalog

Figure 7: ACT-DR5 candidates distributed in richness vs. redshift space. Color
coded are unconfirmed candidates in grey, confirmed systems with
0.1¡$f_{\mathrm{cont}}$¡0.2 in blue and confirmed clusters with
$f_{\mathrm{cont}}$¡0.1 in red.

Figure 8: Left: Redshift distribution of the three largest ICM-selected
cluster catalogs to date. While RASS-MCMF provides the largest number of
clusters at low redshift ($z<0.3$), ACT-DR5 MCMF provides the most clusters at
high redshift ($z>0.5$). For the intermediate redshift range ($0.3z<0.5$)
eRASS1 provides the largest number of clusters. Right: Richness versus tSZE-
based mass of ACT-DR5 candidates and the ACT-DR5 MCMF sample. Color coded are
unconfirmed candidates in grey, confirmed clusters with
0.1¡$f_{\mathrm{cont}}$¡0.2 in blue and confirmed clusters with
$f_{\mathrm{cont}}$¡0.1 in red.

The properties of the ACT-DR5 MCMF cluster catalog and the two example
subsamples are governed by two selections: the tSZE selection threshold in SNR
and the $f_{\mathrm{cont}}$ selection that results in a redshift dependent
richness cut. The SNR selection not only impacts the purity of the tSZE-based
candidate list, but also- due to its correlation with mass- impacts the mass
range of the selected clusters. Because richness also scales with cluster
mass, the completeness with respect to the tSZE selection therefore increases
at fixed richness threshold for increasing SNR thresholds. The SNR-45 sample
therefore benefits two-fold from the increased SNR threshold, in purity and in
completeness. A downside is that the SNR-45 sample is missing a significant
number of low mass clusters compared to the default ACT-DR5 MCMF cluster
sample. Due to the growth of structures over time this increase in effective
mass selection is expected to impact the number of high redshift clusters more
than low redshift clusters.

The MCMF method is designed to maintain the fractional contamination of the
cluster sample constant at all redshifts. The redshift dependent richness cut
implied by the $f_{\mathrm{cont}}$ selection depends to two things: the
original tSZE selection and the properties of noise fluctuations and clusters
in richness-redshift space. In Fig. 7 we show the distribution of ACT-DR5
SNR¿4 candidates in richness-redshift space, highlighting two
$f_{\mathrm{cont}}$-selections in blue and red. The shape in the selection
seen in this figure can be explained as follows. The upturn at low redshift is
due to the properties of the tSZE selection (see Fig.7 in Hilton et al.
(2021)) which leads to few candidates in this region. The lowest value is
reached at $z\approx 0.45$ where the number of real clusters reaches its
maximum. Beyond that redshift the richness threshold increases due to the
limitations of the optical data to disentangle clusters and contaminants. For
reasonably small values of $f_{\mathrm{cont}}$ ($<0.3$) the shape of the
redshift dependent richness selection remains approximately unaltered and the
difference in selection can be approximated by an overall shift in richness
threshold.

Due to the redshift insensitivity of the tSZE based cluster detection, the
ACT-DR5 MCMF cluster catalog contains a large number of clusters at high
redshift. The median redshift of the sample is $z=0.58$. In the left panel of
Fig. 8 we show the redshift distributions of the three largest ICM-selected
cluster samples to date. While the all-sky X-ray based RASS-MCMF cluster
catalog (Klein et al. 2023a) yields the largest number of clusters at $z<0.2$,
the roughly half-sky ACT-DR5 MCMF cluster catalog provides the overall largest
number of clusters at $z>0.6$. The largest current ICM-selected cluster
catalog is the X-ray based eRASS1 (Bulbul et al. 2024) catalog. While covering
a similar sky area as ACT-DR5, it provides the largest number of clusters in
the intermediate redshift regime $0.2<z<0.5$.

In the right panel of Fig. 8 we show the distribution of richness versus a
tSZE-based mass estimate for the ACT-DR5 candidates as well as for the
official ACT-DR5 MCMF and the Purity-95 samples. Compared to the raw tSZE SNR
shown in Fig. 6, we see a tighter richness-mass relation for ACT-DR5 MCMF
clusters. This property of a tSZE selected sample contrasts with an X-ray
selected sample, where the X-ray count rate mass proxy scatters much more
about the mean observable–mass relation (Ghirardini et al. 2024, Singh et al.,
in prep.).

###  4.3 High-z ACT-DR5 MCMF clusters

Figure 9: High redshift galaxy clusters with ongoing star formation. The first
two color composite images show ACT-CL J0431.0-3236 ($z=1.54$, SNR=5.1) in
optical $g,r,z$-bands (left most image) and in WISE $w1,w2$-bands (second from
left). The third and fourth images show the same optical and WISE images for
ACT-CL J1240.4+0534 ($z=1,8$, SNR=5.3). While the BCGs are visible for both
clusters in optical and WISE, we see an unusually high concentration of blue
galaxies near the cluster core.

The ACT-DR5 MCMF cluster catalog represents a major step forward in the number
of high redshift ICM-selected galaxy clusters. Compared to the predecessor
catalog (Hilton et al. 2021), the new sample contains three times more
clusters above $z=1$. In addition, it starts to reach redshifts close to the
peak of cosmic star formation at $z=2$, a redshift where the red sequence of
cluster galaxies is being established. With $\sim$700 clusters at $z>1$, this
sample contains the largest number of high redshift ICM-selected clusters to
date. Current MCMF photo-z’s suggest $\sim$110 to be at even higher redshifts
of $z>1.4$. When looking at clusters at higher redshift, we tend to see more
star forming galaxies in the clusters. Given the rest-frame sampling of the
DECam optical bands, we start to miss passive galaxies in the LS-DR10 imaging
data and see preferentially star forming galaxies. Only in combination with
WISE NIR data can we also study the passive population in the high redshift
regime. In Fig. 9, we show two examples of high redshift clusters. The BCG is
visible as a red galaxy in the optical and is also prominent in the WISE-based
images as the brightest source in the center of the cluster. Aside from the
BCG, few other red galaxies are visible, while both optical images show blue
galaxies potentially associated with the clusters.

