# Star formation history and transition epoch of cluster galaxies based on the
Horizon-AGN simulation

Seyoung Jeon  Department of Astronomy and Yonsei University Observatory, 50
Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea  Sukyoung K. Yi
Department of Astronomy and Yonsei University Observatory, 50 Yonsei-ro,
Seodaemun-gu, Seoul 03722, Republic of Korea  Yohan Dubois  Institut d’
Astrophysique de Paris, Sorbonne Universités, et CNRS, UMP 7095, 98 bis bd
Arago, 75014 Paris, France  Aeree Chung  Department of Astronomy and Yonsei
University Observatory, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of
Korea  Julien Devriendt  Department of Physics, University of Oxford, Keble
Road, Oxford OX1 3RH, UK  San Han  Department of Astronomy and Yonsei
University Observatory, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of
Korea  Ryan A. Jackson  Department of Astronomy and Yonsei University
Observatory, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea
Taysun Kimm  Department of Astronomy and Yonsei University Observatory, 50
Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea  Christophe Pichon
Institut d’ Astrophysique de Paris, Sorbonne Universités, et CNRS, UMP 7095,
98 bis bd Arago, 75014 Paris, France  IPhT, DRF-INP, UMR 3680, CEA, L’Orme des
Merisiers, Bât 774, 91191 Gif-sur-Yvette, France  Jinsu Rhee  Department of
Astronomy and Yonsei University Observatory, 50 Yonsei-ro, Seodaemun-gu, Seoul
03722, Republic of Korea

(Received XX; Revised YY; Accepted ZZ)

###### Abstract

Cluster galaxies exhibit substantially lower star formation rates than field
galaxies today, but it is conceivable that clusters were sites of more active
star formation in the early universe. Herein, we present an interpretation of
the star formation history (SFH) of group/cluster galaxies based on the large-
scale cosmological hydrodynamic simulation, Horizon-AGN. We find that massive
galaxies in general have small values of e-folding timescales of star
formation decay (i.e., “mass quenching”) regardless of their environment,
whilst low-mass galaxies exhibit prominent environmental dependence. In
massive host halos (i.e., clusters), the e-folding timescales of low-mass
galaxies are further decreased if they reside in such halos for a longer
period of time. This “environmental quenching” trend is consistent with the
theoretical expectation from ram pressure stripping. Furthermore, we define a
“transition epoch” as where cluster galaxies become less star-forming than
field galaxies. The transition epoch of group/cluster galaxies varies
according to their stellar and host cluster halo masses. Low-mass galaxies in
massive clusters show the earliest transition epoch of $\sim 7.6$ Gyr ago in
lookback time. However, it decreases to $\sim 5.2$ Gyr for massive galaxies in
low-mass clusters. Based on our findings, we can describe cluster galaxy’s SFH
with regard to the cluster halo-to-stellar mass ratio.

††journal: ApJ

##  1 Introduction

During recent decades, an enormous amount of information on galaxies has been
gathered. Despite the diversity and complexity of observed galaxy properties,
many interesting statistics and relationships have emerged through numerous
observational surveys. Galaxies are often separated into red and blue galaxies
on the color-magnitude plane, forming the red sequence and the blue cloud
(e.g., Strateva et al., 2001; Baldry et al., 2004). For example, the colors
and the morphology of galaxies have a robust correlation with the environment
(e.g., Dressler, 1980; Hogg et al., 2003). While the majority of galaxies in
the Universe are “live” and star-forming, most member galaxies of massive
clusters are mysteriously “red and dead” with significantly depressed star
formation rates (SFR) (e.g., Balogh et al., 1999; Boselli & Gavazzi, 2006).

Various quenching processes have been proposed to explain the aforementioned
diversities (in this study, “quenched” refers to the cessation as well as the
overall decline of star formation in the long-term timescale) and these
processes are classified as internal and external. Internal quenching
regulates star formation mainly via feedback processes inside the galaxy. For
example, strong energy generated by an active galactic nucleus (AGN) heats the
surrounding cold gas and directly transfers momentum, inhibiting the inflow of
cold gas and regulating star formation, especially in the case of massive
galaxies (Croton et al., 2006; Rafferty et al., 2006; Fabian, 2012; Peirani et
al., 2017). In addition, stellar feedback can suppress star formation in
various ways and is suggested to be the main quenching mechanism in low-mass
galaxies in which gas is bound in a shallow potential well. Supernova
explosions provide energy and momentum to the surrounding interstellar medium
(ISM), which can disturb cold gas and prevent it from collapsing (Larson,
1974; Dekel & Silk, 1986; Governato et al., 2010). Moreover, ultraviolet (UV)
radiation generated by young massive stars contributes to star formation
quenching through photoionization, which ionizes the surrounding cold gas and
increases the gas pressure, or by direct radiative pressure (Whitworth, 1979;
Larson & Starrfield, 1971; Hopkins et al., 2020).

Alternatively, environmental effects could affect the SFR of galaxies in dense
regions, where massive dark matter halos are formed. For the galaxies orbiting
in the deep gravitational potential well of a massive halo, their baryon and
dark matter components are disrupted by tidal forces (i.e., tidal stripping;
Richstone, 1976; Gnedin, 2003; Smith et al., 2016). When a galaxy moves in a
cluster environment, the hot intracluster medium (ICM) can strip the ISM of
the galaxy which is the fuel for star formation (i.e., ram pressure stripping;
Gunn & Gott, 1972; Quilis et al., 2000; Chung et al., 2007; Tonnesen et al.,
2007). Otherwise, the ICM can suppress hot gas cooling which consequently
reduces cold gas supply to the galaxy (i.e., starvation or strangulation;
Larson et al., 1980; Balogh et al., 2000). In dense regions, galaxies
experience fly-by encounters and even mergers with nearby galaxies, resulting
in the distortion of galactic structures (i.e., harassment and mergers; Moore
et al., 1996, 1998; Yi et al., 2013; Smith et al., 2015; Sheen et al., 2016).
Even in low-mass host halos, galaxies may already experience environmental
effects before infalling into a cluster halo (i.e., preprocessing; Mihos,
2004; De Lucia et al., 2012; Han et al., 2018; Jung et al., 2018).

