# The Massive and Distant Clusters of WISE Survey XI: Stellar Mass Fractions
and Luminosity Functions of MaDCoWS Clusters at $z\sim 1$

Bandon Decker  Department of Physics and Astronomy, University of Missouri,
5110 Rockhill Road, Kansas City, MO 64110, USA  Mark Brodwin  Department of
Physics and Astronomy, University of Missouri, 5110 Rockhill Road, Kansas
City, MO 64110, USA  Ripon Saha  Department of Physics and Astronomy,
University of Missouri, 5110 Rockhill Road, Kansas City, MO 64110, USA  Thomas
Connor  Jet Propulsion Laboratory, California Institute of Technology, 4800
Oak Grove Drive, Pasadena, CA 91109, USA  Peter R. M. Eisenhardt  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Anthony H. Gonzalez  Department of Astronomy,
University of Florida, 211 Bryant Space Center, Gainesville, FL 32611, USA
Emily Moravec  Astronomical Institute of the Czech Academy of Sciences, Boční
ll 1401/1A, 14100 Praha 4, Czech Republic  Mustafa Muhibullah  Department of
Physics and Astronomy, University of Missouri, 5110 Rockhill Road, Kansas
City, MO 64110, USA  S. Adam Stanford  Department of Physics, University of
California, One Shields Avenue, Davis, CA 95616, USA  Daniel Stern  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Khunanon Thongkham  Department of Astronomy,
University of Florida, 211 Bryant Space Center, Gainesville, FL 32611, USA
Dominika Wylezalek  Astronomisches Rechen-Institut, Zentrum für Astronomie der
Universität Heidelberg, Mönchhofstr. 12-14, D-69120 Heidelberg, Germany  Simon
R. Dicker  Department of Physics and Astronomy, University of Pennsylvania,
209 South 33rd Street, Philadelphia, PA 19104, USA  Brian Mason  National
Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville VA 22903, USA
Tony Mroczkowski  European Southern Observatory, Karl Schwarzschild Straße 2,
D-85748, Garching bei München, Germany  Charles E. Romero  Department of
Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street,
Philadelphia, PA 19104, USA  Green Bank Observatory, Green Bank, WV 24944, USA
Florian Ruppin  Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA

###### Abstract

We present stellar mass fractions and composite luminosity functions (LFs) for
a sample of 12 clusters from the Massive and Distant Clusters of WISE Survey
(MaDCoWS) at a redshift range of $0.951\leq z\leq 1.43$. Using SED fitting of
optical and deep mid-infrared photometry, we establish the membership of
objects along the lines-of-sight to these clusters and calculate the stellar
masses of member galaxies. We find stellar mass fractions for these clusters
largely consistent with previous works, including appearing to display a
negative correlation with total cluster mass. We measure a composite
$3.6~{}\mathrm{\mu m}$ LF down to $m^{*}+2.5$ for all 12 clusters. Fitting a
Schechter function to the LF, we find a characteristic $3.6~{}\mathrm{\mu m}$
magnitude of $m^{*}=19.83\pm 0.12$ and faint-end slope of $\alpha=-0.81\pm
0.10$ for the full sample at a mean redshift of $\bar{z}=1.18$. We also divide
the clusters into high- and low-redshift bins at $\bar{z}=1.29$ and
$\bar{z}=1.06$ respectively and measure a composite LF for each bin. We see a
small, but statistically significant evolution in $m^{*}$ and
$\alpha$—consistent with passive evolution—when we study the joint fit to the
two parameters, which is probing the evolution of faint cluster galaxies at
$z\sim 1$. This highlights the importance of deep IR data in studying the
evolution of cluster galaxy populations at high-redshift.

cosmology: observations — galaxies: clusters: general — galaxies: clusters

##  1 Introduction

The evolution of galaxies in clusters both influences and is influenced by the
partitioning of baryons between the stars in galaxies and the hot gas of the
intracluster medium (ICM). The stellar mass fraction of a cluster—that is, the
fraction of the total mass in stars—$f_{\star}$, offers an in situ measurement
of this partitioning. Measuring $f_{\star}$ and its relation to other cluster
properties can give insight into the feedback processes that drive the cycling
of baryons between states and which affect how galaxies grow and evolve (e.g.,
Lin et al., 2003; Ettori et al., 2006; Conroy et al., 2007). In addition, the
shape of the cluster luminosity function (LF) offers insight into the mass-
assembly history of the cluster. The near-infrared (NIR) LF is a useful proxy
for the stellar mass function, as the luminosity in those bands is tightly
correlated with stellar mass. The NIR LF parameters and how they evolve over
time therefore reflect the mass-assembly history of galaxies in the cluster
(Kauffmann & Charlot, 1998).

Previous studies such as Gonzalez et al. (2013) have analyzed the trend of
$f_{\star}$ with total cluster mass at $z\sim 0.1$ and found an anti-
correlation. This suggests that in the local universe, larger clusters retain
their gas better and are less efficient at forming stars. In Decker et al.
(2019), we also studied the trend of $f_{\star}$ with cluster mass for a
sample of infrared-selected clusters at high redshift and compared this to a
sample of ICM-selected clusters at comparable redshifts to look for
differences due to selection. While we found a larger scatter in $f_{\star}$
in the ICM-selected clusters, there was no significant offset between the two
samples. We also measured a relationship between $f_{\star}$ and total mass
that was consistent with that found by Gonzalez et al. (2013), but the scatter
and systematic uncertainties were too high to draw firm conclusions.