We wish to stress that the MCMF confirmation using LS-DR10 and WISE is
reaching its limits at high redshift and that some increased incompleteness is
expected at the high redshift end because of that. This circumstance might be
highlighted with the example of ACT-CL J1424.7+0642, a previously unknown
cluster that is not confirmed as a member of the ACT-DR5 MCMF cluster sample.
ACT-CL J1424.7+0642 at RA=216.191 DEC=6.715 has a SNR of 7.35, which makes it
very unlikely (0.26%) to be a noise fluctuation (see Figs. 4 & 5). The optical
LS-DR10 image, shown in Fig. 10, does not show any excess of red galaxies, and
only a hint of some blue galaxies at the tSZE position. In the WISE image we
find a bright source, with no optical counterpart close to the tSZE position
and no evidence for additional cluster galaxies. Interestingly, this cluster
has a bright Radio counterpart, as shown in the 1.36 GHz image of the Rapid
ASKAP Continuum Survey (Duchesne et al. 2024) in Fig. 10. The combination of a
bright Radio and bright WISE source indicates the presence of a massive
passive galaxy like the BCG of a galaxy cluster. The lack of optical emission
at the same time suggests that this cluster must be at high redshift. MCMF
estimates a tentative redshift of $z=1.59$ but given that the cluster fails
MCMF confirmation ($f_{\mathrm{cont}}$=0.5) this redshift is more indicative
than definite.

Deeper imaging data and NIR wavelength coverage are needed to confirm and
study these high redshift ACT-DR5 MCMF clusters. In this context, the ongoing
survey by the Euclid mission (Euclid Collaboration et al. 2022) seems to be a
particularly promising source of such data. However, such a sizeable and
uniformly selected high redshift ensemble of clusters would also be well
suited for dedicated, deep observations with other observatories.

Figure 10: ACT-CL J1424.7+0642, a SNR=7.35 ACT-DR5 candidate that is not
confirmed with MCMF. Left: The LS-DR10 $g,r,z$-band color composite does not
show an excess of red galaxies. Middle: WISE $w1,w2$-band color image, showing
a bright WISE source at the location of the tSZE candidate. Right: 1.36 GHz
image of the Rapid ASKAP Continuum Survey (Duchesne et al. 2024), showing a
bright source at the WISE location, indicating the presence of a massive and
passive galaxy. Figure 11: Richness versus tSZE mass of clusters from the
previous ACT-DR5 catalog (Hilton et al. 2021). Richness is either from DES or
SDSS redMaPPer (Rykoff et al. 2016) or CAMIRA (Oguri et al. 2018) and scaled
to account for systematic uncertainties between redMaPPer and CAMIRA. Clusters
also confirmed by MCMF are shown in blue, while those not confirmed with MCMF
(red points) show up largely at low richness and tSZE-based mass.

###  4.4 Comparison to the previous ACT-DR5 cluster catalog

The previous ACT-DR5 cluster catalog (Hilton et al. 2021) employed a mix of
surveys and techniques to obtain redshifts and to confirm cluster candidates.
The main method to obtain redshifts and confirm clusters was a positional
cross-match between the candidate tSZE position and the positions of optically
selected clusters. In the case of sources in the DES (Abbott et al. 2016) and
SDSS (Blanton et al. 2017) regions, cross-matches were performed to optical
clusters identified by the redMaPPer algorithm (Rykoff et al. 2016) with
settings that restricted the fraction of chance matches to a level of 5%. A
similar approach was used over the HSC-SSP (Aihara et al. 2018) footprint,
matching with optically selected clusters from the CAMIRA catalog (Oguri et
al. 2018). Together, this cross-matching of tSZE candidates to optically
selected cluster catalogs resulted in the confirmation of 1,491 clusters
($\sim 35.5$%). For the remaining 64.5% of the previous ACT-DR5 cluster sample
other less controlled methods or surveys were used. Although not explicitly
stated, this suggests that the previous ACT-DR5 catalog can be expected to
have a contamination of $\geq 5$%.

We adopt a maximum distance of 1.5 arcmin to cross match our ACT-DR5 candidate
list with the previous ACT-DR5 cluster catalog (Hilton et al. 2021) and find
12 sources ($0.3$%) from (Hilton et al. 2021) not matched to our candidates.
Most of these sources lie close to masked point sources in the ACT map. We
therefore conclude that mild changes in the source detection pipeline and
point source masking scheme introduces a very minor difference in our
candidate list with respect to the version used in constructing the previous
catalog (Hilton et al. 2021).

From the 4,182 matched sources we find 4,126 (98.7%) to lie simultaneously in
LS-DR10 and WISE unmasked footprints and 4,157 ($99.5$%) to lie at least in
one of the two footprints. We re-confirm 4,007 ($\sim 96.5$%) of the
previously published clusters (Hilton et al. 2021) when adopting a
$f_{\mathrm{cont}}$¡0.2 threshold, leaving 129 of the previous clusters ($\sim
3.5$%) unconfirmed even though they have decent LS-DR10 and WISE coverage.

In Fig. 11 we show the distribution of richness versus tSZE-based mass for the
subset of the sample that have either redMaPPer or CAMIRA measurements
available in the previous catalog (Hilton et al. 2021). We account for an
offset between redMaPPer richness and CAMIRA richness by multiplying the
CAMIRA richness by a factor of 1.5. As visible from the plot, the majority of
the sources not confirmed with MCMF lie at low richness and below the main
scaling relation between richness and tSZE-based mass estimate provided in
Hilton et al. (2021). This is indicative that a significant fraction of the
sources not meeting the MCMF selection threshold are indeed chance super
positions instead of massive clusters. This is further supported by the fact
that the fraction of clusters confirmed with redMaPPer and CAMIRA, for which
Hilton et al. (2021) applied a systematic cut to allow only 5% chance
positions, drops from 63% for MCMF confirmed to 40% for the MCMF unconfirmed
cases. It is worth noting that the cross-matched redMaPPer and CAMIRA clusters
show visibly more scatter between richness and tSZE-based mass than the MCMF-
based richness shown in the right plot of Fig. 8.

