# The CluMPR Galaxy Cluster-Finding Algorithm and DESI Legacy Survey Galaxy
Cluster Catalogue

M. J. Yantovski-Barth 1234, Jeffrey A. Newman 1, Biprateep Dey 1, Brett H.
Andrews 1, Michael Eracleous 5, Jesse Golden-Marx 6, Rongpu Zhou 7

1Department of Physics and Astronomy and Pittsburgh Particle Physics,
Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh,
Pittsburgh, PA 15260, USA
2Department of Physics, Université de Montréal, Complexe des sciences, 1375
Avenue Thérèse-Lavoie-Roux, Montréal (Québec), H2V 0B3, Canada
3Ciela Institute, Université de Montréal, Complexe des sciences, 1375 Avenue
Thérèse-Lavoie-Roux, Montréal (Québec), H2V 0B3, Canada
4Mila, U6666 St-Urbain Street, #200, Montréal (Québec), H2S 3H1, Canada
5Pennsylvania State University, Department of Astronomy and Astrophysics and
Institute for Gravitation and the Cosmos, 525 Davey Laboratory,
The Pennsylvania State University, University Park, PA 16802, USA
6Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China
7Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA E-mail<EMAIL_ADDRESS>0000-0001-5200-4095
0000-0001-8684-2222 0000-0002-5665-7912 0000-0001-8085-5890
0000-0001-8085-5890 0000-0002-6394-045X 0000-0001-5381-4372

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Galaxy clusters enable unique opportunities to study cosmology, dark matter,
galaxy evolution, and strongly-lensed transients. We here present a new
cluster-finding algorithm, CluMPR (Clusters from Masses and Photometric
Redshifts), that exploits photometric redshifts (photo-$z$’s) as well as
photometric stellar mass measurements. CluMPR uses a 2-dimensional binary
search tree to search for overdensities of massive galaxies with similar
redshifts on the sky and then probabilistically assigns cluster membership by
accounting for photo-$z$ uncertainties. We leverage the deep DESI Legacy
Survey grzW1W2 imaging over one-third of the sky to create a catalogue of
$\sim 300,000$ galaxy cluster candidates out to $z=1$, including tabulations
of member galaxies and estimates of each cluster’s total stellar mass.
Compared to other methods, CluMPR is particularly effective at identifying
clusters at the high end of the redshift range considered ($z=0.75$–$1$), with
minimal contamination from low-mass groups. These characteristics make it
ideal for identifying strongly lensed high-redshift supernovae and quasars
that are powerful probes of cosmology, dark matter, and stellar astrophysics.
As an example application of this cluster catalogue, we present a catalogue of
candidate wide-angle strongly-lensed quasars in Appendix C. The five best
candidates identified from this sample include two known lensed quasar systems
and a possible changing-look lensed QSO with SDSS spectroscopy. All code and
catalogues produced in this work are publicly available (see Data
Availability).

###### keywords:

galaxies: clusters: general – catalogues – surveys

††pubyear: 2023††pagerange: The CluMPR Galaxy Cluster-Finding Algorithm and
DESI Legacy Survey Galaxy Cluster Catalogue–LABEL:lastpage

##  1 Introduction

Galaxy clusters are the most massive gravitationally bound structures in the
universe, and therefore they are of great use in cosmology and astrophysics.
One motivation for the creation of catalogues of clusters is to locate likely
areas for observing strongly-lensed time-variable phenomena such as
gravitationally lensed supernovae and quasars (Meyers et al., 2009). The
observation of the time delays between the lensed images of variable
phenomena, coupled with proper modeling of the lensing system, can provide an
independent constraint on the Hubble-Lemaitre parameter (Refsdal, 1964). In
2015, multiple images of a supernova lensed by a galaxy cluster were observed
(Kelly et al., 2015). Recently, the intermediate Palomar Transient Factory and
its successor, the Zwicky Transient Facility (ZTF), have discovered several
additional gravitationally lensed supernovae (Goobar et al., 2017, 2023).
Meanwhile, programs to observe gravitationally lensed quasars are ongoing
(Wong et al., 2020; Napier et al., 2023). Further observations of
gravitationally lensed transients will lead to improved constraints on the
nature of dark matter and the expansion of the universe.

Galaxy cluster catalogues have many other applications. As one example, they
can be used to study the impact of clusters on the evolution of galaxies in
their vicinity (Dressler, 1980). Gravitational magnification of background
galaxies by galaxy clusters enables distant galaxies to be detected and
studied at high resolution (e.g. Bayliss et al., 2014; Livermore et al., 2015;
Johnson et al., 2017; Cornachione et al., 2018; Rivera-Thorsen et al., 2019;
Ivison et al., 2020; Khullar et al., 2021; Bezanson et al., 2022).
Furthermore, they can enable the detection and analysis of merging galaxy
clusters, which are useful for constraining the dark matter scattering cross
section (Wittman et al., 2023). Additionally, measurements of the apparent
abundance of clusters can be a sensitive probe of cosmology (Haiman et al.,
2001; Newman et al., 2002).

A purely photometric galaxy cluster-finding algorithm can take advantage of
the large number of distant sources which can be observed using imaging.
Conveniently, estimates of redshift and stellar mass can be obtained from the
same photometric data. An algorithm which is optimally adapted for photometric
observations will be essential for the upcoming generation of optical and
infrared surveys at facilities such as Rubin Observatory (LSST Science
Collaboration et al., 2009), Euclid Observatory (Euclid Collaboration et al.,
2022), and the Nancy Grace Roman Telescope (Spergel et al., 2015), all of
which will generate immense amounts of photometric data spanning many
passbands.

The redMaPPer algorithm (Rykoff et al., 2014) is an example of a well-tested
photometric galaxy cluster-finding algorithm. The redMaPPer algorithm relies
on a red-sequence model to detect candidate galaxy clusters. Currently
redMaPPer has been applied to data from the Sloan Digital Sky Survey Data
Release 8 (SDSS DR8) (Aihara et al., 2011) and Dark Energy Survey Science
Verification (DES SV) (Dark Energy Survey Collaboration et al., 2016). The
redMaPPer SDSS DR8 catalogue contains 26,311 clusters over the redshift range
0.08<$z$<0.55 (Rykoff et al., 2014), while the redMaPPer DES SV contains 786
clusters from 0.2<$z$<0.9 (Rykoff et al., 2016).

Another well-tested cluster-finding algorithm was developed by Wen et al.
(2009), henceforth WHL. This algorithm has been applied to SDSS (Wen et al.,
2012; Wen & Han, 2015), Subaru Hyper Suprime-Cam (data: Aihara et al. (2018)
and clusters: Wen & Han (2021)), and the Dark Energy Survey (data: The Dark
Energy Survey Collaboration (2005) and clusters: Wen & Han (2022)). The WHL
algorithm, just like the algorithm described in this paper, uses the "nearest-
neighbor" technique to find neighboring galaxies, but it uses a deterministic
slice in redshift rather than the probabilistic approach developed in this
work.

The current largest galaxy cluster catalogue (Zou et al., 2021) was created by
applying the Clustering by Fast Search and Find of Density Peaks (CFSFDP)
algorithm (Rodriguez & Laio, 2014) to the Dark Energy Spectroscopic Instrument
(DESI) Legacy Survey Data Release 8 (Dey et al., 2019). This work identified
540,432 galaxy cluster candidates reaching $z$=1 in the Dark Energy
Spectroscopic Instrument Legacy Survey. Zou et al. (2021) use Sunyaev-
Zel’dovich and X-ray data to calibrate total galaxy masses from richness-mass
relations for this sample. Similarly to WHL, (Zou et al., 2021) use a
deterministic redshift slice, rather than a probabilistic approach.

In this paper we present a new galaxy cluster-finding algorithm, which we call
CluMPR (Clusters from Masses and Photometric Redshifts). This algorithm is
designed to minimize the impact of projection effects caused by uncertainties
in the photometric redshifts (photo-$z$’s) of individual galaxies. In this
work, we optimize the details of this algorithm to enable the selection of a
high-purity sample of high-stellar-mass clusters. These choices aid in
selecting those galaxy cluster candidates which are most likely to cause
strong lensing and help to optimize searches for transients associated with
quiescent galaxies, while simultaneously minimizing contamination of the
sample by lower-mass galaxy groups. Our algorithm determines total cluster
stellar masses probabilistically by incorporating photometric redshift
uncertainties and compensating for redshift-dependent mass incompleteness. We
also provide a catalogue of cluster member galaxies and a catalogue of
candidate gravitationally lensed quasars. Our catalogues are intended to
complement existing galaxy cluster catalogues such as the ones created by Zou
et al. (2021) and Yang et al. (2021) using DESI Legacy Survey data, and can
also be used to compare the results of applying different algorithms to
similar data.