Cluster LFs follow a Schechter distribution (Schechter, 1976) of the form

$\Phi(m)=\frac{\ln(10)}{2.5}\Phi^{*}10^{-0.4(m-m^{*})(\alpha+1)}\mathrm{exp}(-10^{-0.4(m-m^{*})})$

where the overall scaling is paramaterized as $\Phi^{*}$, the characteristic
magnitude ‘knee’ at the bright end is parameterized as $m^{*}$ and the slope
of the faint-end of the function is parameterized as $\alpha$. Many studies
have examined the rest-frame NIR LF of galaxy clusters (e.g., de Propris et
al., 1999; Strazzullo et al., 2006; Muzzin et al., 2007; Mancone et al., 2012;
Wylezalek et al., 2014; Chan et al., 2019) to measure the characteristic
magnitude of the cluster LF at different redshifts. However, measuring $m^{*}$
becomes more difficult at high-redshift because there is a strong degeneracy
between $m^{*}$ and $\alpha$. Therefore meaningfully measuring the former at
high-redshift requires increasingly deep mid-infrared data to also constrain
the latter. Indeed, few previous studies have measured the NIR LF for all
cluster members (i.e., those both on and off the red-sequence) at $z>1$ down
to a depth sufficient to jointly fit both $m^{*}$ and $\alpha$.

To address these problems, we measure stellar mass fractions and LFs for a
sample of clusters from the Massive and Distant Clusters of WISE Survey
(MaDCoWS, Gonzalez et al., 2019). For this work we use MaDCoWS clusters with
previously measured Sunyaev-Zel’dovich (SZ, Sunyaev & Zeldovich, 1970, 1972)
masses, which allows us to measure $f_{\star}$ and compare to the total mass.
We also limit our sample to clusters that have deep mid-infrared photometry.
This allows us to determine the stellar mass more robustly than in our
previous study, Decker et al. (2019), and also allows us to measure the rest-
frame NIR LF down to sufficiently faint magnitudes to fit $m^{*}$ and $\alpha$
jointly. Finally, we limit our sample to clusters that—in addition to the
above criteria—also have optical follow-up data. This allows us to better
determine which objects are true members of the clusters, reducing systematic
errors in our measurements both of $f_{\star}$ and the LF parameters.

We present our cluster sample and describe in more detail the follow-up data
in §2 and describe our analysis in §3. Our results for both $f_{\star}$ and
the LFs are in §4 and we discuss those results in §5. Throughout this paper we
use AB magnitudes in all bands and a concordance $\Lambda$CDM cosmology of
$\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ and
$H_{0}=70~{}\mathrm{km~{}s^{-1}~{}Mpc^{-1}}$. We define $r_{500}$ as the
radius inside which the cluster density is 500 times the critical density of
the universe at the cluster redshift and $M_{500}$ as the mass interior to
that radius.

##  2 Cluster Sample and Data

For this work, we use 12 clusters from the MaDCoWS catalog. These clusters are
drawn from the much larger sample with SZ masses from Brodwin et al. (2015),
Gonzalez et al. (2015), Decker et al. (2019), Di Mascolo et al. (2020), Dicker
et al. (2020), and Orlowski-Scherer et al. (2021) and the SZ masses from
different facilities are generally in good agreement with each other. This
sample of clusters are selected to have previously reported spectroscopic
redshifts, deep follow-up imaging in the mid-infrared, and optical follow-up
photometry. This arrangement of follow-up data was chosen as it allows us to
constrain the membership of clusters using photometric redshift fitting. The
clusters are listed in Table 1, along with their redshifts and information
about the relevant observations. Details of the follow-up data are given below
and details of the SZ observations and mass calculations can be found in the
relevant papers.

###  2.1 Optical Data

All 12 clusters have $r$\- and $z$-band imaging from the Gemini Multi-Object
Spectrograph (GMOS, Hook et al., 2004) on the Gemini Telescopes in Hawai’i and
Chile. These images were taken in several programs: GN-2013A-Q-44,
GN-2013B-Q-8 (both PI: Brodwin), GN-2015A-Q-42 (PI: Perlmutter), GN-2015A-Q-4
(PI: Stalder), GN-2017B-LP-15, GN-2018A-LP-15 (both PI: Stanford), and
GS-2019A-FT-205 (PI: Decker). There was a heterogenous mix of observing
strategies for these programs, partly due to the different sensitivities of
the GMOS CCD during different observing cycles. However they result in a
comparable depth for all the clusters. All of the exposure times are listed in
Table 1.

###  2.2 Infrared Data

These clusters have mid-infrared data from the $3.6~{}\mathrm{\mu m}$ and
$4.5~{}\mathrm{\mu m}$ bands of the Spitzer Space Telescope Infrared Array
Camera (IRAC, Fazio et al., 2004). They were imaged in programs 12101 and
13214 (both PI: Brodwin) and the exposure times for each cluster and each band
are listed in Table 1. Both programs had the same observing strategy, with the
varying exposure times designed to detect galaxies to a relatively uniform
depth relative to $m^{*}$ in different IR background regions.

