# The GOGREEN survey: Dependence of galaxy properties on halo mass at $z>1$
and implications for environmental quenching

Andrew M. M. Reeves1,2, Michael L. Balogh1,2, Remco F. J. van der Burg3,
Alexis Finoguenov4, Egidijus Kukstas5, Ian G. McCarthy5, Kristi Webb1,2, Adam
Muzzin6, Sean McGee7, Gregory Rudnick8, Andrea Biviano9,10, Pierluigi
Cerulo11, Jeffrey C. C. Chan12, M. C. Cooper13, Ricardo Demarco14, Pascale
Jablonka15,16, Gabriella De Lucia9, Benedetta Vulcani17, Gillian Wilson12,
Howard K. C. Yee18, Dennis Zaritsky19

1Department of Physics and Astronomy, University of Waterloo, Waterloo,
Ontario N2L 3G1, Canada
2Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, Ontario,
N2L3G1, Canada
3European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748, Garching,
Germany
4Department of Physics, University of Helsinki, Gustaf Hällströmin katu 2a,
FI-00014 Helsinki, Finland
5Astrophysics Research Institute, Liverpool John Moores University, 146
Brownlow Hill, Liverpool L3 5RF, UK
6Department of Physics and Astronomy, York University, 4700 Keele Street,
Toronto, Ontario, ON MJ3 1P3, Canada
7School of Physics and Astronomy, University of Birmingham, Edgbaston,
Birmingham B15 2TT, England
8Department of Physics and Astronomy, The University of Kansas, 1251 Wescoe
Hall Drive, Lawrence, KS 66045, USA
9INAF – Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11, I-34143
Trieste, Italy
10IFPU – Institute for Fundamental Physics of the Universe, via Beirut 2,
34014 Trieste, Italy
11Departamento de Astronomía, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Concepción, Concepción, Chile
12Department of Physics and Astronomy, University of California, Riverside,
900 University Avenue, Riverside, CA 92521, USA
13Department of Physics and Astronomy, University of California, Irvine, 4129
Frederick Reines Hall, Irvine, CA 92697, USA
14Departamento de Astronomía, Facultad de Ciencias Físicas y Matemáticas,
Universidad de Concepción, Concepción, Chile
15Laboratoire d’astrophysique, École Polytechnique Fédérale de Lausanne
(EPFL), 1290 Sauverny, Switzerland
16GEPI, Observatoire de Paris, Université PSL, CNRS, Place Jules Janssen,
F-92190 Meudon, France
17INAF – Osservatorio astronomico di Padova, Vicolo Osservatorio 5, IT-35122
Padova, Italy
18Department of Astronomy & Astrophysics, University of Toronto, Toronto,
Canada
19Steward Observatory and Department of Astronomy, University of Arizona,
Tucson, AZ 85721, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use photometric redshifts and statistical background subtraction to measure
stellar mass functions in galaxy group-mass ($4.5-8\times
10^{13}~{}\mathrm{M}_{\odot}$) haloes at $1<z<1.5$. Groups are selected from
COSMOS and SXDF, based on X-ray imaging and sparse spectroscopy. Stellar mass
($M_{\mathrm{stellar}}$) functions are computed for quiescent and star-forming
galaxies separately, based on their rest-frame $UVJ$ colours. From these we
compute the quiescent fraction and quiescent fraction excess (QFE) relative to
the field as a function of $M_{\mathrm{stellar}}$. QFE increases with
$M_{\mathrm{stellar}}$, similar to more massive clusters at $1<z<1.5$. This
contrasts with the apparent separability of $M_{\mathrm{stellar}}$ and
environmental factors on galaxy quiescent fractions at $z\sim 0$. We then
compare our results with higher mass clusters at $1<z<1.5$ and lower
redshifts. We find a strong QFE dependence on halo mass at fixed
$M_{\mathrm{stellar}}$; well fit by a logarithmic slope of
$\mathrm{d}(\mathrm{QFE})/\mathrm{d}\log(M_{\mathrm{halo}})\sim 0.24\pm 0.04$
for all $M_{\mathrm{stellar}}$ and redshift bins. This dependence is in
remarkably good qualitative agreement with the hydrodynamic simulation
BAHAMAS, but contradicts the observed dependence of QFE on
$M_{\mathrm{stellar}}$. We interpret the results using two toy models: one
where a time delay until rapid (instantaneous) quenching begins upon accretion
to the main progenitor (“no pre-processing”) and one where it starts upon
first becoming a satellite (“pre-processing”). Delay times appear to be halo
mass dependent, with a significantly stronger dependence required without pre-
processing. We conclude that our results support models in which environmental
quenching begins in low-mass ($<10^{14}M_{\odot}$) haloes at $z>1$.

###### keywords:

galaxies: evolution, galaxies: haloes, galaxies: star formation, galaxies:
groups:general, galaxies: clusters: general, galaxies: high-redshift

††pubyear: 2021††pagerange: The GOGREEN survey: Dependence of galaxy
properties on halo mass at $z>1$ and implications for environmental
quenching–C

##  1 Introduction

It is well established in the standard Lambda Cold Dark Matter ($\Lambda$CDM)
cosmology that gravitationally bound dark matter haloes build up
hierarchically through a combination of smooth accretion of surrounding
matter, as well as merging with smaller structures (White & Rees, 1978;
Navarro et al., 1997; Qu et al., 2017). Complex baryonic physics, such as
radiative cooling and feedback from stars and accreting supermassive black
holes (e.g. White & Frenk, 1991; Finlator & Davé, 2008; Bouché et al., 2010;
Schaye et al., 2010a; Davé et al., 2012; Bower et al., 2017) drives the
formation of galaxies within this gravitationally dominant component.

Up to at least $z\sim 2.5$, galaxy populations exhibit a bimodality in their
star formation rate (SFR) distribution (e.g. Bell et al., 2004b; Brinchmann et
al., 2004; Brammer et al., 2011; Muzzin et al., 2012) and a corresponding
bimodality in the distribution of rest-frame colours (e.g. Strateva et al.,
2001; Baldry et al., 2004; Bell et al., 2004a; Williams et al., 2009; Foltz et
al., 2018; Muzzin et al., 2013b; Taylor et al., 2015). Observations of these
two populations at different redshifts show that the number density and shape
of the stellar mass function (SMF) for quiescent (red) galaxies has evolved
dramatically, while the SMF of star-forming (blue) galaxies has remained
nearly unchanged since $z\sim 3.5$ (e.g. Faber et al., 2007; Muzzin et al.,
2013b; McLeod et al., 2021). This indicates that the latter population
eventually stops forming stars, in a process generically called "quenching".
The distinct bimodality in the colour and SFR distribution indicates that this
quenching must be fairly rapid (e.g. Balogh et al., 2004b; Wetzel et al.,
2012).

The fraction of quiescent galaxies increases strongly with stellar mass, for
$M_{\rm stellar}>10{{}^{9}}~{}\mathrm{M}_{\odot}$ (e.g. Kauffmann et al.,
2003, 2004; Brinchmann et al., 2004; Baldry et al., 2006; Weinmann et al.,
2006; Kimm et al., 2009; Muzzin et al., 2012; van der Burg et al., 2018). At
fixed stellar mass, the quiescent fraction declines with increasing redshift,
and is higher in denser environments. This has been thoroughly demonstrated in
works studying dense environments of rich groups and clusters in the local
$z\sim 0$ universe (Kauffmann et al., 2004; Gómez et al., 2003; Balogh et al.,
2004a; Hou et al., 2014), as well as intermediate $0<z<1$ redshifts (De Lucia
et al., 2004; Wilman et al., 2005; Cooper et al., 2006; McGee et al., 2011;
Giodini et al., 2012; van der Burg et al., 2018; Just et al., 2019) and at
$z>1$ (Muzzin et al., 2012; Balogh et al., 2014; Nantais et al., 2016;
Guglielmo et al., 2019; Strazzullo et al., 2019; van der Burg et al., 2020).
At low redshifts, the dependence of the quiescent fraction on stellar mass and
environment is largely separable (Baldry et al., 2006; Peng et al., 2010;
Kovač et al., 2014; Balogh et al., 2016; van der Burg et al., 2018). This has
been interpreted to indicate that the dominant physical mechanisms are also
separable (e.g. Peng et al., 2010). However, this is not necessarily the case
(De Lucia et al., 2012; Pintos-Castro et al., 2019), and most physically-
motivated models invoked to explain environment quenching have a dependence on
stellar mass (e.g. McGee et al., 2014; Fillingham et al., 2015; Quilis et al.,
2017; Wright et al., 2019).

The environmental dependence of quenching provides a particularly interesting
and discriminating test of models (e.g. Weinmann et al., 2010; McGee et al.,
2014; Hirschmann et al., 2016). Comparison of data with models is challenging,
however, in part because there are many different, commonly used definitions
of environment. These include local density (Cooper et al., 2006; Peng et al.,
2010; Sobral et al., 2011; Darvish et al., 2016; Davidzon et al., 2016; Lemaux
et al., 2019; Kawinwanichakij et al., 2017; Papovich et al., 2018),
group/cluster virial masses and cluster centric distance (Poggianti et al.,
2006; Oman et al., 2013; Muzzin et al., 2014; Paccagnella et al., 2016; van
der Burg et al., 2018; van der Burg et al., 2020), and status as central or
satellite in halo (Muzzin et al., 2012; Mok et al., 2013; Balogh et al.,
2016). The quiescent fraction at fixed stellar mass correlates with
environment in most cases, but the interpretation of the physical mechanisms
behind the observed trends remains elusive, in part because of the difficulty
linking these observations to theoretical models.

To interpret the environmental dependence of the quiescent fraction, numerous
works have used simple accretion models where galaxies take some amount of
time to fully quench, once they enter a new environment. To reproduce the
observations at $z\sim 0$ requires long timescales, of at least $\sim 2-7$ Gyr
(e.g. De Lucia et al., 2012; Wheeler et al., 2014). To reconcile this with the
bimodality in observed SFRs, which requires a rapid transition, it is
frequently assumed to be a long delay time during which the galaxy properties
remain uninfluenced by their environment, before rapid quenching sets in
(Wetzel et al., 2013; Oman & Hudson, 2016). These timescales are shorter at
higher redshift, scaling approximately with the dynamical time (Tinker &
Wetzel, 2010; Tinker et al., 2013; Balogh et al., 2016; Foltz et al., 2018).
However, all these timescales are relative to an assumed starting point
associated with a change in environment. In general this is not well defined,
and depends upon assumptions about the physical mechanisms at work.

A physically relevant definition of environment in $\Lambda$CDM models is the
mass of the host halo, and location within it. The environment of a galaxy
undergoes a large change when it is accreted into a more massive halo and
first becomes a satellite, as it orbits within that new halo. A simple
objective definition of this accretion time is the moment when a galaxy
crosses $R_{200}$111$R_{\Delta}$ can be defined for a halo as the radius
within which the average density is $\Delta$ times either the critical density
of the universe or $\Delta$ times the background density. We use the former
definition in this work. for the first time (Balogh et al., 2000). However,
simulations suggest environment first becomes important when satellites are
cut off from cosmological accretion, which can happen well outside $R_{200}$
(Behroozi et al., 2014; Bahé & McCarthy, 2015; Pallero et al., 2019). On the
other hand, for processes like ram pressure stripping that require a dense
intracluster medium, the more relevant starting point could be well inside
$R_{200}$ (e.g. Muzzin et al., 2014). Moreover, the starting time for
environmental effects depends on when in the merger history hierarchy the
galaxy is accreted, according to one of the above definitions. One
consideration is the accretion onto the main progenitor of the final halo.
Alternatively, the physically relevant definition could be the first time a
galaxy is accreted onto any more massive halo and hence first becomes a
satellite. The latter is often referred to as "pre-processing"; observations
and simulations both indicate that “pre-processing” may be important for a
significant proportion of galaxies and that at least some cluster galaxies had
their star formation quenched in $M_{\mathrm{halo}}\geq
10^{13}\mathrm{M}_{\odot}$ groups prior to being accreted into massive
clusters (Zabludoff & Mulchaey, 1998; Kawata & Mulchaey, 2008; Berrier et al.,
2009; McGee et al., 2009; De Lucia et al., 2012; Hou et al., 2014; Pallero et
al., 2019; Donnari et al., 2021).