Various quenching processes act with different timescales and magnitudes and
are accumulated throughout the history of a galaxy’s evolution (Iyer et al.,
2020). Therefore, the star formation history (SFH) contains cumulative
information about the quenching processes a galaxy has experienced. As
galaxies with different internal properties and external environments may have
taken distinct evolutionary paths, we can connect their physical properties
and SFHs. Massive and quiescent galaxies are believed to have experienced an
early vigorous star formation phase and subsequent rapid quenching, whereas
low-mass galaxies are thought to have a relatively extended SFH (e.g., Thomas
et al., 2005; Ellison et al., 2018). Likewise, cluster galaxies can have
different SFHs compared with field galaxies. Although cluster galaxies are
redder and more quenched than field galaxies at the present epoch, increased
fractions of blue galaxies in high-z clusters exist (Butcher & Oemler, 1978;
Alberts et al., 2013). For example, it has been found that at $z\sim 1$
cluster galaxies have comparable specific SFR (sSFR) and quenched fractions to
field galaxies (Lee et al., 2015; Jian et al., 2020). Moreover, the local SFR-
density relation seems to reverse at high redshifts. This implies that there
was an epoch in which there was no difference in the SFR between the field and
cluster galaxies (Elbaz et al., 2007; Cooper et al., 2008; Hwang et al.,
2019). This “transition epoch”, when derived from the instantaneous SFR or
quenching efficiency either through observations or simulations, seems to
converge at $z\sim 1$ (Alberts et al., 2013; Brodwin et al., 2013; Pintos-
Castro et al., 2019; Hwang et al., 2019; Williams et al., 2022). On the other
hand, considering that there have been many studies indicating various
quenching timescales and processes depending on a galaxy’s internal properties
and environments, it would be mysterious if the transition epoch is universal.
Therefore, it is crucial for our understanding of galaxy formation and
evolution to pin down such epochs and identify their driving mechanisms.

Numerical simulations have emerged as effective tools for studying the history
of galaxy evolution. Since the advent of large-scale N-body simulations
(Springel et al., 2005), cosmological simulations have become dramatically
more sophisticated: to name only a few, Horizon-AGN (Dubois et al., 2014),
Illustris (Vogelsberger et al., 2014), EAGLE (Schaye et al., 2015; Crain et
al., 2015), Mufasa (Davé et al., 2017), IllustrisTNG (Pillepich et al., 2018),
SIMBA (Davé et al., 2019), and New-Horizon (Dubois et al., 2021). Along with
the advances in our understanding of baryonic physics, these simulations have
succeeded in reproducing some key properties of galaxies.

We use Horizon-AGN to investigate star formation quenching histories and
assess the level of the reproduction of observed galaxy properties using the
simulation. We analyze the quenched states and SFHs of simulated galaxies in
various host halo environments and attempt to assess the significance of the
internal and external quenching processes on the SFHs of galaxies. In
addition, we determine whether there is an epoch where the star formation
rates of cluster and field galaxies were equal.

##  2 Methodology

###  2.1 Numerical Simulation

We use the large-scale cosmological hydrodynamic simulation, Horizon-AGN
(Dubois et al., 2014), which was performed using the adaptive mesh refinement
code, RAMSES (Teyssier, 2002). Horizon-AGN uses WMAP7 values to set the
cosmological parameters and includes subgrid astrophysics such as gas cooling
due to primordial and metallic species and heating by the background UV
radiation, star formation, stellar and AGN feedback (Dubois et al., 2014). The
simulation volume is a $100\,{h^{-1}}\,{\rm Mpc}$ on a side cube in comoving
scale and contains $1024^{3}$ dark matter particles. The best spatial
resolution is $\approx 1\,{\rm kpc\,(physical)}$ and the mass resolutions of
the dark matter and star particles are $8\times 10^{7}\ \rm{M_{\odot}}$ and
$2\times 10^{6}\ \rm{M_{\odot}}$, respectively. For a more detailed
description of Horizon-AGN, refer to Dubois et al. (2014). It has been
demonstrated by Kaviraj et al. (2017) that Horizon-AGN shows reasonable
agreement with key observational data such as mass function and cosmic star
formation history, which are crucial in this study.

Figure 1:  Phase space diagram of Horizon-AGN group/cluster member candidates
in the last snapshot ($\rm z=0.018$). The horizontal axis is the distance of
the galaxy to the center of the host halo normalized by the virial radius of
the halo. The vertical axis is the relative velocity of the galaxy normalized
by the circular velocity. The black line indicates the relative velocity
threshold of Han et al. (2018) for selecting satellites. Grey points are
galaxies not identified as members and other color scales show the time-since-
infall of member galaxies (Section 2.2).

###  2.2 Sample Selection

The galaxies and halos in Horizon-AGN are identified using HaloMaker which is
built based on the AdaptaHOP algorithm (Aubert et al., 2004; Tweed et al.,
2009). In total, 124,744 galaxies are identified in the last snapshot
($z=0.018$), and the stellar mass range is $10^{8.2-12.7}\rm{M_{\odot}}$. We
use the galaxies with a stellar mass greater than $10^{9.5}$
$\rm{M_{\odot}}$in the last snapshot to secure the minimum number of stellar
particles (900) for reasonable kinematic stability (see Klypin et al., 1999,
for the issue of particle evaporation). This criterion leaves 53,226 galaxies
as our final sample.