Table 1: Summary of observations of MaDCoWS clusters used in this analysis Cluster ID | RA | Dec. | $z$ |  $t_{\rm exp}$ (s) |  $t_{\rm exp}$ (s) |  $t_{\rm exp}$ (s) |  $t_{\rm exp}$ (s) | SZ Facility
---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) |  |  $r$-band |  $z$-band | $3.6~{}\mathrm{\mu m}$ | $4.5~{}\mathrm{\mu m}$ |
MOO J0105$+$1323 | 01:05:31.5 |  $+$13:23:55 | 1.143 | $13\times 120$ | $9\times 150$ | $14\times 100$ | $33\times 100$ | CARMAaaDecker et al. (2019)
MOO J0319$-$0025 | 03:19:24.4 |  $-$00:25:21 | 1.194 | $5\times 180$ | $12\times 80$ | $10\times 100$ | $24\times 100$ | CARMAbbBrodwin et al. (2015), with mass recalculated in Decker et al. (2019)
MOO J0917$-$0700 | 09:17:04.7 |  $-$07:00:08 | 1.10 | $6\times 180$ | $6\times 60$ | $10\times 100$ | $24\times 100$ | ALMAccDi Mascolo et al. (2020)
MOO J1111$+$1503 | 11:11:42.6 |  $+$15:03:44 | 1.32 | $4\times 300$ | $16\times 80$ | $14\times 100$ | $33\times 100$ | CARMAaaDecker et al. (2019)
MOO J1139$-$1706 | 11:39:28.2 |  $-$17:06:31 | 1.31 | $12\times 120$ | $10\times 60$ | $14\times 100$ | $34\times 100$ | ALMAccDi Mascolo et al. (2020)
MOO J1142$+$1527 | 11:42:45.1 |  $+$15:27:05 | 1.189 | $5\times 180$ | $12\times 80$ | $14\times 100$ | $33\times 100$ | CARMAddGonzalez et al. (2015), with mass recalculated in Decker et al. (2019)
MOO J1155$+$3901 | 11:55:45.6 |  $+$39:01:15 | 1.009 | $5\times 180$ | $12\times 80$ | $12\times 100$ | $26\times 100$ | CARMAbbBrodwin et al. (2015), with mass recalculated in Decker et al. (2019)
MOO J1329$+$5647 | 13:29:50.7 |  $+$56:48:03 | 1.43 | $6\times 180$ | $6\times 60$ | $10\times 100$ | $24\times 100$ | GBTeeDicker et al. (2020)
MOO J1506$+$5136 | 15:06:22.7 |  $+$51:36:45 | 1.09 | $35\times 120$ | $8\times 60$ | $10\times 100$ | $24\times 100$ | GBTeeDicker et al. (2020)
MOO J1514$+$1346 | 15:14:42.7 |  $+$13:46:31 | 1.059 | $5\times 180$ | $12\times 80$ | $10\times 100$ | $24\times 100$ | CARMAbbBrodwin et al. (2015), with mass recalculated in Decker et al. (2019)
MOO J1521$+$0452 | 15:21:04.6 |  $+$04:52:08 | 1.312 | $12\times 120$ | $9\times 150$ | $12\times 100$ | $26\times 100$ | CARMAaaDecker et al. (2019)
MOO J2206$+$0906 | 22:06:28.6 |  $+$09:06:32 | 0.951 | $5\times 180$ | $12\times 80$ | $10\times 100$ | $24\times 100$ | CARMAaaDecker et al. (2019)

###  2.3 Catalogs

For each cluster, we used the optical and infrared imaging described in §2.1
and §2.2 to make four-band ($r$, $z$, $3.6~{}\mathrm{\mu m}$,
$4.5~{}\mathrm{\mu m}$) photometric catalogs. For each cluster, all four
images were transformed onto the same image scale using SWarp (Bertin et al.,
2002). The catalogs were produced by running Source Extractor (SE, Bertin &
Arnouts, 1996) in dual-image mode on all four SWarped images, using the
$3.6~{}\mathrm{\mu m}$ image as the detection image. The SE parameters were
the same as in Decker et al. (2019). The final catalogs used $2\arcsec$
diameter aperture photometry in the optical bands and $4\arcsec$ diameter
corrected to $24\arcsec$ diameter aperture photometry in the infrared bands.
The correction from $4\arcsec$ to $24\arcsec$ used the IRAC aperture
corrections given in Ashby et al. (2009).

For comparison and validation of our fitting (see next section) we also made a
field catalog using $r$\- and $z$\- band images from the Cosmic Evolution
Survey (COSMOS, Scoville et al., 2007) and $3.6~{}\mathrm{\mu m}$ and
$4.5~{}\mathrm{\mu m}$ images from the Spitzer Extended Deep Survey (Ashby et
al., 2013) in the footprint where those two surveys overlap. These catalogs
were made with the same procedure as our cluster catalogs. Because the Spitzer
Extended Deep Survey imaging is deeper than our IRAC imaging, we artificially
degraded the field catalog data in the IRAC bands to match our cluster
catalogs. We do this by adding a small additional scatter to the measured
fluxes. This scatter is randomly drawn from a Gaussian distribution with a
width equal to the quadrature difference of the (higher) error in our
photometry and the error in the Extended Deep Survey.

##  3 Analysis

###  3.1 Cluster Membership

We used EAZY (Brammer et al., 2008) to fit spectral energy distributions
(SEDs) to the four-band photometry for each object in our cluster catalogs and
our field catalog. The result was a best-value and a probability density
function (PDF) of the redshift for each object. We compared the best-value
fitted redshifts in the field catalog to the multi-band photometric redshifts
from the COSMOS catalog (Laigle et al., 2016) to determine the error in our
photometric redshift fitting. After running an iterative $3\sigma$ clipping
routine, we found an error in our photometric redshifts of
$\sigma_{z}=0.17(1+z)$. This error is relatively high, due to the small number
of photometric bands, but is sufficient to isolate cluster members with a low
interloper rate.