To make progress requires observations of galaxies spanning a wide range in
stellar mass and redshift, within a range of well-characterized environments
that can be directly linked to theory. This can be achieved with samples of
groups and clusters with reliable halo mass estimates. The GOGREEN survey
(Balogh et al., 2017, 2021) was designed with this goal, and provides a sample
of 21 galaxy systems at $1<z<1.5$ with deep photometry and extensive
spectroscopy, ranging from groups to the most massive clusters. The groups are
a subset of those identified from the deep X-ray and spectroscopic
observations in the COSMOS and SXDF regions (Finoguenov et al., 2010;
Leauthaud et al., 2012; Giodini et al., 2012; Gozaliasl et al., 2019). COSMOS
groups at $z<1$ have already been studied in some detail. For example, Giodini
et al. (2012) measured stellar mass functions for quiescent and star-forming
galaxies (separated using $NUV$-$R$ colours) in these systems. Among other
things, they found that the fraction of quiescent galaxies increases with halo
mass, and decreases with increasing redshift, though this was not examined as
a function of stellar mass. In the present paper we build upon this work,
taking advantage of deeper X-ray data and additional spectroscopy to extend
the group sample to $1<z<1.5$. We also make use of significantly improved
photometry (deeper/more bands) to separate quiescent/star-forming galaxies and
measure their stellar mass functions and quiescent fractions. Combined with
the GOGREEN sample, and lower redshift comparison samples, this allows for an
improved picture of quenching as a function of both stellar and halo mass,
over the redshift range $0<z<1.5$.

The structure of the paper is as follows. In §2 we describe the spectroscopic
and photometric data sets, as well as the group catalogues, that we use for
the measurements. Results are presented in §3. In §4, we discuss our
measurements in the context of the literature, compare to the BAHAMAS
hydrodynamical simulation, and explore a toy model to constrain pre-processing
scenarios. We then conclude and summarise in §5. In the Appendices, we include
additional details of calibrations and robustness checks, present the
spectroscopically targeted GOGREEN groups and stacked velocity dispersion
measurement, as well as provide supplemental plots to our analysis and
discussion of halo mass trends.

Uncertainties are given at the 1-$\sigma$ (Gaussian) level, unless stated
otherwise. Logarithms with base 10 ($\log_{10}$) are written simply as
“$\log$” throughout this work. All magnitudes are given in the AB magnitude
system, all (RA, DEC) coordinates are given using the J2000 system, and a
Chabrier (2003) initial mass function (IMF) is assumed throughout, unless
specified as otherwise. As well, a flat $\Lambda$CDM cosmology with
$H_{0}=70~{}{\rm km}~{}{\rm s}^{-1}~{}{\rm Mpc}^{-1}$, $\Omega_{m}=0.3$, and
$\Omega_{\Lambda}=0.7$, is assumed. Halo masses and radii are given as either
$(M_{500c},R_{500c})$ or $(M_{200c},R_{200c})$, where $c$ refers to the
critical density of the universe at a given redshift. Conversions of mass and
radius between $500c$ and $200c$ were done using concentration parameters
estimated using the redshift-dependent relation defined in Muñoz-Cuartas et
al. (2011). Finally, whenever the term “field” is used, we are referring to an
average sample of the Universe, which includes all environments.

##  2 Datasets and sample selections

The core analysis of this work is based on 21 X-ray selected groups at
$1<z<1.5$, in the COSMOS and SXDF survey regions. We rely on the excellent
photometric redshifts, calibrated with extensive spectroscopy, and statistical
background subtraction to analyse the galaxy populations in these groups. The
following subsections summarise the data and sample selections, including
comparison samples at lower redshift and higher halo mass.

###  2.1 Photometric data

####  2.1.1 Imaging and catalogues

For COSMOS we use the UltraVISTA (McCracken et al., 2012; Muzzin et al.,
2013a) survey as the source of the photometry and catalogues. The first data
release (DR1, v4.1) provides a $K$-selected catalogue in 38 photometric bands,
covering 1.62 deg2 and with a $5\sigma$ ($2.1^{\prime\prime}$ aperture)
limiting magnitude of $K_{s}=23.9\mathrm{AB}$. The 95% stellar mass-
completeness limit is $10^{10}~{}\mathrm{M}_{\odot}$ at $z=1.5$. The
catalogues include photometric redshifts and rest-frame $U-V$ and $V-J$
colours, computed using EAZY (Brammer et al., 2010). The photometric redshifts
are accurate to $\delta z/(1+z)=0.013$ (68% confidence limits), with a
catastrophic outlier fraction of 1.6% (Muzzin et al., 2013a). The catalogues
also include stellar masses and population parameters, which were obtained
using the spectral energy distribution fitting code FAST (Kriek et al., 2009)
with the Bruzual & Charlot (2003) models. A subset of the COSMOS field is
covered by the ultra-deep stripes of UltraVISTA DR3. This catalogue includes
50 photometric bands and covers a non-contiguous 0.8465 deg2, with a $5\sigma$
($2^{\prime\prime}$ aperture) limiting magnitude of
$K_{s}=24.9\mathrm{AB}$222The official UltraVISTA DR3 data release document
can be accessed here:
https://www.eso.org/sci/observing/phase3/data_releases/uvista_dr3.pdf. A
magnitude $K_{s}=23.5$ is reached at the 90% confidence limit and the 95%
mass-completeness limit is $10^{9.58}~{}\mathrm{M}_{\odot}$ at $z=1.5$. We use
DR3 catalogues for the subset of groups that fall entirely within one of the
stripes of that survey; otherwise we use DR1.

For SXDF we use the SPLASH-SXDF 28-band catalogue (Mehta et al., 2018). The
subset of the SXDF field with all available filters covers 0.708 deg2, with a
5$\sigma(2^{\prime\prime})$ limiting magnitude of $K=25.32$ (Mehta et al.,
2018). By comparing with UltraVISTA DR3, we expect this survey to be 95%
complete above a stellar mass limit of $10^{9.4}~{}\mathrm{M}_{\odot}$ at
$z=1.5$. Photometric redshifts, their uncertainties, and stellar mass
estimates were calculated by the SPLASH team using LePhare (Arnouts et al.,
1999; Ilbert et al., 2006). Photometric redshift uncertainties are reported in
terms of their $\chi^{2}$ fit, $\Delta\chi^{2}=1.0$ for upper and lower 68%
confidence limits. We rely on the rest-frame $U$, $V$, and $J$ colours to
classify galaxies, and we compute these using the SPLASH-SXDF v1.6 photometric
catalogue, with EAZY. Redshifts were fixed to the SPLASH-SXDF photometric
redshifts or spectroscopic redshift, if available.

A small correction ($<0.06$) is made to the photometric redshifts in the range
$1<z<1.5$, to correct a redshift-dependent bias that is observed upon
comparison with a spectroscopic sample, as described in Appendix A. Details of
the various spectroscopic redshift catalogues used for this, as well as
description of how they informed our photometric redshift selection of group
members and groups, can be found in Appendix B.

####  2.1.2 Galaxy selection

We select galaxies from the UltraVISTA DR1 catalogue of Muzzin et al. (2013a),
closely reproducing the selection used by Muzzin et al. (2013b) to calculate
the stellar mass function evolution. Specifically, we select objects
identified as galaxies (rather than stars), with uncontaminated photometry and
$K_{s}<23.4$. We impose an additional cut of ${\rm S/N}>7$ in the $K_{s}$
photometry. We perform a similar selection for UltraVISTA DR3, except to
$K_{s}<24.9$. For galaxies in the SPLASH-SXDF we select galaxies with the same
$S/N>7$ criterion, and $K_{s}<23.7$, corresponding to the $5\sigma$ depth of
that survey. Stars in SXDF are excluded from the sample using the “STAR_FLAG”
parameter, which is based on whether the photometry is best-fit to a stellar
or galaxy template, with an additional restriction that the object does not
belong to the stellar sequence in $BzK$ colour-colour space.

Finally, we make a survey-dependent stellar mass cut to ensure complete,
unbiased galaxy samples. For $1<z<1.5$ groups in UltraVISTA DR1, the
shallowest of the survey regions, we select galaxies with $\log{M_{\rm
stellar}/\mathrm{M}_{\odot}}>z_{\text{group}}+8.5$, corresponding to the mass
completeness limit shown in Figure 2 of Muzzin et al. (2013b). For groups in
the deeper UltraVISTA DR3 and SXDF we conservatively select galaxies with
$M_{\rm stellar}>10^{9.6}~{}\mathrm{M}_{\odot}$, corresponding to the $z=1.5$
completeness limit of DR3.

####  2.1.3 Classification of quiescent and star-forming galaxies

We identify quiescent333Equivalently referred to in some of the literature as
”passive” or ”quenched”. (“red”) and star-forming (“blue”) galaxies using
rest-frame $UVJ$ colour-colour cuts, following Muzzin et al. (2013b). To
ensure consistency between the three photometric catalogues we use, we apply
small systematic shifts to the $U-V$ and $V-J$ colours of galaxies in
UltraVISTA DR1 and SXDF, to match those of UltraVISTA DR3. Specifically, we
calculate the average difference in these colours between surveys, using
galaxies at $1<z<1.5$ and with stellar masses above
$10^{10}~{}\mathrm{M}_{\odot}$. This results in a shift of $\Delta(V-J)=0.08$
and $\Delta(U-V)=-0.05$ for UltraVISTA DR1; for SXDF the corresponding shifts
are 0.10 and 0.15, respectively. We then use the following selection, slightly
modified from Muzzin et al. (2013b) (which worked with the original UltraVISTA
DR1 colours), to identify quiescent galaxies at $1<z<1.5$:

$\displaystyle U-V>1.26,V-J<1.58,$ (1) $\displaystyle U-V>(V-J)\times
0.88+0.47,$ (2)

We illustrate these cuts on the rest-frame $U-V$ vs $V-J$ distribution in
Figure 1.

Figure 1: Rest-frame $U-V$ vs $V-J$ colour distributions for galaxies in the
SXDF (black points), UltraVISTA DR1 (orange contours) and DR3 (blue contours),
with the colour-colour cuts dividing quiescent and star-forming galaxies
(solid black line). Small zeropoint adjustments have been made to SXDF and
DR1, as described in the text. For the purposes of this figure, galaxies are
limited to $1<z<1.5$ and $\log(M_{\mathrm{stellar}}/\mathrm{M}_{\odot})>10$.

We also consider a lower redshift comparison sample of groups, at $0.5<z<0.7$,
selected entirely from UltraVISTA DR1 (see §2.2.2). Noting that the $UVJ$
colour-colour cut is weakly redshift dependent (Williams et al., 2009;
Whitaker et al., 2011; Muzzin et al., 2013b) we instead adopt for these
galaxies exactly the selection of Muzzin et al. (2013b) at $0.0<z<1.0$,
namely:

$\displaystyle U-V>1.3,V-J<1.5,$ (3) $\displaystyle U-V>(V-J)\times
0.88+0.69\,.$ (4)

####  2.1.4 Group selection

The COSMOS groups and SXDF groups we use for our analyses were identified in
Gozaliasl et al. (2019) and Finoguenov et al. (2010), respectively. Each group
in the catalogue has a quality flag ranging from 1 (best) to 5 (worst),
although the precise meaning of these flags is different in the two surveys.
We update quality flags for a subset of groups in the two group catalogues
based on information from our GOGREEN spectroscopy, which increases the number
of available spectroscopically confirmed groups. We then select only groups
with quality flags $<3$, defined in Gozaliasl et al. (2019) to have both X-ray
detections, photometric overdensity of galaxies, and at least one
spectroscopically-confirmed member 444In Table 1, group COSMOS-30317 is listed
as having no spectroscopic members. This discrepancy with Gozaliasl et al.
(2019) may be either due to a different cut in redshift or a difference in the
spectroscopic catalogues being used..

With this selection we have an initial sample of 21 groups at $1<z<1.5$: nine
in UltraVISTA DR1, eight in DR3, and four in SXDF. The properties of these
groups are presented in Table 1. All of these groups have halo masses
estimated to be in the range $13.6<\text{log}(M_{\rm
halo}/\mathrm{M}_{\odot})<14.0$, with an average of $\text{log}(M_{\rm
halo}/\mathrm{M}_{\odot})\approx 13.8$. For each group we calculate
$R_{200c}(z)$ (using Hearin et al., 2017) corresponding to this average mass
(e.g. $R_{200c}=1.044^{\prime}$ in projection at $z=1.25$).