This paper is organised as follows: In Section 2, we describe the data
products we used to generate our galaxy cluster catalogue. Section 3.1
describes the CluMPR algorithm in detail. In Section 3.2, we describe our
process for calibrating galaxy cluster parameters and assigning galaxy cluster
member galaxies. In Section 4, we provide some summary graphs and data
summarizing the characteristics of the CluMPR galaxy cluster catalogue.
Finally, in Section 5 we summarize the results, applications, and potential
future extensions of our work. The Data Availability section provides the
publicly accessible locations of all data and code related to this paper,
including the cluster catalogues and candidate lensed quasar catalogues.
Appendices A and B describe details regarding the calibration of our
algorithm; Appendix C describes a set of candidate gravitationally-lensed
quasars we have identified; and Appendix D describes the columns of all
catalogues assembled in this work. Throughout this paper, we assume a
cosmology with $H_{0}=70\hskip 5.69054ptkm/s/Mpc,\hskip
5.69054pt\Omega_{m}=0.286,\hskip 5.69054pt\Omega_{\lambda}=0.714$ (Bennett et
al., 2014).

##  2 Data

The DESI Legacy Imaging Surveys (Dey et al., 2019) has produced multi-band
images of over a third of the sky (roughly 14,000 deg2) in three optical bands
($g,r,z$) and two infrared bands from WISE ($W1,W2$) (Wright et al., 2010).
The optical component of this survey is composed of three separate programs:
the Beijing-Arizona Sky Survey (BASS) (Zou et al., 2017), the Dark Energy
Camera Legacy Survey (DECaLS) (Flaugher et al., 2015; Dey et al., 2019) with
supplementary data from the Dark Energy Survey (The Dark Energy Survey
Collaboration, 2005), and the Mayall z-band legacy survey (MzLS) (Silva et
al., 2016). A full description of the Legacy Imaging Surveys is available at
Dey et al. (2019). The primary purpose of the DESI Legacy Imaging Surveys is
to obtain targets for spectroscopic follow-up with DESI (Dark Energy
Spectroscopic Instrument, DESI Collaboration et al. (2022)). We obtained
photometric redshifts for our analysis from Zhou et al. (2021) and stellar
mass estimates from Zhou et al. (2023).

We apply several photometric cuts to select galaxies for our analysis with
good measurements of photometric redshifts and stellar masses and with as
little contamination from stars as possible. We restrict ourselves to objects
with

  1. 1.

z magnitude < 21 (photo-z’s are only reliable for z band magnitude < 21, see
Zhou et al. (2021)),

  2. 2.

Either Gaia photometric mean magnitude > 19, Gaia astrometric excess noise >
100.5, or Gaia astrometric excess noise = 0 (in order to remove stars),

  3. 3.

TYPE != PSF (setting TYPE to exclude point spread function objects removes
stars and quasars while keeping the vast majority of galaxies, since bright
galaxies are able to be resolved)

  4. 4.

Photometric redshift > 0.01 (due to photo-z and de-blending unreliability at
low z)

We restrict ourselves to galaxies with a known stellar mass from Zhou et al.
(2023). These are objects which have valid photometry in the $g$, $r$, $z$,
W1, W2 bands and satisfy the stellar rejection cut of
$r-W1>1.75\times(r-z)-1.1$. We limit ourselves to galaxies with a DESI legacy
Survey DR9 bitmask value 0 or 4096, which correspond respectively to unmasked
galaxies and SGA galaxies (Moustakas et al., 2021). The photometric redshifts
and the stellar masses were estimated using the Random Forest algorithm
trained on legacy survey photometry and a carefully vetted training set of
spectroscopic redshifts and stellar population synthesis-based mass estimates.
The errors on the photometric redshifts were generated by perturbing the
original photometry and using the spread in the distributions of the predicted
redshifts as a measure of the error. Detailed descriptions of the methods and
catalogues can be found in Zhou et al. (2021, 2023).

Note: throughout this text, a "sweep" refers to a continuous subsection of
DESI Legacy survey photometry data which is contained in a single file.

##  3 Methods

###  3.1 CluMPR Algorithm Overview

The CluMPR algorithm is comprised of four essential steps:

  1. 1.

Potential Cluster Center Selection: Select galaxies with large stellar mass.

  2. 2.

Membership Probability Calculation: For each potential cluster center galaxy,
evaluate the expected number of neighbouring galaxies within a cylinder
oriented lengthwise along the line of sight ($z$) direction (see Fig. 1). In
the direction of the cylinder’s radius, membership is determined
deterministically using a binary search tree. Along the z direction,
membership is determined probabilistically based on photo-$z$ uncertainties.

  3. 3.

Richness Thresholds: Keep potential cluster center galaxies with a significant
excess of neighbouring galaxies (this is equivalent to defining a galaxy
cluster by setting a minimum richness).

  4. 4.

Aggregation: neighbouring potential cluster center galaxies are aggregated and
the galaxy with the highest local stellar mass is chosen as the cluster
center.

This cylinder-based method is a probabilistic variant of the widely-used
(deterministic) "counts-in-cylinders" technique (Hogg et al., 2004; Blanton et
al., 2006; Kauffmann et al., 2004; Barton et al., 2007; Berrier et al., 2011).
The individual steps of the CluMPR algorithm are explained in greater detail
below.

Figure 1: Search cylinder for CluMPR algorithm, oriented lengthwise along
redshift (line-of-sight) direction and centered on the potential cluster
center galaxy.

####  3.1.1 Potential Cluster Center Selection

Potential cluster center galaxies are chosen from the general pool of galaxies
based on a minimum stellar mass threshold. We set this minimum stellar mass
threshold to be log(stellar mass) > 11.2 [log($M_{\odot}$)]. This mass
threshold was chosen based on visual inspection of galaxy clusters (we chose
to minimize the fraction of low-mass groups and spurious detections in order
to create a high-purity cluster catalogue, but this mass threshold can be
lowered if a lower-purity catalogue is desired).

####  3.1.2 Membership Probability Calculation

For each potential cluster center galaxy chosen in 3.1.1, we count the number
of neighbours in a cylinder oriented lengthwise along the $z$ direction and
centered on the potential cluster center galaxy.

First, galaxies are selected within a fixed physical radius of the cluster
center galaxy (we use three radii: 1 Mpc, 0.5 Mpc, and 0.1 Mpc. The
significance of these radii is explained in the subsequent sections). This
first step is accomplished using a binary search tree; our implementation uses
the cKDTree algorithm from the SciPy Python package (Virtanen et al., 2020).
In order to avoid propagating photo-$z$ errors into the angular positions of
galaxies, we perform this step using angular coordinates. The physical radius
is calculated in angular coordinates using the following equation:

$\mathrm{R_{ang}}(z)=(\mathrm{R_{phys}})\frac{1+z}{D(z)},$ (1)

where $\mathrm{R_{ang}}$ is the radius in angular coordinates (radians),
$\mathrm{R_{phys}}$ is the physical radius in megaparsecs, $z$ is the
redshift, and $D(z)$ is the comoving distance given by

$D(z)=\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})},$ (2)

where $H(z^{\prime})$ is the Hubble-Lemaitre parameter. A graph of Equation 1
for $R_{phys}=1Mpc$ is shown in Figure 2. For each potential cluster center
galaxy, we calculate the radius using Equation 1 and the photo-$z$ of the
cluster center galaxy. To ensure equal area projection of angular coordinates,
we use a sinusoidal (Sanson–Flamsteed) projection. To avoid issues associated
with very small values for angular coordinates, we add an offset of 50 radians
to each coordinate.

Figure 2: Search cylinder radius in angular coordinates (radians) as a
function of redshift. Obtained from Equation 1.

Next, to determine cluster membership along the z-direction (lengthwise along
the cylinder), we apply a redshift cut based on the photo-$z$ of the cluster
center galaxy. The length of this cut (the half-length of the cylinder) is
given by

$L_{z}(z)=2\overline{\sigma_{z}},$ (3)

where $\overline{\sigma_{z}}$ is the interpolated redshift-dependent binned
average of the photo-$z$ error for all galaxies. To train our model for
$\overline{\sigma_{z}}$, we used 100 bins between the lowest and highest
photo-$z$’s in the training sample; we limited $\overline{\sigma_{z}}$ to 0.1
at high z. We used 20 randomly-selected DESI Legacy Survey sweep files (10
from North (BASS/MzLS) and 10 from South (DECaLS)) as our training samples.
The linearly-interpolated trained results for $\overline{\sigma_{z,north}}$
and $\overline{\sigma_{z,south}}$ are shown in Figure 3. $L_{z}$ is
significantly larger than the true redshift extent of a galaxy cluster at all
redshifts, which means that a probabilistic approach to cluster membership
along the z direction is required.