To determine which objects in the cluster catalogs were consistent with being
members of the cluster, we used the full PDFs output by EAZY. For each object,
we first smoothed the output PDF with a Gaussian corresponding to the
$\sigma_{z}=0.17(1+z)$ scatter in our redshift fitting. We then integrated
under this convolved PDF in the range $z_{cl}-\sigma_{z}\leq
z<z_{cl}+\sigma_{z}$, as shown in Figure 1. If this integrated probability was
above 0.3, we considered the object to be a cluster member; everything else
was removed from the catalog. We chose 0.3 as a cutoff to maximize
completeness while still removing the bulk of the line-of-sight interlopers.
Because our cluster masses were only measured out to $r_{500}$, we also
removed from our cluster catalogs objects lying at a projected distance more
than that distance from the cluster center. Since this method would still not
remove every line-of-sight interloper, we also ran this analysis on the field
catalog at each cluster redshift. This provided a set of 12 field catalogs,
each containing the objects from the full field catalog that our analysis
would consider being consistent with cluster members. These ‘interloper’
catalogs provided us a baseline that allowed us to statistically remove line-
of-sight interlopers from our analysis.

Figure 1: Smoothed output PDF from EAZY of a spectroscopically-confirmed (from
Gonzalez et al., 2015) member of MOO J1142$+$1527\. The cluster redshift of
$z=1.19$ is denoted by the black dashed line and the shaded blue region is the
integration range around the cluster redshift. The integrated probability in
that region is 0.65, well above our membership threshold.

###  3.2 Completeness

We measured the photometric completeness of our cluster catalogs by randomly
placing artificial sources into our detection images, running SE, and
recording how many of these artificial sources were detected by SE. For each
cluster we placed a total of one thousand sources per quarter-magnitude bin in
batches of ten sources each. The average $3.6~{}\mathrm{\mu m}$ completeness
curve of the sample is shown as a light blue line in Figure 2. The
completeness reaches a plateau at around 95% at the bright end because of the
high density of infrared sources in the clusters. The vertical dashed line
represents the average $5\sigma$ limit of our data in that cluster. The errors
on the completeness in each bin are Poisson errors and are approximately $3\%$
per bin.

In addition to measuring the photometric completeness of our catalogs, we also
measured the completeness of our cluster member selection algorithm. Each
object in the field catalogs has a redshift from COSMOS, and some
coincidentally lie at the redshifts of our clusters. For each cluster, we
isolated these objects from the field catalog and ran our membership selection
algorithm on them. As with the detection completeness, we split the objects
into quarter-magnitude bins in $3.6~{}\mathrm{\mu m}$. For each quarter-
magnitude bin, we define the membership completeness as the fraction of
objects that our algorithm correctly identified as lying at the cluster
redshift. Since unlike with the artificial sources we used for our detection
completeness, there were a variable number of objects in each bin, we
calculated the error on the membership completeness using bootstrap
resampling. The average membership completeness is shown in Figure 2 as a dark
blue line. Since there were not enough objects at the bright end to have
meaningful statistics, we fixed the completeness in that region to unity.

Figure 2: Mean detection (light blue) and membership (dark blue) completeness
for the clusters in our sample. The error bars on the detection completeness
are Poisson errors and the error bars on the membership completeness are from
bootstrap resampling. At the bright end of the membership completeness curve
($3.6~{}\mathrm{\mu m}$$<18.75$) there were not enough field objects to
produce meaningful statistics, so we fixed the completeness to unity. The
vertical dashed red line at the faint end represents the average $5\sigma$
detection limit.

###  3.3 Stellar Mass

We used another SED-fitting program, FAST (Kriek et al., 2009), on the cluster
catalogs to calculate the stellar masses of the objects along the line of
sight to the cluster. For this, we adopted a Bruzual & Charlot (2003) model, a
Chabrier (2003) initial mass function, a solar metallicity, and we fixed the
redshift of each object in the catalog to the cluster redshift. We only fit to
intrinsic properties of the galaxies, in particular stellar mass. As with the
redshift fitting, we also ran FAST on the field catalog to calibrate the
errors in our fitting and to establish how much field contribution to expect
even after removing interlopers. Comparing the stellar masses we measured in
this way to the stellar masses given in the COSMOS catalog, we adopted a
uniform $0.2~{}\mathrm{dex}$ uncertainty in our stellar mass measurements.

With this final catalog of objects identified as being at the cluster redshift
by EAZY, lying within a projected distance of $r_{500}$ from the cluster
center, and with stellar masses measured from FAST, we calculated the total
stellar mass of the clusters inside $r_{500}$. For each cluster, we first
scaled the stellar mass of each object by the photometric and membership
completeness corrections in §3.2. We then summed these scaled masses to get a
total line-of-sight stellar mass for the cluster. Since this is still expected
to include some small number of interlopers, we measured the total stellar
mass of the statistical interloper catalog in the same way. We scaled that
mass to the area of the cluster catalog and subtracted this expected
interloper contribution—approximately 10% of the line-of-sight mass for most
of the clusters—from the line-of-sight mass to get the total cluster stellar
mass. We calculated the error on the stellar mass of each cluster by
propagating the error of the stellar mass of each individual object in the
cluster and field catalogs, measured in §3.1.

###  3.4 Luminosity Function

We used our membership selection and deep IRAC photometry to produce composite
$3.6~{}\mathrm{\mu m}$ LFs for our cluster sample. For each cluster, we first
evolution-corrected the $3.6~{}\mathrm{\mu m}$ apparent magnitudes of both the
cluster and interloper catalog to the mean redshift of the sample using EZGal
(Mancone & Gonzalez, 2012) and assuming passive evolution after an initial
starburst at $z_{f}=3.0$. We removed the brightest cluster galaxy (BCG) from
the cluster catalog and then binned these evolution-corrected catalogs into
quarter-magnitude bins to produce a line-of-sight LF and a background LF for
each cluster. We then applied both the completeness corrections described in
§3.2 as a function of magnitude to both LFs. Finally, we scaled the background
LF to match the surface area of the cluster and subtracted it off the line-of-
sight LF to produce the individual cluster LF. These individual LFs were
stacked to form the composite LF for the sample. The error on each value in
the individual LFs is from adding in quadrature the Poisson errors of both the
line-of-sight and interloper LFs and the errors on both completeness
corrections. The error on each value in the composite LF is the quadrature sum
of those errors from the individual LFs.