The masses are based on the weak lensing calibrations of the
$L_{\mathrm{X}}-M_{\mathrm{halo}}$ relations in COSMOS (Leauthaud et al.,
2010). The mentioned biases in the Planck 2015 paper (Planck Collaboration et
al., 2016) are relevant only for the hydrostatic mass estimates (see also
Smith et al., 2016, for detailed discussion of the biases in the Planck 2015
paper). For the SZ confirmation on similar galaxy groups (near our redshift
and halo mass range), there is one such measurement, at z=2 (Gobat et al.,
2019). As an alternative indicator of halo mass, we calculate the richness,
$\lambda_{\mathrm{10.2,R<1Mpc}}$, defined as the number of photometrically
background-subtracted galaxies within a 1 Mpc radius above a stellar mass of
$10^{10.2}~{}\mathrm{M}_{\odot}$. In Figure 2 we show the correlation between
these richness values and the $M_{200c}$ masses from X-ray fluxes for our
group sample. This is compared with more massive clusters from van der Burg et
al. (2020), discussed in §2.2.1 below. For that sample, we use halo masses
based on the dynamical analysis of Biviano et al. (2021). Although the
uncertainties on individual richness measurements are large, this comparison
confirms that the group sample is systematically less rich than the cluster
sample, at the level expected from their mass estimates. Only one of the
groups has a richness $\lambda_{10.2,R<1{\rm Mpc}}<1$, significantly lower
than expected of a truly overdense system.

Figure 2: Richness as a function of halo mass for groups in this work (blue
points; mean of sample shown with the square black point) and the GOGREEN
clusters (red points). The richness, $\lambda_{\mathrm{10.2,R<1Mpc}}$, is the
number of background-subtracted galaxies that have stellar masses above
$10^{10.2}~{}\mathrm{M}_{\odot}$ within 1 Mpc, following van der Burg et al.
(2020). Halo masses for the groups are derived from the X-ray fluxes, while
for the clusters they are based on a spectroscopic dynamical analysis from
Biviano et al. (2021). The richness-halo mass relation fit for clusters at
$0.1<z<0.33$ from Simet et al. 2017 is shown for comparison as a dashed line.
Note that one of the groups has a formally negative richness, and lies off the
bounds of this figure.

The subset of $1<z<1.5$ groups with GOGREEN spectroscopy affords the
opportunity to study these groups in some more detail, and to test the
robustness of the statistical background subtraction. This analysis is
presented in Appendix B.3. Where it is possible to calculate a robust velocity
dispersion, we report these values in Table 1. The dynamical halo mass,
$M_{200c,{\rm dyn}}$ (column 8 of Table 1), is then derived using the relation
in Saro et al. (2013). Two groups – SXDF64 and SXDF87 – have dynamical masses
significantly higher than that based on their X-ray emission, and formally
above our arbitrary threshold for low-mass haloes, of
$10^{14}~{}\mathrm{M}_{\odot}$. To be conservative, therefore, we exclude
these two groups from the rest of our analysis, though we have confirmed that
our results are not sensitive to this choice.

Name | RA ($\degree$) | Dec ($\degree$) | z | flag | $\log\big{(}\frac{M_{200c,\text{X}}}{\mathrm{M}_{\odot}}\big{)}$ | Ks limit | $\log\big{(}\frac{M_{200c,\text{dyn}}}{\mathrm{M}_{\odot}}\big{)}$ | $\sigma_{v}(\,{\rm km\,s}^{-1})$ | $N_{\rm spec}$ | $\lambda_{\mathrm{10.2,R<1Mpc}}$
---|---|---|---|---|---|---|---|---|---|---
COSMOS-30221 | 150.56200 | 2.50309 | 1.197 | 0 | 13.80 | 24.9 | 12.90 $\pm$ ${}^{0.30}_{0.39}$ | 200 $\pm$ 50 | 9 | $23.6\pm 5.0$
COSMOS-20267 | 150.44487 | 2.75393 | 1.138 | 1 | 13.91 | 24.9 | – | – | 2 | $20.0\pm 5.3$
COSMOS-30307 | 149.73943 | 2.34139 | 1.028 | 1 | 13.72 | 24.9 | – | – | 3 | $15.4\pm 5.7$
COSMOS-20028 | 149.46916 | 1.66856 | 1.316 | 0 | 13.89 | 24.9 | 13.33 $\pm$ ${}^{0.19}_{0.38}$ | 285 $\pm$ 75 | 10 | $30.0\pm 5.4$
COSMOS-20057 | 150.45229 | 1.91046 | 1.179 | 1 | 13.81 | 24.9 | – | – | 1 | $23.7\pm 5.0$
COSMOS-10155 | 150.59137 | 2.53778 | 1.138 | 1 | 13.89 | 24.9 | – | – | 3 | $12.0\pm 4.1$
COSMOS-30317 | 150.12646 | 1.99926 | 1.019 | 1 | 13.65 | 24.9 | – | – | 0 | 15.0$\pm 4.2$
COSMOS-20072 | 149.86012 | 1.99973 | 1.179 | 1 | 13.92 | 23.9 | – | – | 3 | $6.8\pm 4.6$
COSMOS-20199 | 150.70682 | 2.29253 | 1.095 | 1 | 13.70 | 23.9 | – | – | 5 | $-6.5\pm 2.6$
COSMOS-20198 | 149.59607 | 2.43788 | 1.168 | 1 | 13.81 | 23.9 | – | – | 3 | $14.8\pm 5.8$
COSMOS-20243 | 150.26115 | 2.76857 | 1.315 | 1 | 13.80 | 23.9 | – | – | 1 | $20.0\pm 5.6$
COSMOS-10063 | 150.35902 | 1.93521 | 1.172 | 0 | 13.74 | 23.9 | – | – | 9 | $18.85\pm 5.5$
COSMOS-10105 | 150.38295 | 2.10278 | 1.163 | 1 | 13.79 | 23.9 | – | – | 5 | $12.7\pm 4.9$
COSMOS-20125 | 150.62077 | 2.16754 | 1.404 | 0 | 13.81 | 23.9 | – | – | 8 | $4.4\pm 4.4$
COSMOS-10223 | 150.05064 | 2.47520 | 1.260 | 1 | 13.76 | 23.9 | – | – | 4 | $13.6\pm 3.7$
COSMOS-30323 | 150.22540 | 2.55061 | 1.100 | 2 | 13.71 | 23.9 | – | – | 3 | $5.6\pm 3.6$
SXDF49XGG | 34.49962 | $-5.06489$ | 1.091 | 0 | 13.77 | 25.3 | 13.25 $\pm$ ${}^{0.22}_{0.27}$ | 255 $\pm$ 50 | 14 | $17.0\pm 4.6$
SXDF64XGG∗ | 34.33188 | $-5.20675$ | 0.916 | 0 | 13.76 | 25.3 | 14.20 $\pm$ ${}^{0.18}_{0.21}$ | 530 $\pm$ 80 | 8 | –
SXDF76aXGG | 34.74613 | $-5.30411$ | 1.459 | 0 | 13.93 | 25.3 | 14.06 $\pm$ ${}^{0.38}_{0.54}$ | 520 $\pm$ 180 | 6 | $31.1\pm 5.2$
SXDF76bXGG | 34.74743 | $-5.32348$ | 1.182 | 0 | – | 25.3 | 12.98 $\pm$ ${}^{0.33}_{0.45}$ | 210 $\pm$ 65 | 7 | $3.3\pm 3.3$
SXDF87XGG∗ | 34.53602 | $-5.06303$ | 1.406 | 0 | 13.89 | 25.3 | 14.44 $\pm$ ${}^{0.19}_{0.223}$ | 700$\pm$ 110 | 9 | –
Table 1:  Group names correspond to those in Gozaliasl et al. (2019) and
Finoguenov et al. (2010) for COSMOS and SXDF, respectively; names in boldface
are those included in the GOGREEN (Balogh et al., 2021) spectroscopic survey.
SXDF76XGG has been split into "a" and "b" to identify the foreground group;
the original X-ray mass estimate for $M_{200c,X}$ has been retained only for
the higher redshift system. Group redshifts are taken from the original
catalogues, except where GOGREEN spectroscopy is available to provide an
improved measurement (see Appendix B.3). Column 5 is the group quality flag. A
flag value of 0 denotes a group with confirmed redshift from GOGREEN (Balogh
et al., 2021). Other flag numbers are based on the Gozaliasl et al. (2019)
catalogue: 1 for secure X-ray emission with well-defined centre and at least
one spectroscopic member, and 2 for a system that has some X-ray contamination
from foreground or background systems. Column 6 gives the group halo mass
estimates from the original catalogues, derived from observed X-ray
luminosities. Column 7 gives the $K_{s}$-band limiting magnitude for the
survey from which each group is drawn, as described in the text. Column 9
gives the (spectroscopic) velocity dispersion we determined for our GOGREEN
targeted groups in Appendix B.3, and column 8 shows the corresponding
dynamical masses. Column 9 gives the number of spectroscopic group galaxies in
each group, within a radius of 2$R_{200c}$. The final column gives the
richness for groups in our photometric sample, with richness defined as the
number of group members with $\log(M_{\rm stellar}/\mathrm{M}_{\odot})>10.2$
found within a 1 Mpc circular aperture (see Figure 2 for a comparison of these
values with the GOGREEN clusters sample from van der Burg et al. (2020)). The
two groups indicated with a $\ast$ are excluded from the analysis in this
paper, as their dynamical masses suggest they may exceed our threshold
definition for low-mass haloes.

The mean and median redshift of the group sample (1.179 and 1.170,
respectively) is somewhat lower than that of the field sample (1.236 and
1.39). We have verified that our conclusions are unchanged if we divide the
group and field samples into two redshift bins and conclude that our findings
are not sensitive to this difference.

###  2.2 Comparison samples

####  2.2.1 Higher halo masses at $1<z<1.5$

We contrast our low halo-mass systems at $1<z<1.5$ with eleven higher mass
clusters in the same redshift range from GOGREEN. Our measurements are similar
to those in van der Burg et al. (2020), but recalculated to include only
galaxies within $R_{200c}$. Halo masses are determined dynamically (Biviano et
al., 2021), and we show the correlation between these and the cluster richness
in Figure 2. This sample is divided into two bins of halo mass, though the
highest mass bin contains only two clusters at the lower end of the target
redshift range: SPT-CL J0546-5345 ($z=1.068$) and SPT-CL J2106-5844
($z=1.126$). We note that several clusters (SpARCS-1051, SpARCS-1638,
SpARCS-1034, SpARCS-0219) have low richness values more typical of our groups.

####  2.2.2 Intermediate redshift $0.5<z<0.7$

The galaxy populations in the X-ray selected groups at $z<1$ have been
extensively studied, notably by Giodini et al. (2012). We use a similar sample
but redo the analysis to ensure consistent methodology when comparing with our
higher redshift sample. Specifically, we select fourteen groups in the
redshift range $0.5<z<0.7$ from the COSMOS field, and use the UltraVISTA DR1
catalogue. Since the average rest-frame $UVJ$ colours and photometric
redshifts do not noticeably differ between UltraVISTA DR1 and DR3, we do not
make the adjustments to colours or photometric redshifts that we applied to
the higher redshift sample. The groups are required to be robustly identified
(quality flags $<3$) and in the halo mass range $13.6<\text{log}(M_{\rm
halo}/\mathrm{M}_{\odot})<14.0$. The average halo mass of the sample is
$\text{log}(M_{\rm halo}/\mathrm{M}_{\odot})\approx 13.78$, comparable to the
mass of our higher redshift group sample. The photometric redshift selection,
and statistical background subtraction, is done in an analogous way to that
for the $1<z<1.5$ sample.

For higher mass clusters at this redshift we use the published measurements
and uncertainties in the stellar mass functions for 21 clusters selected based
on their Sunyaev-Zeldovich (SZ) signal, from the Planck all sky survey, from
van der Burg et al. (2018). These clusters span the halo mass range
$14.5<\text{log}(M_{\rm halo}/\mathrm{M}_{\odot})<15.1$. These were analysed
in a very similar way to the clusters from van der Burg et al. (2020) that we
use at higher redshift.

The field sample we compare with at this redshift is comprised of all
UltraVISTA DR1 galaxies with photometric redshifts in the range $0.5<z<0.7$.

####  2.2.3 Low redshift $0.01<z<0.2$

At $0.01<z<0.2$ we use the SDSS-DR7 measurements from (Omand et al., 2014).
Galaxy groups are selected from the Yang et al. (2012) friends-of-friends
catalogue. Halo masses were determined through abundance matching, using the
total group luminosity to rank them. We select haloes in the same mass ranges
as at other redshifts, without any evolution correction; the final sample
includes 13806/3282/483 haloes in the low/intermediate/high mass bins. All
galaxies associated with a halo are included as members, with no additional
selection based on clustercentric radius. Stellar masses are computed
following the procedure described in Brinchmann et al. (2004), with a small
(10 per cent) correction to convert from a Kroupa (2001) to a Chabrier (2003)
initial mass function using Madau & Dickinson (2014). Quiescent galaxies are
identified as those with specific star formation rates
$\mathrm{sSFR}<-0.24\log({M_{\rm stellar}/\mathrm{M}_{\odot}})-8.50$, chosen
to lie below and parallel to the star-forming main sequence identified in
Omand et al. (2014).