We use $L_{z}$ to determine the probability that a galaxy within the search
cylinder’s radius is a true member of the cluster (that is, the probability
that it is truly within the cylinder). In order to determine this probability,
we use statistical resampling, whereby we create many realizations of each
galaxy’s potential true redshift; these realizations are drawn from a Gaussian
distribution centered on the photo-$z$ of the galaxy and with a standard
deviation equal to the photo-$z$ error for the galaxy. The percentage of these
redshift realizations falling within $L_{z}$ of the cluster center galaxy’s
photo-$z$ is the membership probability.

Figure 3: Interpolated redshift-dependent binned average of the photo-$z$
error as a function of redshift. This is the half-length of the search
cylinder. Note that $L_{z}$ is limited to 0.2. South refers to the DECaLS
survey data, while north refers to the data from BASS and MzLS.

####  3.1.3 Richness Thresholds

In order to identify galaxy clusters among the potential cluster center
galaxies selected in Step 3.1.1, we apply a threshold for the minimum number
of galaxies that must be located within the galaxy’s search radius. This is
equivalent to defining what is a galaxy cluster by setting a minimum richness
threshold. We apply a redshift-dependent threshold to both the 1-Mpc cylinder
and the 0.5-Mpc cylinder; only potential cluster center galaxies that satisfy
both thresholds are kept as galaxy clusters. For the 1-Mpc cylinder, the
threshold we used is

$T_{1}>\overline{N}+1.8\sigma_{N},$ (4)

and for the 0.5-Mpc cylinder, the threshold we used is

$T_{0.5}>\overline{N}+1.2\sigma_{N},$ (5)

where $\overline{N}$ is the redshift-binned average number of galaxies within
a cylinder of that size, and $\sigma_{N}$ is the redshift-binned standard
deviation of the number of galaxies within a cylinder of that size. These
thresholds were chosen based on visual inspection for purity of resulting
galaxy clusters. We assume a Poisson distribution for the number of galaxies
within a cylinder, which allows us to define

$\sigma_{N}=\sqrt{\overline{N}}.$ (6)

The richness threshold is interpolated linearly as a function of $z$; the
results of this interpolation are shown in Figure 4. The training data for
these thresholds are the same as in Section 3.1.2, and the redshift bins are
$z$=0.01 apart with a width of $z$ = 0.05, ranging from the lowest to the
highest photo-$z$ in the training samples.

Figure 4: Interpolated redshift-dependent thresholds for 1-Mpc and 0.5-Mpc
cylinders. South refers to the DECaLS survey data, while north refers to the
data from BASS and MzLS.

####  3.1.4 Aggregation

In order to remove duplicate galaxy clusters (several center galaxy candidates
for one galaxy cluster), we use a similar method to Section 3.1.2, except that
the radius of the aggregation cylinder is 1.5 Mpc, and the $L_{z}$ cutoff is
not done probabilistically.

First, all cluster central galaxies are sorted by (uncorrected, see 3.2.2)
total stellar mass within a 0.5 Mpc radius. Beginning with the cluster central
galaxies with the greatest 0.5 Mpc stellar mass, neighbouring cluster central
galaxies within R=1.5 Mpc and $L_{z}$ are labelled as belonging to the same
cluster. The galaxy with the greatest (uncorrected) total stellar mass within
a 0.1 Mpc radius is then chosen as the cluster center galaxy.

####  3.1.5 Data Cleaning

We apply several flags to our galaxy clusters to remove poor objects. First,
galaxy clusters within 0.285 degrees of the edge of the DESI Legacy Survey
footprint are flagged. We flag clusters near large foreground galaxies from
the Siena Galaxy Atlas (SGA) (Moustakas et al., 2021). Our criteria for the
SGA galaxies are either

  1. 1.

D26 > 1.5,

  2. 2.

R_MAG_SB24 < 13.5, R_MAG_SB24>0, and Z_LEDA <0.05

Some of these flagged objects are false detections (which are mostly due to
failed de-blending of foreground galaxies), while others are real galaxy
clusters with significant foreground contamination which could affect the
photo-$z$ and stellar mass estimates. Finally, we remove all clusters located
in the isolated strips of the DESI Legacy Survey footprint with declination <
-10.3 degrees and 110 < Right ascension (degrees) < 250\. We provide a
catalogue which includes all flagged objects as a supplement to the main
CluMPR catalogue.

###  3.2 Cluster Parameters

Each galaxy cluster in our catalogue is labelled with a unique "gid", which is
created from the cluster center galaxy’s coordinates using the following
formula:

round(RA, 6 decimals) * $10^{16}$ \+ round(DEC + 90, 6 decimals) * $10^{6}$

where RA is right ascension and DEC is declination (both in degrees). Each
galaxy cluster has a richness estimate (number of neighbours) for the 1 Mpc,
0.5 Mpc, and 0.1 Mpc search radii. The "gid" is necessary to identify the
galaxy cluster members described in Section 3.2.3. Additionally, we provide
galaxy cluster parameters for redshift and mass as described below.

####  3.2.1 Probability and Mass Weighting of Cluster Redshift Parameters

For each galaxy cluster we calculate the following redshift parameters (in
addition to providing the photo-$z$ information for the cluster center
galaxy):

  1. 1.

Average photo-$z$ of all galaxy cluster member galaxies

  2. 2.

Average photo-$z$ of all galaxy cluster member galaxies (weighted by
membership probability)

  3. 3.

Average photo-$z$ of all galaxy cluster member galaxies (weighted by galaxy
stellar mass times membership probability)

  4. 4.

Standard error of photo-$z$ of all galaxy cluster member galaxies

  5. 5.

Standard error of photo-$z$ of all galaxy cluster member galaxies (weighted by
membership probability)

  6. 6.

Standard error of photo-$z$ of all galaxy cluster member galaxies (weighted by
galaxy stellar mass times membership probability)

The average photo-$z$ estimates for the clusters are biased high, particularly
for certain galaxy clusters at low z. This is due to systematic errors caused
by the higher probability of a line-of-sight galaxy at high z falling within
the cylinder of the cluster (as high-z galaxies have larger photo-$z$ errors
on average). This bias is further amplified by volume effects: the angular
diameter of the cylinder encloses a larger volume at high z. In this work, we
do not attempt to correct for this systematic error. For this reason, we
recommend using the photo-$z$ of the cluster center galaxy as a relatively
unbiased estimate of the galaxy cluster redshift. Discrepancies between
different estimates for the cluster’s average photo-$z$ are useful for
identifying individual clusters with significant line-of-sight effects which
may affect cluster parameters. The standard error estimates are (to first
order) independent of the systematic error described above, so they can be
used as an approximation for the random error uncertainty in cluster redshift.
If more robust estimates (such as Hodges-Lehmann) for cluster redshift and
uncertainty are desired, they can be calculated from the cluster membership
catalogues (see Section 3.2.3).

####  3.2.2 Cluster Richness and Stellar Mass Estimates

We provide a background-corrected richness estimate which is background-
corrected but not corrected for survey incompleteness. This richness is
calculated by summing the membership probabilities of the galaxy cluster
member galaxies and then subtracting the average galaxy background from this
sum. The calculation of the average galaxy background is described in Appendix
A.

We determine galaxy cluster stellar masses by summing the membership
probability-weighted stellar masses of all cluster member galaxies within the
1 Mpc, 0.5 Mpc, and 0.1 Mpc search radii. We exclude galaxies with a mass
below a threshold described in Appendix B (Equation 9). We then subtract the
average galaxy stellar mass background, which is described in Appendix A.
Finally, we apply a redshift-dependent correction factor which compensates for
redshift-dependent mass incompleteness. This factor is described in Appendix
B; the result is an incompleteness-corrected cluster stellar mass containing
all cluster members above log($M_{\star}/M_{\odot}$) = 10.

####  3.2.3 Cluster Membership

For every galaxy cluster, we have compiled a list of member galaxies (galaxies
whose membership probability > 0). For galaxies which are found to belong to
more than one galaxy cluster, we assign the galaxy cluster for which there is
the highest probability of membership. For each galaxy cluster, the cluster
member galaxies can be identified using the "gid" parameter (see Section 3.2).
The membership probability for the cluster member galaxies provided in the
membership catalogue is not the same as the probability defined in Section
3.1.2. Instead, it is the probability of the member galaxy having a true
redshift equal to the photo-$z$ of the galaxy cluster’s central galaxy,
assuming a Gaussian distribution centered on the member galaxy’s redshift with
a standard deviation equal to that member galaxy’s photo-$z$ error.

Figure 5: Examples of galaxy clusters found by the CluMPR algorithm in the
DESI Legacy Survey imaging. The redshifts of these clusters range from $z$ =
0.305 (left column, third row) to $z$ = 0.707 (right column, second row).
Figure 6: Density of galaxy clusters found by the CluMPR algorithm in the DESI
Legacy Survey on the sky.