Although a significant fraction of the systems unconfirmed with MCMF are
indeed noise fluctuations, we do find evidence that some clusters are also
unconfirmed. Those missed are either affected by small scale masking (due to,
e.g., bright stars), are at high redshift (e.g., ACT-CL J0217.7-0345,
$z=1.91$), where we hit the limitations of the optical and NIR imaging surveys
or are at very low redshift (e.g., ACT-CL J2313.9-4243,$z=0.056$). Regarding
the issues at low redshift, it is well known that tSZE detection from surveys
at 90 and 150 GHz becomes increasingly insensitive when clusters reach angular
scales where the primary CMB fluctuations peak. This behavior is also
reflected in the MCMF selection, where the richness threshold
$\lambda_{\mathrm{min}}(z)$ increases toward low redshift. Because this
increase is steep and may depend on the local survey properties, there are
cases where even though the cluster is optically well detected, the cluster
richness does not lie above the MCMF threshold.

Finally, there are 2,226 ACT-DR5 MCMF clusters ($f_{\mathrm{cont}}$¡0.2) that
do not have a match in the previous catalog (Hilton et al. 2021). A certain
fraction of these clusters simply were not in the candidate list used in
Hilton et al. (2021), because in our current analysis we lower the detection
threshold. In addition, the authors of the previous study estimated that $\sim
960$ clusters were missing from their sample. Together then it is not
surprising that we have identified more clusters in the ACT-DR5 MCMF catalog.

The SNR¿5 regime was of special interest in the previous ACT-DR5 catalog
(Hilton et al. 2021) and therefore deserves specific mention. Candidate
clusters in this SNR regime were visually inspected in the previous analysis
to ensure their quality. Furthermore, we expect that almost all SNR¿5 sources
that appear in our candidate list were also present in the previous sample,
although potentially at slightly different SNR values. It is therefore
interesting that we confirm 327 new clusters at SNR¿5, and we fail to confirm
18 clusters despite good imaging data at those locations. These 327 new
clusters correspond to $\sim$12% of all clusters with SNR¿5 and therefore it
indicates a significant incompleteness in the previous ACT-DR5 catalog (Hilton
et al. 2021).

##  5 Comparison of ACT-DR5 MCMF to other catalogs

Given the location and size of the ACT-DR5 footprint, the number of cluster
catalogs with possible overlap with the ACT-DR5 MCMF catalog is too large to
discuss them all. We therefore restrict the comparison to the largest ICM-
selected cluster samples. When comparing to other catalogs, we typically want
to extract the total number of clusters in common as well as the list of
concrete matched clusters. We are also interested in sources that are not
matched but should have been, given their mass and redshift.

To get an estimate of the total number of matched clusters, we first cross
match the samples using a large matching radius of $0.2^{\circ}$. We repeat
the matching procedure after shifting the source positions by $\pm 2^{\circ}$
to estimate the number of chance matches. Using shifted source positions
allows us to include the impact of depth variations with position in the ACT-
DR5 survey. Using the offset distribution of the matched sources for the
normal and the shifted catalog one can measure the number of true matches as a
function of positional offset under the assumption that both offset
distributions are composed of chance matches at large offsets ($\Theta\approx
0.2^{\circ}$). The total number of matches is simply the integral over the
offset distribution minus the integral over the scaled offset distribution of
the chance matches. We can use the same distributions to identify a matching
radius that yields a reasonable compromise between completeness and
contamination by chance matches.

###  5.1 SPT candidate and cluster catalogs

The SPT collaboration has also performed multiple mm-wave surveys, with a
series of receiver upgrades over the years. Most noteworthy are the SPT-SZ
survey (Bleem et al. 2015), the SPT-ECS survey (Bleem et al. 2020) and the
recently published SPTpol 500d survey (Bleem et al. 2024). The MCMF informed
cluster catalog from SPTpol 500d, a new version of the SPT-SZ cluster catalog
(Klein et al. 2023b, SPT-SZ MCMF) and an MCMF augmented version of the SPT-ECS
cluster catalog were then used for the cosmological study presented in Bocquet
et al. (2024).

Of special interest for this work is the SPTpol 500d survey, because it is on
average significantly deeper than the ACT-DR5 dataset and therefore it allows
one to test the purity and completeness of the ACT-DR5 MCMF cluster catalog.
To test for completeness we first match all ACT-DR5 candidates over the SPTpol
500d footprint with the SPTpol 500d candidate list. Given the reasonably small
positional uncertainties and low source densities, we find that a cross-match
radius of 2 arcmin includes all true cluster matches, while only one chance
match is expected. We find 118 of the 339 ACT-DR5 cluster candidates over the
SPTpol 500d field have matches in the SPTpol 500d candidate catalog. From
those 118 matches, 115 (97%) are confirmed in our ACT-DR5 MCMF sample
($f_{\mathrm{cont}}$¡0.2). Of the three missing systems, only one (ACT-CL
J2247.6-5537) is a confirmed cluster in SPTpol 500d, and it suffers from
masking in the optical/NIR survey data due to a nearby star. From the
remaining two clusters, one (ACT-CL J2342.9-5521) shows the largest SPT to ACT
positional offset (99 arcsec) out of all matches. Despite the large offset and
potential small scale masking impeding confirmation, we assume at least two of
the three to be true clusters. Given the number of matches this yields an
estimate of the completeness of the ACT-DR5 MCMF catalog of $\sim$98%, close
to the expected completeness of 97.8% presented in Table 1.

Out of the 221 ACT-DR5 candidates in the common footprint without a match we
find that 52 are members of the ACT-DR5 MCMF cluster catalog
($f_{\mathrm{cont}}$¡0.2). Out of those we find ten that are eRASS1 extent
selected clusters and an additional seven that are eRASS1 point source (see
discussion in Sec. 5.3); using optical, NIR, eRASS1 and ACT-DR5 maps, we
identify another 19 candidates that are likely real clusters. Thus, 36 out of
the 52 ACT-DR5 MCMF clusters ($f_{\mathrm{cont}}$¡0.2) without SPTpol 500d
counterparts are likely real clusters. In total this yields 151 confirmed real
clusters in the SPTpol 500d footprint, out of 167 clusters in the ACT-DR5 MCMF
catalog ($f_{\mathrm{cont}}$¡0.2). This corresponds to a sample purity of
90.4%, close to the expected 89.7% presented in Table 1. With 154 real
clusters out of 339 candidates in the the SPTpol 500d footprint, we can
estimate a contamination level of the ACT-DR5 candidate list to be $\sim$55%,
close to the value estimated in Sec. 3.2 of $53.7\pm 0.5$%.