##  3 Results

###  3.1 Stellar mass functions

The photometric redshift uncertainties in both UltraVISTA and SPLASH-SXDF are
still large enough that galaxies cannot be unambiguously identified as members
of groups or clusters. We therefore rely on statistical background
subtraction, using our representative field sample, to calculate the stellar
mass functions.

The number of group members of a given type (quiescent or star-forming), and
within a given stellar mass bin, is calculated as the number of galaxies
$N_{C}$ within a circular aperture $A_{C}$ around the group centre, and with
photometric redshifts such that $|z-z_{g}|<\Delta z$ relative to the group
redshift, $z_{g}$, minus the corresponding average number of galaxies in the
field within that same aperture and redshift slice. For each galaxy sub-
population (e.g. quiescent or star-forming) the average number of galaxies per
group that we find is described by the following expression:

$\displaystyle\phi(M)=\frac{1}{N_{G}}\sum_{g}\left[N_{C,z_{g}}(M)-N_{\text{survey},z_{g}}(M)\times\left(\frac{A_{C}}{A_{\text{survey}}}\right)\right],$
(5)

where $N_{G}$ is the number of groups, $g$ is a given group,
$N_{\mathrm{survey}}$, $A_{\rm survey}$ are the number of field galaxies and
the total area of the survey from which each group is drawn, respectively. We
use $M=M_{\rm stellar}$ in the above expression for brevity. The aperture size
is chosen to be $R_{200c}$, and a photometric redshift cut of $\Delta z=0.126$
was chosen for all three survey regions (see Appendix A.2 for explanation of
this cut choice). As well, since the area of a group aperture at a given
redshift is negligible relative to the rest of the given survey region at that
redshift, we refer to the overall survey area/volume as the “field”.

The error on the number of background-subtracted galaxies in a group is given
by summing in quadrature the Poisson counting error and the Poisson error term
for the field contribution $(A_{C}/A_{\text{survey}})$, which simplifies to
$\sigma\approx\sqrt{N_{C}(M)}$. The total error for the number of galaxies in
a given mass bin is then

$\displaystyle\sigma(M)=\frac{1}{\phi(M)}\sqrt{\sum_{g}N_{C,z_{g}}^{2}(M)},$
(6)

where $N_{C,z_{g}}$ is the number of galaxies in the circular aperture, $C$,
around a given group, $g$, at redshift $z_{g}$.

Figure 3: Background-subtracted stellar mass function of quiescent (top
panel), star-forming (middle), and total (bottom) group galaxies at $1<z<1.5$.
Overlaid on each plot are the Schechter function fits to the group data (solid
line), normalized to match the number of group galaxies per dex (bin size
$\Delta\log(M_{\mathrm{stellar}}/\mathrm{M}_{\odot})=0.2$), and with shaded
regions indicating the 68% confidence interval on the fit parameters, computed
as described in the text. Error bars shown represent the Poisson shot noise.
Figure 4: We show two of the Schechter function fit parameters for the
$1<z<1.5$ galaxy group population, for the total sample (black), the quiescent
galaxies (red) and the star-forming galaxies (blue). Points represent draws of
parameters within the 68 per cent confidence limits of the fits. The cyan
crosses indicate the best fit parameters. The $\alpha$ parameter for the
quiescent population is effectively unbounded: high values of $\alpha>2$
provide acceptable fits within $2\sigma$. The crosses with solid ellipses
represent the fit parameters and 68 per cent confidence limits for the massive
cluster population at the same redshifts, from van der Burg et al. 2020.

In Figure 3 we present the background-subtracted stellar mass functions for
the full sample of $1<z<1.5$ groups, and separately for the quiescent and
star-forming populations. Each bin is weighted by the number of contributing
groups, such that the resulting values are the average number of galaxies per
group, per dex in stellar mass.

The stellar mass functions are fit with Schechter (1976) functions of the form

$\displaystyle\Phi(M)=\frac{dN}{dM}dM=\Phi^{*}\bigg{(}\frac{M}{M^{*}}\bigg{)}^{\alpha}e^{-M/M^{*}}dM,$
(7)

where $M$ is the stellar mass, $M^{*}$ is the characteristic mass where the
Schechter function transitions between a power law and an exponential cut-off,
and $\alpha$ is the logarithmic slope of the faint-end power-law. We fit all
three parameters, separately for each sample, by minimizing the $\chi^{2}$.
Where needed, we arbitrarily increase the uncertainties to ensure
$\chi^{2}/\nu=1$ for the best fit model, where $\nu$ is the number of degrees
of freedom. This is only important for the quiescent population, for which a
single Schechter function does not provide a good fit ($\chi^{2}/\nu=2.56$);
uncertainties are therefore increased by a factor $\sim 1.6$. With this
adjustment we calculate the 68% confidence limits from the $\chi^{2}$
distribution and determine all parameter combinations that provide a
$\chi^{2}$ within these limits. All points are included in the fits, including
those with contributions from fewer than the maximum number of groups. The
best fit parameters and their uncertainties are given in Table 2, and the mass
functions are shown in Figure 3.

Groups
---
Population | $\log({M^{\ast}/\mathrm{M}_{\odot}})$ | $\alpha$ | $\phi^{\ast}$
|  |  | $[\mathrm{group}^{-1}~{}\mathrm{dex}^{-1}]$
Quiescent | $10.8^{+0.2}_{-0.3}$ | $0.1^{+0.7}_{-0.4}$ | $8.9^{+13.6}_{-6.2}$
Star-forming | $10.6^{+0.2}_{-0.2}$ | $-0.7^{+0.5}_{-0.4}$ | $8.6^{+24.0}_{-6.7}$
Total | $10.9^{+0.1}_{-0.1}$ | $-0.7^{+0.2}_{-0.2}$ | $14.0^{+15.3}_{-7.8}$

Field
---
Population | $\log({M^{\ast}/\mathrm{M}_{\odot}})$ | $\alpha$ | $\phi^{\ast}$
|  |  | $[10^{-3}~{}\mathrm{Mpc}^{-3}~{}\mathrm{dex}^{-1}]$
Quiescent | $10.63^{+0.04}_{-0.03}$ | $0.06^{+0.07}_{-0.07}$ | $1.42^{+0.28}_{-0.24}$
Star-forming | $10.82^{+0.03}_{-0.04}$ | $-1.27^{+0.02}_{-0.02}$ | $1.31^{+0.33}_{-0.27}$
Total | $10.90^{+0.04}_{-0.04}$ | $-1.12^{+0.03}_{-0.03}$ | $1.98^{+0.60}_{-0.46}$

Table 2: Best fit Schechter function parameters and their 68% confidence
limits, for the low-mass halo (group) population and the combined field
(UltraVISTA DR1, DR2, and SXDF). The normalization parameter for the group
galaxies Schechter fits, $\phi^{\ast}$, reproduces the curves in Figure 3; it
has units of number of galaxies per group per dex.  Figure 5: Measured
quiescent fractions for the group (large black squares, offset slightly
horizontally for clarity) and field (small black points). The red shaded
region reflects fits drawn from the 68% confidence limits on the fits to the
quiescent and star-forming stellar mass functions (constrained to be within
the 68% confidence limits on the total stellar mass function fit). The
quiescent fraction for the field derived from Schechter fits for the field is
also shown, as the black dashed curve with yellow shading showing the 68%
confidence limits. Points with $f_{Q}>1$ are a result of uncertainty on the
statistical background subtraction, which can lead to a formally negative
abundance of star-forming galaxies. We also show the quiescent fraction for
the spectroscopic group members (blue diamonds), i.e. within 2$R_{200c}$.

The parameters $M^{\ast}$ and $\alpha$ are compared with the corresponding
fits to the GOGREEN massive cluster population from van der Burg et al.
(2020), in Figure 4. The shape of the total stellar mass function is in
excellent agreement with that measured in more massive clusters. Both the
quiescent and star forming populations prefer a higher $\alpha$ slope than the
clusters, but the Schechter function fit parameter combinations in the two
samples are still consistent at the $2\sigma$ level. Importantly, we do not
observe the excess of low-mass quiescent group members that is seen at low
redshifts (Peng et al., 2010) and we rule out a steep ($\alpha<-1$) low-mass
slope for this population at the 99 per cent confidence level.

###  3.2 Quiescent fraction and quiescent fraction excess

We use the stellar mass functions in the previous section to compute the
quiescent fraction, defined as:

$\displaystyle
f_{Q}(M)\equiv\frac{N_{\text{Q}}(M)}{N_{\text{Q}}(M)+N_{\text{SF}}(M)},$ (8)

where $N_{\text{Q}}$ and $N_{\text{SF}}$ are the number of quiescent and star-
forming galaxies, respectively, as identified in $UVJ$ colour space (see
§2.1.3).

Figure 6: Quiescent fraction excess is shown as a function of stellar mass for
our $1<z<1.5$ galaxy group sample (black squares). The quiescent fraction
excess is significantly nonzero only for $M\gtrsim
10^{11}~{}\mathrm{M}_{\odot}$. The red shaded region represents the 68%
confidence limits derived from the Schechter function fits to our stellar mass
functions. Quiescent fraction excess values (grey triangles) and best fit line
(gray dashed line) are also plotted for the GOGREEN $1<z<1.5$ clusters in (van
der Burg et al., 2020). The clusters show a strong trend in QFE with stellar
mass, particularly above $\log(M^{*}/\mathrm{M}_{\odot})\approx 10.75$.
Additionally, the clusters display significant quiescent fraction excess at
lower stellar masses, with a trend in the data consistent with an
approximately flat relation for
log($M_{\mathrm{stellar}}/\mathrm{M}_{\odot}$)<log($M^{*}/\mathrm{M}_{\odot})$.
Figure 7: Quiescent fraction excess (QFE) shown as a function of halo mass
($M_{200c}$), $M_{\rm halo}/\mathrm{M}_{\odot}$, for two galaxy stellar mass
bins, and for samples at three different redshift ranges, as indicated. Our
new measurements of low-mass haloes at $1<z<1.5$ are shown as the green points
connected by a green solid line. The other samples are described in §2.2. The
halo-mass dependence of the QFE has a similar slope at all redshifts and
stellar mass bins, of $\sim 0.24$ (see text for details). See Figure 19 for a
more detailed breakdown of both $f_{Q}$ and QFE by stellar mass.

We show $f_{Q}$ as a function of stellar mass for our group sample in Figure
5. Uncertainties on the binned data are computed assuming the quiescent and
star-forming stellar mass populations are independent. However, we correctly
account for the covariance when deriving the quiescent fraction from the
Schechter function fits, which are overlaid as the shaded region. We calculate
this by taking random draws from fits within the 68% confidence limits of the
quiescent and star-forming populations, and only keep those for which the sum
is in agreement with the total SMF within the same 68% confidence. There is a
strong dependence on stellar mass, with $f_{Q}$ increasing from near zero to
unity over the full range. For high stellar masses, $M_{\rm
stellar}>10^{11}~{}\mathrm{M}_{\odot}$, $f_{Q}$ is systematically larger in
the group sample than the field. At lower masses the excess is both smaller
and statistically not significant. To complement this comparison, we also
consider the quiescent fraction for the spectroscopic members of the seven
GOGREEN groups (i.e. excluding the two with high dynamical masses, that are
excluded from all our analysis). This has the advantage that it does not rely
on statistical background subtraction. However, the smaller sample size means
we can only consider three stellar mass bins, and we also include all
spectroscopic group members within $2R_{200c}$, a larger aperture than for the
photometric sample. These spectroscopic members are from GOGREEN and any of
the available surveys described in §B.1. As shown in the Figure, the quiescent
fractions for this spectroscopic subsample are fully consistent with our full
group sample.

To better characterize any difference in $f_{Q}$ in groups relative to the
field, we calculate the quiescent fraction excess (QFE)555We choose the
terminology, QFE, for consistency with recent prior works (Wetzel et al.,
2012; Bahé et al., 2017; van der Burg et al., 2020) and a more intuitive
meaning than a variety of other synonymous terms used in the literature. Other
terms synonymous with QFE used in the literature include “transition fraction”
(van den Bosch et al., 2008), “conversion fraction” (Balogh et al., 2016;
Fossati et al., 2017), and “environmental quenching efficiency” (Peng et al.,
2010; Wetzel et al., 2015; Nantais et al., 2017; van der Burg et al., 2018). .
This is defined as

$\displaystyle\text{QFE}\equiv\frac{f_{Q,\text{cluster}}-f_{Q,\text{field}}}{1-f_{\text{Q},\text{field}}},$
(9)

where $f_{\text{Q},\text{field}}$ and $f_{\text{Q},\text{cluster}}$ are the
fractions of quiescent galaxies in the field and cluster, respectively. In a
naive infall interpretation, this quantity represents the fraction of galaxies
accreted from the field that have been transformed into quiescent galaxies to
match the observed group population (e.g. van den Bosch et al., 2008; Peng et
al., 2010; Wetzel et al., 2012; Balogh et al., 2016; Bahé et al., 2017). We
note that QFE$<0$ is a physical solution even in the presence of environmental
quenching processes, since the field is defined as the global population,
including massive haloes; there will therefore be a halo mass scale below
which most “field” galaxies reside in more massive haloes.