##  4 Results

The CluMPR algorithm has returned 309115 candidate galaxy clusters from
0.1<$z$<1\. Galaxy clusters with $z$<0.1 are excluded as our algorithm is not
optimised for those objects, which are frequently contaminated by poorly
deblended foreground galaxies. Figure 5 shows a few examples of galaxy
clusters from our catalogue at various redshifts. The distribution of our
catalogue’s galaxy clusters on the sky is shown in Figure 6. Figure 7 shows
the redshift and stellar mass distribution of CluMPR clusters. For
0.1<$z$<0.3, the sample is volume-limited; above $z$=0.3, the sample is
limited by incompleteness. The stellar mass correction factor causes the
stellar mass distribution to remain relatively flat across z. Low-mass galaxy
clusters are excluded by our thresholds and incompleteness at high z, while
high mass galaxy clusters are volume-limited. The rise in high-stellar-mass
clusters at $z$<0.3 indicates that the catalogue is volume-limited for those
redshifts; the decline in high-stellar-mass clusters at z > 0.6 may be due to
limits imposed by structure formation (relations between our stellar mass
estimates and total halo mass must be established to verify this cause).

Figure 7: 1-Mpc total stellar mass vs photo-$z$ for galaxy clusters found by
the CluMPR algorithm in the DESI Legacy Survey imaging. Cluster stellar masses
have been background-subtracted and corrected for incompleteness with Stellar
Mass Correction Factor. The top histogram shows the redshift distribution of
CluMPR clusters with the redshift distributions of Zou et al. (2021) (black)
and redMaPPer (red) overplotted. The right side histogram shows the stellar
mass distribution of CluMPR clusters.

###  4.1 Cross-matching with Zou et al. 2021

The galaxy cluster catalogue described in Zou et al. (2021) was compiled using
the same DESI Legacy Survey data as our catalogue, so cross-matching the
catalogues yields a direct comparison of our implementation of CluMPR with the
implementation of the CFSFDP algorithm by Zou et al. (2021). For cross-
matching, we apply a simple 2-dimensional cross-matching which ignores line-
of-sight coincidence. The radius for determining a match was 5 arcminutes,
which is sufficient to handle most differences in the choice of cluster center
galaxy while minimizing spurious matches. We found that this radius worked
reasonably well at all redshifts.

Overall, Zou et al. (2021) found 85.4% of the galaxy clusters identified by
CluMPR above $z$=0.1. Figure 8 shows the stellar mass within 1 Mpc for CluMPR
galaxy clusters as a function of redshift; color indicates the fraction of
CluMPR clusters found by Zou et al. (2021) Between $z$=0.1 and $z$=0.75, there
is broad agreement between CluMPR and Zou et al. (2021) on high-richness
objects, but Zou et al. (2021) miss some low-richness candidates identified by
CluMPR. At $z$>0.75, Zou et al. (2021) miss a large fraction of clusters
identified by CluMPR, including high-mass clusters. Visual inspection of the
high-$z$ clusters missed by Zou et al. (2021) indicates a high fraction of
true galaxy clusters. The likely reason for Zou et al. (2021) missing these
objects is that they use a minimum richness of 10 which is not corrected for
incompleteness; at high redshift, many clusters will have a low observed
richness. This effect is best illustrated by Figure 9, which shows the 1-Mpc
CluMPR richness as a function of redshift; color indicates the fraction of
CluMPR galaxies found by Zou et al. (2021). At low-$z$, the CluMPR richness
threshold for a cluster is relatively high, but it drops rapidly with $z$; by
$z~{}0.75$, the richness threshold falls below 10, and the fraction of galaxy
clusters recovered by Zou et al. (2021) falls to 0.5. CluMPR is able to
maintain relatively high purity at high z despite the low richness threshold
by using high-stellar mass galaxies as a reliable starting point for finding
galaxy clusters, which works even in the absence of a large number of cluster
member galaxies.

Figure 8: 1-Mpc total stellar mass vs photo-$z$ for galaxy clusters found by
the CluMPR algorithm in the DESI Legacy Survey imaging (same as Figure 7).
Color represents the fraction of CluMPR clusters found by Zou et al. (2021)
The catalogue by Zou et al. (2021) has broad agreement on high mass clusters.
Figure 9: 1-Mpc richness (expected number of neighbours) vs photo-$z$ for
galaxy clusters found by the CluMPR algorithm in the DESI Legacy Survey
imaging. Color represents the fraction of CluMPR clusters found by Zou et al.
(2021). The lower bound on richness is set by the thresholds in Section 3.1.3
(compare Figure 4).

Overall, CluMPR finds 58.9% of the galaxy clusters identified by Zou et al.
(2021); however, this percentage increases to 87.1% if one limits the Zou et
al. (2021) sample to clusters with richness > 30\. Figure 10 shows 1 Mpc
richness for Zou et al. (2021) galaxy clusters as a function of redshift;
color indicates the fraction of CluMPR clusters found by Zou et al. (2021) As
in Figure 8, there is broad agreement on high-richness objects. At all
redshifts, CluMPR does not find most of the low-richness galaxy clusters
identified by Zou et al. (2021) Thus, it is likely that the CluMPR sample is
better optimised for detecting high-mass clusters with high purity. An
implementation of CluMPR with lower richness thresholds would likely yield
results more similar (for $z<0.75$) to Zou et al. (2021).

Figure 10: 1-Mpc richness (expected number of neighbours) vs photo-$z$ for
galaxy clusters found by Zou et al. (2021) in the DESI Legacy Survey imaging.
Color represents the fraction of CluMPR clusters found by Zou et al. (2021).

###  4.2 Cross-matching with SDSS redMaPPer and SDSS spectroscopy

Both CluMPR and Zou recover most of the objects from the SDSS redMaPPer
cluster catalogue. Using the 5 arcminute radius cross-matching described in
Section 4.1, CluMPR recovers 95% of redMaPPer clusters, while Zou et al.
(2021) recover 99% of redMaPPer clusters. Figure 11 shows the 1-Mpc cluster
stellar mass vs redMaPPer richness. As redMaPPer richness is known to be
correlated with total cluster mass (Simet et al., 2017), the observed
correltion between CluMPR stellar mass and redMaPPer richness is a strong
indicator that CluMPR stellar masses are correlated with total cluster mass.

We provide spectroscopic redshifts for galaxies whose spectra have been
obtained by the Sloan Digital Sky Survey (SDSS) (Blanton et al., 2017). Figure
12 shows the residuals between SDSS spectra and three photometric estimates
for cluster redshift from CluMPR, as well as the residuals for redshift
estimates by redMaPPer and Zou et al. (2021) The bias for average redshifts
obtained by CluMPR is caused by line-of-sight galaxies falling within the
cluster; this bias is increased when photo-$z$ errors are systematically over-
estimated, as this creates an artificially long cylinder and increases the
likelihood for galaxies to fall within the cylinder. Zhou et al. (2021)
indicate that photo-$z$ errors are indeed overestimated, so this bias would be
reduced with an improved redshift uncertainty quantification. However, our
algorithm is still able to reduce the bias by weighting the cluster members by
mass*probability (shown, middle right) or just probability (not shown). This
indicates that our stellar mass and richness estimates are likely also
improved by our probabilistic approach.

Figure 11: 1-Mpc cluster stellar mass vs redMaPPer richness. The redMaPPer
catalogue does not have clusters below richness = 20. The cluster stellar mass
has been background-subtracted and corrected for incompleteness with Stellar
Mass Correction Factor. There is a clear correlation between CluMPR cluster
stellar mass and redMaPPer richness, which indicates that CluMPR stellar
masses are likely correlated with total cluster mass as well. Figure 12:
Residuals between SDSS spectroscopic redshifts and various estimates of
cluster redshift from photo-$z$’s. Plotted intervals contain 68.2% of
clusters. Our catalogue offers several redshift estimates; the photo-$z$ of
the cluster central galaxy is least biased (bottom right). The plotted
weighted average (middle right) uses mass*probability weighting, which reduces
bias compared to an unweighted average redshift (top right). We include a
catalogue of the member galaxies for all clusters, which can be used to
calculate alternative estimates of the cluster redshift. All plotted redshift
estimates underestimate spectroscopic redshift near the high-$z$ limit due to
sharply decreasing galaxy counts at increasing redshifts (attenuation bias,
see Freeman et al. 2009), which leads to a redshift estimate for the cluster
central galaxy which is biased low.

##  5 Conclusions

We have used the CluMPR algorithm to identify over 300,000 galaxy clusters
using the DESI Legacy Survey imaging data. This algorithm provides
incompleteness-corrected stellar masses for each cluster as well as cluster
membership probabilities for potential member galaxies. Our catalogue shows
good correspondence to previous catalogues such as Zou et al. (2021) and SDSS
redMaPPer (Rykoff et al., 2014) in overlapping regimes, and visual inspection
of the detected galaxy clusters indicates that our methods have produced a
high-purity sample.