##  4 Results

###  4.1 Stellar Mass Fractions

The stellar mass fractions we measure for these clusters are given in Table 2
and Figure 3 shows $f_{\star}$ versus $M_{500}$ for the 12 clusters of this
work, plotted as red diamonds. For comparison, we also plot the low-redshift
trend line measured by Gonzalez et al. (2013) as a green dashed line. Clusters
that were also studied in Decker et al. (2019) are indicated in Table 2.

Figure 3: Stellar mass fraction versus total mass for the MaDCoWS clusters in
this analysis, plotted as red diamonds. The error on $f_{\star}$ is from
adding the percent stellar mass error from §3.3 and the percent error of the
total mass of the cluster in quadrature. The green dashed line is the low-
redshift relation from Gonzalez et al. (2013), plotted to provide continuity
with Decker et al. (2019). Table 2: MaDCoWS Stellar Mass Fractions

ID | $M_{500}$ | $M_{\star}$ | $f_{\star}$
---|---|---|---
| $(10^{14}~{}M_{\odot})$ | $(10^{12}~{}M_{\odot})$ | $(10^{-2})$
MOO J0105$+$1323* | $3.92^{+0.46}_{-0.44}$ | $10.6\pm 0.58$ | $2.70^{+0.35}_{-0.34}$
MOO J0319$-$0025* | $3.03^{+0.54}_{-0.46}$ | $6.32\pm 0.49$ | $2.09^{+0.40}_{-0.36}$
MOO J0917$-$0700 | $1.66^{+0.31}_{-0.38}$ | $4.08\pm 0.36$ | $2.48^{+0.51}_{-0.61}$
MOO J1111$+$1503* | $2.02^{+0.30}_{-0.30}$ | $4.95\pm 0.40$ | $2.45^{+0.42}_{-0.42}$
MOO J1139$-$1706 | $2.24^{+0.36}_{-0.52}$ | $8.40\pm 0.66$ | $3.80^{+0.68}_{-0.93}$
MOO J1142$+$1527* | $5.36^{+0.55}_{-0.50}$ | $9.91\pm 0.59$ | $1.85^{+0.22}_{-0.20}$
MOO J1155$+$3901* | $2.53^{+0.50}_{-0.51}$ | $3.21\pm 0.29$ | $1.27^{+0.28}_{-0.28}$
MOO J1329$+$5647 | $3.56^{+0.35}_{-0.33}$ | $7.89\pm 0.64$ | $2.25^{+0.28}_{-0.27}$
MOO J1506$+$5136 | $3.17^{+0.38}_{-0.37}$ | $3.59\pm 0.27$ | $1.16^{+0.16}_{-0.16}$
MOO J1514$+$1346* | $2.39^{+0.51}_{-0.83}$ | $7.34\pm 0.49$ | $3.97^{+1.42}_{-1.67}$
MOO J1521$+$0452* | $3.59^{+1.02}_{-0.92}$ | $7.15\pm 0.60$ | $1.99^{+0.59}_{-0.54}$
MOO J2206$+$0906* | $2.95^{+0.82}_{-0.68}$ | $6.52\pm 0.40$ | $2.52^{+0.90}_{-0.72}$

**footnotetext: Also in Decker et al. (2019)

Figure 4 shows the direct comparison of $f_{\star}$ for the eight clusters
common to both this work and Decker et al. (2019). With only one exception,
MOO J0105$+$1323, the $f_{\star}$ we measure in this work is higher than the
$f_{\star}$ we found in Decker et al. (2019) and for no clusters is it
significantly lower. This is expected as in both works we use IRAC
$3.6~{}\mathrm{\mu m}$ to measure $f_{\star}$—either directly or
indirectly—and the much deeper $3.6~{}\mathrm{\mu m}$ data in this work allow
us to include stellar mass from galaxies that were too faint to be Using
$3.6~{}\mathrm{\mu m}$ luminosity as a proxy for stellar mass, we quantify the
amount of ‘extra’ stellar mass we should expect with these deeper data by
integrating down the composite LF we measure in §4.2. Integrating down to the
depth of our current data versus integrating to the depth of our data in
Decker et al. (2019) shows we are sensitive to approximately 25% more stellar
mass with these deep IRAC data than we were previously. This is consistent
with the change in $f_{\star}$ we see in all but two of the clusters. Further
integrating the LF arbitrarily deep shows this analysis is sensitive to $\geq
95\%$ of the stellar mass in each cluster.

Figure 4: Comparison of the stellar mass fractions of clusters common to this
work and to Decker et al. (2019). The dashed red line represents where the old
and new mass fractions would be equal. Clusters falling above the line have a
higher $f_{\star}$ in this work and clusters falling below the line have a
higher $f_{\star}$ in Decker et al. (2019). The color of each point
corresponds to the mass of the cluster.

MOO J0319$-$0025 and MOO J1142$+$1527 exhibit larger jumps in $f_{\star}$ than
the 25% we expect simply from the deeper data. Those increases likely come
from the improved way we are determining both cluster membership and stellar
mass. Using a fuller sampling of the galaxy SEDs—even in just four bands—gives
us a better and more consistent measurement of the stellar mass of each object
versus what we were able to do in Decker et al. (2019).