The QFE values for our group sample are shown as a function of stellar mass in
Figure 6. The average QFE is significantly greater than zero and shows an
increasing trend with stellar mass, similar to those for the $1<z<1.5$
clusters in van der Burg et al. (2020), also shown in Figure 6. These results
are also similar to those published by Balogh et al. (2016), for groups at a
slightly lower redshift $0.8<z<1$, though we find a larger QFE at the highest
stellar masses. Interestingly, both groups and clusters are consistent with a
mass-independent QFE below $M^{*}$ (QFE $\approx 0$ for groups, QFE $\approx
0.35-0.40$ for clusters) and a significant jump in QFE towards QFE $\approx 1$
above $M^{*}$.

###  3.3 The halo mass dependence of galaxy quenching

We now present QFE as a function of halo mass and stellar mass, in Figure 7.
The group and cluster samples at each redshift are described in §2.2, as are
the definitions of the field samples. Motivated by the results in Figure 6 we
show the results in two stellar mass bins, below and above the jump in QFE.
Our conclusions are unchanged if we use a smaller binning, which we
demonstrate in Appendix C.

In general we find the QFE increases with increasing stellar mass and halo
mass, with at most modest redshift evolution when those parameters are fixed.
Most notably, the dependence of QFE on the logarithm of halo mass appears to
be similar in all stellar mass and redshift bins. To further quantify this, we
fit a linear regression model to QFE as a function of $M_{\rm halo}$ for all
the data, with a single slope but different intercepts for each redshift and
stellar mass bin. We find a slope of $m=0.24\pm 0.04$ with a reduced
$\chi^{2}$ of $\sim 1.06$ indicating an acceptable fit. The points that appear
most discrepant with this simple scaling are for the highest halo masses in
the stellar mass range $10^{10}<M_{\rm stellar}/\mathrm{M}_{\odot}<10^{10.7}$.
The QFE for the GOGREEN data are actually lower than that measured at
intermediate halo masses at the same redshift, though they are consistent at
the $1.6\sigma$ level as determined by a two-tailed split-Gaussian hypothesis
test. Though this appears to differ from the simple scaling derived above, we
note that this approximate independence of QFE on halo masses above $\sim
2\times 10^{14}M_{\rm halo}$ is in fact consistent with what we observe at
other redshift and stellar mass bins, and also with the simulation predictions
discussed below, in §4.1. Although there are only two clusters contributing to
this bin, sample variance is unlikely to be large enough to explain the large
difference between this measurement and the measurement of similarly massive
clusters at lower redshift, given the observed homogeneity of cluster systems
(e.g. Trudeau et al., 2020). On the other hand, the QFE observed in the
massive Planck-selected clusters at $0.5<z<0.7$ is significantly higher than
even the $z=0$ sample at the same mass, and implies a steeper logarithmic
slope than we fit for the sample as a whole. It is possible that this reflects
a bias resulting from the SZ-selection, or a difference in the field samples
near those clusters, but we do not have a good explanation for the result. It
would be useful to include more cluster samples at intermediate redshift in a
future analysis.

##  4 Discussion

It is well known that the quiescent fraction of galaxies in clusters shows a
general decrease with increasing redshift (e.g. Butcher & Oemler, 1984; Haines
et al., 2013) and quiescent populations of galaxies are now well studied for
clusters at $z>1$ (e.g. Brodwin et al., 2013; Nantais et al., 2016; Lee-Brown
et al., 2017; Kawinwanichakij et al., 2017; Foltz et al., 2018; Strazzullo et
al., 2019; Trudeau et al., 2020; van der Burg et al., 2020). In this work we
have used the wide halo mass range of the GOGREEN and COSMOS/SXDF cluster
catalogues to demonstrate (Figure 7) that QFE correlates (increases) with both
stellar mass and halo mass at $1<z<1.5$.

A compilation of QFE values (integrated over a broad stellar mass range) for
low halo mass “groups” at various redshifts was presented by Nantais et al.
(2016). We show an updated version of their figure including our background-
subtracted group measurements in Figure 8, for all galaxies with $M_{\rm
stellar}>10^{10}~{}\mathrm{M}_{\odot}$. Overall, our work is broadly
consistent with the published literature: even within low-mass haloes the
galaxy population exhibits enhanced quenching relative to the field. A
possible redshift trend of QFE with redshift may be apparent in this
compilation. However, we resist drawing any strong conclusions from a further
quantitative comparison, given significant methodological differences between
studies. Moreover, the stellar mass dependence of QFE complicates any physical
interpretation of these integrated values.

Figure 8: Compilation of quiescent fraction excess measurements as a function
of redshift for groups of galaxies, adapted from Nantais et al. (2016). Lines
connect points from the same study. Our sample (green squares) include all
galaxies with $M_{\rm stellar}>10^{10}~{}\mathrm{M}_{\odot}$, the 95% stellar
mass completeness limit at $z=1.5$. Overall the compilation appears to
indicate a gradual redshift evolution of QFE. We caution, however, that the
different analyses shown here are not fully consistent in their methodology or
sample selections.

Our results build on the earlier work of van der Burg et al. (2020), who
measured the stellar mass function of the GOGREEN cluster sample and found
that, while the fraction of quiescent galaxies is much higher in the clusters
than the field, the shape of the stellar mass function for quiescent galaxies
is identical in both environments. The same is true for the star-forming
population. This is a puzzling result and it indicates that, unlike at low
redshift, environmental quenching is not separable from the stellar mass
dependence. This is reflected in the fact that the QFE strongly increases with
increasing stellar mass, from $\sim 30\%$ at $\sim
10^{10}~{}\mathrm{M}_{\odot}$ to $\sim 80\%$ at
$>10^{11}~{}\mathrm{M}_{\odot}$, in contrast with studies in the local
universe. A possible explanation for this, as described in van der Burg et al.
(2020), is that the quenching mechanism in these $z>1$ clusters is an
accelerated version of the same process affecting field galaxies.

However, this interpretation would naively lead to the prediction that cluster
galaxies should be substantially older than field galaxies, which contradicts
the findings of Webb et al. (2020). In that work, SFHs were measured for 331
quiescent galaxies in the same GOGREEN cluster and field samples, using the
PROSPECTOR Bayesian inference code (Leja et al., 2017; Johnson et al., 2019)
to fit the photometric and spectroscopic observations666The full posteriors of
Webb et al. (2020)’s PROSPECTOR fits are available from the Canadian Advanced
Network for Astronomical Research (CANFAR), at
www.canfar.net/storage/list/AstroDataCitationDOI/CISTI.CANFAR/20.0009/data;
DOI:10.11570/20.0009.. They find that galaxies in clusters in the stellar mass
range $10^{10}-10^{11.8}~{}\mathrm{M}_{\odot}$ are older than field galaxies,
but only by $\lesssim 0.3$ Gyr.

As dark matter haloes grow, it is unclear exactly how or when environmental
quenching processes become important (e.g. Bahé et al., 2013). In particular,
if environmental processes are important in low-mass haloes, the quenching of
star formation may take place long before galaxies are finally accreted onto
massive clusters. In the following two subsections, we first explore how well
hydrodynamic simulations reproduce our observations and then we use what we
have learned about the halo mass dependence of the quiescent fraction at
$1<z<1.5$ to explore the extent to which pre-processing could reconcile the
van der Burg et al. (2020) and Webb et al. (2020) results.

###  4.1 Comparison with BAHAMAS hydrodynamic simulation predictions

Figure 9: We show the quiescent fraction excess from the BAHAMAS hydrodynamic
simulation as a function of stellar and halo mass, $M_{\rm
halo}/\mathrm{M}_{\odot}$ at two redshift snapshots as indicated. The
corresponding data from Figure 7 are reproduced here for comparison. In the
simulations, both $f_{Q}$ and QFE decrease with increasing stellar mass, in
stark contrast with the data. However, the correlation with halo mass and
redshift is qualitatively similar to the trends observed in the data. See
Figure 20 for a more detailed breakdown of both $f_{Q}$ and QFE by stellar
mass.

We begin considering the physical implications of our result by determining
the extent to which these correlations are naturally predicted by hydrodynamic
simulations. For this we use the BAHAMAS cluster simulations (McCarthy et al.,
2017; McCarthy et al., 2018), which were run with the standard Planck 2013
cosmology (Ade et al., 2014), using 2$\times 1024^{3}$ particles in a large
cosmological volume, 400 Mpc $h^{-1}$ on a side. Dark matter and (initial)
baryon particles masses of $\approx 4.44\times
10^{9}~{}h^{-1}~{}\mathrm{M}_{\odot}$ and $8.11\times
10^{8}~{}h^{-1}~{}\mathrm{M}_{\odot}$ are used, respectively. These
simulations implement various subgrid physics models, including metal-
dependent radiative cooling in the presence of a uniform photoionising
UV/X-ray background, star formation stellar evolution and chemical enrichment,
and stellar and AGN feedback (see Schaye et al., 2010b, and references therein
for a detailed description of the subgrid implementations). Consistent with
our work and compiled literature results, a Chabrier (2003) IMF is assumed.
The parameters of these prescriptions were adjusted to reproduce the observed
Kennicutt-Schmidt law, the observed galaxy stellar mass function (GSMF), and
the amplitude of group/cluster gas mass fraction–halo mass relation at $z\sim
0$. Thus, these simulations are distinguished from some others by the fact
that they are deliberately calibrated to ensure the correct total baryon
content in haloes. This is important when considering environmental effects on
group scales where, for example, hydrodynamic interactions with the hot gas
may be important.

We select all BAHAMAS groups with halo masses
log$(M_{200c}/\mathrm{M}_{\odot})>13.2$ (no explicit upper halo mass limit).
The group selection is based on a Friends-of-Friends group-finding algorithm
applied to two separate snapshots of the BAHAMAS simulation, at $z=0$ and
$z=1.25$, respectively. For each identified group, all galaxies within
$R_{200c}$ (in 3D space) are considered to be group members. For each group,
the field sample is taken to be all galaxies outside $2.5R_{200c}$. To
separate quiescent from star-forming galaxies, we use the sSFR threshold from
Franx et al. (2008): $\mathrm{sSFR}>0.3/t_{H(z)}$, where $t_{H(z)}$ is the
Hubble time at a given redshift. At $z\sim 1.25$ $t_{H(z)}\sim 5$ Gyr, so this
corresponds to $\mathrm{sSFR}>6\times 10^{-11}\mbox{ yr}^{-1}.$ Although this
is different from the $UVJ$ selection made in the data, we have confirmed that
the qualitative trends of $f_{Q}$ and QFE are stable for large variations in
the choice of star-forming threshold – a multiplicative factor of 2-3 to the
sSFR cut does not qualitatively change our conclusions.

We focus our attention primarily on the QFE trends with halo mass in BAHAMAS,
shown in Figure 9 for the same stellar mass bins as in Figure 6. The quiescent
fractions themselves, and an alternative stellar mass binning, are provided in
Appendix C. We first consider the high stellar mass sample, in the right
panel. The BAHAMAS predictions at $z=0$ are in quite good agreement with the
data, reproducing both the absolute value of the QFE and its dependence on
halo mass. The simulations predict that this halo mass dependence becomes
significantly steeper at $z\sim 1$, in contrast with the observations. There
is reasonable agreement at high halo masses ($M_{\rm halo}>10^{14}~{}{\rm
M}_{\odot}$), though the modest redshift evolution is in the opposite sense to
the observations. On group scales, however, the models predict no significant
QFE at $z\sim 1$, significantly below our measured QFE $=0.48\pm 0.15$.

Turning now to the lower stellar mass bin, $10<\log({M_{\rm
stellar}/\mathrm{M}_{\odot}})<10.7$, the model generally predicts a steep
increase in QFE with halo mass, before flattening around $M_{\rm halo}\sim
2\times 10^{14}~{}{\rm M}_{\odot}$. Over the whole mass range, the dependence
of QFE on halo mass has a logarithmic slope of $\sim 0.3$ at both redshifts
(i.e. a factor of 10 increase in halo mass results in an increase of $\sim
0.3$ in QFE), remarkably similar to our measurement in §3.3. The sense and
magnitude of the redshift evolution is also in good agreement with the
observations. Despite these successes, the absolute value of the QFE itself is
too high, for all halo masses at both $z=0$ and $z=1.25$. This reflects the
difficulties faced by many simulations and models, and is likely due to an
incomplete understanding of feedback (see Kukstas et al. 2021, in
preparation). The result is also sensitive to choices in how quiescent
galaxies are defined, and the aperture within which star formation rates and
masses are measured in the simulations (e.g. Furlong et al., 2015; Donnari et
al., 2019, 2021). The fact that the simulations predict a halo mass dependence
of QFE that is similar to what we observe over a wide range in redshift and
halo mass is encouraging, and suggests that they may be correctly capturing
the relevant physics associated with the impact of large scale structure
growth on galaxy evolution, even if the feedback prescriptions themselves are
not sufficiently accurate to reproduce the observed dependence of QFE on
stellar mass.