CluMPR has several advantages over non-probabilistic cluster-finding
algorithms such as the WHL algorithm (Wen et al., 2009) or the CFSFDP
algorithm (Rodriguez & Laio, 2014) as used by Zou et al. (2021). As an
example, for strong lensing searches, CluMPR enables us to select a sample of
high-stellar-mass clusters that simultaneously has high purity. Higher-
redshift lensing searches are further enhanced as CluMPR is able to more
easily identify galaxy clusters at higher $z$ ($z>0.75$), a regime where
smaller number of cluster galaxies will be brighter than magnitude thresholds
and hence apparent richnesses will be lower. However, by using the presence of
a single massive galaxy ($log(M_{\star})>11.2$) as a signpost to reliably
anchor cluster assignments, CluMPR is able to still identify clusters with
high purity even when the apparent richness drops. In contrast to red-
sequence-based algorithms such as redMaPPer (Rykoff et al., 2014), CluMPR
includes all potential galaxy cluster members in its search, which leads to a
more complete member sample. This difference will become more relevant for
deeper surveys, as we would expect a higher fraction of cluster members which
are not on the red sequence at high $z$ (Schechter, 1976; Dahle et al., 2013).
Finally, our probabilistic approach to cluster membership reduces systematic
errors in cluster properties caused by the incorporation of infalling
foreground or background galaxies.

The CluMPR DESI Legacy Survey catalogue can now be used by astronomers to
search for strongly-lensed time-variable phenomena (such as supernovae and
quasars) as well as other applications. As an example of how this catalogue
may be used, we provide a supplementary catalogue of candidate lensed quasars
in Appendix C. The five best candidates identified from this sample include
two known systems and a possible changing-look lensed QSO with SDSS
spectroscopy. The CluMPR DESI Legacy Survey catalogue can also be used for
galaxy evolution studies and to search for merging galaxy clusters.

The CluMPR sample could be made more useful for cosmological studies by
calibrating the relationship between the cluster stellar masses we have
measured and total cluster masses as derived from Sunyaev-Zel’dovich, X-ray,
or weak lensing measurements. Future datasets from Rubin Observatory (LSST
Science Collaboration et al., 2009), Euclid Observatory (Euclid Collaboration
et al., 2022), the Nancy Grace Roman Telescope (Spergel et al., 2015), and
other next-generation surveys will provide another application for the CluMPR
algorithm. These projects should provide higher-precision photometric
redshifts and useful stellar mass measurements for objects several magnitudes
fainter than the Legacy Survey objects studied here. As a result, these deep
imaging surveys could extend CluMPR’s reach to $z=2$ and beyond.

## Acknowledgements

M. J. Yantovski-Barth would like to acknowledge his PhD advisors Yashar
Hezaveh and Laurence Perreault-Levasseur for their patience and advice as he
finalised the results presented in this paper. Additionally, we thank Gourav
Khullar for a helpful discussion of cluster-lensed quasars. The efforts of M.
J. Yantovski-Barth were supported by three grants from the NASA Pennsylvania
Space Grant Consortium in 2020 and 2021, as well as by the bourse d’excellence
du centenaire from the Fonds du Centenaire of the Department of Physics at the
Université de Montréal. The efforts of J. A. Newman were supported by grant
DE-SC0007914 from the U.S. Department of Energy Office of Science, Office of
High Energy Physics.

The Legacy Surveys consist of three individual and complementary projects: the
Dark Energy Camera Legacy Survey (DECaLS; Proposal ID #2014B-0404; PIs: David
Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID
#2015A-0801; PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy
Survey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS and MzLS
together include data obtained, respectively, at the Blanco telescope, Cerro
Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok telescope, Steward
Observatory, University of Arizona; and the Mayall telescope, Kitt Peak
National Observatory, NOIRLab. The Legacy Surveys project is honored to be
permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a
mountain with particular significance to the Tohono O’odham Nation.

NOIRLab is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation.

This project used data obtained with the Dark Energy Camera (DECam), which was
constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES
Projects has been provided by the U.S. Department of Energy, the U.S. National
Science Foundation, the Ministry of Science and Education of Spain, the
Science and Technology Facilities Council of the United Kingdom, the Higher
Education Funding Council for England, the National Center for Supercomputing
Applications at the University of Illinois at Urbana-Champaign, the Kavli
Institute of Cosmological Physics at the University of Chicago, Center for
Cosmology and Astro-Particle Physics at the Ohio State University, the
Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M
University, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho
de Amparo, Financiadora de Estudos e Projetos, Fundacao Carlos Chagas Filho de
Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de
Desenvolvimento Cientifico e Tecnologico and the Ministerio da Ciencia,
Tecnologia e Inovacao, the Deutsche Forschungsgemeinschaft and the
Collaborating Institutions in the Dark Energy Survey. The Collaborating
Institutions are Argonne National Laboratory, the University of California at
Santa Cruz, the University of Cambridge, Centro de Investigaciones
Energeticas, Medioambientales y Tecnologicas-Madrid, the University of
Chicago, University College London, the DES-Brazil Consortium, the University
of Edinburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, Fermi
National Accelerator Laboratory, the University of Illinois at Urbana-
Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), the Institut de
Fisica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig
Maximilians Universitat Munchen and the associated Excellence Cluster
Universe, the University of Michigan, NSF’s NOIRLab, the University of
Nottingham, the Ohio State University, the University of Pennsylvania, the
University of Portsmouth, SLAC National Accelerator Laboratory, Stanford
University, the University of Sussex, and Texas A&M University.

BASS is a key project of the Telescope Access Program (TAP), which has been
funded by the National Astronomical Observatories of China, the Chinese
Academy of Sciences (the Strategic Priority Research Program “The Emergence of
Cosmological Structures” Grant # XDB09000000), and the Special Fund for
Astronomy from the Ministry of Finance. The BASS is also supported by the
External Cooperation Program of Chinese Academy of Sciences (Grant #
114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant #
11433005).

The Legacy Survey team makes use of data products from the Near-Earth Object
Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet
Propulsion Laboratory/California Institute of Technology. NEOWISE is funded by
the National Aeronautics and Space Administration.

The Legacy Surveys imaging of the DESI footprint is supported by the Director,
Office of Science, Office of High Energy Physics of the U.S. Department of
Energy under Contract No. DE-AC02-05CH1123, by the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility under the
same contract; and by the U.S. National Science Foundation, Division of
Astronomical Sciences under Contract No. AST-0950945 to NOAO.

The Photometric Redshifts for the Legacy Surveys (PRLS) catalogue used in this
paper was produced thanks to funding from the U.S. Department of Energy Office
of Science, Office of High Energy Physics via grant DE-SC0007914.

The Siena Galaxy Atlas was made possible by funding support from the U.S.
Department of Energy, Office of Science, Office of High Energy Physics under
Award Number DE-SC0020086 and from the National Science Foundation under grant
AST-1616414.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS acknowledges support and resources from the
Center for High-Performance Computing at the University of Utah. The SDSS web
site is www.sdss4.org.

SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## Data Availability

The CluMPR galaxy cluster catalogue and cluster member galaxy catalogue are
currently publicly available at DOI:10.5281/zenodo.8157752. The candidate
lensed quasar catalogue is publicly available atDOI:10.5281/zenodo.8164870.
Descriptions of each catalogue’s columns can be found in Appendix D.

These catalogues are also available to NERSC users at
`global/cfs/cdirs/desi/users/mjyb16/CLUMPR_DESI_2023`. The main catalogue can
be found in the folder `clean_data`, the extended catalogue including flagged
objects can be found in the folder `unclean_data`, catalogues of cluster
member galaxies can be found in the folder `members`, and the candidate lensed
quasar catalogues can be found in the folder `lensed_qso_candidates`.

Code from this project can be found on GitHub.

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##  Appendix A Galaxy Count and Stellar Mass Backgrounds

To determine the backgrounds, we run the CluMPR algorithm as described in 3.1,
except that instead of choosing initial points to be the locations of massive
galaxies as in 3.1.1, we use random points on the sky. We then calculate the
number of galaxies falling within the search cylinder as a function of
redshift and take the median across all random points to get a redshift-
dependent median galaxy background. A graph of the background galaxy counts
(with the median overplotted) is shown in Figure 13, and a graph of background
stellar masses (with median overplotted) is shown in Figure 14. We perform
this process separately for the North (BASS/MzLS) and the South (DECaLS)
surveys; the resulting median background galaxy counts and median background
stellar masses are shown in Figure 15 and Figure 16, respectively.