###  4.2 Luminosity Functions

Table 3: MaDCoWS Sample Data and Schechter Parameters  Sample | Ncl | $\bar{z}$ | $\bar{M_{500}}$ | $m^{*}$ | $\alpha$ | $\phi^{*}$
---|---|---|---|---|---|---
|  |  | ($10^{14}M_{\odot}$) |  |  | ($\mathrm{dN}~{}\mathrm{dm}^{-1}~{}\mathrm{cluster}^{-1}$)
All | 12 | 1.18 | $2.96$ | $19.83\pm 0.12$ | $-0.81\pm 0.10$ | $98\pm 13$
High-$z$ | 6 | 1.29 | $3.30$ | $19.86\pm 0.20$ | $-0.95\pm 0.15$ | $89\pm 20$
Low-$z$ | 6 | 1.06 | $2.62$ | $19.69\pm 0.22$ | $-0.77\pm 0.16$ | $96\pm 20$

The composite LF for our full sample of 12 clusters is shown in Figure 5. We
fit a parameterized Schechter function to the measured LF using a Monte Carlo
Markov Chain (MCMC) running a Metropolis-Hastings algorithm. The best-fit
Schechter function is shown as a dashed line in the figure, with the lighter
region showing the $1\sigma$ error on the best fit. The best fit parameters
and the error on them are derived from the mean and standard deviation of the
MCMC posterior chains after discarding the initial ‘burn-in’ period. These
values are listed in Table 3 along with the mean redshift and mass of the
sample.

Figure 5: Composite LF for the full sample of 12 clusters used in this
analysis. The purple points are the average number of objects per magnitude
per cluster in each quarter-magnitude bin and the error bars are derived from
the Poisson error of the individual LFs and the completeness errors. The mean
sample mass and redshift are given, as well as the best fits to the
parameterized Schechter function and their errors, in Table 3. The best-fit
Schechter function itself is shown as a dashed line and the shaded area
represents the $1\sigma$ error on the best fit.

Because $m^{*}$ and $\alpha$ are covariant, in addition to the simple errors
given in Table 3, we also plot the $1\sigma$ (dark) and $2\sigma$ (light)
covariance ellipses for $m^{*}$ and $\alpha$ for the full sample in Figure 6.
This shows the extent of the degeneracy between $m^{*}$ and $\alpha$ as well
as the axis along which our uncertainty is concentrated.

Figure 6: Covariance ellipses showing the $1\sigma$ (dark) and $2\sigma$
(light) errors and covariances in $m^{*}$ and $\alpha$ for the average LF of
all 12 clusters in this work.

Our sample is large enough that we also split our clusters into high- and low-
redshift samples—splitting them at the median redshift of our sample,
$z=1.17$—and measure composite LFs for both of those. The measurement and
fitting for these sub-samples is the same as for the full sample and the mean
masses and redshifts for both sub-samples are also shown in Table 3, along
with the best-fit Schechter function parameters to each LF. These LFs are
shown in Figure 7. As with the full sample, we also plot the covariance
between $m^{*}$ and $\alpha$ for these two sub-samples in Figure 8. This
figure shows that although the error bars for the individual parameters
overlap between the two samples, there is a significant evolution from
$z=1.29$ to $z=1.06$ in the LF as a whole. As discussed in §5.2.3, this is
consistent with being driven by passive evolution in our sample. For
comparison, we also plot in Figure 8 the outlines of the ellipses for the full
sample from Figure 6.

Figure 7: LFs for the high-redshift (upper, red) and low-redshift (lower,
blue) subsamples. The best-fit Schechter functions are plotted as dashed lines
with the $1\sigma$ error on the best fit shown as shaded regions. Each panel
also shows the best fit line from the other panel as a dashed line of the
relevant color. Figure 8: Covariance ellipses showing the $1\sigma$ (dark) and
$2\sigma$ (light) errors and covariances in $m^{*}$ and $\alpha$ for the high-
redshift (red) and low-redshift (blue) sub-samples. For comparison, the whole-
sample covariances ellipses from Figure 6 are outlined underneath.

We also explore fitting a sum of two Schechter function to our LFs, in a
manner similar to Lan et al. (2016). This is motivated particularly by our
high-redshift LF which seems to show an upturn at the faint end that is
possibly more consistent with a second ‘faint’ Schechter function wit a steep
faint-end slope. Although we can’t rule out there being an upturn at the
faint-end of our LFs, to the depth of our data ($\sim m^{*}+2.5$) we find that
this sum of Schechter functions is at best only a marginally better fit to the
data, at a level that is well short of statistical significance.

##  5 Discussion

###  5.1 Stellar Mass Fraction

The stellar mass fractions we compute in this work display many of the same
traits as the stellar mass fractions we calculated in Decker et al. (2019).
The main difference is that the improved measurement of $f_{\star}$ has
resulted in higher values overall, and ten of the twelve now lie slightly
above the Gonzalez et al. (2013) line, nine of them significantly so. They
still appear to follow the slope of the Gonzalez et al. (2013) line, however.
Despite our improved measurements significantly reducing the systematic errors
on our measurements of $f_{\star}$, the statistical errors are still high
enough that we cannot draw any significant conclusions about either the slope
of the trend or its normalization. Much of this is due to the high error on
the total mass of some of the clusters, which factors into the error on
$f_{\star}$. For the well-measured clusters like MOO J1142$+$1527 on the right
side of Figure 3, the $f_{\star}$ error is very small. Deeper SZ imaging on
the MaDCoWS clusters is likely necessary to provide a significant measurement
of properties related to the total mass.

