# Estimating lifetimes of UV-selected massive galaxies at $0.5\leq z\leq 2.5$
in the COSMOS/UltraVISTA field through clustering analyses

Xiaozhi Lin11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei 230026, China<EMAIL_ADDRESS>22affiliation: School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei 230026, China , Guanwen Fang33affiliation: School of Mathematics and
Physics, Anqing Normal University, Anqing 246133, China<EMAIL_ADDRESS>44affiliation: Guanwen Fang and Xiaozhi Lin contributed equally to this work.
, Yongquan Xue11affiliation: CAS Key Laboratory for Research in Galaxies and
Cosmology, Department of Astronomy, University of Science and Technology of
China, Hefei 230026, China<EMAIL_ADDRESS>22affiliation: School of
Astronomy and Space Sciences, University of Science and Technology of China,
Hefei 230026, China , Lulu Fan11affiliation: CAS Key Laboratory for Research
in Galaxies and Cosmology, Department of Astronomy, University of Science and
Technology of China, Hefei 230026, China<EMAIL_ADDRESS>22affiliation:
School of Astronomy and Space Sciences, University of Science and Technology
of China, Hefei 230026, China , Xu Kong11affiliation: CAS Key Laboratory for
Research in Galaxies and Cosmology, Department of Astronomy, University of
Science and Technology of China, Hefei 230026, China<EMAIL_ADDRESS>22affiliation: School of Astronomy and Space Sciences, University of Science
and Technology of China, Hefei 230026, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present an estimation of lifetimes of massive galaxies with distinct UV
colors at $0.5\leq z\leq 2.5$ in the COSMOS/UltraVISTA field. After dividing
the galaxy sample into subsamples of red sequence (RS), blue cloud (BC), and
green valley (GV) galaxies in different redshift bins, according to their
rest-frame extinction-corrected UV colors, we derive their lifetimes using
clustering analyses. Several essentials that may influence the lifetime
estimation have been explored, including the dark matter (DM) halo mass
function (HMF), the width of redshift bin, the growth of DM halos within each
redshift bin, and the stellar mass. We find that the HMF difference results in
scatters of $\rm{\sim 0.2~{}dex}$ on lifetime estimation; adopting a redshift
bin width of $\Delta z=0.5$ is good enough to estimate the lifetime; and no
significant effect on lifetime estimation is found due to the growth of DM
halos within each redshift bin. The galaxy subsamples with higher stellar
masses generally have shorter lifetimes; however, the lifetimes among
different subsamples at $z>1.5$ tend to be independent of stellar mass.
Consistently, the clustering-based lifetime for each galaxy subsample agrees
well with that inferred using the spectral energy distribution modeling.
Moreover, the lifetimes of the RS and BC galaxies also coincide well with
their typical gas depletion timescales attributed to the consumption of star
formation. Interestingly, the distinct lifetime behaviors of the GV galaxies
at $z\leq 1.5$ and $z>1.5$ can not be fully accounted for by their gas
depletion timescales. Instead, this discrepancy between the lifetimes and gas
depletion timescales of the GV galaxies suggests that there are additional
physical processes, such as feedback of active galactic nuclei, accelerating
the quenching of GV galaxies at high redshifts.

###### Subject headings:

galaxies: high-redshift — galaxies: evolution — galaxies: formation

##  1\. Introduction

Understanding complex physical processes involved in star formation of
galaxies is one of the main goals in modern astronomy. On this subject, one of
the key questions is how long star-forming galaxies (SFGs) persist in forming
stars, to which clues have been provided by studying the evolution of stellar
mass functions (SMFs) of SFGs and quiescent galaxies (QGs). In the past
decades, thanks to the large-scale spectroscopic and photometric surveys, such
as the Sloan Digital Sky Survey (SDSS), NMBS, zCOSMOS, UltraVISTA, and
zFOURGE, robust SMFs and luminosity functions of both SFGs and QGs have been
established, which give us an understanding of the evolution of their space
densities (e.g., Moustakas et al. 2013; Muzzin et al. 2013a; Tomczak et al.
2014). Recent extensive surveys, e.g., BOSS that covers a total of over
$\rm{10,000~{}deg^{2}}$ (Eisenstein et al. 2011; Dawson et al. 2013), enable
us to study the SMFs of SFGs and QGs with stellar masses of
$M_{*}>10^{11.5}~{}M_{\odot}$ and at $z<0.7$. Future surveys, such as the
ongoing eBOSS (Dawson et al. 2016; Blanton et al. 2017) and DESI
(Collaboration et al. 2016), would have the potential to study the SMFs of
SFGs and QGs at higher redshifts.