We classify the identified host dark matter halos with the final halo (virial)
mass $M_{\rm halo}\geq 10^{13}\,\rm{M_{\odot}}$ into “group/cluster” halos, as
a conventional way (Dariush et al., 2010; Serra & Diaferio, 2013; Donnari et
al., 2021b). We define the virial radius, $R_{\rm vir}$, as the radius within
which the mean density is 200 times the critical density. Note that the term
“halo” refers to the main (group/cluster) dark matter halo that hosts multiple
satellite galaxies in this study. Using these mass criteria, 428 group/cluster
halos were identified. Subsequently, we define their member galaxies using the
energy criterion introduced by Han et al. (2018), i.e., by finding galaxies
with negative orbital energy within 2.5 times the virial radius. After this
cut, we obtain 8,228 group/cluster galaxies. These member galaxies in the
groups/clusters are our main focus, whilst the remaining galaxies are used as
the control sample. Among the rest, we exclude 3,797 galaxies that are within
3 $R_{\rm vir}$ but not bound to any group/cluster halo from our analysis
because we want to highlight the difference between the field and
group/cluster galaxies. The remaining 41,201 galaxies are labeled as field
galaxies. Figure 1 shows the phase space distribution of the member galaxies
in the Horizon-AGN group/cluster halos at $z=0.018$. The black line shows the
relative velocity ($v$) threshold by the clustocentric distance ($r$)
following Han et al. (2018):

$\frac{v^{2}}{2}+\Phi(r)<\Phi(2.5R_{\rm vir}),$ (1)

where $\Phi$ is the gravitational potential of the group/cluster halo
calculated from the radial density profile (i.e., NFW profile)111Note that the
second condition in this equation was omitted in the original manuscript of
Han et al. (2018).:

$\Phi(s)=\left\\{\begin{array}[]{l l}{-V_{\rm
cir}^{2}}\biggl{[}h(c)g(c)\bigl{(}\frac{\ln(1+cs)}{s}-\ln(1+c)\bigr{)}\\\
\qquad-\frac{1}{2}(1-s^{2})(h(c)-1)+1\biggr{]},&s<1\\\ {-V_{\rm
cir}^{2}}/s,&s\geq 1\end{array}\right.$ (2)

where $s$ is the clustocentric distance of each galaxy in units of $R_{\rm
vir}$ of the host halo ($r/R_{\rm vir}$), $V_{\rm cir}$ is the circular
velocity at $R_{\rm vir}$ of the halo ($\sqrt{GM_{vir}R_{vir}}$), and $c$ is
the concentration index of the halo assuming the NFW profile. Other terms are
defined as follows:

$\displaystyle h(c)$ $\displaystyle=$
$\displaystyle\left[1-\frac{c^{2}g(c)}{3(1+c)^{2}}\right]^{-1}$ (3)
$\displaystyle g(c)$ $\displaystyle=$
$\displaystyle\left[\ln(1+c)-\frac{c}{1+c}\right]^{-1}$ (4)

Figure 1 shows the galaxies within $3R_{\rm vir}$ of our sample group/cluster
halos. The member galaxies are color-coded by “time-since-infall (TSI).”
Inspired by previous studies (Wetzel et al., 2013; Oman et al., 2013; Rhee et
al., 2017, 2020), we define time-since-infall (TSI) by measuring the time
since the galaxy satisfies the aforementioned membership criterion, Equation
(1), in massive halos ($M_{\rm halo}>10^{13}\,\rm{M_{\odot}}$) for the first
time. While there are 787 snapshots in Horizon-AGN with intervals of
$\sim\,17$ Myr, dark matter and gas data are saved only in 61 snapshots with
time intervals of $\sim\,250$ Myr (See also Khim et al., 2020). As we use the
dark matter snapshots to measure the TSI, it is therefore determined with a
time resolution of $\sim\,250$ Myr. The other galaxies are shown in grey.
Member galaxies appear roughly separated in phase space by TSI, which is
consistent with the results of Rhee et al. (2017). We can see that this
criterion saves the galaxies that are temporarily located out of $R_{\rm
vir}$, although they are bound (“back-splashed galaxies”; Gill et al., 2005).

Figure 2:  Quenched fraction distribution of the Horizon-AGN galaxies. The
horizontal axis is the host halo mass, and the vertical axis is the stellar
mass of member galaxies. The black numbers are quenched fractions color-coded
following the key in the bottom and the white numbers are central galaxy
fractions. The mean values of the quenched fractions of field galaxies are
shown in the grey inset. The top and right panels show the number
distributions of the host halos and their member galaxies, respectively. Note
that there are only two halos (clusters) of mass $M_{\rm halo}\geq
10^{14.5}\rm{M_{\odot}}$ because of the limited size of the volume of Horizon-
AGN (140 Mpc on a side).

##  3 Results

###  3.1 Quenched Fraction

The star-formation quenched fraction is observationally well constrained at
least at low redshifts. For example, it is known to strongly depend on the
stellar mass and environment of galaxies (e.g., Baldry et al., 2006; Peng et
al., 2010; Wetzel et al., 2013). Massive galaxies seem to be more quenched
regardless of their environment, whereas low-mass galaxies are more quenched
in higher density regions (Baldry et al., 2006). Such a consensus on the
quenched fractions in observations can be used to test the reliability of
numerical simulations. In this section, we measure the quenched fractions of
our sample of Horizon-AGN galaxies and attempt to understand the impact of
stellar and host halo masses on the quenched fraction.

Various methods have been used to measure quenched fractions: double Gaussian
fitting on color-magnitude distributions (Baldry et al., 2004; Balogh et al.,
2004), color-color (e.g., UVJ) distributions (Labbé et al., 2005; Wuyts et
al., 2007; Williams et al., 2009), specific SFR (sSFR) (Kauffmann et al.,
2004; Santini et al., 2009), and distance to the star-forming main sequence
(Tacchella et al., 2016; Fang et al., 2018; Pillepich et al., 2019). Donnari
et al. (2021a) have reported that the quenched fraction as a function of
stellar mass significantly varies with different measurement choices. Bearing
such uncertainties in mind, we use sSFR to define quenched galaxies, that is,
when $\rm{sSFR}<\frac{1}{6t_{\rm H}(z)}$, where $t_{\rm H}(z)$ is the age of
the Universe at the epoch of the target redshift ($z$), corresponding to $\sim
10^{-11}\rm yr^{-1}$ at the present epoch (Tacchella et al., 2019).

