# Satellite Content and Halo Mass of Galaxy Clusters: Comparison between Red-
Sequence and Halo-based Optical Cluster Finders

Jesse B. Golden-Marx,1 Ying Zu,1,2,3 Jiaqi Wang1, Hekun Li1, Jun Zhang1,2,3,
Xiaohu Yang1,2,3,4
1Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China
2Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao
Tong University, Shanghai 200240, China
3Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of
Education, Shanghai Jiao Tong University, Shanghai 200240, China
4Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240,
China E-mail<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS>
(May 2023)

###### Abstract

Cluster cosmology depends critically on how optical clusters are selected from
imaging surveys. We compare the conditional luminosity function (CLF) and weak
lensing halo masses between two different cluster samples at fixed richness,
detected within the same volume ($0.1{<}z{<}0.34$) using the red-sequence and
halo-based methods. After calibrating our CLF deprojection method against mock
galaxy samples, we measure the 3D CLFs by cross-correlating clusters with SDSS
photometric galaxies. As expected, the CLFs of red-sequence and halo-based
finders exhibit redder and bluer populations, respectively. The red-sequence
clusters have a flat distribution of red galaxies at the faint end, while the
halo-based clusters host a decreasing faint red and a boosted blue population
at the bright end. By comparing subsamples of clusters that have a match
between the two catalogues to those without matches, we discover that the CLF
shape is mainly caused by the different cluster centroiding. However, the
average weak lensing halo mass between the matched and non-matched clusters
are consistent with each other in either cluster sample for halos with
$\lambda>30$ ($M_{h}^{\mathrm{WL}}{>}1.5\times 10^{14}h^{-1}M_{\odot}$). Since
the colour preferences of the two cluster finders are almost orthogonal, such
a consistency indicates that the scatter in the mass-richness relation of
either cluster sample is close to random. Therefore, while the choice of how
optical clusters are identified impacts the satellite content, our result
suggests that it should not introduce strong systematic biases in cluster
cosmology, except for the $\lambda<30$ regime.

###### keywords:

galaxies: evolution — galaxies: formation — galaxies: abundances — galaxies:
statistics — cosmology: large-scale structure of Universe — gravitational
lensing: weak

††pubyear: 2023††pagerange: Satellite Content and Halo Mass of Galaxy
Clusters: Comparison between Red-Sequence and Halo-based Optical Cluster
Finders–References

##  1 Introduction

Galaxy clusters are one of the most sensitive probes of dark energy and cosmic
growth (see section 6 of Weinberg et al., 2013, for a comprehensive review),
but cluster cosmology in the optical is currently in a state of minor crisis.
The most recent cosmological constraints using galaxy clusters (e.g., Abbott
et al., 2020; Costanzi et al., 2021; Park et al., 2021) favored lower
$\Omega_{m}$ and higher $\sigma_{8}$ compared against the constraints inferred
from the cosmic microwave background (Planck Collaboration et al., 2020),
large-scale structures traced by galaxies (Alam et al., 2017; Beutler et al.,
2011; Abbott et al., 2018), and clusters detected with Sunyaev-Zeldovich
(Bocquet et al., 2019) and X-ray (Mantz et al., 2015; Chiu et al., 2022)
observations. To resolve this discrepancy, much work has been done to
determine which systematic uncertainties may be responsible. For instance,
projection effects are believed to boost the richness (i.e., the number of
galaxies above some luminosity threshold) and the large-scale weak lensing
(WL) signal of optical clusters (Zu et al., 2017; Busch & White, 2017), and
therefore potentially bias cosmological constraints (e.g., Abbott et al.,
2020; Sunayama et al., 2020; Myles et al., 2021; Wu et al., 2022; Zeng et al.,
2022).

Although the projection effects in optical cluster cosmology are problematic,
they may mask a larger overall issue. The methods currently used to optically
identify galaxy clusters are imperfect and have existing biases that must be
taken into account. For the past twenty years, the two primary methods used to
identify optical clusters from photometric surveys are the red sequence-based
cluster finder, which uses overdensities of “red and dead” galaxies on the sky
to select cluster candidates (e.g., Gladders & Yee, 2000, 2005; Miller et al.,
2005; Rykoff et al., 2012, 2014, 2016) and the halo-based method (Yang et al.,
2005, 2007, 2021), which instead relies on the overdensities of all galaxies
above some magnitude threshold, including the blue galaxies that are often
excluded by the red sequence (also see Wen et al., 2012; Tinker, 2021; Zou et
al., 2021; Tinker, 2022). Although these methods greatly differ, both have
been used to successfully identify large samples of cluster candidates with
reasonably high completeness and purity.

Both methods of identifying galaxy clusters unavoidably have their own
preferences for the satellite content. Identifying clusters using the red
sequence works well at $z{<}1$ where clusters have long been observationally
dominated by strong red sequences (Gladders & Yee, 2000). Under the assumption
that the red sequence forms as a result of halo quenching (i.e., the
probability of a galaxy being quiescent depends only on halo mass (Birnboim &
Dekel, 2003; Zu & Mandelbaum, 2015)) red sequence-based cluster finders should
be efficient at sifting out the most massive systems without introducing any
biases due to the colour selection. However, red-sequence clusters may be more
strongly impacted by halo assembly bias (e.g., Gao et al., 2005; Gao & White,
2007; Jing et al., 2007; Dalal et al., 2008), as these overdensities of
quiescent galaxies may be older clusters with higher concentrations that more
efficiently formed at early times (Zu et al., 2021, 2022). Additionally, this
method does not account for the blue galaxies that observations have found
exist abundantly within the cluster environment, which may impact the
observational properties of the cluster.

Initially developed with the spectroscopic galaxies in mind (Yang et al.,
2005, 2007), the halo-based method has been shown to work across a wide
redshift range and have a high completeness of clusters even when applied to
imaging data (Yang et al., 2021). This method evaluates the membership
probabilities for all galaxies regardless of colour, and relies on the total
luminosity of all member galaxies as a proxy of halo mass, which may result in
weighting blue galaxies more favorably than the red ones. The primary bias for
this method results from photometric-redshift estimates, which are poorer for
blue galaxies, leading to contamination in the membership due to stronger
projection effects. In the current work, we exploit the fact that the
detection preferences associated with the red-sequence and halo-based finders
are roughly orthogonal, and investigate the differences not just between the
two cluster catalogues, but also between clusters that are co-detected by both
finders and those detected only once.

One primary statistical measurement used to characterize the galaxy content
within clusters is the conditional luminosity function (CLF; Yang et al.,
2003), defined as the luminosity function of all the cluster member galaxies
at fixed halo mass. Traditionally, the CLFs of photometric clusters are
measured by directly counting the number of member galaxy candidates
identified by the cluster finder, sometimes weighted by the inferred
membership probabilities (e.g., To et al., 2020). However, such a direct
method is incapable of recovering the true underlying CLFs of the haloes,
primarily due to biases in the colour selection and background contamination
caused by projection effects (Zu et al., 2017). Instead of using the
membership provided by cluster catalogues, one could take advantage of the
excellent photometric redshifts of clusters, and measure the CLFs by cross-
correlating clusters with galaxies from imaging data (Hansen et al., 2009; Lan
et al., 2016; Meng et al., 2022). This approach accounts for the CLF
contribution of all galaxies regardless of colour, and the contamination from
interloper galaxies from distant, uncorrelated structures can be statistically
removed by subtracting the measurement around random points on the sky.
Moreover, the contribution of galaxies that are outside the 3D radius of the
cluster but in a physically correlated structure along the line-of-sight
(e.g., in the filaments that connect to the cluster) can be modelled
theoretically (Lan et al., 2016). As a result, instead of measuring the CLF
properties of individual clusters, this cross-correlation or “stacking”
approach measures the parameters associated with ensembles of clusters that
are stacked together based on a given richness. In this work, we follow this
stacking methodology to explore the different CLFs of clusters detected by
red-sequence and halo-based cluster finders.

Robust measurements of CLFs depend critically on the correct identification of
central galaxies, which is unfortunately challenging to achieve for current
optical cluster finders using just the imaging data (e.g., Johnston et al.,
2007; George et al., 2012; Hollowood et al., 2019). Miscentring would result
in a partial overlap between the measurement aperture and the true cluster
region, leading to an underestimation of the CLF amplitudes and, more
problematic, biases in the shape of the CLFs due to the clumpiness of
satellite distribution within clusters. This issue can be statistically
remedied in cluster weak lensing studies by introducing a miscentring kernel
that can be calibrated against the X-ray centres (Zhang et al., 2019a) and
convolved with a theoretical dark matter density profile (Navarro et al.,
1997). However, the correct underlying CLFs cannot be reconstructed with the
same procedure, primarily because the true CLF model is unknown (especially at
the faint end). In this work, for the first time we investigate the impact of
miscentring on the measured shape of the CLF, by examining the CLFs of the
same clusters but centred differently by the two cluster finders.

Discrepancies in the galaxy content between clusters detected with different
cluster finders do not necessarily introduce significant biases in
cosmological analyses. For a volume-complete cluster sample above some true
halo mass $M_{h}$, the red sequence-based and halo-based cluster finders may
both detect all the clusters but assign them richnesses with differing amounts
of logarithmic scatter at fixed halo mass. In the ideal scenario, such a
scatter in the mass-richness relation could be purely statistical in nature,
and thus can be self-calibrated when constraining cosmology with clusters
(Rozo et al., 2010; Tinker et al., 2012; Zu et al., 2014). However, if the
cluster finder preferentially assigns higher richnesses to haloes with a
particular CLF, such as having a higher red fraction due to older formation
time, or higher blue fraction due to stronger projection effects within dense
environments, the scatter is no longer random and would systematically bias
the constraint on cosmological parameters. In this work, we take advantage of
the orthogonal CLF preferences between the red-sequence and halo-based
finders, and divide each catalogue into two subsamples based on whether the
clusters are also detected by the other cluster finder. Therefore, by
examining whether there exists any significant discrepancy in the average halo
mass measured from weak lensing between the two subsamples, we can better
understand the nature of the scatter in the mass-richness relation for
different cluster finders.

The structure of the paper is as follows. In Section 2, we describe the
photometric galaxy catalogue and cluster samples selected for this analysis.
In Section 3, we describe how the CLFs are measured in both samples and
compare the galaxy content revealed by their CLFs. In Section 4 we describe
the cluster weak lensing measurements and compare the weak lensing halo masses
between the matched vs. non-matched clusters within each sample. We conclude
the paper and look to the future in Section 5. Throughout this analysis, we
assume a flat $\Lambda$CDM characterized by Planck Collaboration et al. (2020)
cosmology with $\Omega_{M}$=0.315 and $h$=0.674. Additionally, we note that
unless explicitly stated, we are using $\log\,M_{h}$ for the logarithm with
base 10 of halo mass in unit of $h^{-1}M_{\odot}$, defined by the halo radius
within which the mean halo density is $\Delta{=}200$ times the mean density of
the Universe. Finally, all distances measured are co-moving distances in units
of $h^{-1}\mathrm{Mpc}$.

##  2 Data

###  2.1 The SDSS Photometric Galaxy Sample

In this analysis, we measure the CLF by statistically counting the galaxies
from a photometric catalogue constructed using Sloan Digital Sky Survey Data
Release 16 (SDSS DR16; York et al., 2000; Ahumada et al., 2020). We select all
galaxies with clean photometry (CLEAN=1) with an SDSS r-band Galactic
extinction-corrected model-magnitude greater than 21
(modelmag_r$-$extinction_r<21.0) for completeness. In total, this yields a
sample of about 47 million galaxies within a sky coverage of approximately
8500 deg2.

For measuring the CLF contribution from any given cluster at redshift, $z$, we
estimate the absolute magnitude Mr for each galaxy assuming it is at the
redshift of that cluster as $M_{r}{=}m_{r}{-}\mathrm{DM}(z)$, where $m_{r}$ is
the apparent magnitude after being corrected for Galactic extinction and
DM($z$) is the distance modulus at the redshift of the cluster. Interloper
galaxies that are not physically associated or correlated with the cluster,
hence assigned the wrong redshift, will be statistically removed by the
background subtraction (as discussed in Section 3.3).