###  5.2 Comparison to Other LF Studies

A comparison of our measurements of $m^{*}$ and $\alpha$ to other works across
a range of redshifts is shown in Figures 9 and 10. To facilitate comparison
with some other studies, we again use EZGal to convert our apparent
$3.6~{}\mathrm{\mu m}$ magnitudes to absolute $H$-band magnitudes. We use a
Bruzual & Charlot (2003) model with a formation redshift of $z_{f}=3.0$, a
Chabrier (2003) initial mass function, and a solar metallicity to calculate
the $k$-correction, but because we are already probing the rest-frame $H$-band
with our $3.6~{}\mathrm{\mu m}$ observations, the $k$-correction is almost
entirely model-independent and changing the model parameters does not affect
the $k$-correction by more than $0.01$ at any of our redshifts. Doing this, we
find $M_{H}^{*}=-23.91\pm 0.12$ for our overall LF and $M_{H}^{*}=-24.09\pm
0.20$ and $M_{H}^{*}=-23.80\pm 0.22$ for the high- and low-redshift LFs,
respectively. Figure 9 shows this comparison for $M_{H}^{*}$.

Figure 9: Absolute $H$-band characteristic magnitude versus redshift for a
number of cluster LF studies. The error on the Lin et al. (2004) point is
0.02, smaller than the data marker. The dashed purple line shows the expected
passive evolution in $M^{*}_{H}$ Figure 10: Faint-end slope versus redshift
for a number of cluster LF studies. The labels for the points are the same as
in Figure 9. The error on the Lin et al. (2004) point is 0.02, smaller than
the data marker.

For both $M^{*}$ and $\alpha$ we plot the results from our full sample as a
solid diamond and the results from the high- and low-redshift samples as red
and blue open diamonds, respectively. In Figure 9, we also plot as a dashed
line the expected trend of $M^{*}_{H}$ with redshift, assuming passive
evolution. We discuss the comparison works below.

####  5.2.1 Comparisons at Similar Redshift

Owing to the difficulty in getting deep enough mid-infrared data, few studies
have previously been able to measure both $m^{*}$ and $\alpha$ simultaneously
at this redshift range. One such study is Mancone et al. (2012), who measured
composite LFs in $3.6~{}\mathrm{\mu m}$ and $4.5~{}\mathrm{\mu m}$ for seven
IRAC Shallow Cluster Survey (ISCS) clusters at a median redshift of $z=1.35$,
slightly higher than our mean redshift overall, but a good match to our high-
redshift sub-sample. Converting the apparent Vega magnitudes they report into
absolute AB magnitudes, they found best fit parameters for the
$3.6~{}\mathrm{\mu m}$ LF of $M_{H}^{*}=-23.85\pm 0.30$ and $\alpha=-0.97\pm
0.14$. The faint-end slope we measure for the high-redshift clusters matches
this result almost exactly and the characteristic magnitude we find is also
consistent with their results.

Another similar study is Chan et al. (2019), who used IRAC $3.6~{}\mathrm{\mu
m}$ imaging to measure rest-frame $H$-band LFs for red-sequence galaxies in
seven clusters from the infrared-selected Gemini Observations of Galaxies in
Rich Early Environments (GOGREEN, Balogh et al., 2017) survey at a mean
redshift of $\bar{z}=1.15$. They report their results in terms of absolute
$H$-band magnitudes and find $M_{H}^{*}=-23.52^{+0.15}_{-0.17}$, which is
somewhat fainter than our value of $M_{H}^{*}=-23.91\pm 0.12$ at $z=1.18$.
They also find a much more steeply falling faint-end slope than we do, with a
value of $\alpha=-0.35\pm 0.15$. This difference in $\alpha$ may be a result
of their only including red-sequence galaxies, whereas we include everything
with a photometric redshift consistent with being a cluster member. Other
works (e.g., Muzzin et al., 2007; Strazzullo et al., 2006; De Propris, 2017)
have found that the faint-end of the cluster LF is dominated by blue galaxies,
which would explain the discrepancy between our results and red-sequence-only
results. Alternatively, Connor et al. (2019) found that the traditional model
of the red sequence as a sloped line does not hold at fainter magnitudes. This
could cause a drop-off in the measured red-sequence LF that is unconnected to
the galaxy population of the cluster.

####  5.2.2 Comparisons at Other Redshifts

To put our results into a wider context, we also compare to studies at other
redshifts and with different cluster selection and fitting methods. To make as
fair a comparison as possible to $m^{*}$ at other redshifts, we use the model
described in §5.2 to convert all the magnitudes into the absolute $H$-band and
limit ourselves to studies where the reported values still probe either the
rest-frame $H$-band or nearby $K$-band. These comparisons are also shown in
Figures 9 and 10.

At the lowest-redshift of our comparisons, de Propris et al. (1998) looked at
the Coma cluster at $z=0.023$ to a depth much fainter than $m^{*}$. They
jointly fit $m^{*}$ and $\alpha$ at magnitudes brighter than $m^{*}+3$, though
at magnitudes fainter than $m^{*}+3$, they found a sharp rise in the number of
galaxies and fit this with a power law. At a similar redshift, Lin et al.
(2004) calculated a stacked $K$-band LF for a sample of 13 Abell clusters with
X-ray follow-up. We compare to their joint fits of $m^{*}$ and $\alpha$,
though they also attempt to fix $\alpha$ due to the uncertainty in the faint-
end slope. Their errors on both parameters are $0.02$, which is small enough
it does not appear on Figures 9 and 10. At higher redshift, de Propris et al.
(1999) looked at a heterogenous selection of clusters in the $K$-band in
redshift bins up to $z=0.92$. They only fit to $m^{*}$, fixing the faint-end
slope at $\alpha=-0.9$. We compare $z=0.25$, $z=0.32$ and $z=0.4$ bins, where
the conversion from $K$-band into rest-frame $H$-band has a minimal
$k$-correction. Similarly, Muzzin et al. (2007), measured the observed-frame
$K$-band for clusters at a mean redshift of $\bar{z}=0.296$. In addition to
reporting a composite LF for all galaxies in their clusters, Muzzin et al.
(2007) also split their galaxies by whether they were quiescent or star-
forming/recently-quenched. They found a relatively flat faint-end slope for
their overall LF, $\alpha=-0.84\pm 0.08$ and a much steeper faint-end slope of
$\alpha=0.17\pm 0.18$ for red-sequence galaxies. At similar redshift, Andreon
(2001) studied a single cluster at $z=0.3$ in the $K_{S}$-band. He measured a
LF down to $m^{*}+5$ in various areas of the cluster, but the comparison we
show is to the global values he reported. At higher redshifts, we also compare
to some of the results from Wylezalek et al. (2014), who measured the
$4.5~{}\mathrm{\mu m}$ and $3.6~{}\mathrm{\mu m}$ LF for clusters from the
Clusters Around Radio Loud AGN program (CARLA, Wylezalek et al., 2013) in
several redshift bins in the range $1.3<z<3.1$. We show two of these in
Figures 9 and 10, again where the conversion from $3.6~{}\mathrm{\mu m}$ or
$4.5~{}\mathrm{\mu m}$ has a minimal $k$-correction.