Among these findings, it has usually been found that changes of star-formation
rates (SFRs) of galaxies are accompanied by that of their colors (e.g., Baldry
et al. 2004; Bell et al. 2004; Faber et al. 2007). According to a bimodality
in the color-magnitude diagram and thereby defining three distinct galaxy
populations, i.e., red sequence (RS), blue cloud (BC), and green valley (GV)
galaxies, astronomers have systematically studied multiple physical properties
of these three galaxy populations. Though some dust-reddened SFGs may
contaminate the red population, most RS galaxies generally exhibit low levels
of star formation and old stellar populations, while the BC population is
usually dominated by those galaxies with high SFRs and young galaxies with
plenty of gas. The nature of GV galaxies lies between the other two galaxy
populations (e.g., Gu et al. 2018, 2019).

Apart from the galaxy colors (e.g., Bell et al. 2004; Baldry et al. 2004;
Borch et al. 2006; Xue et al. 2010; Salim 2014; Lee et al. 2015; Wang et al.
2017), there are other indicators characterizing the star-formation nature of
galaxies, e.g., the morphology (Strateva et al. 2001; Pan et al. 2013; Barro
et al. 2013, 2014) and some spectral features such as the Balmer absorption
line (e.g., Kuntschner et al. 2002; Kim et al. 2013; Kim & Yoon 2017) and the
$4000~{}\AA$ break (e.g., Kauffmann et al. 2003; Lambas et al. 2012; Rowlands
et al. 2018; Angthopo et al. 2019). Using these indicators, a number of
studies have found that the color or SFR transition of galaxies is associated
with the consumption of their gas content (e.g., Kereš et al. 2005; Dekel &
Birnboim 2006; Kruijssen 2015; Nelson et al. 2018). If this “quenching”
scenario is correct, the timescales of the galaxy color/SFR transition and
their gas depletion must be consistent. Some authors have studied galaxy
quenching by quantifying the evolution of SMFs (or luminosity functions) of
SFGs and QGs at high redshifts (e.g., Fritz et al. 2014; Rowlands et al.
2018). The others, with the help of some infrared/sub-millimeter surveys
(e.g., Hershel and ALMA), have managed to study the cosmic evolution of the
gas content as well as the gas depletion rate of galaxies with different SFRs
(e.g., Lagos et al. 2011; Geach et al. 2011; Tacconi et al. 2018; Liu et al.
2019; Castignani et al. 2020; Magnelli et al. 2020). These studies have
revealed that both the galaxy quenching and gas depletion timescales increase
with decreasing redshifts. The typical quenching timescale of GV galaxies can
be as long as several Gyrs in the local universe (Rowlands et al. 2018;
Phillipps et al. 2019; Correa et al. 2019). Methods such as fitting the
spectral energy distributions (SEDs) of galaxies (e.g., Phillipps et al. 2019;
Zick et al. 2018; Belfiore et al. 2018) and using cosmological simulations
(e.g., Feldmann et al. 2017; Nelson et al. 2018; Correa et al. 2019; Donnari
et al. 2019) have been widely used to derive the star-formation histories of
galaxies and their lifetimes.