The SPT-SZ MCMF and SPT-ECS catalogs are shallower than SPTpol 500d and
therefore do not allow us to test completeness and purity to the same extent
as SPTpol 500d. On the other hand, their large footprints of $\sim$2,500 deg2
provide more clusters in common that could be used for cross-calibration
studies. We find that using the cross-match radius of 2 arcmin yields 435
matches between SPT-SZ MCMF confirmed clusters and the ACT-DR5 cluster
candidates, from which 431 (99%) are members of the ACT-DR5 MCMF cluster
catalog ($f_{\mathrm{cont}}$¡0.2). Repeating the same exercise with SPT-ECS
clusters, we find 290 matches with the ACT-DR5 candidate list, of which 288
(99.3%) are members of the ACT-DR5 MCMF cluster catalog. Taking the overlap
between the different SPT-based catalogs into account, we find a total of 768
matches between those three SPT cluster catalogs and our ACT-DR5 MCMF cluster
catalog. This sample is sufficiently large to allow for cross checks of, for
example, the tSZE inferred mass or the survey selection function models.

In summary, the cross-comparison with the deeper SPTpol 500d survey confirms
the completeness and contamination levels we present for the ACT-DR5 MCMF
catalog as well as our measurement of the initial contamination level of the
ACT-DR5 candidate list.

###  5.2 RASS-MCMF cluster catalog

The RASS-MCMF cluster catalog (Klein et al. 2023a) covers 25,000 deg2 of
extragalactic sky and contains 8,449 clusters. The optical data used for
cluster confirmation is identical to that used in the current analysis, and we
expect only minor departures from complete coverage of our ACT-DR5 footprint
due to RASS low exposure time regions. We find 1,061 of the RASS-MCMF clusters
in our ACT-DR5 MCMF catalog. The offset distribution of matches shows an
excess over chance matches out to $\sim$5.5 arcmin. Using a matching radius of
4 arcmin yields 1,066 matches with an expected contamination by chance matches
of 1.7% and an incompleteness (i.e., true matches with offsets ¿4 arcmin) of
1.3%. This number of matches corresponds to $\sim$16% of the full ACT-DR5 MCMF
sample. Conversely, $\sim$29% of the RASS-MCMF sample within the ACT-DR5
footprint is also present in the ACT-DR5 MCMF cluster catalog.

The fraction of matched clusters is a strong function of redshift, mainly due
to the different redshift sensitivity of the candidate selection in the X-ray
and the tSZE. The fraction of RASS-MCMF clusters with an ACT-DR5 MCMF match
rises steeply with redshift and flattens out at z¿0.5, reaching fractions of
$\sim$80-85%. This match fraction is close to the expected purity of the RASS-
MCMF sample of 90%. We do not expect to match the full 90% of the RASS-MCMF
sample that consists of real clusters, because there is significant scatter in
the X-ray and tSZE observable-mass relations and there are variations of the
survey depths over the sky in both surveys. Considering those effects, one can
take the 80% match fraction for clusters at z¿0.5 to be a lower limit for the
sample purity of the RASS-MCMF catalog and therefore a confirmation of the
expected purity.

As mentioned in Section 3.1.1, there are two important changes in the MCMF
algorithm for this analysis that impact the cluster richness estimates.
Namely, we exclude the $w1-w2$ color from our MCMF richness estimate, and we
improve the red sequence model for the $z-w1$ color. These changes weaken the
rather strong redshift trends in the richness-mass relation seen in RASS-MCMF
clusters, which are caused by the v-shaped redshift dependency of the $w1-w2$
color and the uncertainty in the redshift variation of the scatter in the
$z-w1$ color (e.g. see Fig.1 in Klein et al. (2023b)). Because the richness
algorithms for RASS-MCMF and ACT-DR5 MCMF differ, the resulting richness
measurements for the same cluster appearing in the two catalogs also differ.
We performed richness measurements using both richness algorithms for RASS-
MCMF and ACT-DR5 MCMF clusters to model the differences in richness, and we
provide a transformation (see Table 2) that can be used to calculate updated
richnesses for the RASS-MCMF clusters that are equivalent to those one would
obtain using the current MCMF settings.

We have examined the richnesses coming from the previous and current MCMF
algorithms, finding that their scatter about the ICM based mass estimates is
similar. That is, both types of richness estimates are valid, internally self-
consistent, and can be used as mass proxies with similar scatter. The
remaining differences between RASS-MCMF and ACT-DR5 MCMF richnesses are then
just the consequence of using different ICM-based priors on cluster center and
mass.

###  5.3 eRASS1 cluster catalog

Figure 12: eRASS1 matched ACT-DR5 MCMF clusters in mass versus redshift.
Matches to the extent selected cluster and group catalog are highlighted in
yellow. ACT-DR5 MCMF clusters matched to sources in the eRASS1 point source
sample are shown in blue. One famous ACT-DR5 cluster missed in the eRASS1
cluster sample is ACT-CL J2344.7-4243 (the Phoenix cluster).

Figure 13: Optical LS-DR10 image (top) and smoothed eROSITA count rate image
(bottom) of ACT-CL J1416.5+0534 (z=0.23, SNR=4.9) provide one example of a low
redshift ACT-DR5 MCMF cluster only matched to the eRASS1 point source catalog.

Figure 14: Optical LS-DR10 image (top) and smoothed eROSITA count rate image
(bottom) of ACT-CLJ0840.5+0543 (z=0.28, SNR=9.75). This is one of 924 ACT-DR5
MCMF clusters with matches in the eRASS point source catalog. This particular
cluster, along with $\sim$70 other clusters in the common eRASS1-ACT-DR5
footprint, is missing from the eRASS1 cluster sample but is included in the
RASS-MCMF cluster sample.