Figure 2 shows the quenched fraction distribution for different stellar and
host halo masses. The galaxies are divided into twelve stellar mass bins each
with a 0.25 dex size, and group/cluster galaxies are additionally divided into
eight halo mass bins of size 0.25 dex as well. The number distributions of the
group/cluster halos and member galaxies are shown in the top and right panels
respectively, with the quenched fractions of field galaxies shown in the left
inset. Field galaxies, which are hardly affected by environmental effects,
show a gradually increasing quenched fraction with increasing stellar mass.
This is often known as the “mass quenching” i.e., more massive galaxies
exhibit a higher quenched fraction. A relatively early and short SFH, perhaps
associated with feedback effects, is often considered an important process
behind mass quenching. Furthermore, we notice three features of the
group/cluster galaxies. Firstly, the most massive members
($M_{*}>10^{11.5}\,\rm{M_{\odot}}$) are virtually all quenched regardless of
the host halo mass. This is similar to what has been found for field galaxies
and consistent with observations (Peng et al., 2010; Wang et al., 2017, 2018).
Hence, it seems that the most massive galaxies are mass quenched whether they
are in the field or in a cluster. We write the fraction of central galaxies in
each bin in white letters. The most massive bins are dominated by central
galaxies. But, whether including or excluding centrals makes little difference
to the trend. Secondly, the satellite galaxies in the most massive halos
($M_{\rm halo}>10^{14.5}\,\rm{M_{\odot}}$) are also predominantly quenched,
regardless of the galaxy stellar mass. This implies that environmental effects
dominate the quenching process in massive clusters. Lastly, there is a
diagonally-increasing trend in the quenched fraction (from top left to bottom
right in the galaxy mass range of $M_{*}\rm<10^{11}\,\rm{M_{\odot}}$). The
quenched fraction is higher in a more massive halo and lower mass galaxy. This
combined trend is largely consistent with various observations (e.g., Baldry
et al., 2006; Peng et al., 2010) as well as with the theoretical expectation
based on the ram pressure stripping mechanism (Jung et al., 2018).

The high quenched fraction in low-mass satellite galaxies may call for
attention. The non-monotonic trend of the quenched fraction with respect to
the stellar mass is a natural feature of the diagonal trend combined with the
mass quenching. However, the observational view of this issue is still
unclear. For example, Peng et al. (2010, see their Figure 6) reported that
intermediate-mass galaxies in dense environments exhibit lower values of red
galaxy fractions compared with those of the surrounding mass bins, which
appears to be consistent with our findings in the Horizon-AGN galaxies.
However, Baldry et al. (2006) did not find such a non-monotonic mass trend,
despite being based on similar data. If the Horizon-AGN galaxies are deemed
unrealistic owing to the high values of quenched fractions in the low-mass
regime, this could be related to the so-called satellite overquenching
problem. Kimm et al. (2009) reported that low-mass satellite galaxies exhibit
substantially higher quenched fractions than their observational counterparts
in several semi-analytic models. Moreover, some hydrodynamical simulations
report a similar issue, perhaps due to resolution effects. For example, Dickey
et al. (2021) investigated resolution effects using the SIMBA and EAGLE
simulations, and found that the choice of resolution causes significant
differences in the star-formation properties of low-mass galaxies.
Interestingly, Donnari et al. (2021a, b) identified a larger variation in
rather more massive galaxies from resolution tests in the TNG simulation.

In summary, Horizon-AGN reproduces the two most robust observational facts:
mass quenching at the high-mass end regardless of the environment and severe
environmental quenching in massive clusters regardless of galaxy stellar mass.
The diagonally-increasing trend in the quenched fraction from massive galaxies
in low-mass halos to low-mass galaxies in massive halos can be attributed to
the ram pressure stripping phenomenon; however, determining whether it is
consistent with observations requires further investigation.

Figure 3:  Mean SFHs of the Horizon-AGN galaxies of different masses. Other
mass bins not shown here are presented in Appendix A. Different colors and
line styles represent the different classifications of environments, and the
number of galaxies for each subsample is indicated in between brackets. The
group galaxies (green) mean the galaxies in the host halos of mass
$10^{13-14}\rm{M_{\odot}}$ and the cluster galaxies (red) are in the halos
more massive than $10^{14}\rm{M_{\odot}}$. The shaded area shows the 1$\sigma$
standard deviation. All the lines are smoothed using a Savitzky-Golay filter
and a Gaussian kernel.

###  3.2 Star formation history

We have demonstrated that Horizon-AGN reproduces the observed quenched
fractions of galaxies well at a given snapshot. Now we focus on the entire SFH
of galaxies and their relationship with stellar mass and host halo mass. There
are two conventional methods for building the SFH of a simulated galaxy. One
is to track the SFH of the main progenitors in the merger tree (Pandya et al.,
2017; Iyer & Gawiser, 2017; Zheng et al., 2022). An advantage of this method
is that it helps directly compare with observations at various redshifts
because progenitors are observationally identifiable at various redshifts.
Another method is to build the SFH based on the age distribution of stellar
particles which belong to the galaxy at the last snapshot (Sparre et al.,
2015; Diemer et al., 2017; Joshi et al., 2021). This allows for the fact that
in-situ formed stars may leave the host galaxy while ex-situ formed stars may
join the galaxy later. The star particle age distribution approach therefore
encodes more complete information about the SFH and the current composition of
the galaxy, therefore we use this approach.

Figure 3 shows the SFHs of the Horizon-AGN galaxies in different mass ranges.
The top panel shows the case of low-mass galaxies, the shapes of their SFHs
are substantially different from each other due to their environments. Cluster
galaxies (the red line) are more star-forming at the beginning but quickly
quenched. On the other hand, field galaxies (the blue line) exhibit more
extended star formation over time. The bottom panel shows the massive galaxy
case where there is little difference between group/cluster and field
galaxies. These results are consistent with previous observational studies
(Thomas et al., 2005; Ellison et al., 2018), and moreover, the differences in
SFRs near the last snapshot also agree with the results of the quenched
fraction (Figure 2). The mass ranges which are missing in Figure 3 show a
continuous trend among the presented mass bins (see Appendix A). Thus, the
environment significantly affects the SFH of galaxies in the case of low-mass
galaxies. When the SFR is measured at a certain epoch (e.g., the present
epoch), it is not likely a transient feature but a result of the whole SFH
containing the mass quenching prior to the infall into the host cluster and
the environmental quenching after the infall. When the mean properties are
compared, cluster galaxies have been more passive than field galaxies for a
substantial period (see the top panel of Figure 3). We call the point at which
they cross the “transition epoch” and attempt to quantify it in the following
sections.