Although great efforts have been made, there are still some challenges to
understand the distinct star-formation histories of different galaxy
populations. For example, it is difficult to obtain high-quality spectra for
galaxies with wide ranges of SFR and redshift. While there have been many
robust optical spectroscopic surveys in the local universe (e.g., Abolfathi et
al. 2018; Aguado et al. 2019a, b; Westfall et al. 2019), obtaining high-
resolution spectra is still a challenging task when going to higher redshifts,
where the galaxy sample may suffer from selection effects to a large degree
(e.g., Gruppioni & Pozzi 2019; Zheng & Shen 2020). Besides, using photometric
colors to separate SFGs from the quiescent population may also suffer from
contamination of dust-reddened galaxies. Moreover, it is difficult to
systematically measure gas properties, e.g., through the 21 cm and carbon
monoxide molecule (CO) emission lines, for a large number of high-redshift
galaxies over a wide range of SFR. To estimate the gas content of high-
redshift galaxies, researchers usually rely on several assumptions, including,
e.g., a constant CO-to-$\rm{H_{2}}$ conversion factor (Bolatto et al. 2013;
Tacconi et al. 2013; Shangguan et al. 2020), a constant gas-to-dust mass ratio
as found in the local universe (e.g., Leroy et al. 2011), and multiple dust
natures (e.g., temperature and optical properties) (Scoville et al. 2016,
2017).

Considering the aforementioned challenges and uncertainties, here we choose to
take another route, i.e., using clustering analyses, to calculate the
lifetimes of galaxies with different levels of star formation and at a broad
range of redshift (0.5$\leq z\leq$2.5). Hickox et al. (2012) used their
clustering results to estimate the lifetimes of sub-millimeter galaxies based
on the assumption that for each DM halo with a mass similar to that of the
sub-millimeter galaxies, the galaxy reside in it would undergo the sub-
millimeter phase. This assumption is based on the scenario of co-evolution of
galaxy/AGN and DM halo (e.g., Hickox et al. 2009) that the behaviors of AGN
host galaxies are accompanied by the growth of their host dark matter (DM)
halos. When the halo mass reaches a certain critical value, the star-
formation/AGN feedback will be triggered, and thus the state of the AGN host
galaxy changes (see their Figure 16). This scenario is supported by a number
of studies, where the DM halo plays an important role in shaping the star-
formation activity and regulating the central black hole activity (e.g.,
Moster et al. 2013; Cai et al. 2013; Behroozi et al. 2013; Vogelsberger et al.
2014; Chen et al. 2019). This method has also been applied to estimate the
duty cycle of high-redshift AGNs (e.g., Martini & Weinberg 2001; Haiman & Hui
2001; Eftekharzadeh et al. 2015; He et al. 2017).

In this work, we make a follow-up estimation of the lifetimes of BC, GV, and
RS galaxies based on the clustering results of the three galaxy populations in
Lin et al. (2019). We inherit the classification of the three galaxy
populations in Wang et al. (2017), where they demonstrated that their
extinction-corrected UV-color separation of the BC, GV, and RS galaxies can be
a good representation of the intrinsic star-forming, transition, and quiescent
galaxies. Apart from Lin et al. (2019), the higher clustering (larger halo
masses) of quiescent galaxies than star-forming galaxies had also been
uncovered by a number of studies(e.g., Norberg et al. 2002; Coil et al. 2004,
2008; Zehavi et al. 2011; Bray et al. 2015; Coil et al. 2017), thus we would
expect a clear difference in their estimated lifetimes according to the
galaxy-halo co-evolution scenario (e.g. Hickox et al. 2009) and the assumption
of Hickox et al. (2012).

The method we apply for the lifetimes of UV-selected galaxies is similar to
that of Hickox et al. (2012) for sub-millimeter galaxies. But in Hickox et al.
(2012), they only used a single halo mass function (HMF) to estimate the
clustering-based lifetime of galaxies. In this work, we further check and
evaluate to which extent different factors, such as the choices of the HMF,
the redshift bin width, the growth of DM halos within each redshift bin, and
the stellar mass of galaxy sample would affect the lifetimes of different
galaxy populations. Besides, it is meaningful to compare our lifetime result
with that derived from SED modeling, since clustering measurement and SED
modeling are two entirely different routes. Moreover, recent mid-to-far
infrared observations have enabled us to make rough estimation of the
molecular gas amounts and gas depletion timescales through the consumption of
star formation in different types of galaxies (e.g., Popping et al. 2017;
Tacconi et al. 2018; Catinella et al. 2018; Fluetsch et al. 2019; Decarli et
al. 2019; Magnelli et al. 2020). Thus, comparing the lifetimes with the
typical gas depletion timescales in these galaxies would shed light on the
physical connections between galaxy color transformation and gas consumption
mechanisms as well.