Figure 13: Number of galaxies within 1 Mpc at random sky pointings vs
photometric redshift. The rise in galaxies above $z$=1.2 is likely caused by
misclassified stars, but they do not affect the median which remains at 0. The
data in this graph are for the South (DECaLS) portion of the DESI Legacy
Survey; the exact same process was applied for the North (BASS/MzLS) portion
and yielded similar results. A comparison between the resulting median
backgrounds for North and South are shown in Figure 15. Figure 14: 1-Mpc total
stellar log(mass) at random sky pointings vs photometric redshift. Pointings
which do not contain any galaxies (stellar mass = 0) are plotted at log(mass)
= 9.2. For galaxies at $z$>1.2, stellar masses become unreliable. The data in
this graph are for the North (BASS/MzLS) portion of the DESI Legacy Survey;
the exact same process was applied for the South (DECaLS) portion and yielded
similar results. A comparison between the resulting median backgrounds for
North and South are shown in Figure 16. Figure 15: Median background galaxy
counts within 1 Mpc vs photometric redshift. The differences in backgrounds
between North and South are due to differing survey characteristics and
different models for redshift for each survey. Figure 16: Median background
1-Mpc total stellar log(mass) vs photometric redshift. For galaxies at
$z$>1.2, stellar masses become unreliable. The differences in mass backgrounds
between North and South are due to differing survey characteristics and
different models for stellar mass and redshift for each survey.

##  Appendix B Stellar Mass Incompleteness Correction

The galaxy sample used by our algorithm is magnitude-limited (see 2). Low-
stellar-mass galaxies are thus excluded from our catalogue at a rate which
increases with redshift. Thus, in order to obtain a redshift-independent total
stellar mass estimate for each cluster, we must correct for the stellar mass
incompleteness of our data. To correct for stellar mass incompleteness, we
apply a redshift-dependent scaling factor to the galaxy cluster stellar
masses. This factor is determined by creating a stellar mass model based on
the Schechter function (Schechter, 1976):

$\Phi(M)dM=\Phi^{\star}\left(\frac{M}{M_{\star}}\right)^{\alpha}exp\left(\frac{-M}{M_{\star}}\right)\frac{dM}{M_{\star}},$
(7)

where $M$ is the galaxy stellar mass, $M_{\star}$ is the characteristic galaxy
stellar mass where the power-law form of the function ends, $\Phi(M)$ is the
number density of galaxies with a mass $M$, and $\Phi_{\star}$ is the
normalization factor. We fit this model to the high-mass end of the mass
distribution histograms of galaxies observed to be in a cluster. We are able
to use this method since we observe the histograms at all redshifts to roughly
follow the Schechter function at high masses, though the number count
approaches 0 for low mass galaxies (an example of a stellar mass histogram is
shown in Figure 18).

As the first step of building our model, we use the Schechter function to
approximate the true mass distribution of galaxies, given by the double
Schechter function (Davidzon et al., 2017)

$\Phi(M)dM=\left(\Phi_{1}^{\star}\left(\frac{M}{M_{\star}}\right)^{\alpha_{1}}+\Phi_{2}^{\star}\left(\frac{M}{M_{\star}}\right)^{\alpha_{2}}\right)exp\left(\frac{-M}{M_{\star}}\right)\frac{dM}{M_{\star}}$
(8)

. This approximation is necessary in order to reduce the number of parameters
to be fitted. A demonstration of the sufficiency of a single Schechter
function to fit to a double Schechter function above $log(M)=10$ is shown in
Figure 17. We fit $\alpha$ in our model to the double Schechter for the values
shown in Table 1. $M_{\star}$ and $\Phi(M)$ in our model are fitted to the
stellar mass distribution of galaxies in our clusters.

We fit $M_{\star}$ and $\Phi^{\star}$ to the high-mass end of the galaxy mass
distribution histogram for several redshift bins (as an example, see Figure
18). We use 22 redshift bins between 0.025<$z$<1.025 with bin width $z$=0.05.
The high-mass ends of each histogram are located by fitting a piecewise linear
function of the form

$Limit\hskip 2.84526ptlog(M_{\odot})=minimum(a*z+b,11.2)$ (9)

to the peaks of the histograms, where z is redshift bin. The maximum limit of
$log(M_{odot})=11.2$ is required in order to ensure that the mass of the
cluster central galaxy is not excluded from the mass calculation in the main
algorithm (see section 3.2). We find the best-fit values to be a = 1.36202 and
b = 9.96855. This model, along with an example of actual histogram peak
locations for a small sample of galaxies, is shown in Figure 19.

The contribution of missing low-mass galaxies to the total stellar mass is
determined by taking the ratio between the integral of Equation 7 from
$log(M)=10$ to infinity and the integral of Equation 7 from the mass limit 9
to infinity. The latter integral represents the incomplete stellar mass
calculated by the main algorithm. The ratio between the integrals is thus

$factor=M_{tot}/M{inc},$ (10)

where $M_{tot}$ is the total stellar mass above $log(M)=10$ and $M{inc}$ is
the incomplete (observed) stellar mass. Multiplying the observed galaxy
cluster stellar masses by this redshift-dependent mass correction factor
allows us to estimate the true stellar masses above $log(M)=10$ within the 1
Mpc, 0.5 Mpc, and 0.1 Mpc search radii. We find that a quadratic equation is a
good fit for the observed log(stellar mass) correction factor as a function of
redshift:

$log(factor(z))=az^{2}+bz+c,$ (11)

where a = 0.19347, b = 0.23392, and c = 0.00694. A plot of the stellar mass
correction factor as a function of redshift is shown in Figure 20; we trained
this model on a dataset consisting of 150 sweep files randomly chosen from
DESI Legacy Survey North and South. The stellar mass correction factor is
provided for all galaxy clusters in our catalogue so that the incomplete
stellar mass can be easily retrieved.

Figure 17: Comparison of the Double Schechter function with the best-fit
Schechter function as an approximation. The best-fit Schechter is a very good
approximation to the high-mass end of the Double Schechter function, down to
approximately $log(M_{\odot})=10$. The parameters for the Double Schechter in
this example are $\Phi_{1}^{\star}=0.098,\hskip
5.69054pt\alpha_{1}=-1.3,\hskip 5.69054pt\Phi_{2}^{\star}=1.58,\hskip
5.69054pt\alpha_{2}=-0.39,\hskip 5.69054ptlog(M_{\star})=10.83$. The
parameters for the best-fit Schechter function are $\Phi^{\star}=1.663,\hskip
5.69054pt\alpha=-0.459,\hskip 5.69054ptlog(M_{\star})=11.043$. Assuming
different parameters for the Double Schechter function results in similarly
well-fitting Schechter approximations. Figure 18: An example of fitting a
Schechter function to the observed galaxy stellar mass histogram for the
redshift bin centered on z = 0.66. The histogram follows a Schechter function
above log(mass) = 11, but below this mass the number counts decrease due to
incompleteness. The Stellar Mass Correction Factor (Equation 10) is the ratio
between two integrals along the green line (Schechter function): the total
mass is the integral above log(mass) = 10, and the incomplete mass is the
integral above the peak of the histogram (in this case, above log(mass) = 11).
This ratio is designed to compensate for the missing galaxy counts for
galaxies with masses below log(mass) = 11. Figure 19: The histogram peak-
finding function used to determine the log(mass) above which stellar mass
histograms should roughly follow a Schechter function. The red line is the
location of histogram peaks for one example sweep; there is good
correspondence between the model and the example. In practice, we use many
sweeps simultaneously when training the Stellar Mass Correction Factor
(Equation 10), which reduces the random variation in histogram peak location.
Figure 20: Stellar Mass Correction Factor (Equation 10) as a function of
redshift. Mass incompleteness increases with redshift (higher mass limit), so
the correction factor increases correspondingly. Table 1: Schechter parameters
used to derive Stellar Mass Correction Factor Double Schechter Parameters
---
$Redshift$ | $log(M_{\odot})$ | $\alpha_{1}$ | $\Phi_{1}(e5)$ | $\alpha_{2}$ | $\Phi_{2}(e5)$
$0.1<z<0.5$ | $10.97$ | $-0.75$ | $79.53$ | $-2.04$ | $0.18$
$0.5<z<0.8$ | $10.90$ | $-0.50$ | $92.06$ | $-2.14$ | $0.09$
$0.8<z<1.1$ | $10.87$ | $-0.44$ | $102.63$ | $-2.14$ | $0.07$
Best-fit Schechter Parameters |  |
$Redshift$ | $M_{\odot}$ | $\alpha$ | $\Phi$ |  |
$0.1<z<0.5$ | varies with $z$ | $-0.7535$ | varies with $z$ |  |
$0.5<z<0.8$ | varies with $z$ | $-0.5015$ | varies with $z$ |  |
$0.8<z<1.1$ | varies with $z$ | $-0.441$ | varies with $z$ |  |

##  Appendix C Gravitationally Lensed Quasar Candidates

As a supplement to our galaxy cluster catalogue and galaxy cluster member
catalogue, we provide a catalogue of candidate gravitationally lensed quasars.
Gravitationally lensed quasars are valuable as probes of the Hubble parameter
for the expansion of the universe (Suyu et al., 2017). As data, we used
quasars from the DESI Bright and Dark survey target selection (Myers et al.,
2023). We compiled the candidate lensed quasar catalogue by counting the
number of quasars within the Einstein radius of each galaxy cluster identified
by CluMPR. We use two Einstein radii: one corresponds to wide angle lensing by
the entire cluster ($M=10^{15}M_{\odot}$), and the other corresponds to
lensing by the core of the cluster ($M=0.25*10^{15}M_{\odot}$). Since there is
contamination of QSO targets by stars, particularly in dense star fields, we
remove galaxy clusters which have counts within 3 Einstein radii >= 6 times
the counts within one Einstein radius. We rate candidates based on how many of
the quasars have similar colors in bands g-r, g-z, and r-W1. If at least two
quasars are within 1 magnitude of each other in all 3 colors, we rate the
candidate at Grade C. If at least three quasars are within 1 magnitude of each
other in all 3 colors, we rate the the candidate at Grade B. If either 4
quasars or two combinations of 3 quasars are within 1 magnitude of each other
in all 3 colors, we rate the candidate as Grade A. Spectroscopic follow-up of
the lensed quasar candidates and cluster lens modeling will enable
confirmation of true lensed quasars.