We note that because our goal for this analysis is a comparison of the CLF
measured in clusters identified in the Rykoff et al. and Yang et al.
catalogues within a relatively narrow redshift range ($0.1{<}z{<}0.34$), we
choose not to account for the K-correction, which systematically shifts the
CLFs by the same amount for both measurements. Moreover, the K-correction to
$z=0.1$ is of the order of $\sim$0.1 magnitudes across our redshift range and
given our average bin size (0.4 magnitudes) when measuring the CLF would only
have a minimal impact on our presented results.

Figure 1: The richness distributions of the Yang (thin green curve) and
redMaPPer (thick magenta curve) samples. The richness estimates of Yang
clusters are measured by counting the total number of member galaxies brighter
than $M_{r}{=}{-}20.4$ and enclosed within an $1h^{-1}\mathrm{Mpc}$ aperture,
and then abundance-matched to the richness distribution of redMaPPer clusters,
hence the consistency between the two curves. Vertical dashed lines indicate
the edges of the four richness bins adopted in our analysis.

###  2.2 Red Sequenced-based Cluster Sample: redMaPPer

The red sequenced-based cluster catalogue we employ is the SDSS redMaPPer
v6.3111https://risa.stanford.edu/redmapper/ cluster catalogue (Rykoff et al.,
2014) derived by applying a red-sequence-based photometric cluster finding
algorithm to the SDSS DR8 imaging (Aihara et al., 2011). Briefly, redMaPPer
iteratively self-trains a model of red-sequence galaxies calibrated by an
input spectroscopic galaxy sample, and then attempts to grow a galaxy cluster
centred on every photometric galaxy. Once a galaxy cluster has been identified
by the matched-filters, the algorithm iteratively solves for a photometric
redshift based on the calibrated red-sequence model, and recentres the
clusters about the best candidates for central galaxies.

Therefore, each redMaPPer cluster is a conglomerate of red-sequence galaxies
on the sky, with each galaxy assigned a membership probability
$p_{\mathrm{mem}}$ and a probability of being the central $p_{\mathrm{cen}}$.
For each cluster, the richness $\lambda_{\mathrm{RM}}$ is computed by summing
the $p_{\mathrm{mem}}$ of all member galaxy candidates, and roughly
corresponds to the number of red-sequence satellite galaxies brighter than
$0.2\,L_{*}$ within an aperture of ${\sim}1\,h^{-1}\mathrm{Mpc}$ (with a weak
dependence on $\lambda_{\mathrm{RM}}$). At $\lambda_{\mathrm{RM}}{\geq}20$,
the SDSS redMaPPer cluster catalogue is approximately volume-complete up to
$z{\simeq}0.34$ (Groenewald et al., 2017), with cluster photometric redshift
uncertainties as small as $\delta(z)=0.006/(1+z)$ (Rykoff et al., 2014; Rozo
et al., 2015).

We use the $\lambda{\geq}20$ sample between $z{=}0.1$ and $0.34$. For this
analysis, our initial sample of redMaPPer clusters consists of the 9226
clusters between the redshift range $0.10<z<0.34$. For each cluster we
identify the most likely central galaxy candidate provided by redMaPPer and
treat its location as the centre for the CLF measurement.

###  2.3 Halo-based Cluster Sample of Yang et al. (2021)

The halo-based clusters used in this analysis are detected by the Yang et al.
(2021)222https://gax.sjtu.edu.cn/data/DESI.html cluster finder (hereafter
referred to as the Yang clusters). This cluster finder is an extension of the
spectroscopic group finder previously developed by Yang et al. (2005, 2007)
and has been updated to incorporate both photometric and spectroscopic
information. The photometric data used to construct the Yang et al. (2021)
catalogue comes from the DESI Legacy Imaging Survey DR9 (Dey et al., 2019),
which covers the redshift range $0.0<z\leq 1.0$.

We briefly summarise the halo-based methodology used in the Yang cluster
finder below and refer interested readers to Yang et al. (2021) for the
technical details. The halo-based algorithm attempts to build groups or
clusters centred on every galaxy within the legacy imaging data by assigning
initial membership to galaxies using a friends-of-friends approach assuming
some small linking length (Yang et al., 2007). Using the total luminosity
within these initial overdensities, the mass-to-light ratio of each cluster is
determined via abundance matching, which allows each tentative cluster to be
assigned a halo mass. Cluster membership is then determined assuming the
galaxy distribution follows the same phase-space distribution as the dark
matter particles. This requirement selects all member galaxies above a
background level within $R_{180}$ from the luminosity-weighted cluster centre.
This process is then repeated iteratively (3-4 times) until the mass-to-light
ratios converge. Using mock galaxy samples, Yang et al. (2021) demonstrated
that this adaptive halo-based algorithm can reliably detect systems with mass
above a few ${\times}10^{13}h^{-1}M_{\odot}$ with a purity higher than 90%.

Figure 2: The matched fraction as a function of richness for the Yang (left
panel) and redMaPPer (right panel) cluster samples, using three different
search radii of 0.3$R_{h}$ (circles), 0.5$R_{h}$ (triangles), and 0.7$R_{h}$
(squares). We adopt the cross-matching results using the 0.7$R_{h}$ search
radius as our fiducial split between matched vs. non-matched clusters in the
paper.

The Yang catalogue provides an abundance-matched halo mass,
$M_{180}^{L_{\mathrm{tot}}}$, defined with $\Delta{=}180$ instead of $200$ and
based on the total luminosity of galaxies in the cluster (with DESI z-band
apparent magnitude ${z_{\rm DESI}}$<21.0 within $R_{180}$). Yang et al. (2021)
showed that the scatter about the $M_{180}^{L_{\mathrm{tot}}}$-$M_{h}$
relation is about 0.2 dex above $10^{14}h^{-1}M_{\odot}$, similar to that in
redMaPPer. For centring, instead of computing $p_{\mathrm{cen}}$ as done in
redMaPPer, the Yang catalogue adopts the luminosity-weighted centre as the
fiducial centre of a cluster. For this analysis, we do not cover the entire
redshift range of the Yang catalogue, but instead use data only from the
redshift range over which redMaPPer is complete as well as only those clusters
with $\log M_{180}^{L_{\mathrm{tot}}}{\geq}14.0$. These cuts limit the Yang
sample to 29014 clusters. We also note that although this catalogue was
developed using the DESI imaging, we measure the CLF for Yang clusters using
the same SDSS photometric data as for redMaPPer.

###  2.4 redMaPPer and Yang Clusters within the Same Volume

The purpose of our CLF analysis is to compare the galaxy populations within
haloes identified by either redMaPPer or the Yang cluster finder. However,
these two cluster catalogues are identified using different galaxy surveys and
cover different spatial and redshift regions. To account for this, we only
select clusters in the overlapping footprint between the two samples and
within the redshift range $0.1<z<0.34$ where both samples are reasonably
complete at the high mass end. The DESI legacy imaging footprint is larger
than the region covered by SDSS, so much of the data removed is from the Yang
catalogue. In total, the number of redMaPPer clusters was reduced to 8770, a
reduction of less than 5%. In contrast, the number of Yang clusters was
reduced to 15063, a reduction of 48.1%. Moreover, to accurately compute the
covering fraction of any regions on the sky, we require that each cluster be
within the SDSS BOSS spectroscopic footprint, for which convenient window and
mask polygon files333https://data.sdss.org//sas/dr12/boss/lss/ are available.
Removing clusters outside the BOSS window (minus the bright star masks)
function slightly reduces the number of available clusters in each footprint.
The redMaPPer cluster sample then consists of 8300 clusters and the Yang
sample consists of 14155 clusters, a reduction of 5.4% and 6.0%, respectively.

To facilitate the comparison between these two cluster samples, we need to
account for the difference in the aperture sizes adopted by the two cluster
finders when defining their respective halo mass proxy. In particular,
redMaPPer provides a richness ($\lambda_{\mathrm{RM}}$) measured within
roughly 1$h^{-1}\mathrm{Mpc}$, while the $M_{180}^{L_{\mathrm{tot}}}$ of Yang
clusters are estimated within $R_{180}$, which is usually larger than
1$h^{-1}\mathrm{Mpc}$. Therefore, for each Yang cluster, we redefine a new
richness $\lambda^{\mathrm{new}}_{\mathrm{Y}}$ by measuring the total number
of member galaxies (with equal weights) with SDSS r-band absolute magnitudes
$M_{r}{<}{-}20.4$, enclosed within a fixed $1h^{-1}\mathrm{Mpc}$ aperture
centred on the luminosity-weighted centre. The member galaxies are directly
adopted from the Yang membership catalogue, which is roughly volume-complete
above $M_{r}{>}-20.4$ between $0.1{<}z{<}0.34$.

To further bring the two richness measurements
($\lambda^{\mathrm{new}}_{\mathrm{Y}}$ vs. $\lambda_{\mathrm{RM}}$) to the
same scale, we perform a standard abundance matching (assuming zero scatter)
to ensure that the richness distribution of the top 8300 Yang clusters (ranked
by $\lambda^{\mathrm{new}}_{\mathrm{Y}}$) is the same as the
$\lambda_{\mathrm{RM}}$ distribution of the 8300 redMaPPer clusters. After
abundance matching, we reassign a richness value $\lambda_{\mathrm{Y}}$ to
each of the top 8300 Yang clusters, and adopt $\lambda_{\mathrm{Y}}$ for the
Yang clusters for the remainder of the paper. Figure 1 shows the richness
distributions of the redMaPPer (thick magenta curves) and Yang (thin green
curves) clusters within the same volume, using the $\lambda_{\mathrm{RM}}$ and
$\lambda_{\mathrm{Y}}$ estimates, respectively. Going forward, unless
otherwise noted we will refer to both richnesses as $\lambda$.

Using the two abundance matched samples, we follow the approach used in Lan et
al. (2016) and only select clusters into our final analysis samples with a
covering fraction within 1$h^{-1}\mathrm{Mpc}$ from the cluster centre higher
than 95%. This cut is needed to remove clusters located at the edges of the
survey as well as those near regions that are masked by bright foreground
stars. This criteria reduces the number of clusters in our final sample to
7582 for redMaPPer and 7562 for Yang, reductions of 8.7% and 8.9%,
respectively.

Finally, for the CLF and weak lensing analyses we divide each cluster sample
into four subsamples based on $\lambda$, indicated by the vertical dashed
lines in Figure 1 — the four richness bins are $\lambda=[20,\,30]$,
$[30,\,40]$, $[40,\,55]$, and $[55,\,200]$, respectively.

Figure 3: Validation of the CLF deprojection method using mock photometric
galaxies in three different halo mass bins. In each panel, the black solid
line is the input CLF from Yang et al. (2008), open circles are the 2D
projected CLF measured from the stacking method, and filled circles are the 3D
deprojected CLF from the 2D results, using the magnitude-dependent correction
factor at each halo mass bin $f_{\mathrm{corr}}(M_{r})$. There is excellent
agreement between the deprojected 3D CLF and the input CLF.

####  2.4.1 Cross-Matching Cluster Samples

As discussed in Section 1, we also investigate the properties between the
clusters that are co-detected by both cluster finders and those detected in
one but not in the other. To cross-match between the two catalogues, for a
cluster from one catalogue, we identify from the other catalogue any cluster
detected within some projected 2D radial aperture and with a redshift within
3$\sigma_{z}$ as a match, where $\sigma_{z}$ is the uncertainty on the
photometric redshift estimate. For redMaPPer this is $\delta(z)/(1+z)$ and
$\approx$0.006 at $z=0.1$ and $\approx$0.015 at $z=0.3$ (Rykoff et al., 2014;
Rozo et al., 2015); For Yang clusters this is $\sigma_{z}\approx 0.008$ (Yang
et al., 2021). For this cross-matching, we only use the two final samples, so
a cluster classified as “non-matched” might have a matched counterpart with
$\lambda<20$ in the other cluster catalogue. Were we to use the entire samples
(down to much lower $\lambda$), as well as extend out to arbitrarily larger
radii, the matched fraction would increase significantly, thereby making the
comparison less useful.