The outline of this paper is the following. We briefly describe our adopted
data and sample selection in Section 2. In Section 3, we introduce the
clustering method to calculate the lifetimes of the UV-selected galaxies based
on their clustering properties presented in Lin et al. (2019). Then we
investigate several factors affecting the lifetime estimation in Section 4. We
further compare our clustering-based lifetimes with those derived from the SED
modeling as well as the typical gas depletion timescales of galaxies in
Section 5. Finally, Section 6 summarizes our major results.

Throughout this paper, we adopt a flat cosmology with $\Omega_{\rm M}=0.3$,
$\Omega_{\Lambda}=0.7$, $H_{0}=70~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$, and a
normalization of $\sigma_{8}=0.84$ for the matter power spectrum.

##  2\. Data

The data analyzed in this work are from the COSMOS/UltraVISTA survey, which
covers $\rm{\sim 1.62~{}deg^{2}}$ in the COSMOS/UltraVISTA field. We utilize
the Muzzin et al. (2013b) catalog, which is based on the UltraVISTA
$K_{s}$-band imaging (McCracken et al. 2012) and includes multi-wavelength
photometries, i.e., 30 bands from $\sim 0.15$ to $\sim 24~{}\mu$m. The
photometric redshift, rest-frame colors, and stellar population parameters,
including the stellar mass and star-formation history, are estimated for each
galaxy by the SED-fitting method (Muzzin et al. 2013b).

###  2.1. Redshift, Stellar Mass, and $A_{V}$

In the following, we directly adopt the photometric redshift, stellar mass,
star-formation history, attenuation, and rest-frame colors estimated by Muzzin
et al. (2013b). Only relevant data are briefly introduced as follows (see
Muzzin et al. 2013b for more details).

The Muzzin et al. (2013b) photometric catalog contains two bands from GALEX
(FUV and NUV), one band from CFHT/MegaCam ($u^{*}$), six broad bands ($g^{+}$,
$r^{+}$, $i^{+}$, $z^{+}$, ${\rm B}_{j}$, and ${\rm V}_{j}$) and twelve
optical medium bands (IA427–IA827) from Subaru/Suprime-Cam, four near-infrared
broad bands ($Y$, $J$, $H$, and $K_{\rm s}$) as well as the 3.6, 4.5, 5.8,
8.0, and 24 $\mu$m channels from Spitzer’s IRAC and MIPS. The observed-frame
wavelength coverage of this catalog is from $\sim 0.15~{}\mu$m to $\sim
24~{}\mu$m, and then for the maximal redshift of $\sim 2.5$ considered in this
paper, the rest-frame wavelength coverage is $\sim 0.04~{}\mu$m to $\sim
6.8~{}\mu$m. Therefore, the UltraVISTA data have a good wavelength coverage in
the redshift desert at $z\sim 1.4-2.5$.

The photometric redshifts and rest-frame colors in the COSMOS/UltraVISTA
catalog were computed through fitting the multiwavelength SEDs using EAZY
software (Brammer et al. 2008). The templates adopted by Muzzin et al. (2013b)
include seven initial templates in EAZY (i.e., six from the PEGASE models of
Fioc & Rocca-Volmerange 1999 and one red template from the model of Maraston
2005) and two additional templates, i.e., a one-gigayear-old single-burst
Bruzual & Charlot (2003) model for strong post-starburst-like features at
$z>1$, and a slightly dust-reddened young population for the UV bright
galaxies at $1.5<z<3.5$ to improve the accuracy of photometric redshift. All
the above templates were considered in the derivation of photometric redshifts
of the COSMOS/UltraVISTA galaxies. During the computation of the photometric
redshifts, they ran the EAZY software by fixing the photometric redshifts to
the best observed spectroscopic redshifts to determine the zero point offsets
in each photometric band. Then the photometry in each band was adjusted by the
corresponding offset and EAZY was re-run, until the median offset in every
filter converges.