Figure 4:  SFHs (the shaded area) of four sample Horizon-AGN galaxies and fits
using Equation (5) (the lines). The different colors and line-styles represent
the cases of $\tau\sim\ $1, 2, 4, and 10.6 Gyr, respectively. Each line and
shaded area are normalized to the integrated SFR for clarity.

###  3.3 $e$-folding Timescale

The SFH of a galaxy may fluctuate with different timescales due to various
physical processes. However, environmental quenching, which is our focus, is
rather a long-term process. Thus, we aim to derive simple parameterized
functions over time that fit the overall shape of the SFH. We select the
“delayed tau-model” (see Wetzel et al., 2013; Rhee et al., 2020, for
reference) in this study because it is intuitively consistent with the
expectation from simple closed-box systems (e.g., Schmidt, 1959) yet versatile
enough to allow a delay in the onset of star formation. The following equation
is used:

$\psi(t)=\left\\{\begin{array}[]{l
l}A(t-T_{0})\exp{\bigl{(}-\frac{t-T_{0}}{\tau}\bigr{)}}&\quad t>T_{0}\\\
0&\quad\text{otherwise,}\\\ \end{array}\right.$ (5)

where $\psi$ is the SFR, $t$ is the age of the Universe, $A$ is the amplitude,
and $T_{0}$ is the delay in SFH. We consider the e-folding timescale $\tau$ a
proxy of the quenching timescale. Figure 4 shows the schematic trends for
various values of $\tau$ compared to a sub-sample of simulated galaxies. The
e-folding timescale, $\tau$, governs the overall shape of the SFH of a galaxy.
Moreover, as a larger $\tau$ represents more extended star formation, we can
use this model to also fit galaxies with a rising SFR (see the black line in
Figure 4).

Figure 5:  Distribution of the e-folding timescales ($\tau$) of field galaxies
in the Horizon-AGN simulation. The grey-scale histogram shows the column-
normalized number density, and the white contours show the number of galaxies
with labels marked. The median values are presented for four mass bins.
Figure 6:  Residual values of $\tau$ (cluster galaxies - field galaxies) for
different host mass bins, color-coded according to stellar mass bins in the
same manner as illustrated in Figure 5. The median value with its standard
error of each bin is shown as a shaded line. A small absolute value of
$\Delta\tau$ indicates that the environmental effect to the SFH is negligible,
while a negative value indicates the existence of star formation suppression
by environmental effects.  Figure 7:  The residual values of $\tau$ as a
function of the host mass, as in Figure 6 but divided into different TSI bins.
From the left to right, the TSIs of subsamples have been increased. The color
code is similar to the codes shown in Figures 5 and 6 (the red and blue colors
denote massive and less massive member galaxies, respectively).

Now we present how the $\tau$ of each galaxy depends on its stellar mass and
the environment, as expected in Figure 3. Figure 5 exhibits the values of the
field galaxies in the Horizon-AGN simulation. The white contours indicate
number density, and the background grey-scaled histogram is the column-
normalized number density of field galaxies in each bin. The median values are
color-coded for each stellar mass bin. The e-folding timescales for the
Horizon-AGN field galaxies range from 2.7 to 1.7, which agrees with the toy
model results of Peng et al. (2010). There is a mild but systematic trend in
which more massive galaxies exhibit smaller values of $\tau$, i.e., there is a
rapid decline in SFH. Consequently, more massive galaxies are more likely to
be quenched early. This explains the quenched fractions of field galaxies
(Figure 2). Note that more than half of massive
($M_{*}>10^{11}\,\rm{M_{\odot}}$) field galaxies are quenched at $z\sim 0$.
Now, we take the field galaxies as a control sample to be compared with
group/cluster satellite galaxies.

Figure 6 shows the difference in $\tau$ between the field and cluster galaxies
($\Delta\tau$). A negative value of $\Delta\tau$ indicates that cluster
galaxies exhibit smaller values of $\tau$, such that their star formation has
experienced a more rapid decline than field galaxies for a fixed stellar mass.
Most satellite galaxies exhibit negative values of $\Delta\tau$, this means
that their SFRs have been further decreased by a massive halo environment.
This is particularly true for low-mass satellites
($M_{*}<10^{10.5}\,\rm{M_{\odot}}$) in more massive groups/clusters, where
additional environmental quenching appears to be more significant. Conversely,
the most massive member galaxies (red line) exhibit little difference
($\Delta\tau\sim 0$) from their field counterparts. It seems that the most
massive galaxies are less impacted by their environment. These results are all
consistent with the quenched fractions in Figure 2.

It is clear that both stellar and cluster halo masses affect the quenching
timescale. However, not all cluster satellites exhibit an environmental effect
if they have lived in a massive halo for only a short period of time.
Consequently, we want to check the significance of the time spent by a
satellite galaxy in its host halo (group/cluster). Herein, we adopt the time-
since-infall (TSI) introduced in Section 2.2. We separate galaxies into
several TSI bins and recalculate $\Delta\tau$. Figure 7 shows the satellite
galaxies for three different ranges of TSI: recent, intermediate, and ancient
infallers arriving into their host halos between 0 and 2 Gyr ago, 2 and 5 Gyr
ago, and 5 Gyr or more ago, respectively. Most importantly, there is a clear
trend with TSI. The values of $\Delta\tau$ decrease from recent to ancient
infallers, indicating the high significance of environmental effects on
ancient infallers as expected. However, it is somewhat unexpected that the
environmental effect (significantly negative values of $\Delta\tau$) is
clearly visible in low-mass galaxies despite the small values of the TSI (blue
line in the left panel). This phenomenon can be explained by the combination
of two effects. Lower-mass satellites are more easily affected by their host
environment in the first place. In addition, when they arrive in massive halos
(e.g., $M_{\rm halo}>10^{14}\,\rm{M_{\odot}}$), they are likely to have
experienced preprocessing quenching in smaller halo environments before the
cluster infall (De Lucia et al., 2012; Han et al., 2018; Jung et al., 2018).
Although it may not seem statistically significant, it is interesting to note
that the most massive ancient infallers (red line in the right panel) exhibit
negative values of $\Delta\tau$, hinting for additional environmental
quenching. In our simple quenched fraction analysis (Section 3.1), and when we
ignore the TSI difference (Figure 6), these massive galaxies seemed to remain
unaffected by environmental quenching. However, when we delve into the TSI
effect, we see a hint of additional environmental effects even in such massive
galaxies.