###  4.2 Toy models

As described earlier, simple toy models of galaxy clusters at $1<z<1.5$, in
which environmental quenching occurs after accretion onto the main progenitor,
are unable to simultaneously match the observed quiescent fractions and
relative ages of quiescent galaxies (van der Burg et al., 2020; Webb et al.,
2020). In particular, van der Burg et al. (2020) note that the stellar mass-
dependence of environmental quenching needs to be similar to that of the
quenching process in the general field to result in the observed SMFs, and
they propose that clusters experience an early accelerated form of that same
phenomenon during the protocluster phase. However, Webb et al. (2020) find
mass-weighted ages in cluster galaxies that are only slightly older than field
galaxies. Webb et al. (2020) then use a simple infall model to demonstrate
that neither a simple head-start to formation time for the cluster galaxies
nor a simple quenching time delay (i.e. time since infall into a cluster)
alone can explain both the enhanced quiescent fraction and very similar mass-
weighted ages (MWAs) for the cluster and field populations.

Figure 10: The time delay parameter values for our pre-processing and no pre-
processing toy models are plotted as a function of stellar mass (points with
error bars) for both groups (green) and clusters (orange). These time delays
are directly constrained by the group/cluster quiescent fractions in each
stellar mass bin; the uncertainties are a result of uncertainties on the
observed quiescent fractions (i.e. propagated through the model).

Our results, which show how galaxy populations correlate with environment in
haloes with masses well below that of massive clusters, suggest that “pre-
processing” is likely to be important. The BAHAMAS simulations include a more
complete treatment of halo growth and hydrodynamic processes, including pre-
processing in a physically motivated way. The fact that those simulations
predict a similar trend of QFE with halo mass to that observed is encouraging,
and suggests that the failures of the toy models discussed above may lie in
their simplified definitions of accretion time. We aim in this section to fit
and contrast two toy models, one with pre-processing and one without, using
the quiescent fraction excess in a range of halo masses at $1<z<1.5$. We can
then use these models to predict group/cluster galaxies’ average stellar mass-
weighted ages and compare to values derived from GOGREEN observations.

####  4.2.1 Toy model descriptions

The infall-based quenching model we use here is the same as that in Webb et
al. (2020), but with changes to the accretion history. More explicitly, we use
the Schreiber et al. (2015) SFR evolution for star-forming galaxies. When
galaxies are quenched their SF is immediately truncated. We track the number
of star-forming and quiescent galaxies from $z=10$ to $z=1.2$ and then compare
galaxies which have stellar masses between
$10^{10}$–$10^{12}~{}\mathrm{M}_{\odot}$ at $z=1.2$. Galaxies that are self-
quenched are modelled using the self-quenching efficiency proposed by Peng et
al. (2010), i.e.: the quenching probability is $\propto$
SFR/$M_{\mathrm{stellar}}$, using the SFR for a galaxy of stellar mass
$M_{\mathrm{stellar}}$. We also assume that the shape of star-forming SMFs in
our model do not differ between field and clusters, as found by van der Burg
et al. (2020). Finally, it is assumed that all galaxies can undergo satellite
quenching, with star-forming galaxies in clusters completely quenching after a
$t_{\mathrm{delay}}$ amount of time has elapsed from the time of first
“infall”, which we will define below.

Figure 11: Using our fit $t_{\mathrm{delay}}$ values (Figure 10), we predict
the difference in average formation time between the groups/high-mass clusters
and the field. For a given galaxy in a given sample, $t_{\mathrm{form}}$ is
defined as $t_{\mathrm{obs}}-$MWA, where MWA is the mass-weighted age from the
stellar population synthesis modelling of quiescent galaxies in the GOGREEN
spectroscopic sample (Webb et al., 2020). Model prediction values are shown
with simple points with errorbars and connecting lines for the groups and
clusters in green and orange, respectively. Observationally-derived average
values are also shown, with the high-mass clusters running average shown as
the dashed orange line (shaded regions for the boot-strapped standard
deviation on the mean) and the groups are shown as a single point (black star)
with green shaded region. The horizontal width of the green groups shaded
region is the bootstrapped standard deviation on the mean stellar mass.
Quiescent galaxies in groups are younger than those in clusters, ruling out
the predictions of our simple model without pre-processing. Although the time
delay exhibits a strong dependence on stellar mass, the dependence of mean MWA
on stellar mass in our models is weak. The MWA has a significant halo mass
dependence only for the no pre-processing model. For a plot showing the
$t_{\mathrm{obs}}-$MWA values for individual galaxies in all samples, see
Figure 18.

For this quenching model there are therefore two parameters. The self-
quenching efficiency normalization is set by reproducing the observationally
measured field quiescent fraction (i.e. with $t_{\mathrm{delay}}$ fixed to
$t_{\mathrm{delay}}=0$). The $t_{\mathrm{delay}}$ parameter is then
iteratively fit to reproduce the fraction of quiescent galaxies in a given
cluster (see specifics described further in §4.2.2). As this represents
quenching in excess of the field population, the delay time corresponds most
directly to the QFE. If $t_{\mathrm{delay}}=0$, all cluster galaxies would be
quenched; increasing the parameter reduces the QFE as fewer galaxies have been
in the cluster long enough to quench. Finally, we neglect mergers, which are
included in van der Burg et al. (2020), for simplicity.

To explore whether simple pre-processing alleviates tension between quiescent
fraction and ages, we consider two extreme definitions of galaxy “infall
time”. The first defines infall as the first time a central galaxy becomes a
satellite, using the models of McGee et al. (2009) (which were applied to
N-body dark matter simulations) as published in Balogh et al. (2016). In their
findings, this amounts to a roughly constant accretion rate, with clusters
starting their accretion $\sim 0.5$ Gyr before groups. For the other extreme
we assume galaxies are only accreted once they cross the virial radius of the
most massive progenitor halo, using the halo mass accretion rate in Bouché et
al. (2010). We assume that the number of infalling galaxies in a given
timestep is proportional to the mass accreted in a given timestep. We will
refer to these two models as the “pre-processing model” and “no pre-processing
model”, respectively.

For simplicity, we only compare our “groups” sample (lowest halo-mass bin)
with the high-mass clusters (highest halo mass bin) at $1<z<1.5$ presented in
our results section (see §3.3).

####  4.2.2 Toy model results

We start by considering the stellar mass dependence of the $t_{\text{delay}}$
parameter, in Figure 10. This parameter is effectively calculated from the
observed quiescent fraction excess (see §4.2.1), which has a strong stellar
mass trend; thus, $t_{\text{delay}}$ similarly shows a strong trend with
stellar mass, such that $t_{\text{delay}}$ decreases with increasing stellar
mass in both models. As well, we observe the expected difference between the
pre-processing and no pre-processing infall models. Pre-processing models
require a longer $t_{\text{delay}}$ to reproduce the same quiescent fraction,
given that first accretion happens earlier. The values of $t_{\text{delay}}$
that we find are broadly consistent with similar work at these redshifts (e.g.
Balogh et al., 2016), and with measurements at lower redshift (e.g. Wetzel et
al., 2013) assuming they evolve proportionally to the dynamical time. As we
are primarily interested in the trends with stellar and halo mass, we do not
comment further on the absolute value of this parameter.

Most relevant to our discussion here, we find that the halo mass dependence of
$t_{\text{delay}}$ depends on the accretion model. In the no pre-processing
case there is a significant dependence on halo mass. Shorter delay times are
needed in higher mass clusters, to reproduce our observations that the
quiescent fractions are higher in those systems. In the pre-processing model,
galaxies accreted into a cluster effectively get a head-start, and this
largely accounts for the difference in quiescent fraction. Thus, we find that
in a pre-processing model, the observed dependence of QFE on halo mass (at
fixed stellar mass) can be explained with a $t_{\text{delay}}$ that has at
most a weak dependence on halo mass. In this case the variation in QFE with
halo mass derives primarily from the fact that the accretion time distribution
is a function of halo mass (see e.g. De Lucia et al. (2012) for further
discussion of this).

We now try to break this degeneracy by considering the mass-weighted ages of
the quiescent galaxies in the two models. To show this, we define a formation
time, $t_{\mathrm{form}}=t_{\text{obs}}-\text{MWA}$, which is the difference
between the age of the universe (at a given galaxy’s observed redshift) and
the determined MWA value. This value reflects how long after the Big Bang it
took until the mass-weighted bulk of stars had formed. We show our MWA
predictions in the form of the difference in $t_{\mathrm{form}}$ between
groups/clusters and the field in Figure 11. The no pre-processing model
predicts $t_{\mathrm{form}}$ should increase (ie: MWA decreases) with
increasing halo mass, by about $\sim 150-200$ Myr between the lowest and
highest halo masses shown here. In contrast, the pre-processing model predicts
no significant dependence of MWA on halo mass.

We now compare these predictions directly with observed
measurements777Specifically, here we show the medians of the posteriors. of
MWA from Webb et al. (2020), shown in Figure 11. The running mean
$t_{\mathrm{form}}$ for the highest halo mass sample (“high-mass clusters”) is
shown; for the group sample, which includes only 15 quiescent galaxies, we
show the mean and standard deviation (green shaded region) of the whole
stellar mass range ($10<\log(M_{\mathrm{stellar}}/\mathrm{M}_{\odot})<12$). A
more detailed version of this Figure, showing age measurements for individual
galaxies in all samples, is given in Appendix B.4.

The no pre-processing model predicts that quiescent galaxies in groups should
be $\sim 200$ Myr older than galaxies in clusters. Although our sample of
group galaxies is small, it is inconsistent with this prediction: if anything,
the quiescent group galaxies are younger than their counterparts in more
massive systems. The data are more consistent with the pre-processing model.
In this case, dependence on halo mass is weak, but in the observed direction
for the highest stellar mass bin we consider, which also corresponds most
closely to the mean stellar mass of our data.

In summary, including pre-processing does a reasonable job of explaining the
halo mass dependence of quiescent galaxy ages, with a $t_{\rm delay}$
parameter that is nearly independent of halo mass. This is broadly consistent
with a picture where the environmental quenching is caused by the same
physical mechanism in groups and clusters. The data suggest, however, that
quiescent galaxies in clusters may be even older than can be explained in the
pre-processing model. One possible explanation for this would be if galaxies
in rich proto-cluster environments undergo earlier quenching than primordial
environments for group galaxies, as discussed in van der Burg et al. (2020)
and Webb et al. (2020). It seems increasingly likely that a significant
portion of the GOGREEN cluster galaxy population was subject to an accelerated
quenching mechanism at $z\sim 3-4$. This is additionally compatible with
recent high redshift work showing that quiescent galaxies exist at redshifts
as high as e.g. $z\sim 3-5$ (Valentino et al., 2020; Forrest et al., 2020).

##  5 Conclusions

We use photometric redshifts and statistical background subtraction to measure
stellar mass functions of galaxies in low mass haloes at $1<z<1.5$ (“groups”).
These groups are selected from COSMOS and SXDF surveys, based on X-ray and
sparse spectroscopy. We compute the quiescent fraction ($f_{Q}$) and quiescent
fraction excess (QFE) for these systems, as a function of stellar mass. The
result is then compared with higher mass clusters at $1<z<1.5$ from the
GOGREEN survey (Balogh et al., 2021), and a compilation of lower redshift
samples at $0<z<0.2$ and $0.5<z<0.7$ that span a similar range of halo mass as
our $1<z<1.5$ samples.

Observationally, we find:

  * •

Excess quenching in $1<z<1.5$ groups relative to the field, with an overall
QFE of $\sim 20\%$ for galaxies with
$\log(M_{\mathrm{stellar}}/\mathrm{M}_{\odot})>10$.

  * •

Unlike at low redshift, environmental quenching is not separable from the
stellar mass dependence. This can be seen as an increase of the QFE in groups,
from $\sim 10\%$ below $M^{*}$ to $\sim 100\%$ above $M^{*}$. A similar trend
is present in more massive clusters, where the QFE increases from $\sim 40\%$
to $\sim 85\%$.