The eROSITA collaboration recently released the first survey data covering the
western Galactic hemisphere (Merloni et al. 2024). In addition to the catalog
of all detected X-ray sources, there is an extent selected cluster and group
catalog (Bulbul et al. 2024). The cluster and group catalog is of special
interest, because it serves as the basis of the eROSITA eRASS1 cosmology
analysis (Ghirardini et al. 2024), in which the eROSITA team reports an
unusually high value of $S_{8}$ compared to the majority of other recent
cluster and large scale structure analyses (e.g. Mantz et al. 2015; Lesci et
al. 2022; Chiu et al. 2023; Bocquet et al. 2024; Dark Energy Survey and Kilo-
Degree Survey Collaboration et al. 2023; Madhavacheril et al. 2024).

We first match the ACT-DR5 MCMF sample to the cluster and group catalog. We
find 1,306 matches, almost all (¿99%) showing offsets smaller than 3.5 arcmin.
Imposing a maximum offset of 3.5 arcmin yields 1,326 matches with an
expectation of $\sim$20 chance matches, corresponding to a contamination level
of 1.5%. We then remove matches to the eRASS1 cluster and group catalog and
match the remaining ACT-DR5 MCMF clusters to the eRASS1 point source catalog,
obtaining 965 matches in excess to the expected number of chance matches. In
total then, of the 2,271 ACT-DR5 MCMF clusters that match to eRASS1 sources,
43% are classified as point sources rather than clusters and groups.

In Fig. 12 we show the distribution of eRASS1 matches in mass versus redshift.
The eRASS1 X-ray extent and extent likelihood selected sample of clusters and
groups results in a dramatic loss of clusters compared to the sample that
matches both X-ray point source and extended source catalogs. This impacts
clusters with masses $M_{500}\approx 2\times 10^{14}$ M⊙ at intermediate
redshifts z$\sim$0.35 and beyond. The X-ray extent selection suffers from
incompleteness when there is strong emission from the cluster center. One
clear example of this is one of the most massive clusters missed in the eRASS1
cluster and group catalog, which is ACT-CL J2344.7-4243 (the Phoenix cluster),
which is visible in Fig. 12 at $z\approx 0.6$ and $M_{500}\approx 10^{15}$ M⊙.

Examining some of the low-z ACT-DR5 MCMF clusters that appear in the eRASS1
point source sample, we find two additional reasons for clusters to be lost
from the eRASS1 cluster and group catalog: source splitting and low surface
brightness. In Fig. 13 we show the region around ACT-CL J1416.5+0534
($z=0.23$, SNR=4.9, $M_{500}=2.1\times 10^{14}$ M⊙). The smoothed eRASS1 map
shows clear evidence for extended emission at the cluster location, but this
eRASS1 source is classified as point like. Given the clearly apparent extent
of the cluster, the most likely reason for this classification is that the
extent likelihood is low.

In Fig. 14 we show another case of an ACT-DR5 MCMF cluster missed in the
eRASS1 cluster and group catalog: ACT-CLJ0840.5+0543 (z=0.28, SNR=9.75,
$M_{500}=4.8\times 10^{14}$ M⊙). The X-ray emission of the cluster is
apparent, but it seems to be distributed in different patches and potentially
impacted by a real point source in the northern clump. This cluster is in fact
bright enough to be one out of $\sim$70 ACT-DR5 MCMF clusters that are listed
as eRASS1 point sources and simultaneously are confirmed RASS-MCMF clusters
(Klein et al. 2023a).

Despite the fact that we identify numerous cases of clusters only appearing in
the point source sample, it is not immediately clear if this could lead to a
bias in the estimated $S_{8}$ parameter found in Ghirardini et al. (2024). In
fact the large amount of intrinsic scatter between the X-ray mass-proxy
observable and the cluster mass found in Ghirardini et al. (2024) results in a
very broad selection function in cluster mass, allowing for clusters being
missed (or perhaps more accurately– mis-classified) even at very high masses.
For example the cluster ACT-CLJ0840.5+0543 at $z=0.28$ has an eRASS1 exposure
time of $150$ seconds, yielding only a 40% detection probability even with a
mass of $M_{500}=4.8\times 10^{14}$ M⊙ (see Figure D.1 in Ghirardini et al.
2024). What can be concluded from the cross matching is that a significant
number of clusters are appear only in the eRASS1 point source sample, and that
this is happening even at lower redshift and at masses greater than
$M_{500}\approx 2\times 10^{14}$ M⊙.

It is worth noting that the eRASS1 cluster cosmological analysis (Ghirardini
et al. 2024) is the first ICM-based cluster study that consciously adopts a
significantly contaminated cluster sample ($\sim$15% of the systems are not
expected to be real clusters) and that attempts to account for the impact of
non-cluster contaminants. An important ingredient in the so-called mixture
model of real clusters and contaminants is the richness distribution along
random lines of sight and along lines of sight towards eRASS1 point sources.
The latter is meant to represent the richness distribution of AGN and other
point like sources contaminating the extent selected cluster and group
catalog. As we have shown here, there are in fact a large number of massive
clusters in the point source catalog, which will populate the high richness
regime in the richness distribution along lines of sight toward eRASS1 point
sources. That in turn biases the mixture model.

This is supported by the eRASS1 simulations (Seppi et al. 2022) that indicate
that $\sim$4.3% of all eRASS1 sources are galaxy clusters. Given the
$\sim$930,000 detections in eRASS1 (Merloni et al. 2024) this translates into
$\sim$40,000 clusters in eRASS1 from which only one quarter ($\sim$10,000) are
found in the cluster and group catalog (Bulbul et al. 2024). The richness
distribution of point sources therefore contains three times more clusters
than the actual eRASS1 cluster and group catalog. This may bias the richness
distribution of the AGN model (Kluge et al. 2024), making it appear quite
different from the richness distribution along random lines of sight. We note
that in the RASS-MCMF catalog a similar analysis showed no difference in the
richness distribution along random lines of sight and the distribution along
lines of sight toward AGN (Klein et al. 2023a). Estimating the impact of this
potentially biased modeling of the contamination on the cosmological results
(Ghirardini et al. 2024) requires further attention, but is beyond the scope
of this paper.

Figure 15: Left & middle: Stacked ACT-DR6 CMB lensing map for the full ACT-DR5
MCMF catalog (left) and and for the $z>1$ subsample (middle). Black contours
show lensing convergence starting at $\kappa=0.03$ in steps of 0.01. The field
of view is $25\times 25$ arcmin. Right: Projected CMB lensing mass within 5
arcmin radius $M_{\mathrm{5\,arcmin}}$ in bins of tSZE derived mass. The red
line shows the best fit power law, suggesting a slope of $1.15\pm 0.15$.