As shown in Figure 2, we use three different search radii, 0.3Rh, 0.5Rh and
0.7Rh. For the sake of simplicity we use $R_{h}$ that corresponds to the
original apertures adopted by the cluster finders. We see that for the most
massive clusters ($\lambda{\simeq}55$), the matched fraction is greater than
80% for a search radius greater than 0.5Rh. Unsurprisingly, the matched
fraction decreases with decreasing richness, as the matches are increasingly
more likely to be found in the $\lambda<20$ systems. Besides, the completeness
and purity of both cluster catalogues also drop rapidly with richness. We note
that a more complete analysis of the comparison between the clusters and
groups identified by redMaPPer and the Yang cluster finder is needed, but
beyond the scope of this analysis. For the analysis in this paper, we adopt
0.7Rh as the fiducial search radius for our matched vs. non-matched split.

##  3 The Conditional Luminosity Functions

###  3.1 Stacked 2D CLF Measurement

To compare the satellite populations within clusters from both catalogues, we
measure the CLF for clusters within each $\lambda$ bin. The standard approach
(e.g., Hansen et al., 2009; Lan et al., 2016) is to separate the central and
satellite components of the CLFs. However, because the Yang clusters are not
centered on the central galaxy and the central is included when measuring the
CLF within 1$h^{-1}\mathrm{Mpc}$ of the luminosity weighted center, we include
the central galaxy component in the measurements of all CLFs in this analysis.
Despite this inclusion of centrals in the CLFs, we only focus on the magnitude
range dominated by the satellite galaxies and refer to the member galaxies at
those magnitudes simply as “satellites” during our CLF comparison. In an
upcoming analysis, we plan to build on this work and use the central galaxy to
investigate the impact of the magnitude gap, a tracer of BCG hierarchical
growth (Golden-Marx & Miller, 2018), on the CLF.

Unlike some previous analyses (e.g., To et al., 2020), we do not directly use
either the redMaPPer or the Yang membership candidates to construct the CLF
(though we present that direct result as well). Instead, we closely follow the
stacking procedures of Lan et al. (2016) to measure the CLF of both cluster
catalogues using the SDSS photometric galaxies. Using the centroid positions
identified in those catalogues, for each galaxy cluster, we identify all
galaxies within a projected distance of 1$h^{-1}\mathrm{Mpc}$. We use a
constant radial aperture to mitigate the impact of uncertainties in the halo
mass estimates. We then convert their apparent magnitudes into absolute
magnitudes assuming the cluster redshift. However, as previously noted because
this is a photometric measurement, we must account for contamination from
interloper galaxies. For each cluster, we assign the redshift and halo mass to
ten random points within the survey footprint and use the same
1$h^{-1}\mathrm{Mpc}$ aperture to determine the background contribution. By
subtracting off the background from the measurement centred on the cluster
centre, we obtain the 2D CLF measurement for each individual cluster.

We next employ the BOSS window function (with bright stars masked out) to
estimate the covering fraction of each cluster region. More specifically, we
determine what fraction of the sky within a 1$h^{-1}\mathrm{Mpc}$ aperture is
not masked as a result of bright stars or edge effects. The covering fractions
are generally high as our final analysis sample includes only those clusters
with covering fraction greater than 95%. We multiply the measured CLF by the
inverse of this fraction to correct for masking. Additionally, when stacking
the measurements, we use the limiting magnitude of the SDSS observations
($m_{r}{=}21$) to calculate the absolute magnitude limit for each cluster and
correct for this difference across our redshift range $0.1{<}z{<}0.34$. We
then calculate the stacked 2D CLF by measuring the mean 2D CLF within
1$h^{-1}\mathrm{Mpc}$ of the cluster centre (with the mean random background
contribution subtracted) for each of the four richness bins described in
Section 2.4.

At this point, we are left with a 2D CLF, which is representative of the
average galaxy population projected within a 1$h^{-1}\mathrm{Mpc}$ 2D aperture
about the cluster centre. However, our goal is to estimate the 3D isotropic
CLF within a 1$h^{-1}\mathrm{Mpc}$ radius, for which we need to further remove
the CLF contribution from galaxies in the correlated structures but physically
outside the 1$h^{-1}\mathrm{Mpc}$ radius.

###  3.2 Deprojecting 2D CLF to 3D: Mock Test using N-body Simulation

Figure 4: For four bins in $\lambda$, we compare the CLF measured directly
from the membership catalogues (circles with errorbars) to those measured from
cross-correlating with the photometric galaxies (curves with uncertainty
bands) in both the Yang (top row) and redMaPPer (bottom row) catalogues. In
the lower panels, we denote $0.2L_{*}$ as the gray band.

We compute the correction factor to deproject the CLF measurements from 2D to
3D by following the approach outlined in Lan et al. (2016). Although such a
deprojection approach has been applied to real data by Lan et al. (2016), we
verify our capability of accurately measuring the average CLF per cluster
within a 3D radius using a sample of simulated clusters and a mock catalogue
of photometric galaxies. In particular, we determine whether any additional
calibrations are needed to accurately remove the contribution from correlated
structures to the CLF when deprojecting the 2D measurements to 3D.

Using the ELUCID N-body constrained cosmological simulation (Wang et al.,
2016), we construct the mock photometric galaxy catalogue by populating dark
matter haloes according to the halo mass-dependent CLF parameters measured by
Yang et al. (2008). For a continuous coverage of the ELUCID halo mass function
with galaxies down to $m_{r}{=}21$, we interpolate the CLF parameters across
the three most massive bins of Yang et al. (2008) and extrapolate the CLFs of
Yang et al. (2008) to fainter magnitudes assuming fixed faint-end slopes. We
further assume the galaxy distribution follows an isotropic NFW profile inside
the dark matter haloes with the same concentration and velocity dispersion as
the dark matter particles. Following Salcedo et al. (2022), we populate the
halos within the SDSS main sample volume inside the ELUCID constrained
simulation.

Following the approach outlined in the Appendix B2 of Lan et al. (2016), we
calculate a correction factor $f_{\mathrm{corr}}$ to account for the
discrepancy that exists between the CLF projected within a 2D aperture and
that enclosed within a 3D halo radius, which results from galaxies associated
with the correlated structure outside the clusters. We briefly describe the
deprojection method below and refer interested readers to Lan et al. (2016)
for the analytic derivations. To determine this correction factor at any given
magnitude for each halo mass/richness bin, $f_{\mathrm{corr}}(M_{r})$, we
measure the normalized surface number density of galaxies at a given magnitude
between 0.1R200 and 2.5R200 and fit the power-law slope, $\gamma$, to this
profile. We then use equation B7 from Lan et al. (2016) to convert the power-
law slope $\gamma$ to the correction factor $f_{\mathrm{corr}}$. We note that
this process differs from Lan et al. (2016) in that we adopt a magnitude-
dependent $\gamma$ to remove the contribution from the correlated structure
instead of measuring the median value of $\gamma$ within each richness bin.

As shown in Figure 3, the 2D CLF overpredicts the number of galaxies
associated with each cluster. However, after applying the correction factors
to the 2D CLF, our deprojected 3D CLF is in excellent agreement with the
theoretical input from Yang et al. (2008). Thus, the magnitude-dependent
correction factors are needed to accurately reproduce the input CLF.
Therefore, in our observational analysis, we measure the surface density
profile out to 2.5R200 at fixed $M_{r}$ for each cluster and measure the
correction factor as a function of $M_{r}$ for clusters of each richness bin.

###  3.3 Stacked 3D CLFs: Comparison with Direct CLFs

After testing the stacking and deprojection method with mock galaxy samples,
we are now ready to apply the method to the redMaPPer and Yang cluster samples
using the SDSS photometric catalogue. We apply the correction factor
$f_{\mathrm{corr}}(M_{r}|\lambda)$ to each 2D CLF to remove the effect of
galaxies that are outside of our 1$h^{-1}\mathrm{Mpc}$ radius when measured in
3D, thereby converting our previously measured 2D CLFs to the deprojected 3D
CLFs.

Although both the redMaPPer and Yang cluster catalogues provide membership
catalogues to identify the galaxies associated with each cluster, the
uncertainties associated with the membership probabilities are generally
large, with significant biases due to the redshift uncertainties and
projection effects (Zu et al., 2017). Therefore, CLFs constructed directly
from those membership catalogues (hereafter referred to as “direct” CLFs) are
expected to be incomplete. Figure 4 presents the results of the direct CLF
when measured within 1$h^{-1}\mathrm{Mpc}$ for each membership catalogue,
compared to the 3D CLF measured from the stacking method (band). For the
direct CLFs of Yang clusters, we apply a constant shift of 0.07 magnitudes to
convert the DESI magnitudes of the member galaxies to SDSS r-band magnitudes
$M_{r}$. We note that redMaPPer also provides a membership probability
$p_{\mathrm{mem}}$, which we use in our estimate of the direct CLF.

As shown in Figure 4, we find very similar trends between the direct and
stacked CLFs for both the Yang (top row) and redMaPPer (bottom row) clusters
at the bright end ($M_{r}{<}{-}21.5$), indicating that both cluster finders
are highly effective in identifying the bright member galaxies. At the faint
end, we note that redMaPPer uses a limiting magnitude of $0.2L_{*}$ (gray
vertical bands) to determine its membership catalogue. As a result, it is
unsurprising that we see the redMaPPer direct CLF (filled circles) drop off
beyond that limit. Meanwhile, the Yang catalogue imposes an apparent magnitude
cut on the membership galaxies, so we do not observe a steep drop-off in the
direct measurement like what is seen in redMaPPer, and their direct and
stacked CLFs are in reasonable agreement until $M_{r}{\sim}{-}19.5$.
Interestingly, the direct CLFs of both Yang and redMaPPer exhibit a modest
deficit compared to their respective stacked CLFs at ${-}20{>}M_{r}{>}{-}21$,
suggesting that both cluster finders are relatively conservative in assigning
memberships to galaxies with intermediate luminosities between sub-$L_{*}$ and
$L_{*}$. However, we note that the deficit should be caused by the
incompleteness of different types of galaxies, (relatively redder galaxies in
Yang and bluer galaxies in redMaPPer).

###  3.4 CLF Comparison between the Yang and redMaPPer Clusters

####  3.4.1 CLF Amplitudes

Figure 5: Comparison between the total number of member galaxies
$N_{\mathrm{mem}}$ between the Yang (green triangles) and redMaPPer (magenta
circles) clusters in four different richness bins. The satellite numbers are
computed from integrating the stacked CLFs above $M_{r}{=}{-}20.4$. The lower
panel shows the ratio of the number of members in the redMaPPer clusters
divided by that in the Yang clusters.

We compute the total number of member galaxies brighter than $M_{r}{=}-20.4$,
$N_{\mathrm{mem}}$, by integrating the stacked CLFs of both cluster samples at
fixed $\lambda$ (as shown in Figure 4) above ${-}20.4$. Figure 5 compares the
$N_{\mathrm{mem}}$ between the Yang (green triangles) and redMaPPer (magenta
circles) clusters in the four richness bins, with the ratios
$N_{\mathrm{RM}}/N_{\mathrm{Y}}$ (circles) shown in the bottom panel. The
number of member galaxies at the lower richness end ($\lambda$$\leq$55) are in
good agreement (within $10\%$). In contrast, at the higher richness end the
redMaPPer clusters contain $\sim$20% more member galaxies. By using a fixed
$1h^{-1}\mathrm{Mpc}$ aperture, we eliminate the impact of halo definitions.
Therefore, the discrepancy is likely driven by the combination of different
centring and scatters in the mass-richness relation between the two
catalogues. In particular, it is likely that 1) the centrals identified by
redMaPPer are located in denser regions than the luminosity-weighted centre
from the Yang catalogue; and 2) the new $\lambda_{\mathrm{Y}}$ richness that
we redefined for the Yang catalogue has a slightly larger scatter (at fixed
halo mass) than $\lambda_{\mathrm{RM}}$ and $M_{180}^{L_{\mathrm{tot}}}$
(i.e., the original halo mass proxy of the Yang catalogue) at the high mass
end, thereby finding intrinsically lower mass clusters at the same $\lambda$.
However, as discussed further in Section 4, our cluster weak lensing analysis
suggests that the relatively larger scatter in our new richness
$\lambda_{\mathrm{Y}}$ is likely the more dominant cause, which is expected
because $\lambda_{\mathrm{Y}}$ is not optimized for minimising the scatter in
the mass-richness relation. Additionally, we note that when using the BCGs
identified by Yang (not shown in this analysis), this $N_{\mathrm{mem}}$
discrepancy is enlarged, suggesting that the luminosity-weighted centres are
indeed better choices for the CLF measurement.