###  C.1 Recovery of known cluster-lensed quasars

To date, six widely-separated gravitationally lensed quasars (corresponding to
a cluster-scale lens mass) have been discovered (Inada et al., 2003, 2006;
Dahle et al., 2013; Shu et al., 2018, 2019; Martinez et al., 2023). Of the six
known cluster-lensed quasars, we recover three: SDSS J1004+4112 (Grade A core-
lensed), SDSS J1029+2623 (Grade C core-lensed), and COOL J0542-2125 (Grade A
core-lensed). We do not recover SDSSJ2222+2745 or SDSS J1326+4806 due to only
one of the quasar images being in the DESI target catalogues as a QSO, and we
do not recover SDSS J0909+4449 due to the lensing cluster being absent from
both the CluMPR and Zou et al. (2021) catalogues (it is more accurately
described as a galaxy group).

###  C.2 New Grade A core-lensed candidates

Of the five core-lensed candidates ranked Grade A, two are known lensed
quasars. We discuss the remaining Grade A core-lensed candidates below.

####  C.2.1 CluMPR J1278885160133433856: A candidate lensed changing-look
quasar

The candidate lensed quasars located near the cluster CluMPR
J1278885160133433856 are shown in Figure 21. The galaxy cluster has an SDSS
spectroscopic redshift of $z$=0.525, and the cluster center galaxy is located
at RA = 127.8885, DEC = 43.4339 (J2000). Two of the candidate lensed images,
QSO 1 and QSO 2, are confirmed quasars with SDSS redshifts of $z=1.3056\pm
0.0002$ and $z=1.3021\pm 0.0004$, respectively. The spectra of these quasars
are shown in Figure 22, with zoomed-in views of the C III] and Mg II broad
emission lines (Figures 23 and 24, respectively). The alignment between
emission lines in the two spectra indicates that QSO 1 and QSO 2 are at the
same or nearly the same redshift ($z\approx 1.3$), but the noise in the
spectra and the absence of narrow emisison lines makes evaluating the accuracy
of the assigned SDSS spectroscopic redshifts challenging. There are notable
differences between the spectra: QSO 2 appears brighter and has a redder color
(negative slope) compared to QSO 1, and the broad emission lines appear more
prominent in QSO 2 than in QSO 1.

We offer two potential explanations for the differences in the spectra: either
these are two distinct quasars that happen to be at very similar redshifts, or
these are two images of the same highly-variable ("changing-look") quasar.
Changing-look quasars are quasars whose luminosity and colors (i.e., continuum
slopes) change quickly by a large factor over the course of hundreds of days
to several years (e.g. Bian et al., 2012; MacLeod et al., 2016, 2019). The
luminosity changes are accompanied by the appearance and disappearance of
broad emission lines (e.g. LaMassa et al., 2015; Runnoe et al., 2016; Ruan et
al., 2016). Changing-look quasars have recently been discovered at redshifts
beyond 1 (Ross et al., 2020; Guo et al., 2020). The variability timescale of
changing look quasars is on the same order as the time delays between lensed
images for a cluster-scale lens, so it is possible that the difference in
spectra between QSO 1 and QSO 2 is due to a gravitational time delay.

To evaluate the changing look quasar interpretation further, we consider the
following:

  * •

According to the formal uncertainties, the redshifts of the two quasars ($\sim
30$–$50~{}{\rm km~{}s}^{-1}$) differ by $\approx 8\sigma$. However, the
detailed discussion of redshift determination methods used by the SDSS by Lyke
et al. (2020, see their Section 4.6 and Figure A1) suggests a much larger
redshift uncertainty of $\sim 300~{}{\rm km~{}s}^{-1}$. Viewed in this light,
the quasar redshifts differ only by $1.5\sigma$.

  * •

We measured the rest-frame equivalent widths of the broad emission lines and
found them to be ${\rm EW(Mg\;II)}=34\pm 2$ Å and ${\rm EW(C\;III])}=33\pm 3$
Å in QSO 1 and ${\rm EW(Mg\;II)}=26\pm 1$ Å and ${\rm EW(C\;III])}=25\pm 1$ Å
in QSO 2. These measurements reflect the Baldwin effect (anti-correlation
between EW and luminosity; e.g., Green et al., 2001, and references therein)
and follow the behavior seen in extremely variable quasars by Ren et al.
(2022).

  * •

A caveat to the above is that, the relative luminosities of the two quasars
cannot be readily evaluated as the observed brightness of quasar images
depends on the magnification of the lens (Bauer et al., 2011), so the true
luminosity of the quasar images cannot be estimated without a well-constrained
mass model for the lens.

The above considerations suggest that the the lensed changing-look quasar
hypothesis remains plausible for QSO 1 and QSO 2. To test this hypothesis
further we suggest the following.

  * •

Improving the redshift determination. – This can be done by obtaining spectra
with higher signal-to-noise ratio in the near-IR or sub-mm bands to detect
narrow emission lines. Spectra in the wavelength range 7800–8600 Å can catch
the narrow [Ne V] $\lambda\lambda$3346,3426 and [O II]
$\lambda\lambda$3726,3729 doublets. The wavelengths of these lines and their
relative strengths (the [Ne V]/[O II] ratio) can test the hypothesis that QSO
1 and QSO 2 are lensed images of the same quasar. In the same spirit,
spectroscopy in the J band can catch the H$\beta$ line and [O III]
$\lambda\lambda$4959,5007 doublet and allow a similar test. Alternatively,
spectroscopy with ALMA may detect the CO(2-1) line from the quasar host
galaxies (rest frequency of 230.5 GHz) that will allow for an exquisite
redshift determination.

  * •

Monitoring for spectroscopic variability. – If the spectra of QSO 1 and QSO 2
do indeed reflect a change in the state of a single quasar, we should expect
that one of the two spectra will change to match the other in the next few
years. Therefore, the hypothesis can be tested by repeated spectroscopic
observations of the two quasars.

There are two more possible quasar images (C1 and C2) which have been selected
for DESI spectroscopy. Finally, there is a dim object resembling a lensing
arc. Further spectroscopy and deeper imaging are necessary to determine
whether these objects are related to QSO 1 and QSO 2.

Figure 21: Candidate gravitationally lensed quasar system near galaxy cluster
CluMPR J1278885160133433856 (cluster center galaxy in green). QSO 1 and QSO 2
are confirmed quasars with SDSS redshifts of 1.306 and 1.302, respectively.
Objects C1 and C2 are QSO targets for DESI. C3 is a very faint object which
resembles a lensing arc. Figure 22: SDSS spectra for QSO 1 and QSO 2 in the
candidate gravitationally lensed quasar system near galaxy cluster CluMPR
J1278885160133433856. The broad emission lines (CIII] and MgII) are redshifted
to very similar values near $z=1.3$, but the slopes of the spectra are very
different. This indicates that these are either two different quasars located
at the same redshift, or two images of the same highly variable (changing-
look) quasar. Figure 23: Close-up of the CIII] broad emission line. QSO 1 has
an SDSS redshift estimate of 1.306, while QSO 2 has an SDSS redshift estimate
of 1.302. The spectrum for QSO 2 has been shifted down by a constant of 3 flux
units in order to improve legibility. The black lines are cubic spline fits to
the spectra. Figure 24: Close-up of the MgII broad emission line. QSO 1 has an
SDSS redshift estimate of 1.306, while QSO 2 has an SDSS redshift estimate of
1.302. The black lines are cubic spline fits to the spectra.

####  C.2.2 CluMPR J2439482900096790528

The candidate lensed quasars located near the cluster CluMPR
J2439482900096790528 are shown in Figure 25. The galaxy cluster’s central
galaxy is located at the photometric redshift $z=0.333$ with RA = 243.9483,
DEC = 6.7904. There are no publicly available spectra for the candidate
quasars at this time, but they are part of the DESI dark time targets.
Spectroscopy and deeper imaging are necessary to determine whether this system
is a quasar-lensing cluster.