##  6 Cross-correlation with ACT-DR6 CMB lensing

We utilize the recently published ACT-DR6 CMB lensing map, (Madhavacheril et
al. 2024) which has $\sim$68% overlap with our ACT-DR5 MCMF sample, to measure
the CMB lensing signal associated with the MCMF confirmed cluster sample.
Notable here is that the difference in footprints reflect a more restrictive
masking applied for the CMB lensing study.

The CMB lensing map was constructed from the spherical harmonic
$a_{\mathrm{l,m}}$ convergence coefficients, where the recommended multipole
range for the maps is $600<l<2100$, resulting in an angular resolution of
$\sim$5 arcmin (Madhavacheril et al. 2024). The lensing map contains the
convergence $\kappa=\Sigma/\Sigma_{\mathrm{crit}}$ along each line of sight.
We use the ACT CMB lensing convergence maps to explore the significance of the
detection of the ACT-DR5 MCMF sample. Using the Purity-95 sample (see Tab. 1),
which provides a nice match of completeness and purity, we show the mean CMB
lensing convergence maps of the full sample (Fig. 15 left panel) and for the
$z>1$ cluster subsample (right panel of same figure). The field of view shown
is $25\times 25$ arcmin, corresponding to $\sim$5$\times$5 resolution elements
in the $\kappa$ maps. We obtain a clear, positive mean convergence signal for
both cluster samples with signal-to-noise ratios of 16.4 and 4.3,
respectively, within an aperture of 5 arcmin. The estimated signal-to-noise
ratios are based on measurements of 10,000 representative mock samples where
clusters are positioned randomly within the CMB lensing convergence maps. We
find that the values for the full ACT-DR5 MCMF cluster sample are consistent
with the Purity-95 subsample.

The detection significance of 16.4 $\sigma$ for the full ACT-DR5 MCMF cluster
catalogue is the strongest CMB lensing signal found to date for a cluster
sample. Additionally, the significance as well as the mean redshift of the
high-z subsample (¡z¿=1.2) exceeds that of the optical+IR selected MadCows
sample (Madhavacheril et al. 2020) and currently represents the highest signal
to noise high-redshift CMB lensing measurement for clusters. The strong
detection of lensing convergence around the high redshift subsample highlights
the fact that this ACT-DR5 MCMF sample includes many massive, high-redshift
halos, in accordance with expectation.

For lines of sight containing a massive cluster, the weak gravitational
lensing convergence is typically dominated by the surface mass density
$\Sigma$ of the cluster. Because $\kappa$ is the ratio of the cluster surface
mass density to the critical surface mass density $\Sigma_{\mathrm{crit}}$,
there is also a dependence on the cluster and source redshift. Because the
surface density of a galaxy cluster depends on the line of sight through the
cluster, which scales as cluster mass $M_{500}^{1/3}$, the weak gravitational
lensing convergence (or shear) provides mass information about the cluster
that has long been used to calibrate cluster masses in ICM selected samples
(e.g., Applegate et al. 2014; Hoekstra et al. 2015; Dietrich et al. 2019;
Klein et al. 2019b; Bocquet et al. 2019, 2023; Grandis et al. 2024, Singh et
al, in prep).

We therefore use the ACT-DR6 lensing convergence maps to explore the mass
information that is present for the ACT-DR5 MCMF sample. Namely, we divide the
Purity-95 sample (see Tab. 1) into mass bins, using the mass estimates
following the original ACT-DR5 cluster catalog paper (Hilton et al. 2021).
Similar to the previous analysis step, we create mean CMB lensing convergence
maps but now for each mass bin. Using the mean $\Sigma_{\mathrm{crit}}$ of the
clusters contributing to a given mass bin, we calculating
$\sum_{i}\kappa_{i}*\Sigma_{\mathrm{crit}}$ on the mean CMB convergence map
for all pixels $i$ within 5 arcmin from the cluster center. We call the
resulting mass estimate $M_{\mathrm{5arcmin}}$, corresponding to the cluster
mass projected within a 5 arcmin cyclinder around the cluster center. Within
each mass bin the cluster sample spans a broad range of redshift, but at the
median redshift of the sample the 5 arcmin angular aperture corresponds to a
physical radius of 2 Mpc. Thus, the $M_{\mathrm{5\,arcmin}}$ notionally
corresponds to the mass of the cluster within a cylinder with a radius of 2
Mpc. This 2 Mpc is close to the virial radius of the clusters in ACT-DR5 MCMF,
and so the $M_{\mathrm{5\,arcmin}}$ can be considered to be a rough estimate
of the total cluster mass.

In the right-most panel of Fig. 15 we plot $M_{\mathrm{5\,arcmin}}$ as a
function of tSZE-based halo mass estimate $M_{500}$ (Hilton et al. 2021). In
producing the mean convergence maps, we account for the impact of
contamination in the cluster sample. For this we use a catalog extracted along
random lines of sight with the same selection cuts as those used for the real
sample. After scaling the random catalog to the expected amount of
contamination, we correct for the contribution of contaminants in each mass
bin.

Correcting for the contaminating (i.e., non-cluster) systems requires
accounting for the fact that the MCMF selection involves a redshift dependent
richness cut. This implies that each object in our cluster sample has an
associated overdensity of passive galaxies, even in the rare cases where the
object is associated with a tSZE noise fluctuation. Instead of assuming a
value of zero we therefore use the average convergence around sources from the
catalog of random lines of sight that share the same richness selection used
for the real catalog. In fact we find a mean convergence for the sources
identified along random lines of sight that is above zero with $2\sigma$
significance and that would correspond to a tSZE-based mass of $M_{500}\approx
6\times 10^{13}$ M⊙.