####  3.4.2 Dependence of the CLF Shape on Galaxy Colour and Cluster Finders

Figure 6: Comparison of the CLF shapes between the Yang (green) and redMaPPer
(magenta) clusters in four richness bins. In each upper panel, the total CLF
(solid curves) is decomposed into the contributions from red (dashed curves)
and blue (dotted curves) galaxies. The lower panels show the red fractions as
functions of the absolute r-band magnitude.

We now focus on the shapes of the stacked CLFs to compare the galaxy content
at different absolute magnitudes between the redMaPPer and Yang clusters.
Additionally, to examine the colour difference in the galaxy content, we split
our photometric galaxies by colour to measure the stacked CLFs of the red and
blue satellites separately. Unlike Lan et al. (2016), who used $u{-}r$ colour,
or Meng et al. (2022), who used $g{-}z$ colour, we use the redshift-dependent
redMaPPer red-sequence defined in $g{-}r$ to identify all red galaxies across
the redshift range of interest. This approach also accounts for the shallow
slope of the red sequence on the colour-magnitude diagram at any given
redshift. We then identify galaxies with $g{-}r$ colours 2$\sigma$ below the
red sequence as blue and all other galaxies as red.

Figure 6 compares the CLFs between the redMaPPer and Yang clusters using the
two overall samples (i.e., regardless of whether a cluster in one sample has a
matched counterpart in the other sample). At the bright end ($M_{r}<-20.4$),
where the massive red galaxies dominate the CLFs, the shapes are in excellent
agreement. However, the faint-end slopes of the overall CLF (solid curves)
differ, with $\alpha_{\mathrm{Yang}}{\simeq}{-}0.90$ (green) and
$\alpha_{\mathrm{RM}}{\simeq}{-}1.1$ (magenta). This discrepancy appears to be
driven by the lower luminosity red galaxies (dashed curves), where redMaPPer
clusters exhibit a relatively constant population, compared to the slightly
decreasing population in Yang clusters. Meanwhile, we detect a boost in the
population of blue galaxies in the Yang clusters at $M_{r}{<}{-}21$ compared
to the redMaPPer clusters, and the amplitude of the boost declines with
increasing richness, exhibiting little difference between the two blue CLFs in
the highest-$\lambda$ bin. Both discrepancies in the red and blue CLFs are
consistent with our expectation that the colour preferences of the red-
sequence and halo-based cluster finders are largely orthogonal. The bottom
panels show the red fractions as functions of $M_{r}$, further demonstrating
that the redMaPPer clusters are systematically redder than the Yang clusters.

Naively, one might expect the shape discrepancy between the redMaPPer and Yang
CLFs shown in Figure 6 to be caused mainly by the mismatched clusters (i.e.,
clusters detected in one catalogue but not in the other). However, from Figure
2 we know that the mismatched fraction rapidly decreases as a function of
$\lambda$, but that the shape discrepancies in the overall and red CLFs shown
in Figure 6 are consistent across the four richness bins. Therefore, while the
mismatched clusters are likely the culprit for the discrepancy in the blue
CLFs (the $\lambda$-dependent boost at $M_{r}{<}{-}21$), the discrepancy in
the red CLFs (hence overall CLFs) is mainly due to the different centring
algorithms between the two cluster finders. To test this hypothesis, we
disentangle the effect of centring from that of mismatching by splitting the
Yang clusters at each $\lambda$ bin into two subsamples, one with matched
counterparts in the redMaPPer sample based on our fiducial cross-matching
criteria (hereafter referred to as “matched”), and the other one without
(“non-matched”). In Figure 7, we show the stacked CLFs of the matched clusters
using both the original Yang luminosity-weighted centres (solid curves) and
the highest-$p_{\mathrm{cen}}$ galaxies of their matched counterparts in
redMaPPer (dashed curves), as well as the stacked CLF of the non-matched
clusters using their original Yang centres (dotted curves). Similar to Figure
6, we decompose each CLF into the contributions from red (red curves) and blue
(blue curves) galaxies, and show the red fractions in the bottom panels.

Figure 7: Comparison of the CLF shapes between the Yang clusters that have
matched counterparts in redMaPPer using luminosity-weighted centers (solid
curves with uncertainty bands), the same matched clusters using redMaPPer
centrals as centers (dashed curves), and the Yang clusters with no matched
counterparts in redMaPPer (dotted curves) in four different richness bins. In
each upper panel, the overall CLF of each subsample (black) is decomposed into
contributions from the red (red) and blue (blue) satellite galaxies. The lower
panels show the red fractions as functions of r-band absolute magnitude.
Figure 8: Similar to Figure 7, but for redMaPPer clusters. The lower panels
show the blue fractions as functions of r-band absolute magnitude.

By comparing the CLFs of the matched clusters measured with different centres,
we find that the shape discrepancy induced by switching centres closely mimics
that between the Yang and redMaPPer CLFs seen in Figure 6. The flat
(decreasing) population of all (red) faint satellites observed around Yang
centres (solid black/red curves) becomes increasing (flat) at the faint end
when switching to the redMaPPer centrals (dashed curves). Therefore, we
conclude that these shape differences between the overall (as well as red)
CLFs measured for the two cluster catalogues are primarily caused by the
different centring algorithms. The redMaPPer algorithm tends to identify the
location of the most densely populated pocket of red galaxies as the centre of
a cluster, while the luminosity-weighted centre found by Yang likely sits more
closely to the geometric centre of a larger cluster region but may miss the
true centre (defined by the minimum of the potential) by a large margin in
morphologically-disturbed systems. As a result, neither centring algorithm
works perfectly in all scenarios, and it is thus more plausible that the true
CLF of a correctly-centred cluster sample (e.g., by X-ray observations) is
closer to the average of the two versions. We plan to test this hypothesis by
measuring the CLF for clusters with well measured X-ray centres in the future.

For the Yang clusters with no matched counterparts in redMaPPer (dotted blue
curves), their blue galaxy CLFs are similar to those of the matched clusters
(solid blue curves) in shape, but with a slightly larger amplitude at the
bright end. However, for non-matched clusters, the red galaxy CLFs (dotted red
curves) are significantly lower than the red galaxy CLFs of the matched
clusters (solid red curves), by an amount ranging from 20% in the highest
richness bin to a factor of two in the lowest richness bin. Consequently, the
red fraction of the non-matched clusters is systematically lower than that of
the matched clusters in all richness bins (bottom panels). Since those non-
matched clusters are significantly less dominated by the red galaxies than the
matched clusters, it is unsurprising that the redMaPPer algorithm does not
detect them as clusters with $\lambda{>}20$. Meanwhile, the Yang cluster
finder detects progressively more systems with bluer satellite population than
redMaPPer, especially in low-richness systems.

We perform a similar test using the redMaPPer sample by splitting the
redMaPPer clusters into matched vs. non-matched subsamples based on the
existence of matched counterparts in the Yang sample. The result is shown in
Figure 8. Likewise, the shape discrepancy persists in the matched clusters
(solid black curves), but promptly disappear after we adopt the same centres
for these clusters (dashed black curves). The increasing (relatively flat)
population of all (red) faint galaxies when centered on the redMaPPer central
transforms into the relatively flat (slightly decreasing) population of all
(red) low-luminosity galaxies when centred on the Yang luminosity weighted
center. Interestingly, the red galaxy CLFs of the matched vs. non-matched
clusters are consistent across all richness bins, as the redMaPPer algorithm
regards them as indistinguishable by counting the red galaxies. However, the
blue CLFs of the non-matched clusters (dotted blue curves) are much lower than
those of the matched clusters (solid blue curves) in the two low-richness
bins, while the two sets of blue CLFs are consistent in the two high-richness
bins, mirroring the trend seen for the Yang sample in Figure 7. As a result,
the bottom panels show that the blue fraction of the non-matched clusters
(dotted blue curves) is generally lower than that of the matched clusters
(solid blue curves).

To summarise the results from Figures 7 and 8, the shape of the overall CLFs
are primarily sensitive to the centroiding of clusters. In particular, the
faint-end slope could exhibit an increasing or a decreasing trend, depending
on whether the cluster centres are located within a dense clump of red
galaxies or a luminosity-weighted centre averaged over a large spread of both
red and blue galaxies. In addition, some of the discrepancies in the CLF
shapes are caused by the orthogonal colour preferences between the two cluster
finders, resulting in the non-detection of the extremely red-dominated
clusters by Yang and the relatively blue clusters by redMaPPer, respectively.

###  3.5 CLF Parameters and Comparison to Previous Results

Since the introduction of the Schechter Function (Schechter, 1976), the
observed CLF has been characterised by its amplitude ($\phi^{*})$,
characteristic luminosity ($M^{*}$), and faint-end slope ($\alpha$), all
information which provides insight into the physical processes that
characterise galaxy formation and evolution. To measure the three parameters
associated with our CLF for each of the samples shown in Figures 6, 7, and 8,
we fit our measured CLF to a function of the form

$\phi(M_{r})=\frac{\ln(10)}{2.5}\phi^{*}10^{0.4(M^{*}-M_{r})(\alpha+1)}\exp[{-10^{0.4(M^{*}-M_{r})}}],$
(1)

using a standard Bayesian inference method using the Markov Chain Monte Carlo
(MCMC) ensemble sampler PyMC. The results are given in Table 1 along with the
redMaPPer results when the central is not included. In addition, the CLF
parameters for the (non-) matched subsamples of the Yang and redMaPPer
clusters are provided in the Appendix in Tables 3 and 4, respectively.