In summary, we have derived the e-folding timescales ($\tau$) for the SFH of
galaxies in the Horizon-AGN simulation. The e-folding timescales exhibit a
clear dependence on the galaxy mass and host halo mass. The most massive
galaxies seem to have an early and quick SFH almost regardless of the
environment. Except for the most massive galaxies (e.g.,
$M_{*}>10^{11}\,\rm{M_{\odot}}$), all the satellite galaxies exhibit
environmental effects. The magnitude of environmental effects seems larger in
lower mass galaxies and in more massive halos, as expected from the ram
pressure stripping process. Moreover, we find that galaxies that have spent
more time in their host halo have a shorter e-folding timescale, indicating a
more significant environmental effect. These findings on the simulated
galaxies are largely in agreement with previous observational studies and
consistent with basic physical expectations.

##  4 Discussion

###  4.1 Transition epoch

Figure 8:  Transition epoch for the Horizon-AGN satellite galaxies in a bubble
chart. The horizontal axis is the host halo mass, and the vertical axis is the
stellar mass of satellite galaxies. The redder color means an earlier
transition, and the bluer color means a later transition (color scales in the
bottom). We set the size of bubbles based on the distributional uncertainties
of transition epochs. Larger symbol size indicates that the transition clearly
exists, whereas a small size indicates that there is no transition. The size
indicators are shown in the left inset, and we color-coded only bins that
exhibited errors smaller than 0.75 Gyr. The black dashed line is a guideline
that distinguishes the reliability of the existence of the transition epoch.
The top and right panels show numbers of halos and galaxies shown in Figure 2.

In the previous section, we established that today’s “red and dead” cluster
galaxies exhibited higher SFRs at an earlier epoch (Figure 3 (a)) and smaller
e-folding timescales of SFH decay than field galaxies. It is natural that
cluster galaxies switched from more active to less active at some point in
history, compared with field galaxies (Gerke et al., 2007). We call this
moment the “transition epoch.” Several studies have suggested similar
concepts, that there was an epoch where the relation between SFR and density
was reversed (Elbaz et al., 2007; Brodwin et al., 2013; Hwang et al., 2019).
While the average SFR of galaxies decreases towards the center of a cluster at
low redshifts (Balogh et al., 2000), the trend becomes faint or even reversed
at high redshifts. We now attempt to determine whether the Horizon-AGN
simulation presents a transition epoch as expected and, if so, whether it is
universal or a function of the galaxy stellar mass or group/cluster host halo
mass. It is not trivial to uniquely determine the transition epoch in practice
mainly because the actual SFHs of individual simulated galaxies are rather
noisy, resulting in multiple crossings between the SFRs of the cluster and
field galaxies. We thus use the cumulative SFHs to mitigate complications. A
full description of the process is provided in Appendix B. In addition, the
reliability of an estimated transition epoch mainly depends on how
significantly the SFHs of the cluster galaxies are different from those of
their field counterparts. Therefore, we perform bootstrapping to estimate the
uncertainty in determining the transition epoch.

Figure 8 presents the results in a bubble chart color-coded by the measured
transition epochs for each stellar mass and host halo mass bin, in a manner
similar to that used for Figure 2. The size of the symbol represents the
inverse of the dispersion in the transition epoch measured from bootstrapping;
the larger the circle, the more reliable the measurement. The transition epoch
measurements exhibit small values of uncertainty ($<0.75$ Gyr, filled circles
with color) in most low-mass galaxies and therefore are reasonably robust. The
first notable feature is that transition epochs are reliably measured only for
relatively low-mass galaxies of $M_{*}<10^{10.5}\rm{M_{\odot}}$. This is
consistent with the results obtained from the e-folding timescales (Figure 6).
More massive galaxies than that exhibit negligible environmental effects.
Next, we notice a diagonal trend from blue to red in the low stellar mass
regime, i.e., as galaxy stellar mass decreases and as host halo mass
increases, the transition epoch occurs earlier (larger values in lookback
time). This is again consistent with what was learned from the quenched
fraction (Figure 2). We present the line for $\log(M_{\rm halo}/M_{*})=3.25$
in the figure to guide the eyes. It appears that the transition epochs are
clearly defined below this line. This will be discussed further in the
following section.

The transition epoch ranges from 5.2 to 7.6 Gyr in lookback time depending on
the stellar and cluster halo masses. This corresponds to $z\approx 0.56$ –
1.02 according to the standard LCDM cosmology. Considering the age of the
Universe ($\approx 13.8$ Gyr), this range of transition epoch (2.4 Gyr or 17%
of the age of the Universe) may be viewed as narrow, in which case one may
think that there is a “universal” transition epoch. Whether we consider it
universal or not, it may clearly be the case in which the most apparent
relation between the SFR and environment found in today’s galaxies is valid
only for half of the cosmic history. It would be interesting to robustly
determine the transition epoch in terms of internal and external properties of
galaxies in future observations.

###  4.2 Halo-to-Stellar Mass Ratio Dependence

Figure 9:  Upper panel (a) shows the quenched fraction of the Horizon-AGN
group/cluster galaxies as a function of the halo-to-stellar mass ratio. The
red line is the quenched fraction of the more ancient infaller galaxy, and the
blue line is that of the more recent infaller galaxy. The shaded areas show
95% confidence intervals of the estimated quenched fractions. The markers and
the errorbars are same as the lines and the shades but the low number bins
($N_{\rm gal}\leq 50$). The black dashed line at the $\log(M_{\rm
halo}/M_{*})=2.75$ indicates the arbitrary demarcation where the trend is
reversed. The lower panel (b) shows the results of $\tau$ as a function of the
halo-to-stellar mass ratio for the same galaxies. The color code and detailed
format are the same as the upper panel.