Performing a detailed observable–mass scaling relation analysis is beyond the
scope of this paper and would require a cluster optimized CMB lensing map
(Saha et al. 2024; Ansarinejad et al. 2024). Nevertheless, it is still
interesting to perform a power law fit to the measurements to estimate their
potential constraining power. In doing so we find that we can constrain the
normalization of the scaling relation to a level of 6.7%, and we find a slope
of $1.15\pm 0.15$. The slope is consistent with the expectation of unity
between $M_{500}$ and $M_{\mathrm{5\,arcmin}}$ inlimit where the cluster is
unresolved and $M_{\mathrm{5\,arcmin}}$ probing the total mass of the cluster.
We note that interpretation of the normalization of the scaling relation is
not straight forward at this point due to the assumptions made on
$M_{\mathrm{5\,arcmin}}$ as well as uncertainties on the scaling relation
connecting tSZE-based mass estimate to true halo mass.

Because the CMB lensing maps used here are not optimised for cluster lensing
studies, the derived values need to be viewed with some caution. Imperfect
modeling of the tSZE signal can bias the CMB lensing maps low, and the $\sim$5
arcmin resolution of the map blurs the cluster signal somewhat. Both effects
reduce the signal-to-noise ratios provided here. A proper cluster focused CMB
lensing analysis of the ACT-DR6 data set will therefore likely yield even
better signal-to-noise than we present here.

##  7 Conclusions

In this work we present the ACT-DR5 MCMF cluster catalog. Using publicly
released mm-wave maps and tSZE cluster finding code from the ACT collaboration
(Naess et al. 2020; Hilton et al. 2021), we create a cluster candidate list of
12,671 sources with SNR¿4. A systematic follow-up of 97% of all these ACT-DR5
candidates using LS-DR10 and WISE imaging data, allows us to exclude the noise
fluctuations from the candidate list and create a cluster catalog with
redshift and richness measurements.

In contrast to the previously published ACT-DR5 catalog (Hilton et al. 2021)
we extract the ACT-D5 candidate list to lower SNR (SNR=3) than our
subsequently imposed catalog threshold of SNR=4. This ensures a stable SNR-
selected sample at SNR¿4 and avoids additional selection effects that are
apparent in the candidate list near the extraction threshold. Using MCMF
derived richnesses of the candidates and along random lines of sight within
the survey region, we characterize the ACT-DR5 candidate list and measure the
contamination by noise fluctuations as a function of SNR threshold SNRmin. We
find excellent agreement between the measured contamination and that expected
from pure Gaussian noise over almost four orders of magnitude in dynamic range
of the candidate sample size. The tSZE candidate list has an initial
contamination of 53%.

We use the MCMF algorithm to construct a cluster catalog with controlled
purity and completeness and define our baseline ACT-DR5 MCMF catalog of 6,237
clusters with a purity of $\sim$90% and completeness with respect to the
originial tSZE selection of 98%. Our cluster catalog contains $\sim 51$%
(2,080) more clusters than the previous ACT-DR5 catalog. The number of high-z
($z>1$) clusters exceeds 700, tripling the cluster sample at high redshifts
compared to the previous catalog constructed with the same mm-wave data.

Overall, in comparison with the original ACT-DR5 cluster catalog (Hilton et
al. 2021), the ACT-DR5 MCMF catalog provides 1) a true SNR-selected sample at
SNR¿4, 2) is $\sim 51$% larger overall and contains three times more z¿1
clusters, 3) enables control over final catalog purity and 4) delivers higher
completeness with respect to the tSZE selection. The last element is
highlighted by the fact that our analysis leads to the confirmation of 327 new
clusters at SNR¿5. This catalog opens new opportunities for cosmological
studies as well as the study of high redshift clusters and their components
such as the ICM, galaxies and AGN.

In addition to the baseline ACT-DR5 MCMF cluster catalog described above, we
explain how subsets with much higher purity (lower contamination) can be
selected to support, for example, a robust cosmological analysis.

We explore the properties of the ACT-DR5 MCMF cluster sample by comparing it
to a deeper tSZE cluster sample from SPT and to the two largest ICM selected
cluster catalogs currently available: the all-sky RASS-MCMF catalog and the
eROSITA eRASS1 western Galactic hemisphere catalog. Matching the ACT-DR5 MCMF
catalog with the tSZE-based but deeper SPTpol 500d survey (Bleem et al. 2024)
allows us to test and confirm the sample purity and completeness of the ACT-
DR5 MCMF candidate list and cluster sample. Over 1,000 ACT-DR5 MCMF clusters
are matched to RASS-MCMF (Klein et al. 2023a), with the matched fraction
rising rapidly to z$\sim$1 due to the differences in the X-ray and tSZE
selections. The matched fraction of RASS-MCMF sources within the ACT-DR5 MCMF
catalog reaches 80-85% at z¿0.5, reflecting the expected contamination of the
X-ray selected cluster sample. Just over 1,300 ACT-DR5 MCMF clusters match to
X-ray sources from eROSITA eRASS1 (Merloni et al. 2024). We find that 43% of
the ACT-DR5 MCMF clusters with eRASS1 counterparts are classified as X-ray
point sources rather than clusters and groups. This underscores the strong
impact of the X-ray extent selection employed for deriving the eRASS1 group
and cluster sample (Bulbul et al. 2024). Our analysis underscores that within
the context of the modest $\sim$0.5 arcmin angular resolution of the eROSITA
sky survey (Predehl et al. 2021), an X-ray extent selected group and cluster
catalog suffers from high incompleteness.

The average weak gravitational lensing convergence signal extracted from ACT-
DR6 CMB lensing maps (Madhavacheril et al. 2024) around ACT-DR5 MCMF clusters
yields a detection significance of 16.4 $\sigma$, the highest value reported
so far for CMB lensing in a galaxy cluster sample. In addition, we find a 4.3
$\sigma$ detection of CMB lensing for the $z>1$ ACT-DR5 MCMF cluster
subsample. With a mean redshift of $z\approx 1.2$, this subsample is currently
the highest redshift cluster sample with a significant CMB lensing detection.
As those significances are based on CMB lensing maps that have not been
optimised for cluster lensing, the significances we present may be seen as
conservative estimates. We present the mean CMB lensing mass within 5 arcmin
($M_{\mathrm{5\,arcmin}}$) to halo mass relation for the sample, where the
halo masses are estimated using a previously calibrated tSZE observable-mass
relation (Hilton et al. 2021). This exercise indicates that
$M_{\mathrm{5\,arcmin}}$ scales as halo mass $M_{500}^{1.15\pm 0.15}$ with the
amplitude constrained with $\sim$6% precision, underscoring the promise of the
ACT-DR6 CMB lensing dataset for calibrating the masses of these ACT-DR5 MCMF
clusters.