Table 1: Best-fitting parameters of the CLFs in Figure 6 . bin | M∗ | $\alpha$ | $\phi^{*}$
---|---|---|---
All galaxies of Yang clusters
1 | -22.02 $\pm$ 0.04 | -0.99 $\pm$ 0.02 | 2.44 $\pm$ 0.09
2 | -21.65 $\pm$ 0.04 | -0.83 $\pm$ 0.03 | 4.60 $\pm$ 0.16
3 | -21.64 $\pm$ 0.05 | -0.88 $\pm$ 0.03 | 5.40 $\pm$ 0.24
4 | -21.65 $\pm$ 0.06 | -0.89 $\pm$ 0.04 | 6.66 $\pm$ 0.35
Red galaxies of Yang clusters
1 | -21.93 $\pm$ 0.03 | -0.83 $\pm$ 0.02 | 2.48 $\pm$ 0.07
2 | -21.61 $\pm$ 0.04 | -0.70 $\pm$ 0.03 | 4.41 $\pm$ 0.14
3 | -21.61 $\pm$ 0.05 | -0.77 $\pm$ 0.03 | 5.16 $\pm$ 0.22
4 | -21.59 $\pm$ 0.06 | -0.78 $\pm$ 0.04 | 6.65 $\pm$ 0.34
Blue galaxies of Yang clusters
1 | -21.63 $\pm$ 0.14 | -1.47 $\pm$ 0.05 | 0.29 $\pm$ 0.05
2 | -21.29 $\pm$ 0.17 | -1.30 $\pm$ 0.09 | 0.55 $\pm$ 0.12
3 | -21.34 $\pm$ 0.20 | -1.30 $\pm$ 0.11 | 0.58 $\pm$ 0.15
4 | -21.37 $\pm$ 0.27 | -1.43 $\pm$ 0.13 | 0.52 $\pm$ 0.18
All galaxies of redMaPPer clusters
1 | -22.13 $\pm$ 0.04 | -1.15 $\pm$ 0.02 | 2.14 $\pm$ 0.11
2 | -21.96 $\pm$ 0.05 | -1.10 $\pm$ 0.02 | 3.35 $\pm$ 0.16
3 | -21.84 $\pm$ 0.05 | -1.11 $\pm$ 0.03 | 4.63 $\pm$ 0.26
4 | -21.78 $\pm$ 0.06 | -1.06 $\pm$ 0.03 | 6.83 $\pm$ 0.44
Red galaxies of redMaPPer clusters
1 | -22.03 $\pm$ 0.04 | -1.01 $\pm$ 0.02 | 2.34 $\pm$ 0.09
2 | -21.82 $\pm$ 0.04 | -0.95 $\pm$ 0.02 | 3.72 $\pm$ 0.16
3 | -21.75 $\pm$ 0.05 | -0.99 $\pm$ 0.03 | 4.89 $\pm$ 0.24
4 | -21.71 $\pm$ 0.05 | -0.95 $\pm$ 0.03 | 7.05 $\pm$ 0.40
Blue galaxies of redMaPPer clusters
1 | -21.20 $\pm$ 0.17 | -1.57 $\pm$ 0.07 | 0.32 $\pm$ 0.07
2 | -21.46 $\pm$ 0.24 | -1.57 $\pm$ 0.09 | 0.32 $\pm$ 0.10
3 | -21.43 $\pm$ 0.29 | -1.61 $\pm$ 0.10 | 0.38 $\pm$ 0.16
4 | -21.55 $\pm$ 0.30 | -1.67 $\pm$ 0.10 | 0.36 $\pm$ 0.15
All galaxies of redMaPPer clusters - no BCG
1 | -21.50 $\pm$ 0.03 | -0.94 $\pm$ 0.02 | 3.64 $\pm$ 0.14
2 | -21.59 $\pm$ 0.04 | -0.98 $\pm$ 0.02 | 4.54 $\pm$ 0.19
3 | -21.58 $\pm$ 0.04 | -1.00 $\pm$ 0.03 | 5.83 $\pm$ 0.28
4 | -21.59 $\pm$ 0.05 | -0.99 $\pm$ 0.03 | 8.01 $\pm$ 0.48
Red galaxies of redMaPPer clusters - no BCG
1 | -21.38 $\pm$ 0.03 | -0.75 $\pm$ 0.02 | 3.83 $\pm$ 0.11
2 | -21.46 $\pm$ 0.04 | -0.80 $\pm$ 0.02 | 4.90 $\pm$ 0.17
3 | -21.50 $\pm$ 0.04 | -0.89 $\pm$ 0.03 | 6.00 $\pm$ 0.25
4 | -21.51 $\pm$ 0.05 | -0.87 $\pm$ 0.03 | 8.32 $\pm$ 0.44
Blue galaxies of redMaPPer clusters - no BCG
1 | -21.19 $\pm$ 0.17 | -1.57 $\pm$ 0.07 | 0.32 $\pm$ 0.08
2 | -21.46 $\pm$ 0.24 | -1.57 $\pm$ 0.09 | 0.32 $\pm$ 0.10
3 | -21.43 $\pm$ 0.29 | -1.61 $\pm$ 0.10 | 0.38 $\pm$ 0.16
4 | -21.55 $\pm$ 0.29 | -1.68 $\pm$ 0.10 | 0.36 $\pm$ 0.15

The primary takeaway from measuring the parameters, given in Table 1, is that
only $\phi^{*}$ depends on $\lambda$, in agreement with the results from
Hansen et al. (2009). However, the comparison between the parameters measured
for the redMaPPer and Yang clusters is more insightful. As given in Table 1,
the underlying parameters within the CLF slightly differ, as suggested by
Figure 6. We find that the $M^{*}$ for redMaPPer and Yang clusters are largely
consistent, which reflects the location of the bright-end turn-over in Figure
6. Moreover, as shown in Figure 6, we see that the faint-end slope, $\alpha$
for the redMaPPer catalogue is slightly steeper, for both red, blue, and
overall galaxies. Lastly, as previously mentioned, we unsurprisingly find that
the amplitude $\phi^{*}$ is generally larger for the redMaPPer sample, in
agreement with the results from Figure 5.

Since the galaxy cluster CLF is well studied, there are many results we can
compare our stacked measurements to. However, these parameters are mildly
covariant, so the inclusion of the central galaxy, which impacts the
measurement of M∗ also impacts $\alpha$ and $\phi$. Based on the inclusion of
the central, we do not compare our measured M∗ to previous results. However,
we also present the redMaPPer results without the central galaxy to make a
comparison to the $\alpha$ measurements from Hansen et al. (2009) and Zhang et
al. (2019b).

One of the best comparison samples is from Zhang et al. (2019b), who measured
the CLFs of clusters identified by redMaPPer in the Dark Energy Survey (DES)
using the DES Science Verification data (Rykoff et al., 2016). Although this
analysis covers both a larger range in halo mass and higher redshift, we find
strong agreement between our $\alpha$ value for red galaxies when the BCG is
excluded with their measurement, $\alpha{\simeq}{-}0.7-0.8$, for DES-
redMaPPer selected clusters in our redshift range. Of note, Zhang et al.
(2019b) also detect a similar trend in $\alpha$, such that the value becomes
slightly steeper with richness.

We also compare our results to Hansen et al. (2009), which cover a similar
redshift and halo mass range, but instead use the MaxBCG catalogue (Koester et
al., 2007). We note that Hansen et al. (2009) measure the CLF within R200 as
opposed to a 1$h^{-1}\mathrm{Mpc}$ aperture; however, as shown in their Figure
3, their average aperture size is comparable to ours. Another difference is
that Hansen et al. (2009) fix $\alpha$ for the blue galaxies, while we allow
it to be a free parameter. Given our use of a similar aperture, it is
unsurprising that we measure similar values $\phi^{*}$ for both the red and
blue samples and the overall sample. Additionally, the $\alpha$ value for red
galaxies ($\alpha{\simeq}{-}0.7$) and overall galaxy sample
($\alpha{\simeq}{-}0.9$) show general agreement with our results excluding the
BCG, though our values are $\sim$0.1 steeper. For both of our samples, the
blue galaxy $\alpha$ is much steeper than what Hansen et al. (2009) assume,
$\alpha_{blue}{=}{-}1.0$. Some of these discrepancies may result from the
difference in cluster finding algorithm and how red and blue galaxies are
defined. Hansen et al. (2009) uses a fixed bimodal definition based on $g{-}r$
colours, while we use the redMaPPer defined red sequence which is redshift
dependent.

Although measured for lower redshift galaxy clusters, we compare our results
to those presented in Lan et al. (2016) ($0.01<z<0.05$) and Meng et al. (2022)
($0.01<z<0.08$), which use clusters identified in the Yang et al. (2007)
spectroscopic group catalogue. The primary differences between our measurement
approaches are the different apertures within which we measure the CLF
(1$h^{-1}\mathrm{Mpc}$ vs R200), how we identify red and blue galaxies, and
our inclusion of the central galaxy. However, given that we both use the halo-
based cluster finder, this is a relevant comparison. We note that we only
compare our results to the bright portion of the measurements presented in Lan
et al. (2016) and Meng et al. (2022), because we do not observe the magnitude
range fainter than $M_{r}{=}{-}18$) due to the higher redshift nature of our
cluster samples.

For the total sample of galaxies, our measurements for $\alpha$ are in
agreement with the results from both Lan et al. (2016) and Meng et al. (2022).
For the red galaxies, our values of $\alpha$ are in agreement at the
lowest-$\lambda$ end, while at the high-richness end, our $\alpha$ is
shallower by 0.15-2 than both Lan et al. (2016) and Meng et al. (2022). For
blue galaxies, we find that our measurements for $\alpha$ are in agreement
with the values presented in Lan et al. (2016) and Meng et al. (2022) for the
entire $\lambda$ range. We note that the difference between our measurements
and those from Lan et al. (2016) and Meng et al. (2022) is primarily due to
the different performances of the photometric and spectroscopic halo-based
cluster finders as well as our inclusion of the central galaxy.

In addition to the CLF parameters, we also compare our measured red fractions
to those from Hansen et al. (2009) and Meng et al. (2022). Hansen et al.
(2009) measure a red fraction within the entire cluster of $\sim$80% for
galaxy clusters, in strong agreement with the results we find for both the
redMaPPer and Yang samples. In a more direct comparison, Meng et al. (2022)
find that the red fraction increases slightly with $\lambda$ and that it also
increases from 60% to 100% with $M_{r}$ over the range $-18>M_{r}>-23$, which
is in excellent agreement with our measurement shown in Figure 6.

In Lan et al. (2016), one of their primary conclusions is that the CLF can be
characterised by a faint-end slope that steepens at $M_{r}=-18.0$. This same
faint-end slope was similarly detected by Tinker et al. (2021) and Meng et al.
(2022) for CLF samples measured at low redshifts ($z<0.08$). Although the
higher average redshift of our sample prevents us from directly making this
measurement, we address this comparison by extrapolating our faint-end
behaviors to fainter magnitudes. In particular, Lan et al. (2016) note that
the increase in the faint-end slope is driven predominately by the red
galaxies, where $\alpha=-1.8$. Moreover, both Lan et al. (2016) and Meng et
al. (2022) find that the fraction of red galaxies greatly increases in the
magnitude range $-12<M_{r}<-16$, and thus results in that faint end slope.
However, it is worth noting that this trend was identified using groups and
clusters identified in the Yang et al. (2007) spectroscopic group catalogue,
using the luminosity-weighted centres. Interestingly, Figure 6 suggests we may
see that the faint-end population starts to modestly rise at $M_{r}{=}{-}19$
for some richness bins. However, even though we do not analyse the faintest
$M_{r}$ range of interest, we find that the CLF shape is sensitive to the
centring algorithm used and we expect this conclusion to hold at fainter
magnitudes. Therefore, while Lan et al. (2016) and Meng et al. (2022) suggest
that this steeper faint-end slope is a result of the different formation
processes behind red and blue galaxies in the cluster environment, our results
suggest that this effect may be impacted by the choice of cluster finder,
particularly the way the centres of the galaxy clusters are identified.

To summarise the results from our stacked CLF comparison, we confirm the
expectation that the redMaPPer cluster finder generally detects clusters with
a higher fraction of red galaxies and vice versa for the Yang finder. However,
the shape discrepancy between the CLFs of the two cluster samples is, somewhat
unexpectedly, mainly due to the difference in cluster centring between the two
finders. For a cluster cosmology analysis, however, this is very encouraging
news because cluster miscentring can be robustly calibrated using X-ray data
(Zhang et al., 2019a) and accurately modelled out in the cluster weak lensing
analysis when measuring halo mass (e.g., Zu et al., 2021; Wang et al., 2022),
the key quantity for cluster cosmology.

##  4 Weak Lensing Halo Mass of Clusters

###  4.1 Weak Lensing of Matched vs. Non-Matched Clusters

After examining the discrepancy in the galaxy content between the two cluster
samples, we now explore whether a significant systematic uncertainty exists as
a result of any cluster detection bias associated with the galaxy content
preferred by each finder. As mentioned in Section 1, it is plausible that the
detection bias associated with a particular cluster finder would drive the
otherwise random scatter in the mass-richness relation to be systematic,
sometimes with strong non-Gaussian tails. For example, if a cluster finder
preferentially assigns higher richnesses to haloes with a higher red fraction
due to an older formation time, while misdetecting young haloes with higher
blue fraction but the same halo mass. The scatter in the mass-richness
relation would have a systematic contribution, so that the halo mass
distribution at fixed richness would include a long tail into the low-mass
regime. By the same token, another cluster finder could assign higher
richnesses to systems with an excess number of blue galaxies projected within
the 2D aperture due to the larger photo-z errors of blue galaxies, thereby
producing a similar long tail at the low-mass end. Unfortunately, it is
observationally challenging to ascertain the existence of such a systematic
component in the scatter, as the underlying halo mass distribution of the
clusters is unknown.

Table 2: Parameter descriptions and priors for WL halo mass inference.