At $z<1.5$, the photometric redshifts are well consistent with the
spectroscopic redshifts from zCOSMOS (Lilly et al. 2007, 2009) and other
spectroscopic surveys (Onodera et al. 2012; Bezanson et al. 2013; van de Sande
et al. 2013), with a $3\sigma$-outlier fraction of $1.56\%$ and a low rms
scatter of $\delta z/(1+z)=0.013$. At $z>1.5$, the photometric redshifts agree
well with that from the NOAO Extremely Wide-Field Infrared Imager (NEWFIRM)
Medium-Band Survey (NMBS, Whitaker et al. 2011), with the fraction of the
UltraVISTA sources considered to be $5\sigma$ outliers of the NMBS survey
being $2.0\%$ and the rms dispersion of UltraVISTA photometric redshifts being
$\delta z/(1+z)=0.026$. We note that, when available, the Muzzin et al.
(2013b) catalog adopted the spectroscopic redshifts rather than the
photometric ones.

Although Muzzin et al. (2013b) provided a $\chi^{2}$ value to evaluate each
SED-fitting result in their catalog, we do not apply any cut on this value,
since nearly $\sim 90\%$ of the sources have a reduced $\chi^{2}<2$.

We have removed stars and quasars using the flags in the catalog and further
limit the redshifts of galaxies to be within $0.5\leq z\leq 2.5$ in our study.
We obtain 126,222 sources in total, which are further divided into four
smaller redshift bins, i.e., $0.5\leq z<1.0$, $1.0\leq z<1.5$, $1.5\leq
z<2.0$, and $2.0\leq z\leq 2.5$, in order to study the lifetime evolution of
different galaxy populations.

Muzzin et al. (2013b) had also provided the information of stellar population
of COSMOS/UltraVISTA galaxies in their catalog. The stellar population
parameters, including the stellar mass ($M_{\ast}$), visual attenuation
($A_{V}$), and the stellar age of galaxies, were computed using the FAST code.
They directly adopted the redshifts they derived above as calibrations and
inputs. Then the observed SEDs of COSMOS/UltraVISTA galaxies were fitted with
the assembly of stellar population templates in the Maraston (2005) model,
assuming a solar metallicity, a Chabrier initial mass function (Chabrier
2003), and a Calzetti et al. (2000) dust extinction law. To construct the SED
templates, Muzzin et al. (2013b) assumed exponentially-declining star-
formation histories for all COSMOS/UltraVISTA galaxies with the form of ${{\rm
SFR}\propto\exp(-t/\tau)}$, where ${t}$ is the time since the onset of star
formation and $\tau$ is the e-folding star-formation timescale that varies
between $10^{7}\hskip 1.42262pt{\rm yr}$ and $10^{10}\hskip 1.42262pt{\rm
yr}$, respectively.

###  2.2. Sample Construction

Table 1Best-fit clustering properties of galaxy subsamples in the
COSMOS/UltraVISTA field