As shown in the case of ACT-CL J1424.7+0642, an ACT-DR5 candidate with SNR=7.3
that is not confirmed by MCMF, we do expect to miss clusters– especially at
high redshift– due to the limitations of the available follow-up data. Survey
data from the operating Euclid mission (Euclid Collaboration et al. 2024) in
near infrared will enable significant improvements in cluster followup in the
next years. Independent of that, the current ACT-DR5 MCMF catalog, including
$\sim$700 clusters at z¿1, is already an important step forward, opening a new
large-sample window into the high redshift cluster population and the physical
processes shaping the associated galaxy and AGN populations.

###### Acknowledgements.

This work was initiated in part at the Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-2210452 Our thanks go out
to Matt Hilton for supportive comments and the encouragement to undertake this
project. In addition, we thank the ACT Collaboration for their generosity in
making their excellent dataset and toolkit available for the broader
community. We acknowledge financial support from the MPG Faculty Fellowship
program at MPE and from the Ludwig-Maximilians-Universität (LMU Munich). We
gratefully acknowledge the NSF supported Legacy Survey program (see also
https://www.legacysurvey.org/acknowledgment for detailed acknowledgements).
This research has made use of “Aladin sky atlas” developed at CDS, Strasbourg
Observatory, France (Bonnarel et al. 2000) and topcat (Taylor 2005).

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##  Appendix A Richness transformation factors

Table 2 shows the first eleven entries of the transformation factor between
$g,r,z,w1,w2$-based MCMF richness used in Klein et al. (2023a) and the updated
$g,r,z,w1$-based richness presented in this work. The richness in Klein et al.
(2023a) needs to be multiplied with the factor given in Table 2 to transform
to the $g,r,z,w1$-based richness used for ACT-DR5 MCMF. The full table will be
provided on CDS after acceptance of the paper.

Table 2: Transformation factors between $g,r,z,w1,w2$-based MCMF richness used in Klein et al. (2023a) and $g,r,z,w1$-based richness used in this work for a given redshift z.  z | GRZW1W2_TO_GRZW1
---|---
0.0 | 0.8719359466775638
0.001 | 0.8712321542454213
0.002 | 0.8704275823591408
0.003 | 0.8700663818489788
0.004 | 0.8697660905334137
0.005 | 0.8692880596168335
0.006 | 0.8688464482133338
0.007 | 0.8678339705471051
0.008 | 0.8666583418494209
0.009 | 0.865665771149266
0.01 | 0.8649257435343979
… | …

##  Appendix B Column description of the ACT-DR5 MCMF catalog

In Table 3 we describe the columns included on the released ACT-DR5 MCMF
catalog. All entries related to tSZE-based measurements follow closely those
provided in the original ACT-DR5 catalog presented Hilton et al. (2021) and we
kindly refer the interested reader to that work for more detailed discussion
of those entries. Entries related to redshift, richness and confirmation are
provided from MCMF and correspond to the best result from combining LS DR-10
and unWISE MCMF runs.

Table 3: Column names and description of the ACT-DR5 MCMF catalog Column name | Description
---|---
NAME | Cluster name in the format ACT-CL JHHMM.m ± DDMM
RADEG | R.A. in decimal degrees (J2000) of the SZ detection by ACT
DECDEG | Decl. in decimal degrees (J2000) of the SZ detection by ACT
SNR | Signal-to-noise ratio, optimized over all filter scales
TEMPLATE | Name of the optimal matched filter template for this cluster
TILENAME | Name of the ACT map tile in which the cluster was found
FIXED_SNR | Signal-to-noise ratio at the reference 2.4 arcmin filter scale
Y_C | Central Comptonization parameter ($10^{-4}$) of the optimal matched filter template
ERR_Y_C | Uncertainty on column Y_C
FIXED_Y_C | As Y_C but using measured at the reference filter scale of 2.4 arcmin
FIXED_ERR_Y_C | Uncertainty on column FIXED_Y_C
Z_1_COMB | Redshift of the best optical counterpart
Z_2_COMB | Redshift of the second best counterpart
L_1_COMB | Richness of the best counterpart
L_2_COMB | Richness of the second best counterpart
EL_1_COMB | Uncertainty on column L_1_COMB
EL_2_COMB | Uncertainty on column L_2_COMB
EZ_1_COMB | Uncertainty on column Z_1_COMB
EZ_2_COMB | Uncertainty on column Z_2_COMB
F_CONT_1_COMB |  $f_{\mathrm{cont}}$of best counter part
F_CONT_2_COMB |  $f_{\mathrm{cont}}$of second best counter part
M500_1_COMB | tSZE-based estimate of $M_{\mathrm{500c}}$ in units of $10^{14}M_{\odot}$ using Z_1_COMB
| (corresponds to M500c in Hilton et al. (2021))
M500_1_COMB_ERRPLUS | positive uncertainty to M500_1_COMB
M500_1_COMB_ERRMINUS | negative uncertainty to M500_1_COMB
M500_2_COMB | As M500_1_COMB but for Z_2_COMB
F_CONT_1_COMB_SNR45 |  $f_{\mathrm{cont}}$of best counter part for SNR¿4.5 sample
F_CONT_2_COMB_SNR45 |  $f_{\mathrm{cont}}$of second best counter part for SNR¿4.5 sample
M500_1_COMB_CAL | Similar to M500_1_COMB but rescaled with an calibration factor of $0.71\pm 0.07$
| (corresponds to M500_cal in Hilton et al. (2021))
M500_1_COMB_CAL_ERRMINUS | positive uncertainty to M500_1_COMB_CAL
M500_1_COMB_CAL_ERRPLUS | negative uncertainty to M500_1_COMB_CAL
M500_2_COMB_CAL | As M500_1_COMB_CAL but using Z_2_COMB
Z_SPEC_1 | Spectroscopic redshift for the best optical counter part
Z_SPEC_TYPE | Type of spec-z. 1: literature redshift, 2: ¿2 spec-z galaxies, 3: $\leq 2$ spec-z galaxies