Parameter | Description | Prior
---|---|---
$\log M_{h}^{\mathrm{WL}}$ | Log-halo mass |  $\mathcal{U}$(13.0,16.0)
c | Halo concentration |  $\mathcal{N}$(5,1.52)
$f_{\rm off,RM}$ |  redMaPPer offset fraction |  $\mathcal{N}$(0.30,0.042)
$\sigma_{\rm off,RM}$ |  redMaPPer characteristic offset |  $\mathcal{N}$(0.18,0.022)
$f_{\rm off,Y}$ |  Yang offset fraction |  $\mathcal{U}$(0.0, 1.0)
$\sigma_{\rm off,Y}$ |  Yang characteristic offset |  $\mathcal{U}$(0.0,1.0)

$\mathcal{N}(a,b)$ refers to a Normal distribution with mean and variance of a
and b and $\mathcal{U}(a,b)$ refers to a uniform distribution with upper and
lower limits a and b.

Figure 9: The $\Delta\Sigma$ profiles of the Yang (green squares with
errorbars) and redMaPPer (magenta circles with errorbars) clusters in four
different richness bins. Solid green and magenta lines are the best-fitting
model curves for the Yang and redMaPPer clusters, respectively.

However, by taking advantage of the two cluster samples within the same volume
detected by complementary approaches, we can circumvent the issue by exploring
whether a significant discrepancy exists in the average halo masses estimated
from weak lensing ($M_{h}^{\mathrm{WL}}$), in particular between the matched
and the non-matched clusters at fixed $\lambda$ of each cluster catalogue. The
rationale is as follows. Assuming cluster catalogues $\mathcal{A}$ and
$\mathcal{B}$ have detection biases that are roughly orthogonal, at fixed
$\lambda$ of catalogue $\mathcal{A}$, clusters that have matched counterparts
in $\mathcal{B}$ are likely the systems that are least affected by the
detection bias of $\mathcal{A}$, while the $\mathcal{A}$ clusters that do not
have any matches in $\mathcal{B}$ are potentially the ones that are
intrinsically low-mass systems but promoted above the richness/detection
threshold of $\mathcal{A}$. Therefore, if we observe a significant discrepancy
in the weak lensing halo mass of the matched vs. non-matched clusters from
$\mathcal{A}$, this would suggest the existence of a strong systematic
component in the scatter, which would then lead to severe bias in the cluster
cosmology analysis.

We follow a similar approach to estimate $M_{h}^{\mathrm{WL}}$ as what is
described in depth in Zu et al. (2021) and use the same shear
catalogue444https://gax.sjtu.edu.cn/data/DESI.html as in Wang et al. (2022).
Therefore, here we describe the data used for our measurements and how we fit
the $\Delta\Sigma$, excess surface density profile. A more detailed
description of our cluster weak lensing formalism is provided in the Appendix.
The weak gravitational lensing signal is measured using the weak lensing shear
catalogue derived from the Dark Energy Camera Legacy Survey (DECaLS) DR8 (Dey
et al., 2019), which is part of the DESI Legacy Imaging Survey. For this
analysis, our source galaxy photometry and photometric redshifts come from Zou
et al. (2019). These redshifts are supplemented with the photometric redshifts
estimates from (Zhou et al., 2021). For each richness bin, the shear
catalogues are constructed from the DECaLS DR8 data using the Fourier_Quad
method (Zhang et al., 2015, 2019c; Zhang et al., 2022).

###  4.2 Bayesian Modelling of Cluster Weak Lensing on Small Scales

In this analysis, we use a Bayesian framework to infer the weak lensing halo
masses directly from the $\Delta\Sigma$ profiles, following the same method as
outlined in Zu et al. (2021). Since our goal is to measure the halo mass
$M_{h}^{\mathrm{WL}}$, we only use the $\Delta\Sigma$ profiles measured at
radii below the transition scale between the so-called one-halo and two-halo
terms, which occurs at $\sim$2$h^{-1}\mathrm{Mpc}$ (Zu et al., 2014).

For each of the four $\lambda$ bins, we have four parameters that we are
fitting for the $\Delta\Sigma$ profile, including the halo mass
$M_{h}^{\mathrm{WL}}$, halo concentration $c$, fraction of miscentred clusters
with offset from the true halo centre $f_{\rm off}$, and the characteristic
offset of miscentred clusters $\sigma_{\rm off}$. We note that the miscentring
effects are highly degenerate with halo concentration if $\sigma_{\rm off}$ is
allowed to freely vary. We mitigate this degeneracy for the redMaPPer clusters
using the results from Zhang et al. (2019a) which use Chandra X-ray
observations to constrain the offset and average miscentring fractions as
$\langle\sigma_{\rm off}\rangle$=0.18$\pm$0.02$h^{-1}\mathrm{Mpc}$ and
$\langle f_{\rm off}\rangle$=0.30$\pm$0.04. As of the time of this work, no
such analysis has been done for the Yang clusters, so we adopt uniform,
uninformative priors on the miscentring parameters for the $\Delta\Sigma$ of
Yang clusters.

We determine our priors following the approach in Zu et al. (2021) and
describe them in Table 2. To determine the posterior distribution for each of
these four parameters, we again use the MCMC ensemble sampler PyMC. We run the
sampler for 500,000 steps to ensure covariance and derive the posterior
constraints after a burn-in of 100,000 steps. The median values for the
$M_{h}^{\mathrm{WL}}$ and the 68% confidence limits of the 1D posteriors for
the halo masses are provided in Figures 9, 11, and 12 and the posterior
constraints on all other parameters are provided in Tables 5 and 6 for the
Yang and redMaPPer samples, respectively.

Figure 10: Similar to Figure 5, but for the weak lensing halo masses.

###  4.3 Halo Mass Comparison between Matched vs. Non-Matched Clusters

We start by measuring the overall $M_{h}^{\mathrm{WL}}$-$\lambda$ relations of
the redMaPPer and Yang samples. Figure 9 shows the $\Delta\Sigma$ profiles of
the redMaPPer (magenta circles with errorbars) and Yang (green squares with
errorbars) clusters, binned by $\lambda$ in the same manner described in
Section 2.4. In each panel, we show our fits (solid curves) over the radial
range used from the $\Delta\Sigma$ profile to estimate the weak lensing halo
mass, which is given at the bottom of each panel and shown in Figure 10. Our
$M_{h}^{\mathrm{WL}}$ estimates are consistent with the similar measurements
from Wang et al. (2022) and Simet et al. (2017) for the Yang and redMaPPer
samples, respectively. Similar to Figure 5, the WL halo mass of the redMaPPer
clusters is generally higher than that of the Yang clusters at fixed
$\lambda$, though the difference is rather small at at $\lambda{<}30$. Since
the effect of miscentring should be largely mitigated in the $\Delta\Sigma$
modelling, the discrepancy between the two $M_{h}^{\mathrm{WL}}$-$\lambda$
relations in Figure 10 should be primarily induced by the slightly larger
scatter in our newly defined richness $\lambda_{\mathrm{Y}}$ for the Yang
clusters compared to the original Yang richness and the redMaPPer richness.

Figure 11: Comparison between the weak lensing profiles of the Yang clusters
with matched counterparts in redMaPPer (dark circles with errorbars) and those
without (gray triangles with errorbars) in four different richness bins. Dark
and gray solid curves are the best-fitting model curves for the matched vs.
non-matched clusters, respectively. The lower panels show the ratio between
the two $\Delta\Sigma$ profiles. Figure 12: Similar to Figure 11, but for
redMaPPer clusters that are split into matched vs. non-matched subsamples.

We now turn to the $\Delta\Sigma$ profiles of the matched vs. non-matched Yang
clusters in the four bins of $\lambda$, as shown in Figure 11. In each panel,
circles and triangles are the $\Delta\Sigma$ profiles of the matched and non-
matched clusters, respectively, while dark and gray solid curves are the
corresponding best-fitting model predictions. The WL halo mass estimates and
the percentage of matched clusters within each bin are listed in the bottom of
each top panel, with each bottom panel showing the ratios between the two
$\Delta\Sigma$ profiles. The two sets of WL halo masses are largely consistent
within $1\sigma$ for Yang clusters above $\lambda{=}30$, but differ by
$\sim$1.4$\sigma$ in the lowest richness bin. In particular, the average log-
halo mass of the matched clusters is $14.12{\pm}0.14$, $0.32$ dex higher than
that of the non-matched clusters ($13.80{\pm}0.10$). This suggests that in the
lowest-richness bin of the Yang catalogue, the redder clusters are likely more
massive than the systems dominated by blue satellites. In the three higher
richness bins, we do not detect any significant discrepancy between the two
subsamples. Therefore, we expect that the Yang clusters should be largely free
of systematic biases induced by the detection method if limited to clusters
with $\lambda>30$ ($M_{h}^{\mathrm{WL}}{>}1.5\times 10^{14}h^{-1}M_{\odot}$).

Similarly, Figure 12 shows the result of the same analysis using the
$\Delta\Sigma$ profiles of the matched vs. non-matched redMaPPer clusters.
Across the three largest $\lambda$ bins, the average WL halo masses of the two
subsamples are consistent with each other within 1$\sigma$, indicating that
the preference of redMaPPer to select red-dominated clusters does not incur
any significant systematic uncertainties in the scatter for clusters with
$\lambda>30$ ($M_{h}^{\mathrm{WL}}{>}1.5\times 10^{14}h^{-1}M_{\odot}$), but
differ by $\sim$1.8$\sigma$ in the lowest richness bin. In particular, the
log-halo mass of matched clusters is $14.18{\pm}0.05$, $0.22$ dex higher than
that of the non-matched clusters ($13.96{\pm}0.07$). This suggests that in the
lowest-richness bin of the redMaPPer catalogue, there may exist a population
of clusters with richnesses that have been artificially inflated as a result
of projection effects in agreement with previous results (e.g., Wu et al.,
2022; Wetzell et al., 2022). Interestingly, in the two low-$\lambda$ bins we
find that the large-scale bias of the matched clusters is lower than that of
the non-matched ones. This bias discrepancy at fixed WL halo mass is similar
to the cluster assembly bias effect (Zu et al., 2021), although it could also
be induced by a stronger projection effect in the non-matched subsample (Zu et
al., 2017).

Overall, Figures 11 and 12 illustrate that when splitting the clusters into
the matched and non-matched subsamples, the two WL halo masses at fixed
$\lambda$ are always consistent with each other within $1\sigma$ for clusters
with mass above $M_{h}^{\mathrm{WL}}{\simeq}1.5\times 10^{14}h^{-1}M_{\odot}$.
This is encouraging news — despite the recognizable differences in the CLF
shape between the Yang and redMaPPer cluster catalogues, our cluster weak
lensing analysis exhibits no evidence for a strong systematic uncertainty in
the scatter of the mass-richness relations in either cluster catalogue.

##  5 Conclusions

In this analysis, we focus on characterising the CLF and the WL halo masses
for galaxy clusters identified using two different methodologies, the red-
sequence (Rykoff et al., 2014) and halo-based method (Yang et al., 2021),
using the overlapping volume at $0.1{<}z{<}0.34$ where both catalogues are
complete. To mitigate the impact of different halo radius estimates, we adopt
the same aperture ($1h^{-1}\mathrm{Mpc}$) to measure the richness and the CLFs
for both cluster catalogues. In addition, we perform an abundance matching
scheme over the two catalogues, to make sure the richness distributions of the
two samples are the same above $\lambda=20$.

After thoroughly testing our stacking approach of measuring CLFs using a mock
sample of photometric galaxies, we apply the method to the redMaPPer and Yang
clusters by cross-correlating clusters with the SDSS photometric galaxy
sample. We find mild differences in the total number of member galaxies with
$M_{r}{<}-20.4$ between the two cluster catalogues, which is largely induced
by the different random scatters in the respective mass-richness relations.

Focusing on the shape comparison between the CLFs of the two cluster samples
at fixed $\lambda$, we observe modest differences in the behaviors of the two
samples toward the faint end; redMaPPer clusters host more low-luminosity red
satellites with a relatively flat population and Yang clusters contain a
slightly declining population of fainter red galaxies. Meanwhile, Yang
clusters exhibit a boosted population of blue galaxies at $M_{r}{<}{-}21$
compared to the redMaPPer clusters at the low richness. As a result, the red
(blue) fraction of the redMaPPer clusters is higher (lower) than the Yang
ones, especially at the low richness end.