Galaxy subsample | $N_{\rm source}^{\rm a}$ | $\bar{z}^{\rm b}$ | $\log(M_{\ast}/M_{\odot})^{\rm c}$ | $A^{\rm d}_{\omega}$ | $b_{\rm gal}^{\rm e}$ | $\log(M_{\rm halo}/[h^{-1}{M_{\odot}}])^{\rm f}$ |
---|---|---|---|---|---|---|---
$0.5\leq z<1.0$ |  $\log(M_{\ast}/M_{\odot})\geq 10$ & 80% mass-completeness |
$\rm RS$ | 6051 | 0.823 | 10.53 | 0.204$\pm$0.034 | 1.87$\pm$0.16 | $12.98_{-0.16}^{+0.14}$ |
$\rm GV$ | 4116 | 0.789 | 10.39 | 0.107$\pm$0.013 | 1.38$\pm$0.08 | $12.40_{-0.15}^{+0.13}$ |
$\rm BC$ | 3774 | 0.843 | 10.21 | 0.092$\pm$0.011 | 1.23$\pm$0.07 | $12.03_{-0.18}^{+0.15}$ |
| $\log(M_{\ast}/M_{\odot})\geq 10.5$ & 98% mass-completeness |
$\rm RS$ | 3234 | 0.819 | 10.80 | 0.201$\pm$0.034 | 1.80$\pm$0.15 | $12.92_{-0.17}^{+0.14}$ |
$\rm GV$ | 1546 | 0.800 | 10.67 | 0.115$\pm$0.020 | 1.41$\pm$0.12 | $12.43_{-0.22}^{+0.19}$ |
$\rm BC$ | 537 | 0.856 | 10.59 | 0.123$\pm$0.030 | 1.36$\pm$0.17 | $12.28_{-0.37}^{+0.27}$ |
$1.0\leq z<1.5$ |  $\log(M_{\ast}/M_{\odot})\geq 10$ & 80% mass-completeness |
$\rm RS$ | 5099 | 1.223 | 10.51 | 0.149$\pm$0.031 | 2.09$\pm$0.22 | $12.76_{-0.20}^{+0.17}$ |
$\rm GV$ | 2143 | 1.219 | 10.44 | 0.158$\pm$0.022 | 2.13$\pm$0.15 | $12.80_{-0.13}^{+0.12}$ |
$\rm BC$ | 5798 | 1.270 | 10.26 | 0.096$\pm$0.011 | 1.67$\pm$0.10 | $12.27_{-0.13}^{+0.12}$ |
| $\log(M_{\ast}/M_{\odot})\geq 10.5$ & 98% mass-completeness |
$\rm RS$ | 2602 | 1.222 | 10.76 | 0.188$\pm$0.043 | 2.29$\pm$0.26 | $12.92_{-0.21}^{+0.18}$ |
$\rm GV$ | 921 | 1.216 | 10.69 | 0.176$\pm$0.025 | 2.20$\pm$0.16 | $12.85_{-0.13}^{+0.12}$ |
$\rm BC$ | 1192 | 1.297 | 10.63 | 0.183$\pm$0.026 | 2.26$\pm$0.16 | $12.85_{-0.13}^{+0.11}$ |
$1.5\leq z<2.0$ |  $\log(M_{\ast}/M_{\odot})\geq 10$ & 80% mass-completeness |
$\rm RS$ | 3354 | 1.729 | 10.50 | 0.168$\pm$0.032 | 3.04$\pm$0.29 | $12.91_{-0.16}^{+0.14}$ |
$\rm GV$ | 1393 | 1.727 | 10.38 | 0.112$\pm$0.016 | 2.52$\pm$0.18 | $12.61_{-0.13}^{+0.12}$ |
$\rm BC$ | 4971 | 1.733 | 10.29 | 0.119$\pm$0.014 | 2.44$\pm$0.14 | $12.55_{-0.11}^{+0.10}$ |
| $\log(M_{\ast}/M_{\odot})\geq 10.5$ & 98% mass-completeness |
$\rm RS$ | 1719 | 1.735 | 10.74 | 0.221$\pm$0.053 | 3.27$\pm$0.39 | $13.02_{-0.20}^{+0.16}$ |
$\rm GV$ | 506 | 1.735 | 10.69 | 0.151$\pm$0.037 | 2.73$\pm$0.33 | $12.74_{-0.22}^{+0.18}$ |
$\rm BC$ | 1316 | 1.780 | 10.67 | 0.144$\pm$0.015 | 2.58$\pm$0.13 | $12.64_{-0.09}^{+0.09}$ |
$2.0\leq z\leq 2.5$ |  $\log(M_{\ast}/M_{\odot})\geq 10$ & 80% mass-completeness |
$\rm RS$ | 1251 | 2.231 | 10.54 | 0.294$\pm$0.042 | 5.50$\pm$0.39 | $13.36_{-0.10}^{+0.09}$ |
$\rm GV$ | 1029 | 2.243 | 10.46 | 0.147$\pm$0.022 | 3.57$\pm$0.27 | $12.75_{-0.12}^{+0.11}$ |
$\rm BC$ | 2239 | 2.215 | 10.31 | 0.211$\pm$0.025 | 3.59$\pm$0.21 | $12.78_{-0.10}^{+0.09}$ |
| $\log(M_{\ast}/M_{\odot})\geq 10.5$ & 98% mass-completeness |
$\rm RS$ | 697 | 2.264 | 10.71 | 0.374$\pm$0.084 | 5.67$\pm$0.64 | $13.40_{-0.15}^{+0.13}$ |
$\rm GV$ | 477 | 2.224 | 10.69 | 0.286$\pm$0.057 | 4.90$\pm$0.49 | $13.21_{-0.14}^{+0.12}$ |
$\rm BC$ | 662 | 2.194 | 10.70 | 0.259$\pm$0.040 | 4.38$\pm$0.34 | $13.07_{-0.11}^{+0.10}$ |