Figure 25: Candidate gravitationally lensed quasar system near galaxy cluster
CluMPR J2439482900096790528 (cluster center galaxy in green). Objects in white
circles are QSO targets for DESI, and there is a possibility that they are all
images of the same gravitationally lensed quasar.

####  C.2.3 CluMPR J2420498300113853952

The candidate lensed quasars located near the cluster CluMPR
J2420498300113853952 are shown in Figure 26. The galaxy cluster’s central
galaxy is located at the photometric redshift $z=0.750$ with RA = 242.0498,
DEC = 23.8542. One of the candidate quasars (QSO 1) has SDSS spectroscopy with
a spectroscopic redshift of 2.205. There are no publicly available spectra for
the other candidate quasars at this time, but they are part of the DESI dark
time targets. Spectroscopy and deeper imaging are necessary to determine
whether this system is a quasar-lensing cluster.

Figure 26: Candidate gravitationally lensed quasar system near galaxy cluster
CluMPR J2420498300113853952 (cluster center galaxy in green). Objects in white
circles are QSO targets for DESI, and there is a possibility that some of them
are images of the same gravitationally lensed quasar (it is relatively
unlikely that they are all the same quasar due to their configuration). QSO 1
is a confirmed quasar with SDSS spectroscopic redshift of 2.205.

##  Appendix D catalogue descriptions

Below we provide descriptions of the column names and contents for the CluMPR
DESI Legacy Survey catalogue, associated cluster member catalogue, and the
candidate gravitationally lensed quasar catalogues. The public location of the
catalogues can be found in the Data Availability) section.

Table 2: Main CluMPR Galaxy Cluster catalogue Columns | Description
---|---
Index | Sequential number in catalogue
RA_central | Right ascension of central cluster galaxy (deg)
DEC_central | Declination of central cluster galaxy (deg)
z_spec | Spectroscopic redshift of central cluster galaxy (-1 if unavailable)
z_spec_err | Estimated error in spectroscopic redshift of central cluster galaxy (-1 if unavailable)
z_median_central | Median photometric redshift of central cluster galaxy
z_average_no_wt | Mean photometric redshift of all cluster galaxies
z_average_prob | Mean photometric redshift of all cluster galaxies, weighted by membership probability
z_average_mass_prob | Mean photometric redshift of all cluster galaxies, weighted by membership probability times mass
z_std_central | Std. deviation of photometric redshift of central cluster galaxy
z_std_no_wt | Std. deviation of photometric redshift of all cluster galaxies
z_std_prob | Std. deviation of photometric redshift of all cluster galaxies, weighted by membership probability
z_std_mass_prob | Std. deviation of photometric redshift of all cluster galaxies, weighted by membership probability times mass
gid | Unique identifier of the central cluster galaxy. See Section 3.2 for definition.
mass_central | Log(stellar mass) of the central cluster galaxy
cluster_mass_onempc | Log(stellar mass) of the cluster within a 1-Mpc physical radius, corrected for incompleteness and galaxy background
cluster_mass_halfmpc | Log(stellar mass) of the cluster within a 0.5-Mpc physical radius, corrected for incompleteness and galaxy background
cluster_mass_tenthmpc | Log(stellar mass) of the cluster within a 0.1-Mpc physical radius, corrected for incompleteness and galaxy background
richness_onempc | Expectation number of observed galaxies in cluster within a 1-Mpc physical radius, corrected for galaxy background
richness_halfmpc | Expectation number of observed galaxies in cluster within 0.5-Mpc physical radius, corrected for galaxy background
richness_tenthmpc | Expectation number of observed galaxies in cluster within 0.1-Mpc physical radius, corrected for galaxy background
Table 3: Extended CluMPR Galaxy Cluster catalogue Columns | Description
---|---
Index | Sequential number in catalogue
RA_central | Right ascension of central cluster galaxy (deg)
DEC_central | Declination of central cluster galaxy (deg)
z_spec | Spectroscopic redshift of central cluster galaxy (-1 if unavailable)
z_spec_err | Estimated error in spectroscopic redshift of central cluster galaxy (-1 if unavailable)
z_median_central | Median photometric redshift of central cluster galaxy
z_average_no_wt | Mean photometric redshift of all cluster galaxies
z_average_prob | Mean photometric redshift of all cluster galaxies, weighted by membership probability
z_average_mass_prob | Mean photometric redshift of all cluster galaxies, weighted by membership probability times mass
z_std_central | Std. deviation of photometric redshift of central cluster galaxy
z_std_no_wt | Std. deviation of photometric redshift of all cluster galaxies
z_std_prob | Std. deviation of photometric redshift of all cluster galaxies, weighted by membership probability
z_std_mass_prob | Std. deviation of photometric redshift of all cluster galaxies, weighted by membership probability times mass
RELEASE | Integer denoting the camera and filter set used for the central galaxy, which will be unique for a given processing run of the data,
| from Tractor catalogues
BRICKID | A unique Brick ID for the central galaxy, from Tractor catalogues
OBJID | catalogue object number within this brick for the central galaxy, from Tractor catalogues
MASKBITS | Bitwise mask indicating that the central galaxy touches a pixel in the coadd/*/*/*maskbits* maps, from Tractor catalogues
gid | Unique identifier of the central cluster galaxy. See Section 3.2 for definition.
mass_central | Log(stellar mass) of the central cluster galaxy
cluster_mass_onempc | Log(stellar mass) of the cluster within a 1-Mpc physical radius, corrected for incompleteness and galaxy background
cluster_mass_halfmpc | Log(stellar mass) of the cluster within a 0.5-Mpc physical radius, corrected for incompleteness and galaxy background
cluster_mass_tenthmpc | Log(stellar mass) of the cluster within a 0.1-Mpc physical radius, corrected for incompleteness and galaxy background
mass_bkgd_onempc | Mass background (which has been subtracted from cluster_mass_onempc) for one-mpc radius
mass_bkgd_halfmpc | Mass background (which has been subtracted from cluster_mass_onempc) for one-mpc radius
mass_bkgd_tenthmpc | Mass background (which has been subtracted from cluster_mass_onempc) for one-mpc radius
correction_factor | Correction factor for mass incompleteness
neighbours_onempc | Expectation number of observed galaxies in cluster within a 1-Mpc physical radius, not corrected for galaxy background
neighbours_halfmpc | Expectation number of observed galaxies in cluster within a 0.5-Mpc physical radius, not corrected for galaxy background
neighbours_tenthmpc | Expectation number of observed galaxies in cluster within a 0.1-Mpc physical radius, not corrected for galaxy background
richness_onempc | Expectation number of observed galaxies in cluster within a 1-Mpc physical radius, corrected for galaxy background
richness_halfmpc | Expectation number of observed galaxies in cluster within 0.5-Mpc physical radius, corrected for galaxy background
richness_tenthmpc | Expectation number of observed galaxies in cluster within 0.1-Mpc physical radius, corrected for galaxy background
flag_foreground | 1 indicates the presence of a major foreground galaxy (cluster parameters may be unreliable in such cases).
| Value 0 should indicate good clusters.
edge_mask | 0 indicates that the cluster is located close enough to the edge of the footprint that parameters may be affected.
| Value 1 should indicate good clusters.
flag_footprint | 1 indicates a galaxy cluster which is located in an isolated area of the footprint (these are excluded from the main dataset).
Table 4: CluMPR Galaxy Cluster Membership catalogue Columns | Description
---|---
Galaxy | Unique galaxy id, constructed as gid above. See Section 3.2 for definition
Cluster | Unique galaxy id of cluster central galaxy
Galaxy Mass | Stellar mass of galaxy
Galaxy Redshift | Photometric redshift of galaxy
Cluster Redshift | Median photometric redshift of cluster central galaxy
Galaxy Redshift Uncertainty | Uncertainty in the photometric redshift of this galaxy
Cluster Membership Probability | Cluster membership probability of this galaxy
Table 5: Candidate Lensed Quasar catalogues Columns | Description
---|---
lens_RA_central | Right ascension of central cluster galaxy (deg)
lens_DEC_central | Declination of central cluster galaxy (deg)
lens_z_spec | Spectroscopic redshift of central cluster galaxy (-1 if unavailable)
lens_z_spec_err | Estimated error in spectroscopic redshift of central cluster galaxy (-1 if unavailable)
lens_z_median_central | Median photometric redshift of central cluster galaxy
cluster_mass_onempc | Log(stellar mass) of the cluster within a 1-Mpc physical radius, corrected for incompleteness and galaxy background
lens_gid | Unique identifier of the central cluster galaxy. See Section 3.2 for definition.
QSO_RA | Right ascension of each QSO (deg)
QSO_DEC | Declination of each QSO (deg)
g_r | g-r color of each QSO
g_z | g-z color of each QSO
r_w1 | r-w1 color of each QSO