To investigate the origin of the CLF shape discrepancy between the two
clusters finders, we split the clusters from each catalogue at fixed $\lambda$
into two subsamples, one with matched counterparts in the other catalogue and
the other without. The two sets of matched subsamples retain the same shape
discrepancy exhibited by the overall sample, indicating that the discrepancy
results largely from the clusters being assigned different centres. In
particular, after we switch to the centres found by redMaPPer for the Yang
clusters, we reproduce the flat/slightly increasing faint-end slope shown by
the overall redMaPPer clusters. Therefore, for the first time we demonstrate
that the overall shape and the faint-end slope of the CLF is sensitive to the
centroiding algorithm employed by the optical cluster finders. Therefore, a
robust scheme for mitigating miscentring is a prerequisite for any future
study of the galaxy formation physics inside clusters using CLFs, whether it
be direct or stacked measurements.

Strong colour preference in optical cluster finders does not necessarily
translate into systematic biases in the scatter of the mass-richness relation
of clusters. We investigate the latter using the same sets of matched vs. non-
matched clusters in each catalogue, by taking advantage of the orthogonality
in colour preference between the two cluster finders. In particular, if the
average halo mass of the matched clusters is significantly different from that
of the non-matched clusters at fixed $\lambda$, we would regard it as a strong
evidence for the existence of systematic biases in the cluster catalogue. We
estimate the average halo masses using the cluster weak lensing profiles
measured from the DECaLS shear catalogue using Fourier_Quad (Zhang et al.,
2015, 2019c; Zhang et al., 2022), by modelling the signal below
$2h^{-1}\mathrm{Mpc}$ as the convolution between a projected NFW profile and a
miscentring kernel function informed by the X-ray data (Zu et al., 2021). We
do not find significant discrepancies between the weak lensing halo masses
between the matched vs. non-matched clusters at halo mass above
$M_{h}{\simeq}1.5*10^{14}h^{-1}M_{\odot}$ ($\lambda>30)$ for either cluster
catalogue, suggesting that any systematic trend in the scatter of the mass-
richness relation must be weak. However, at lower richnesses, we find
discrepancies likely caused by either the inclusion of a large population of
blue galaxies or the impact of projection effects (e.g., Zu et al., 2017; Wu
et al., 2022; Wetzell et al., 2022) Therefore, the choice of how optical
clusters are identified, whether by a red-sequence based or halo-based
approach, should not impact cluster cosmological analyses, if the clusters
with $M_{h}<1.5*10^{14}h^{-1}M_{\odot}$ are excluded.

Although we do not find strong evidence of systematic uncertainties due to the
CLF preference of cluster finders at $z{<}0.34$, it is unclear whether
detection bias would plague future cluster surveys at higher redshifts. In
particular, the Rubin Observatory Legacy Survey of Space and Time (LSST;
Ivezić et al., 2019), the China Space Station Telescope (CSST; Gong et al.,
2019), and the Roman Space Telescope (Roman; Spergel et al., 2015) will allow
us to detect clusters out to $z\approx 1.5$ (Ivezić et al., 2019). Although
high-redshift clusters and proto-clusters have been identified out to $z>1.5$
(e.g., Cerulo et al., 2016; Cooke et al., 2016; Golden-Marx et al., 2019;
Willis et al., 2020; Li et al., 2022) many of the clusters identified at $z>1$
will look different than their low-redshift counterparts, and not all will be
characterized by a strong red sequence (Brodwin et al., 2013). Therefore, it
is important to keep pushing our finder comparison to higher redshifts, in
order to thoroughly understand the systematics and nuances of the different
methods of identifying galaxy clusters for LSST, CSST, and Roman.

Going forward, there are many avenues of inquiry that we aim to explore
building off the analysis presented here. In the near future, we plan to
continue to investigate the CLF, including its central galaxy component, as we
investigate whether the CLF can inform BCG hierarchical growth. We plan to
split the CLF by magnitude gap, the difference in brightness between the BCG
and nth brightest cluster member. As found in Golden-Marx & Miller (2018) the
magnitude gap is an observational proxy for hierarchical assembly so this
split may allow us to infer the properties of the progenitors of BCGs.
Additionally, we plan to perform radial splits to determine whether there
exists an observational signature of the BCG “sphere of influence” identified
by Chen et al. (2022) in the galaxy population. In the future, we plan to
improve the CLF and WL measurements by incorporating the spectroscopic data
from DESI (Abareshi et al., 2022) and centring information from eROSITA
(Predehl et al., 2021), thereby allowing us to better constrain cluster
identification and properties as we enter the era of CSST, LSST, and Roman.

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## Acknowledgements

We thank Emmet Golden-Marx, Yuanyuan Zhang, and Zhiwei Shao for stimulating
discussions about this work. JGM and YZ acknowledge the support by the
National Key Basic Research and Development Program of China (No.
2018YFA0404504), the National Science Foundation of China (11873038, 11621303,
11890692, 12173024), the science research grants from the China Manned Space
Project (No. CMS-CSST-2021-A01, CMS-CSST-2021-A02, CMS-CSST-2021-B01). JZ
acknowledges the support by the National Science Foundation of China
(11621303, 11890691, 12073017). YZ, XHY, and JZ acknowledge the support from
the “111” project of the Ministry of Education under grant No. B20019. YZ
ackonwleges the generous sponsorship from Yangyang Development Fund, and
thanks Cathy Huang for her hospitality during the pandemic at the Zhangjiang
Hi-Technology Park where he worked on this project.

##  Appendix A CLF Parameters of Matched vs. Non-Matched Subsamples

In Table 3, we present the CLF parameters for the Yang subsamples with and
without a cross-match. In Table 4, we present the CLF parameters for the
redMaPPer subsamples with and without a cross-match.

Table 3: Best-fitting CLF parameters of the Yang cluster subsamples in Figure 7. Ycen or RMcen indicates that the Yang or redMaPPer centres are adopted for the CLF measurement.  bin | M∗ | $\alpha$ | $\phi^{*}$
---|---|---|---
All galaxies of Matched Yang clusters (Ycen)
1 | -22.05 $\pm$ 0.06 | -1.07 $\pm$ 0.03 | 2.42 $\pm$ 0.13
2 | -21.73 $\pm$ 0.05 | -0.91 $\pm$ 0.03 | 4.28 $\pm$ 0.21
3 | -21.63 $\pm$ 0.06 | -0.87 $\pm$ 0.04 | 5.49 $\pm$ 0.29
4 | -21.65 $\pm$ 0.06 | -0.90 $\pm$ 0.04 | 6.68 $\pm$ 0.40
Red galaxies of Matched Yang clusters (Ycen)
1 | -21.95 $\pm$ 0.05 | -0.93 $\pm$ 0.03 | 2.58 $\pm$ 0.12
2 | -21.65 $\pm$ 0.05 | -0.78 $\pm$ 0.03 | 4.28 $\pm$ 0.18
3 | -21.62 $\pm$ 0.06 | -0.79 $\pm$ 0.04 | 5.15 $\pm$ 0.25
4 | -21.60 $\pm$ 0.06 | -0.79 $\pm$ 0.04 | 6.56 $\pm$ 0.36
Blue galaxies of Matched Yang clusters (Ycen)
1 | -21.62 $\pm$ 0.23 | -1.56 $\pm$ 0.08 | 0.24 $\pm$ 0.07
2 | -21.48 $\pm$ 0.25 | -1.42 $\pm$ 0.11 | 0.40 $\pm$ 0.13
3 | -21.36 $\pm$ 0.23 | -1.33 $\pm$ 0.12 | 0.54 $\pm$ 0.15
4 | -21.22 $\pm$ 0.30 | -1.35 $\pm$ 0.16 | 0.65 $\pm$ 0.24
All galaxies of Matched Yang clusters (RMcen)
1 | -22.28 $\pm$ 0.08 | -1.21 $\pm$ 0.04 | 1.95 $\pm$ 0.18
2 | -21.91 $\pm$ 0.07 | -1.07 $\pm$ 0.04 | 3.43 $\pm$ 0.26
3 | -21.79 $\pm$ 0.07 | -1.01 $\pm$ 0.04 | 4.43 $\pm$ 0.32
4 | -21.72 $\pm$ 0.08 | -0.97 $\pm$ 0.05 | 5.97 $\pm$ 0.53
Red galaxies of Matched Yang clusters (RMcen)
1 | -22.21 $\pm$ 0.07 | -1.11 $\pm$ 0.04 | 2.05 $\pm$ 0.17
2 | -21.86 $\pm$ 0.06 | -0.95 $\pm$ 0.04 | 3.50 $\pm$ 0.23
3 | -21.79 $\pm$ 0.07 | -0.94 $\pm$ 0.04 | 4.16 $\pm$ 0.29
4 | -21.63 $\pm$ 0.08 | -0.86 $\pm$ 0.05 | 6.11 $\pm$ 0.49
Blue galaxies of Matched Yang clusters (RMcen)
1 | -21.51 $\pm$ 0.31 | -1.62 $\pm$ 0.11 | 0.23 $\pm$ 0.10
2 | -21.25 $\pm$ 0.28 | -1.43 $\pm$ 0.13 | 0.44 $\pm$ 0.16
3 | -21.32 $\pm$ 0.25 | -1.41 $\pm$ 0.12 | 0.46 $\pm$ 0.15
4 | -21.58 $\pm$ 0.36 | -1.54 $\pm$ 0.14 | 0.35 $\pm$ 0.18
All galaxies of Non-Matched Yang clusters
1 | -21.96 $\pm$ 0.05 | -0.90 $\pm$ 0.03 | 2.52 $\pm$ 0.11
2 | -21.54 $\pm$ 0.06 | -0.69 $\pm$ 0.05 | 4.87 $\pm$ 0.25
3 | -21.65 $\pm$ 0.12 | -0.85 $\pm$ 0.08 | 5.33 $\pm$ 0.60
4 | -21.57 $\pm$ 0.22 | -0.79 $\pm$ 0.17 | 6.70 $\pm$ 1.35
Red galaxies of Non-Matched Yang clusters
1 | -21.86 $\pm$ 0.04 | -0.72 $\pm$ 0.03 | 2.53 $\pm$ 0.09
2 | -21.50 $\pm$ 0.06 | -0.52 $\pm$ 0.05 | 4.54 $\pm$ 0.20
3 | -21.59 $\pm$ 0.12 | -0.69 $\pm$ 0.09 | 5.02 $\pm$ 0.49
4 | -21.39 $\pm$ 0.22 | -0.61 $\pm$ 0.18 | 7.11 $\pm$ 1.18
Blue galaxies of Non-Matched Yang clusters
1 | -21.67 $\pm$ 0.17 | -1.44 $\pm$ 0.06 | 0.31 $\pm$ 0.06
2 | -20.98 $\pm$ 0.22 | -1.03 $\pm$ 0.17 | 0.87 $\pm$ 0.21
3 | -21.14 $\pm$ 0.44 | -1.21 $\pm$ 0.27 | 0.93 $\pm$ 0.45
4 | -20.94 $\pm$ 0.96 | -1.03 $\pm$ 0.68 | 0.75 $\pm$ 1.20
Table 4: Best-fitting CLF parameters of the redMaPPer cluster subsamples in Figure 8. Ycen or RMcen indicates that the Yang or redMaPPer centres are adopted for the CLF measurement.  bin | M∗ | $\alpha$ | $\phi^{*}$
---|---|---|---
All galaxies of Matched redMaPPer clusters (RMcen)
1 | -22.00 $\pm$ 0.05 | -1.06 $\pm$ 0.03 | 2.71 $\pm$ 0.15
2 | -21.93 $\pm$ 0.05 | -1.08 $\pm$ 0.02 | 3.57 $\pm$ 0.20
3 | -21.85 $\pm$ 0.06 | -1.09 $\pm$ 0.03 | 4.63 $\pm$ 0.29
4 | -21.77 $\pm$ 0.06 | -1.05 $\pm$ 0.03 | 6.85 $\pm$ 0.48
Red galaxies of Matched redMaPPer clusters (RMcen)
1 | -21.94 $\pm$ 0.05 | -0.94 $\pm$ 0.03 | 2.77 $\pm$ 0.13
2 | -21.86 $\pm$ 0.05 | -0.96 $\pm$ 0.03 | 3.66 $\pm$ 0.18
3 | -21.75 $\pm$ 0.06 | -0.97 $\pm$ 0.03 | 4.99 $\pm$ 0.29
4 | -21.72 $\pm$ 0.06 | -0.95 $\pm$ 0.03 | 7.00 $\pm$ 0.45
Blue galaxies of Matched redMaPPer clusters (RMcen)
1 | -21.57 $\pm$ 0.26 | -1.56 $\pm$ 0.10 | 0.25 $\pm$ 0.09
2 | -21.23 $\pm$ 0.26 | -1.41 $\pm$ 0.13 | 0.48 $\pm$ 0.16
3 | -21.56 $\pm$ 0.33 | -1.67 $\pm$ 0.11 | 0.30 $\pm$ 0.14
4 | -21.44 $\pm$ 0.33 | -1.59 $\pm$ 0.12 | 0.42 $\pm$ 0.19
All galaxies of Matched redMaPPer clusters (Ycen)
1 | -21.93 $\pm$ 0.04 | -0.96 $\pm$ 0.02 | 3.12 $\pm$ 0.13
2 | -21.91 $\pm$ 0.05 | -0.99 $\pm$ 0.03 | 3.74 $\pm$ 0.19
3 | -21.74 $\pm$ 0.06 | -0.99 $\pm$ 0.03 | 5.09 $\pm$ 0.30
4 | -21.58 $\pm$ 0.07 | -0.93 $\pm$ 0.04 | 7.36 $\pm$ 0.48
Red galaxies of Matched redMaPPer clusters (Ycen)
1 | -21.84 $\pm$ 0.04 | -0.81 $\pm$ 0.02 | 3.22 $\pm$ 0.12
2 | -21.83 $\pm$ 0.05 | -0.87 $\pm$ 0.03 | 3.81 $\pm$ 0.16
3 | -21.68 $\pm$ 0.05 | -0.87 $\pm$ 0.03 | 5.14 $\pm$ 0.26
4 | -21.45 $\pm$ 0.07 | -0.77 $\pm$ 0.04 | 7.85 $\pm$ 0.45
Blue galaxies of Matched redMaPPer clusters (Ycen)
1 | -21.46 $\pm$ 0.18 | -1.42 $\pm$ 0.08 | 0.36 $\pm$ 0.08
2 | -21.43 $\pm$ 0.25 | -1.44 $\pm$ 0.11 | 0.41 $\pm$ 0.13
3 | -21.68 $\pm$ 0.30 | -1.56 $\pm$ 0.11 | 0.33 $\pm$ 0.13
4 | -21.40 $\pm$ 0.29 | -1.56 $\pm$ 0.11 | 0.48 $\pm$ 0.19
All galaxies of Non-Matched redMaPPer clusters
1 | -22.15 $\pm$ 0.05 | -1.19 $\pm$ 0.02 | 1.92 $\pm$ 0.12
2 | -21.96 $\pm$ 0.08 | -1.11 $\pm$ 0.03 | 3.25 $\pm$ 0.26
3 | -21.75 $\pm$ 0.09 | -1.10 $\pm$ 0.04 | 4.91 $\pm$ 0.47
4 | -21.75 $\pm$ 0.11 | -1.08 $\pm$ 0.05 | 6.90 $\pm$ 0.76
Red galaxies of Non-Matched redMaPPer clusters
1 | -22.06 $\pm$ 0.05 | -1.05 $\pm$ 0.02 | 2.13 $\pm$ 0.11
2 | -21.80 $\pm$ 0.07 | -0.95 $\pm$ 0.04 | 3.67 $\pm$ 0.25
3 | -21.64 $\pm$ 0.08 | -0.97 $\pm$ 0.04 | 5.19 $\pm$ 0.55
4 | -21.64 $\pm$ 0.10 | -0.95 $\pm$ 0.05 | 7.40 $\pm$ 0.93
Blue galaxies of Non-Matched redMaPPer clusters
1 | -20.75 $\pm$ 0.24 | -1.51 $\pm$ 0.11 | 0.49 $\pm$ 0.16
2 | -21.43 $\pm$ 0.50 | -1.67 $\pm$ 0.13 | 0.27 $\pm$ 0.18
3 | -21.00 $\pm$ 0.43 | -1.39 $\pm$ 0.19 | 0.77 $\pm$ 0.44
4 | -21.00 $\pm$ 0.64 | -1.59 $\pm$ 0.24 | 0.78 $\pm$ 0.70