The main result of our analysis is that the quenched fractions and transition
epochs are sensitive to both the stellar mass and host (cluster or group) halo
mass: the “diagonal trend.” Baldry et al. (2006) expressed the quenched
fraction using a unified formula that includes the environmental density and
stellar mass (see their Equation (10) and Figure 12). Herein, we inspect the
quenched fractions and e-folding timescales in terms of the halo-to-stellar
mass ratio.

Figure 9 (a) shows the quenched fractions of all satellite galaxies as a
function of the cluster halo-to-stellar mass ratio. We divide the satellite
galaxies into three bins in TSI based on the same criteria used in Figure 7.
For galaxies in halos with $\log(M_{\rm halo}/M_{*})\gtrsim 2.75$, a clear
trend of the quenched fraction increasing with the halo-to-stellar mass ratio
exists. This seems to be the main condition for environmental effects to
become strong. However, galaxies with low halo-to-stellar mass ratios (i.e.,
$\log(M_{\rm halo}/M_{*})\lesssim 2.75$) exhibit an opposite trend. This is a
result of mass quenching. These are mostly massive galaxies in small halos;
hence, mass quenching is more effective than environmental quenching.
Moreover, ancient infallers in general show higher values of quenched
fractions, which illustrates the significance of the TSI effect on the
quenched states, as expected.

Similarly, we present the relation between $\tau$ and the mass ratio in Figure
9 (b). Panel (b) is almost a mirror image of Panel (a). The environmental
effect on the entire SFH of the galaxy is clearly visible at $\log(M_{\rm
halo}/M_{*})\gtrsim 2.75$. As the halo-to-stellar mass ratio increases, the
values of $\tau$ go down to 1 Gyr.

Figure 10:  e-folding timescale as a function of the halo-to-stellar mass
ratio. Each panel shows the position on the two-dimensional phase space
diagram. The projected distance ($r_{\rm proj}$) and line-of-sight velocity
($v_{\rm LOS}$) of galaxies are measured from 300 random line-of-sights. The
grey dashed lines and shades show the median values and their 16-84 percentile
uncertainties, and the black solid lines show the best fits to them. The bins
with less than 300 projected galaxies are not displayed. The fitting
parameters in Equation (6) are given in each panel. A relatively-more robust
region of the cluster is roughly marked by the thick box that includes 12
panels.

Based on these results, one may want to infer the e-folding timescale of SFH
decay using the mass ratio and TSI. However, TSI is difficult to determine for
observed galaxies in reality (Rhee et al., 2017). Hence, we derive $\tau$ for
galaxies in various locations of phase space instead. Figure 10 presents the
same information as in Figure 9 (b) (i.e., the relation between the mass ratio
and $\tau$) but sub-divided in different regions of projected position and
line-of-sight velocity phase space. To generate this diagram, we randomly
generate 300 line-of-sight positions and velocities for each satellite galaxy
following the exercise of Rhee et al. (2017) so that the observed data can be
easily compared. The phase space of this diagram goes beyond the virialized
region marked by the curve in Figure 1. We fit the data (grey dashed lines)
using the parameterized formula (black curves):

$\tau(q)=\alpha\ln{(q/2.75)}+\beta(q-2.75)+\tau_{*}$ (6)

where $\tau$ is the e-folding timescale, $q$ is the logarithmic of the mass
ratio ($\log(M_{\rm halo}/M_{*})$), $\alpha$ is the slope for the low-mass-
ratio region, $\beta$ is the slope for the high-mass-ratio region, and
$\tau_{*}$ is the characteristic e-folding timescale at $q=2.75$. We first
notice the $n$-shape in the fits of many panels, as shown in Figure 9 (b). The
relatively more robust satellite members of the group/cluster halos are in the
regions inside $r_{\rm proj}/R_{\rm vir}=1.5$ and $|v_{\rm LOS}|/V_{\rm
cir}=1.5$, i.e., the nine panels marked by the thick black box. Inside this
region, the $n$-shape is apparent, and it is more apparent in the satellites
close to the phase space center (i.e., closer to the halo center and with
lower values of $|v_{\rm LOS}|/V_{\rm cir}$). The slope of the fit for the
high mass ratios ($\beta$) gradually increases from $-1.9$ in the central
region of Panel (f) to $-1.3$ in the outer region of Panel (l). This is all
consistent with the results shown in Figure 9 (b). In the regions well outside
the halos ($R_{\rm vir}>1.5$, Panels (o) through (q)), there are no or few
galaxies with mass ratios lower than the pivot value and therefore the fits
are monotonic. It is natural because such massive galaxies are likely to be
brightest cluster/group galaxies that are rather centrally located in the
halo. The galaxies well outside the halos ($R_{\rm vir}>1.5$, Panels (r)
through (u)) exhibit significantly larger values of $\beta$, indicating a
smaller environmental effect on them. The fast-moving satellites in Panels
(a), (b), and (c) are likely recent infallers in group-size halos, which have
not been sufficiently affected yet by environmental effects; hence they
exhibit larger values of $\beta$ than those of the robust members. The values
of $\tau_{*}$ are remarkably similar among the panels, hinting for a universal
demarcation between mass quenching and environmental quenching in the
group/cluster galaxies.

##  5 Conclusion

Using the hydrodynamical cosmological simulation, Horizon-AGN, we have
investigated how the SFHs of galaxies vary with their internal and external
properties. We fit the SFH of each galaxy with a parameterized form and
measure the e-folding timescale, $\tau$, which quantifies the pace of star
formation quenching. Distinct quenching features have been found in the
Horizon-AGN galaxies, and our main results are as follows:

  * •

Massive galaxies exhibit higher values of star formation quenched fractions
and smaller e-folding timescales in their SFHs than low-mass galaxies,
regardless of their environments. This implies that the current low star
formation rates in massive galaxies are caused by a vigorous star-forming
phase and subsequent rapid quenching in the early universe. As expected, mass
quenching, which is a dominant process for massive galaxies, is clearly seen
in the Horizon-AGN galaxies.