  * •

When controlling for stellar mass, both $f_{Q}$ and QFE correlate (increase)
with halo mass. Observations at all redshifts and stellar masses are
consistent with a single logarithmic slope of
$\mathrm{d}(\mathrm{QFE})/\mathrm{d}\log(M_{\mathrm{halo}})\sim 0.24\pm 0.04$.

In our discussion, we compare to the BAHAMAS hydrodynamical simulation and
also with toy models in which galaxies quench star formation upon infall,
after some time delay. For the latter we contrast a pre-processing model,
where galaxies begin this quenching time delay upon infall into any larger
halo, and a no pre-processing model in which the time delay only begins when
the galaxy is accreted into the main progenitor.

From this analysis, we find:

  * •

The BAHAMAS hydrodynamic simulation reproduces the trend of quiescent fraction
excess with halo mass seen in the data. Specifically they show a steep
increase in QFE with halo mass, which then flattens to a near constant value
for halo masses $M_{\rm halo}\gtrsim 2\times 10^{14}~{}\mathrm{M}_{\odot}$.
When fit with a straight line, the average trend is
$\mathrm{d}(\mathrm{QFE})/\mathrm{d}\log(M_{\mathrm{halo}})\sim 0.30$, which
compares well with the observed $0.24\pm 0.04$. This suggests the simulation
may be capturing the relevant physics behind the role of large scale structure
growth on galaxy evolution.

  * •

Both the quiescent fraction and the quiescent fraction excess predicted by
BAHAMAS decreases with increasing stellar mass, opposite to what is observed.
This probably indicates an incomplete model of subgrid feedback and/or star
formation at galaxy scales in the BAHAMAS simulation.

  * •

From the toy models, we find the time delay until quenching begins must depend
on stellar mass, reflecting the strong dependence of group/cluster quiescent
fractions on stellar mass. In the absence of pre-processing, this delay time
also has a strong dependence on halo mass, decreasing with increasing mass.

  * •

We find pre-processing reduces the discrepancy with the observed halo mass
dependence of quiescent galaxy mass-weighted ages. Specifically, assuming
quenching occurs when a galaxy first becomes a satellite increases the average
age of quiescent cluster galaxies, relative to a model without pre-processing.
However, the data suggest that quiescent galaxies in clusters at $1<z<1.5$ may
still be older than can be explained in this simple pre-processing model.

These observations further demonstrate that galaxy evolution depends on more
than just stellar mass, in a nontrivial way that is still not fully captured
by models. The environment, at least through the host halo mass, plays an
important role at all redshifts $z<1.5$. This effect, however, is not
separable from the dependence on stellar mass. Moreover, it is important even
in low-mass haloes at $z\sim 1$, and thus likely not solely due to extreme
physics like ram pressure stripping of cold gas reservoirs. The most natural
physical mechanism that is expected to operate on all scales probed in this
work is the shutoff of cosmological accretion onto satellites, and the
subsequent overconsumption of gas reservoirs (e.g. McGee et al., 2014). This
physics should be included with reasonable fidelity in hydrodynamic
simulations, and it is encouraging that the BAHAMAS simulations are able to
reproduce the observed halo-mass dependence, even while there remain problems
on small-scales. Forthcoming, homogeneous surveys with large telescopes –
particularly those with highly multiplexed spectroscopy – will make these
statements much more precise and useful for constraining models. Observations
of high redshift protoclusters, with JWST and other facilities, will determine
whether or not there are additional effects that accelerate star formation
quenching in these environments, as hinted at indirectly by our data.

## Acknowledgements

We thank the native Hawaiians for the use of Mauna Kea, as observations from
Gemini, CFHT, and Subaru were all used as part of our survey.

Data products were used from observations made with ESO Telescopes at the La
Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data
products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf
of the UltraVISTA consortium. As well, this study makes use of observations
taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ESA
HST, which is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS5-26555. MB gratefully acknowledges
support from the NSERC Discovery Grant program. BV acknowledges financial
contribution from the grant PRIN MIUR 2017 n.20173ML3WW_001 (PI Cimatti) and
from the INAF main-stream funding programme (PI Vulcani). GW acknowledges
support from the National Science Foundation through grant AST-1517863, HST
program number GO-15294, and grant number 80NSSC17K0019 issued through the
NASA Astrophysics Data Analysis Program (ADAP). Support for program number
GO-15294 was provided by NASA through a grant from the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Incorporated, under NASA contract NAS5-26555. GR thanks the
International Space Science Institute (ISSI) for providing financial support
and a meeting facility that inspired insightful discussions for team “COSWEB:
The Cosmic Web and Galaxy Evolution”. GR acknowledges support from the
National Science Foundation grants AST-1517815, AST-1716690, and AST-1814159,
NASA HST grant AR-14310, and NASA ADAP grant 80NSSC19K0592. GR also
acknowledges the support of an ESO visiting science fellowship. This work was
supported in part by NSF grants AST-1815475 and AST-1518257.

We thank M. Salvato for allowing us to use the master spectroscopic catalog
used within the COSMOS collaboration.

This work made use of data products derived from Prospector (Leja et al.,
2017; Johnson et al., 2019), python-fsps and FSPS (Conroy & Gunn, 2010; Conroy
et al., 2009). We also used the following Python (Van Rossum & Drake Jr, 1995)
software packages: Astropy (Robitaille et al., 2013; Price-Whelan et al.,
2018), matplotlib (Hunter, 2007), scipy (Virtanen et al., 2020), ipython
(Pérez & Granger, 2007), numpy (Harris et al., 2020), pandas (pandas
development team, 2020; Wes McKinney, 2010), and emcee (Foreman-Mackey et al.,
2013).

##  6 Data availability

Group catalogues are publicly available for both the COSMOS and SXDF group
samples at
https://academic.oup.com/mnras/article/483/3/3545/5211093?login=true#supplementary-
data and in Finoguenov et al. (2010), respectively.

The photometric datasets used for the $1<z<1.5$ groups analysis in this work
are from sources in the public domain; UltraVISTA DR1 and DR3 at
http://ultravista.org/, and SPLASH SXDF at
https://homepages.spa.umn.edu/~mehta074/splash/.

Spectroscopic release information and datasets are also available in the
public domain, 3D-HST at https://archive.stsci.edu/prepds/3d-hst/, UDSz at
https://www.nottingham.ac.uk/astronomy/UDS/UDSz/, XMM-LSS survey at
https://heasarc.gsfc.nasa.gov/W3Browse/all/xmmlssclas.html, VANDELS at
https://www.eso.org/sci/publications/announcements/sciann17248.html. The
compilation of published redshifts in the COSMOS field was obtained from a
catalogue curated by M. Salvato; further information about a future public
release of this catalogue can be found at
https://cosmos.astro.caltech.edu/page/specz.

Access to the GOGREEN and GCLASS data release of spectroscopy, photometry, and
derived data products, including jupyter python3 notebooks for reading and
using the data, is available at the CADC (https://www.cadc-ccda.hia-iha.nrc-
cnrc.gc.ca/en/community/gogreen), and NSF’s NOIR-Lab
(https://datalab.noao.edu/gogreendr1/). Future releases, science results and
other updates will be announced via the GOGREEN website at
http://gogreensurvey.ca/. The full posteriors of Webb et al. (2020)’s
PROSPECTOR fits are available from the Canadian Advanced Network for
Astronomical Research (CANFAR), at
www.canfar.net/storage/list/AstroDataCitationDOI/CISTI.CANFAR/20.0009/data;
DOI:10.11570/20.0009.

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##  Appendix A Photometric redshift calibration and selection

###  A.1 De-biasing photometric redshifts using available spectroscopic
redshifts

Figure 12: The difference between catalogued photometric and spectroscopic
redshifts for UltraVISTA DR1 (top) and SPLASH-SXDF (bottom) is shown as a
function of spectroscopic redshifts. The final quadratic fit (solid blue
curve) is based on the points within the shaded region, which we determine by
an iterative sigma-clipping procedure.

In Figure 12 we show the correlation between $\Delta z/(1+z_{\mathrm{spec}})$
and $1+z_{\mathrm{spec}}$, for all galaxies in COSMOS and SXDF with available
spectroscopic redshifts, where $\Delta z=z_{\mathrm{phot}}-z_{\mathrm{spec}}$.
These include deep spectroscopy focused on the $1<z<1.5$ regime by GOGREEN
(Balogh et al., 2021). For UltraVISTA DR3 (not shown), there is no significant
bias, up to $z\sim 2$. Both UltraVISTA DR1 and SXDF show a small bias, such
that the photometric redshift is lower than the spectroscopic redshift for
$1<z<2$. This difference increases modestly with redshift.

To correct for this we fit a quadratic function to this correlation. We find a
fit of
$z_{\mathrm{phot}}=-0.085z_{\mathrm{spec}}^{2}+1.121z_{\mathrm{spec}}-0.044$
for UltraVISTA DR1 and
$z_{\mathrm{phot}}=0.087z_{\mathrm{spec}}^{2}+0.737z_{\mathrm{spec}}+0.150$
for SPLASH-SXDF. We remove the bias by subtracting off the difference of this
relation from a linear one-to-one relation between spectroscopic and
photometric redshifts. Galaxies with photometric redshifts outside 2-4
$\sigma$ (depending on dataset) are considered outliers, and iteratively
removed from the fit. The lower and upper 68% photometric redshift
uncertainties are also corrected to reflect this shift in photometric
redshift. We note that this does not entirely eliminate the bias from SPLASH-
SXDF, as photometric redshifts at $z\sim 1.3$ are still too small, on average,
by $\sim 0.04$ after the correction.

For each galaxy, we also apply a small redshift-dependent stellar mass
correction, to account for the corresponding change in luminosity distance.
This adjustment is by a factor of
$\big{[}D_{L}(z_{\text{pert}})/D_{L}(z_{\text{true}})\big{]}^{2}$. $D_{L}(z)$
is the luminosity distance at a given redshift and $z_{\text{pert}}$ and
$z_{\text{true}}$ are the redshifts of the galaxy originally and after being
perturbed, respectively 888To see how this factor comes about, consider that
the original “$M_{\mathrm{true}}$” of the galaxy in the UltraVISTA dataset was
$M_{\mathrm{true}}=4\pi D_{L}^{2}(z_{\mathrm{true}})F[M/L],$ with $F$ being
the observed flux of the galaxy and $M/L$ being the mass/luminosity ratio of
the galaxy. The recovered stellar mass of that galaxy after having its
photometric redshift perturbed will be $M_{\mathrm{rec}}=4\pi
D_{L}^{2}(z_{\text{pert}})F[M/L]$. Taking the ratio of
$M_{\mathrm{rec}}/M_{\mathrm{true}}$ then gives the desired result..

###  A.2 Photometric redshift selection

COSMOS: UltraVISTA DR1  COSMOS: UltraVISTA DR3  SPLASH-SXDF

Figure 13: Verification of the photometric redshift cuts for COSMOS
(UltraVISTA DR1 and DR3 UltraDeep) and SXDF using the spectroscopic redshifts.
The COSMOS: UltraVISTA DR1 subplot shown here contains all 23 $1<z<1.5$
groups, regardless of quality flag, to maximize the spectroscopic and
photometric redshift matches. Dashed horizontal lines show the photometric
redshift cut that was chosen.

As described in §3.1, group members are first selected to lie within a
photometric redshift range of width $\Delta z=0.126$, prior to applying a
statistical background subtraction. Plots of available spectroscopic redshifts
matched to photometric redshifts (for UltraVISTA DR1 and SPLASH-SXDF) are
shown in Figure 13, following van der Burg et al. (2013). This Figure
demonstrates how the size of this cut compares with the photometric redshift
scatter, for galaxies with available spectroscopic redshifts.

The choice of $\Delta z=0.126$ is made as it corresponds to $\Delta
z=2\times$median$(z_{u68}-z_{\text{peak}})$ for UltraVISTA DR1 and DR3. Here,
$z_{u68}$ is the 84th percentile of the photometric redshift probability
distribution and $z_{\text{peak}}$ is the peak of that distribution (i.e.
$z_{u68}$ is the upper 1-sigma confidence interval) . Although SPLASH-SXDF has
smaller photometric redshift uncertainties, thanks to greater depth in several
bands, there is still a remaining bias at $z\sim 1.2-1.3$ that is not fully
removed from the debiasing described in Appendix A.1. Therefore we
conservatively adopt the same $\Delta z=0.126$ for all systems, to help
mitigate this.