##  Appendix B Theoretical Model of $\Delta\Sigma$

In this analysis, we use a Bayesian framework to infer $M_{h}^{\mathrm{WL}}$
directly from the $\Delta\Sigma$ profiles, following the same mathematical
formalism as outlined in Zu et al. (2021), which are computed as

$\Delta\Sigma(r_{p})=\overline{\Sigma}({<}r_{p})-\Sigma(r_{p}),$ (2)

where $\overline{\Sigma}({<}r_{p})$ and $\Sigma(r_{p})$ are the average
surface matter density within and at the projected radius rp. Without
miscentring, $\Sigma(r_{p})$ can be computed from the 3D isotropic halo-matter
cross-correlation function $\xi_{\rm hm}(r)$,

$\Sigma(r_{p})=\rho_{m}\int_{-\infty}^{+\infty}\xi_{\rm
hm}(r_{p},r_{\pi})dr_{\pi},$ (3)

where $\rho_{m}$ is the mean density of the Universe.

To model the effect of miscentring, we assume that the fraction of miscentred
BCGs is $f_{\rm off}$ with an offset of $\sigma_{\rm off}$ from the true
cluster centre and that this distribution follows a shape-2 Gamma distribution
$p(r_{\rm off})$ with a characteristic offset $\sigma_{\rm off}$,

$p(r_{\rm off})=\frac{r_{\rm off}}{\sigma_{\rm
off}^{2}}\exp\left(-\frac{r_{\rm off}}{\sigma_{\rm off}}\right).$ (4)

Thus, the observed surface matter density profile when accounting for
miscentring can be written as

$\Sigma^{\rm obs}(r_{p})=f_{\rm off}\Sigma^{\rm off}(r_{p})+(1-f_{\rm
off})\Sigma(r_{p}),$ (5)

where

$\begin{split}\Sigma^{\rm off}(r_{p})=&\frac{1}{2\pi}\int_{0}^{\infty}dr_{\rm
off}p(r_{\rm off})\\\
&\times\int_{0}^{2\pi}d\theta\Sigma(\sqrt{r_{p}^{2}+r_{\rm
off}^{2}-2r_{p}r_{\rm off}^{2}}\cos{\theta})\end{split}\\\ $ (6)

The purpose of this weak lensing analysis is to compare the average halo
masses of different cluster samples, which are primarily inferred from
$\Delta\Sigma$ on scales below $2h^{-1}\mathrm{Mpc}$ (Zu et al., 2014).
Therefore, we only model $\xi_{\rm 1h}$, the “1-halo” term of $\xi_{\rm hm}$,
for predicting $\Delta\Sigma$ at $r_{p}{<}2h^{-1}\mathrm{Mpc}$ Zu et al.
(2014)

$\xi_{\rm 1h}=\frac{\rho_{\mathrm{NFW}}(r|M_{h})}{\rho_{m}}-1,$ (7)

where $\rho_{\mathrm{NFW}}$ is the NFW density profile. Additionally, we fit
to the observed $\Delta\Sigma$ down to a minimum scale of
$r_{p}=0.1h^{-1}\mathrm{Mpc}$, below which the extra lensing effect caused by
the stellar mass of the BCG needs to be incorporated.

###  B.1 WL Parameters

In Tables 5 and 6, we present the 1D posterior distributions for the fitting
to the $\Delta\Sigma$ profiles used to estimate $M_{h}^{\mathrm{WL}}$.

Table 5: Best-fitting WL model parameters of the Yang cluster subsamples in Figures 9 and 11. bin | log $M_{h}^{\mathrm{WL}}$ | c | $\sigma_{\mathrm{off}}$ | foff
---|---|---|---|---
Overall Yang clusters
1 | 13.89 $\pm$ 0.09 | 5.06 $\pm$ 1.15 | 0.43 $\pm$ 0.30 | 0.23 $\pm$ 0.13
2 | 14.14 $\pm$ 0.08 | 4.58 $\pm$ 1.30 | 0.43 $\pm$ 0.31 | 0.35 $\pm$ 0.18
3 | 14.34 $\pm$ 0.11 | 4.08 $\pm$ 1.52 | 0.31 $\pm$ 0.25 | 0.52 $\pm$ 0.24
4 | 14.53 $\pm$ 0.12 | 4.79 $\pm$ 1.27 | 0.50 $\pm$ 0.31 | 0.42 $\pm$ 0.17
Matched Yang clusters
1 | 14.12 $\pm$ 0.14 | 5.18 $\pm$ 1.21 | 0.52 $\pm$ 0.29 | 0.31 $\pm$ 0.19
2 | 14.05 $\pm$ 0.17 | 5.36 $\pm$ 1.29 | 0.52 $\pm$ 0.29 | 0.31 $\pm$ 0.19
3 | 14.37 $\pm$ 0.12 | 4.50 $\pm$ 1.46 | 0.26 $\pm$ 0.25 | 0.52 $\pm$ 0.22
4 | 14.55 $\pm$ 0.10 | 4.00 $\pm$ 1.03 | 0.42 $\pm$ 0.26 | 0.34 $\pm$ 0.15
Non-Matched Yang clusters
1 | 13.80 $\pm$ 0.10 | 5.02 $\pm$ 1.22 | 0.45 $\pm$ 0.30 | 0.21 $\pm$ 0.14
2 | 14.01 $\pm$ 0.14 | 5.31 $\pm$ 1.20 | 0.52 $\pm$ 0.31 | 0.17 $\pm$ 0.15
3 | 14.28 $\pm$ 0.19 | 4.50 $\pm$ 1.36 | 0.39 $\pm$ 0.29 | 0.29 $\pm$ 0.21
4 | 14.90 $\pm$ 0.19 | 4.76 $\pm$ 1.36 | 0.49 $\pm$ 0.30 | 0.59 $\pm$ 0.23
Table 6: Best-fitting WL model parameters of the redMaPPer cluster samples in Figures 9 and 12. bin | $\log M_{h}^{\mathrm{WL}}$ | c | $\sigma_{\mathrm{off}}$ | foff
---|---|---|---|---
Overall redMaPPer clusters
1 | 13.95 $\pm$ 0.04 | 7.09 $\pm$ 0.90 | 0.18 $\pm$ 0.02 | 0.27 $\pm$ 0.04
2 | 14.28 $\pm$ 0.06 | 5.24 $\pm$ 0.88 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04
3 | 14.53 $\pm$ 0.07 | 3.65 $\pm$ 0.75 | 0.18 $\pm$ 0.02 | 0.31 $\pm$ 0.04
4 | 14.70 $\pm$ 0.06 | 4.88 $\pm$ 0.82 | 0.18 $\pm$ 0.02 | 0.17 $\pm$ 0.04
Matched redMaPPer clusters
1 | 14.18 $\pm$ 0.05 | 5.81 $\pm$ 0.90 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04
2 | 14.34 $\pm$ 0.07 | 4.62 $\pm$ 0.95 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04
3 | 14.43 $\pm$ 0.09 | 3.84 $\pm$ 1.01 | 0.18 $\pm$ 0.02 | 0.31 $\pm$ 0.04
4 | 14.65 $\pm$ 0.06 | 4.98 $\pm$ 0.84 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04
Non-Matched redMaPPer clusters
1 | 13.96 $\pm$ 0.07 | 5.00 $\pm$ 0.94 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04
2 | 14.29 $\pm$ 0.10 | 4.61 $\pm$ 1.16 | 0.18 $\pm$ 0.18 | 0.31 $\pm$ 0.04
3 | 14.43 $\pm$ 0.15 | 3.51 $\pm$ 1.10 | 0.18 $\pm$ 0.02 | 0.28 $\pm$ 0.04
4 | 14.53 $\pm$ 0.14 | 5.40 $\pm$ 1.02 | 0.18 $\pm$ 0.02 | 0.30 $\pm$ 0.04

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