a The source number of each galaxy subsample.
---
b The median redshift of each galaxy subsample.
c The median stellar mass of each galaxy subsample.
d The clustering amplitude of the angular correlation function of each galaxy
subsample.
e The bias of each galaxy subsample.
f The halo mass approximately converted from the bias.

Out of the above 126,222 sources, we only include the 41,218 galaxies with
$\log(M_{\ast}/M_{\odot})\geq 10$ as our final sample for further analyses.
This sample is the same as that of Lin et al. (2019), which consists of 12
subsamples of galaxies that have three distinct UV colors in four redshift
bins. The adopted stellar-mass limit corresponds to a mass completeness of 80%
near $z\sim 2.5$, while the mass-completeness limit reaches 98% for
$\log(M_{\ast}/M_{\odot})\geq 10.5$ (see Lin et al. 2019 for more details).

Utilizing the color-separation criteria proposed by Wang et al. (2017), we
separate the RS, GV, and BC galaxies according to their rest-frame extinction-
corrected UV colors. Although there are some other color-separation criteria
classifying galaxies with different SFRs, a number of them are based on the
observed colors and do not consider the effect of extinction correction. Thus
a proportion of RS galaxies may be imposters that are actually BC galaxies
with severe dust extinction. In Brammer et al. (2009), they showed that by
using extinction-corrected $\rm U-V$ colors, dust-reddened BC galaxies can be
better separated from the intrinsic RS galaxies. Later, Wang et al. (2017)
refined the old criteria by Bell et al. (2004) and Borch et al. (2006) such
that they are more consistent from local to $z\sim 2.5$.

The new separation criteria are as follows:

$\begin{split}{\rm\leftline{\hbox{RS:}}}\\\ {\rm(U-V)_{rest}-0.47A_{v}\geq
0.126\log(M_{\ast}/M_{\odot})+0.58-0.286z,}\\\ {\rm\leftline{\hbox{GV:}}}\\\
{\rm(U-V)_{rest}-0.47A_{v}<0.126\log(M_{\ast}/M_{\odot})+0.58-0.286z}\\\
\centerline{\hbox{\&}}\\\ {\rm(U-V)_{rest}-0.47A_{v}\geq
0.126\log(M_{\ast}/M_{\odot})-0.24-0.136z,}\\\ {\rm\leftline{\hbox{BC:}}}\\\
{\rm(U-V)_{rest}-0.47A_{v}<0.126\log(M_{\ast}/M_{\odot})-0.24-0.136z},\end{split}$
(1)

where ${\rm(U-V)_{rest}}$ is the rest-frame UV color and the value of 0.47 is
the correction factor computed for the Calzetti et al. (2000) extinction law.
Brammer et al. (2009) showed that dust-reddened SFGs can be better separated
from the intrinsic old and red galaxies by applying this correction factor.
According to the above criteria, we classify each galaxy into one of the three
galaxy populations, i.e., RS, GV, or BC. Then each final galaxy subsample is
constructed such that the redshifts of its galaxies fall within one of the
four redshift intervals between $0.5\leq z\leq 2.5$, their stellar masses
satisfy $\log(M_{\ast}/M_{\odot})\geq 10.0$ and their rest-frame extinction-
corrected UV colors belong to one of the three populations. We have also
selected galaxy subsamples of $\log(M_{\ast}/M_{\odot})\geq 10.5$ to study the
influence of stellar mass on the resulting lifetimes. The source number and
median redshift of each galaxy population within each redshift bin are
tabulated in Table 1, and the distribution of the COSMOS/UltraVISTA galaxies
in the color-mass diagram as well as the rough separations of the galaxy
subsamples are shown in Figure 1.