####  5.2.3 Evolution of the LF

Figure 9 shows the clusters in this work fall into a larger pattern of passive
evolution in $M_{H}^{*}$ going out to $z\sim 2$. Similarly, Figure 10 shows
very little change in the faint-end slope over cosmic time when looking only
at $\alpha$ over a range of studies. This suggests that the evolution in the
parameters shown in Figure 8 is primarily driven by passive evolution.

To confirm this, we evolution-correct the galaxies in the high- and low-
redshift sub-samples to $z=1.18$, the mean redshift of the full sample,
assuming passive evolution. We then re-make the LFs and run the same joint fit
as above. The results of this are shown in Figure 11. With passive-evolution
‘baked-in’ to the fit, the LF parameters are now consistent within two sigma,
supporting the interpretation that these clusters are evolving passively,
consistent with other studies at this redshift.

Figure 11: Covariance ellipses showing the $1\sigma$ (dark) and $2\sigma$
(light) errors and covariances in $m^{*}$ and $\alpha$ for the high-redshift
(red) and low-redshift (blue) sub-samples, assuming passive evolution. For
comparison, the whole-sample covariances ellipses from Figure 6 are outlined
underneath.

##  6 Conclusions

We have presented stellar mass fractions and $3.6~{}\mathrm{\mu m}$ LFs for a
sample of 12 infrared-selected clusters from the MaDCoWS catalog. We used
optical and deep mid-IR follow-up data to fit SEDs to objects along the lines-
of-sight to the clusters. This allowed us to more precisely identify cluster
members, measure more thorough stellar masses for the clusters and measure the
faint-end slope of the LF.

The stellar mass fractions we report for these clusters are in good agreement
with previous works, and are consistent with the Gonzalez et al. (2013) trend
line with respect to total mass. For the individual clusters previously
studied in Decker et al. (2019), the new values of $f_{\star}$ reported here
are consistent with—but mostly higher than—the previous values, with much of
the difference being attributable to the deeper data set we use here.

The composite $3.6~{}\mathrm{\mu m}$ LF we fit for all 12 clusters has a best-
fit characteristic magnitude and faint-end slope of $m^{*}=19.83\pm 0.12$ and
$\alpha=-0.81\pm 0.10$, respectively. Both are consistent with other works
that have attempted to measure the rest-frame NIR LF for all cluster members
at these redshift ranges. When we split our sample into a high-redshift bin at
$\bar{z}=1.29$ and a low-redshift bin at $\bar{z}=1.06$ we find that there is
a significant evolution in the best-fit Schechter function parameters,
consistent with passive evolution. This significance is only seen in the
covariance ellipse for $m^{*}$ and $\alpha$ jointly. This highlights the need
to study $m^{*}$ and $\alpha$ jointly. Comparing to works at other redshifts,
our results are consistent with passive evolution since $z\sim 2$.

In future, follow-up data on more MaDCoWS clusters will allow us to better
identify trends with redshift and other cluster parameters. In addition,
deeper infrared data—such as will be attainable from the next generation of IR
space telescopes—will allow us to more definitively answer questions about the
evolution of the faint galaxy population in clusters at $z>1$.

The work of T.C., P.E., and D.S. was carried out at the Jet Propulsion
Laboratory, California Institute of Technology, under a contract with NASA.
F.R. acknowledges financial supports provided by NASA through SAO Award Number
SV2- 82023 issued by the Chandra X-Ray Observatory Center, which is operated
by the Smithsonian Astrophysical Observatory for and on behalf of NASA under
contract NAS8-03060. This work is based in part on observations made with the
Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA. Support for
this work was provided by NASA through an award issued by JPL/Caltech. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. This work was
based in part on observations obtained at the Gemini Observatory, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under a cooperative agreement with the NSF on behalf of the Gemini
partnership: the National Science Foundation (United States), the National
Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, Tecnología
e Innovación Productiva (Argentina), and Ministério da Ciência, Tecnologia e
Inovação (Brazil). Funding for this program is provided by NASA through the
NASA Astrophysical Data Analysis Program, award 80NSSC19K0582. Support for
this work was provided by the National Aeronautics and Space Administration
through Chandra Award Number GO7-18123A issued by the Chandra X-ray Center,
which is operated by the Smithsonian Astrophysical Observatory for and on
behalf of the National Aeronautics Space Administration under contract
NAS8-03060. Based on observations with the NASA/ESA Hubble Space Telescope
obtained at the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Incorporated, under
NASA contract NAS5-26555. Support for program number HST-GO-14456 was provided
through a grant from the STScI under NASA contract NAS5-26555.

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