  * •

Massive cluster environments additionally suppress the star formation of their
member galaxies. The galaxies in the most massive cluster halos exhibit
significantly higher quenched fractions and substantial drops of $\tau$
compared with their field counterparts. This environmental quenching becomes
stronger in more massive halos.

  * •

The overall star formation activities, including the quenched fraction and the
drop of $\tau$, exhibit a diagonal trend in the stellar mass versus halo mass
plane. The additional quenching of satellite galaxies fades when the host halo
is not sufficiently massive enough to suppress the star formation of its
satellites, which is consistent with theoretical expectations related to ram
pressure stripping.

  * •

Time-since-infall (TSI), i.e., the total time duration of a satellite galaxy
residing in a group/cluster halo, has a significant impact. The longer a
satellite galaxy resides in a group/cluster halo, the more it is affected by
environmental quenching. Ancient infallers with larger TSI values exhibit
smaller values of $\tau$ than recent infallers. Even the most massive
satellites, whose SFH appears to be predominantly determined by mass
quenching, show a moderate drop of $\tau$ when they have spent a long time in
massive halos.

  * •

In Horizon-AGN, the transition epoch, where cluster galaxies become less star-
forming than field galaxies, occurs 5.2 – 7.6 Gyr ago in lookback time. This
is largely consistent with the previous results of the “reversal epoch,” in
which SFR-density relations are reversed (Elbaz et al., 2007; Hwang et al.,
2019). However, the transition epoch can only be found for low-mass
satellites, because massive galaxies do not show much dependence on the
environment. It ranges from 5.2 to 7.6 Gyr, displaying a diagonal trend on the
stellar mass versus halo mass plane (low mass satellites in massive clusters
exhibit early transition epoch), which is consistent with the trend in the
e-folding timescale as well.

  * •

The cluster halo-to-stellar mass ratio and TSI may be key parameters for
inferring the SFHs of satellite galaxies. Satellite galaxies with different
quenching processes are well separated by the mass ratio. With an increase in
the mass ratio, the quenched fractions gradually approach unity, and $\tau$
decreases to 1 Gyr. This trend is reversed at $\log(M_{\rm
halo}/M_{*})\lesssim 2.75$ because, in that domain, mass quenching is more
important than environmental quenching. We detect a systematic variation in
the relation between $\tau$ and the mass ratio of the satellite galaxies in
the observable phase space.

Through this investigation, we have attempted to understand the origin of the
star formation properties of galaxies with respect to stellar mass and
environment. Horizon-AGN galaxies reproduce the most fundamental features such
as mass quenching and environmental quenching. Mass quenching is probably a
result of stellar and AGN feedback. A more massive galaxy forms stars more
efficiently and consumes cold gas in the early part of cosmic history. The
mass of the central black hole scales with the dynamical mass of the galaxy,
and thus, the feedback energy from the black hole must play a gradually more
important role with increasing stellar mass (Volonteri et al., 2016). The
pattern of environmental quenching seems consistent with the expectation from
the ram pressure stripping mechanism. As a result of collaboration or
competition between the two quenching mechanisms, galaxies do develop and
exhibit different properties at different epochs to the extent that some of
the most apparent relations are reversed. We are encouraged by the level of
detail of the models that reproduce some key properties of present-day
galaxies and provide predictions for the high-redshift Universe.

## Acknowledgments

We thank the referee for constructive criticism that clarified a few important
issues in the original manuscript. S.K.Y. acknowledges the support from the
Korean National Research Foundation (2020R1A2C3003769). A.C. acknowledges the
support from the Korean National Research Foundation (2022R1A2C100298211).
T.K. was supported by the National Research Foundation of Korea
(2020R1C1C1007079). R.A.J. was supported by the Yonsei University Research
Fund (Yonsei Frontier Lab, Young Researcher Supporting Program) of 2021. This
study was partially supported by the Spin(e) grant ANR-13-BS05-0005 of the
French Agence Nationale de la Recherche and the Center for Galaxy Evolution
Research (2022R1A6A1A03053472). It also relied on the HPC resources of the
Horizon Cluster hosted by the Institut d’Astrophysique de Paris. We warmly
thank S. Rouberol for running the cluster on which the simulation was post-
processed.

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##  Appendix A Star formation histories of entire samples

We exhibited the mean SFH of Horizon-AGN galaxies in Figure 3 for
representative stellar mass bins. Here we present the same results of all mass
bins in Figure 11.

Figure 11:  Mean SFHs of the Horizon-AGN galaxies of different masses in the
same manner as Figure 3.

##  Appendix B Transition epoch determination

We determine the transition epoch as follows. First, we separate our simulated
galaxies into stellar mass bins of size 0.25 dex and member galaxies into
another 0.25 dex bins in host halo mass. We derive the mean SFH of field and
cluster galaxies in each bin. A simple solution would then be to determine the
intersection point of the SFHs of the field and cluster galaxies. However, as
star formation is often bursty and SFHs are very noisy, we decide to use
cumulative SFHs. We then calculate the residual cumulative SFH between cluster
and field galaxies. We regard the epoch when these residual values reach a
maximum as the transition epoch. The procedure is schematically illustrated
for models in Figure 12. Furthermore, we can obtain the uncertainty of the
estimated transition epoch through bootstrap sampling. Based on 1000
bootstrapped pairs of SFHs, we estimate the mean and its error.

Figure 12:  Procedure to determine the transition epoch. We generate two mock
SFHs with different $\tau$ considering their intrinsic noise. The red and blue
lines show the mock SFHs of cluster and field galaxies, respectively. Panel
(a) shows the mean SFH of both samples. Panel (b) shows cumulative SFHs. Panel
(c) presents the residuals of the two cumulative SFHs. The grey vertical lines
in Panel (a) show all the crossing points that could be misidentified as the
transition epoch. The black dashed line shows the determined transition epoch,
corresponding to the maximum point of the residual.