###  A.3 Field stellar mass functions

In Figure 14 we show our field stellar mass functions, which were fit and
described in §3.1. We plot Schechter fits using the best fit parameters in
Table 2 and contrast our field fits with those from Muzzin et al. (2013b),
which exclusively measured stellar mass functions using the UltraVISTA survey
region. We find similar stellar mass functions and our fits are consistent
with theirs, within 2$\sigma$. The similarity in fit is expected, given that
much of our survey area is UltraVISTA. We note that the Schechter fit for the
quiescent population doesn’t quite fit the very high mass end as well as the
Muzzin et al. (2013b) fits. This explains why the quiescent fraction curves,
as plotted in Figure 5, turn over rather than flattening out for the two
highest stellar mass bins. As well for the quiescent population, there is some
deviation from the fit in the lowest stellar mass points. These small
discrepancies in fit do not significantly affect any of our results,
discussion, or conclusion.

Figure 14: Stellar mass function of quiescent (top panel), star-forming
(middle), and total (bottom) field galaxies at $1<z<1.5$. Error bars shown
represent the Poisson shot noise. Overlaid on each plot are the Schechter
function fits (solid line), normalized to match the number of field galaxies
per Mpc3 per dex (bin size
$\Delta\log(M_{\mathrm{stellar}}/\mathrm{M}_{\odot})=0.2$), and with shaded
regions indicating the 68% confidence interval on the fit parameters, computed
as described in the text. The best fit from Muzzin et al. (2013b) is shown as
a dashed line.

##  Appendix B Spectroscopy and GOGREEN spectroscopic groups

Nine of the groups in our $1<z<1.5$ redshift range were observed with Gemini-
GMOS by the GOGREEN spectroscopic survey (Balogh et al., 2021): four from
COSMOS and five from SXDF, as listed in Table 1. The names given to the
GOGREEN-identified groups are the same as those used in the original COSMOS
(Gozaliasl et al., 2019) and SXDF (Finoguenov et al., 2010) group catalogues.
For simplicity we will refer to individual groups using a shortened name
comprised of the last three digits in the formal name appended to the
appropriate catalogue (e.g. COSMOS-28 instead of COSMOS-20028). Here we
present additional details of the spectroscopic datasets used (Appendix B.1),
definition of our field sample at $1<z<1.5$ (Appendix B.2), as well as
analysis of the dynamics (in particular, confirming the group membership and
dynamical masses for a subset of our groups in Appendix B.3), and description
of the spectroscopically-determined mass-weighted ages used for a subset of
our group sample (Appendix B.4).

###  B.1 Spectroscopy

Our primary source for redshifts is the GOGREEN survey (Balogh et al., 2021).
This sample was selected from galaxies with $z^{\prime}<24.25$ within a
$5.5\arcmin\times 5.5\arcmin$ area around nine targets in these two fields. A
broad colour selection was applied to reduce foreground and background
galaxies. The survey provides redshifts for an average $\sim 45\%$ of the
parent cluster population within 500 kpc, unbiased with respect to galaxy type
for stellar masses $M_{\rm stellar}>10^{10.2}~{}\mathrm{M}_{\odot}$ (Balogh et
al., 2021). We use 173 robust redshifts in the COSMOS field, and 198 in the
SXDF field, from GOGREEN. For more details we refer to Balogh et al. (2021).

In addition, for COSMOS we use the master spectroscopic redshift catalogue of
publicly available redshifts in use within the COSMOS collaboration, curated
by M. Salvato (priv. comm.). We also use available 3D-HST redshifts (Skelton
et al., 2014; Brammer et al., 2012), but only for the purpose of checking and
debiasing the photometric redshifts in Appendix A.1.

For the SXDF field we supplement the GOGREEN redshifts with data from UDSz
(Bradshaw et al., 2013; McLure et al., 2013), the XMM-Large Scale Structure
(XMM-LSS) survey (Melnyk et al., 2013; Chiappetti et al., 2013) and VANDELS
(Pentericci et al., 2018). 3D-HST data are again used only for debiasing the
photometric redshifts.

###  B.2 Field sample at 1<_z_ <1.5

For comparison with our sample of overdense galaxy systems, we define a
reference “field” galaxy sample that is representative of the average galaxy
population. We simply define our field sample as all galaxies in the
UltraVISTA and SPLASH-SXDF catalogues with photometric redshifts in the range
of interest. This includes galaxies that make up our group sample, but as they
only make up $\sim 1$ per cent of the total, this has a negligible effect on
our analysis. Calculated field values are an area-weighted average. Such a
sample includes overdense regions, and thus provides a lower contrast to our
group sample than would a comparison with low-density regions (Peng et al.,
2010; Kawinwanichakij et al., 2017; Papovich et al., 2018) or samples of
“central” galaxies. It will also be influenced by cosmic variance (Kovač et
al., 2010; Moster et al., 2011), though this is small given the relatively
large area of the combined surveys.

###  B.3 Group membership, dynamics and masses

In Figure 15 we show the spatial and redshift distribution of galaxies with
spectroscopic redshifts, in each of the 5′ GMOS fields of view, with respect
to the X-ray contours. In addition to GOGREEN, we include spectroscopy from
available public sources as described in §B.1. We calculate a new weighted
centre for each group within 1 Mpc ($\sim 2R_{200c}$) of the catalogued
position, and then rerun an iterative 2.5$\sigma$-clipping routine to
calculate the final redshift and velocity dispersion. This procedure did not
converge for two groups. In the case of COSMOS-125, the member candidates are
spatially concentrated but redshift distribution does not show a clear peak.
For COSMOS-63 it is the opposite, with a strong redshift overdensity but no
spatial concentration of sources. Finally, for SXDF-76 we find two distinct
groups along the line of sight. We keep both in the catalogue, and we have
labelled them SXDF-76a and SXDF-76b.

Figure 15: Groups in the COSMOS field spectroscopically targeted by GOGREEN.
Left subplots: spatial distribution of galaxies with a spectroscopic redshift
centred on the identified group centre and within the iterative velocity
dispersion cut. GOGREEN targets are indicated with green diamonds. The solid
grey contours indicate smoothed X-ray emission. Right subplots: distribution
of spectroscopic redshifts within the circular field of view (solid line)
shown on the left. The dashed vertical lines indicates the redshift selection
of galaxies that are displayed on the corresponding left subplot.

Figure 16: SXDF GOGREEN spectroscopically targeted groups. Analogous to Figure
15.

We estimate dynamical halo masses from these velocity dispersion values using
the relation presented in Saro et al. (2013). All galaxies within the
2.5$\sigma$ velocity cut and within 2$R_{200c}$ count as spectroscopic members
for the purposes of Table 1, where we present all of these values. We also
compute an average velocity dispersion by stacking all group galaxies in an
ensemble. We find $\sigma_{v}=352\pm 32\,{\rm km\,s}^{-1}$, which has been
corrected by subtracting in quadrature the estimated individual redshift
uncertainty of $280\,{\rm km\,s}^{-1}$ observed-frame, from Balogh et al.
(2021). This corresponds to a halo mass of
log$(M_{200c}/M_{\text{solar}})=13.61\pm^{+0.11}_{-0.12}$, again using the
Saro et al. (2013) relation. The distribution of the velocities in this
ensemble is shown in 17.

Figure 17: Distribution of rest-frame galaxy velocities in the ensemble for
all 83 spectroscopic group members in 9 GOGREEN groups. Vertical lines
indicate the mean of $0\,{\rm km\,s}^{-1}$ (solid black), velocity dispersion
(vertical dashed), and final 2.5$\sigma$ cut (black dotted). Thirty evenly
sized bins were used, corresponding to a bin width of $\sim 56\,{\rm
km\,s}^{-1}$. The curved blue line indicates a Gaussian with mean 0 and
standard deviation set to the velocity dispersion, normalized so the area
corresponds to the total number of spectroscopic members.

###  B.4 Mass-weighted ages

Formation times of quiescent galaxies in the GOGREEN spectroscopic sample,
from Webb et al. (2020), are shown as a function of stellar mass in Figure 18.
This figure complements Figure 11, which only shows differences with the field
and omits the individual measurements for the sake of clarity. Formation times
are computed from the time of observation and the mass-weighted age, as
$t_{\mathrm{form}}=t_{\mathrm{obs}}-$MWA. The running mean is additionally
shown for the intermediate halo-mass bin used in the QFE vs halo mass
analysis. The field, low-mass clusters, and high-mass clusters all show a
declining trend of $t_{\mathrm{form}}$ with stellar mass. Both cluster sub-
samples display ages about 200-300 Myr older than the field, as noted by Webb
et al. (2020) for the entire cluster sample. The group sample, on the other
hand, appears younger than the field by $\sim 150-200$ Myr, as noted in
§4.2.2.

The small group sample has a high average galaxy stellar mass. To check that
our results are not dominated by the most massive group galaxies, which could
be central galaxies with a different formation history, we highlight these as
green diamonds with black borders on Figure 18. There is no evidence that the
ages of these galaxies are significantly different from other group members.

Figure 18: This figure is a more detailed complement to Figure 11 in the main
body of the paper. The points show measurements of $t_{\mathrm{obs}}-$MWA as a
function of stellar mass, from stellar population synthesis modelling of
quiescent galaxies in the GOGREEN spectroscopic sample (Webb et al., 2020).
Individual $t_{\mathrm{obs}}-$MWA and stellar mass values are plotted for the
field (blue dots), group (green diamonds), low-mass cluster (yellow dots), and
high-mass cluster samples (small orange crosses). Groups, low-mass clusters,
and high-mass clusters are the three halo mass bins explored at $1<z<1.5$ in
§3.3. The most massive galaxy in each group (’BCGs’) are indicated with a
black diamond (two groups had only one quiescent galaxy). As well, a running
mean with bootstrapped standard deviation on the mean is shown for the field
and cluster samples. For the groups, there are only 15 quiescent spectroscopic
members so we simply plot the mean $t_{\mathrm{obs}}-$MWA and mean stellar
mass of the full sample with a black star and a shaded region reflecting the
bootstrapped errors. Quiescent galaxies in groups are not older than those in
clusters, ruling out the predictions of our simple model without pre-
processing.

##  Appendix C Sensitivity of results to stellar mass binning

In §3.3 we consider the halo mass dependence of the QFE in two broad stellar
mass bins (Figure 7), motivated by the qualitative change in QFE shown in
Figure 6. Here we subdivide the lower mass bin into two, to demonstrate
explicitly that the results are similar in both bins. We show both the QFE,
and the $f_{Q}$ values from which it is derived, in this binning in Figure 19.
As claimed, the halo mass dependence observed in the two lowest stellar mass
bins is very similar. In particular, the unusually high QFE observed for the
Planck clusters at intermediate redshifts persists in both stellar mass bins.
As discussed in §3.3, we find the data in all three stellar mass bins are
consistent with a QFE that depends on
$\log(M_{\mathrm{halo}}/\mathrm{M}_{\odot})$ with a slope of $m\approx 0.24\pm
0.04$.

In Figure 20 we show the BAHAMAS simulation results (originally shown in
Figure 9) for the same quantities and binning. It is readily apparent here
that, at fixed halo mass, the predicted $f_{Q}$ and QFE in the simulations
decreases with increasing stellar mass, in contrast with the observations.
This behaviour is well-known and discussed further in Kukstas et al. (in
preparation). Despite this, the correlation with halo mass is similar in both
lower stellar mass bins – it increases up to a halo mass of $\sim 2\times
10^{14}\mathrm{M}_{\odot}$ and becomes much shallower as halo mass increases
further. This behaviour is not inconsistent with what we observe, with the
possible exception of the intermediate redshift Planck-selected clusters.

Figure 19: Quiescent fraction (top row) and quiescent fraction excess (bottom
row) are shown as a function of halo mass ($M_{200c}$), $M_{\rm
halo}/\mathrm{M}_{\odot}$, for three galaxy stellar mass bins (one stellar
mass bin per column), and for samples at three different redshift ranges, as
indicated. Our new measurements of low-mass haloes at $1<z<1.5$ are shown as
the green points connected by a green solid line. The other samples are
described in §2.2. Horizontal lines represent the field at a given redshift
range (errors are not significantly larger than the line widths on this plot).
The bottom row is analogous to Figure 7, with a further subdivision of that
Figure’s lowest stellar mass bin.

Figure 20: We show results from the BAHAMAS hydrodynamic simulation, for the
quiescent fraction (top) and quiescent fraction excess (bottom) as a function
of stellar and halo mass, $M_{\rm halo}/\mathrm{M}_{\odot}$ at two redshifts
as indicated, with the same stellar mass binning as in Figure 19. The
corresponding data from that Figure are shown, omitting the intermediate
redshift sample for clarity. In the simulations, both $f_{Q}$ and QFE decrease
with increasing stellar mass, in contrast with the data. However, the
correlation with halo mass and redshift is qualitatively similar to the trends
observed in the data. The bottom row is analogous to Figure 9 in the main body
of the paper, with the lowest stellar mass bin subdivided into two.