Figure 1.—  The distribution of COSMOS/UltraVISTA galaxies in the extinction-
corrected rest-frame UV color versus stellar mass diagram. The separations of
the RS, GV and BC galaxies are roughly shown as the solid lines. We have also
selected subsamples of higher stellar masses (i.e.,
$\log(M_{\ast}/M_{\odot})\geq 10.5$), shown as dash lines to study the
influence of stellar mass on the resulting lifetimes.

In Lin et al. (2019), we have measured the clustering amplitudes of the three
galaxy populations through the two-point angular correlation functions. Then
by properly comparing their correlation functions with that of the DM halos at
the respective redshifts, we have derived the bias factors of all galaxy
populations. After that, the bias of each galaxy population is converted into
the corresponding host halo mass, $M_{\rm halo}$, following the prescription
of Sheth et al. (2001). The clustering amplitudes ($A_{\omega}$), bias factors
as well as the DM halo masses of all galaxy populations are also tabulated in
Table 1. More details on the clustering analyses are elaborated in Lin et al.
(2019). Note that, for easy reference, Table 1 shown in this work is a simple
replication of part of the Table 1 in Lin et al. (2019).

##  3\. Method to calculate lifetime

In this section, we introduce the clustering method to calculate the galaxy
lifetime following Hickox et al. (2012). Similar to the assumption in Hickox
et al. (2012) (see their Section 4.4), we assume that for every DM halo with a
mass similar to that of the measured host halo mass of RS, BC or GV galaxies
in Lin et al. (2019) (tabulated in Table 1 in this paper), the galaxy residing
in that DM halo would go through the corresponding RS, BC or GV phase during
its evolution. In this assumption, we have pre-assumed a one to one halo
occupation relation for the host DM halo in the halo mass range of the
subsamples we concern. And we define the mass of a halo similar to the
measured $M_{\rm halo}$ of one of the RS, BC or GV galaxy subsamples as the
mass falling within the measured $M_{\rm halo}$ error bar of that galaxy
population.

We note that fixing the mean halo occupation to one may not always be true, if
we consider more complex clustering properties for these galaxies. In some
halo occupation distribution (HOD) models (e.g., Zheng et al. 2005, 2007; Ross
et al. 2009; Zehavi et al. 2011; Wake et al. 2011; Kim et al. 2015; Beltz-
Mohrmann et al. 2020), galaxies are assumed to occupy all DM halos above some
minimum halo mass. Thus in these models, the mean occupation of DM halo is no
longer equal to one, but it is a function of $M_{\rm halo}$, i.e., more
massive halos have the potential to host more galaxies. This is also
consistent with some semi-analytic prescriptions based on large cosmological
simulations (e.g., Berlind & Weinberg 2002; Manera et al. 2013). But in Lin et
al. (2019), we only adopted a simple linear bias relative to dark matter to
convert the clustering amplitude into $M_{\rm halo}$, and in this simplified
case, all galaxies in a given subsample reside in DM halos of similar masses.
At the low mass end (e.g., $\log(M_{\rm halo}/[h^{-1}{M_{\odot}}])\leq 13.0$),
a one to one occupation relation between galaxy and DM halo may slightly over-
estimate the mean number of galaxies occupying a DM halo predicted by HOD
models and semi-analytic prescriptions based on cosmological simulations
(e.g., Zheng et al. 2005; Manera et al. 2013; Wechsler & Tinker 2018). While
at the high mass end (e.g., $\log(M_{\rm halo}/[h^{-1}{M_{\odot}}])>13.0$),
the situation is the contrary, the one to one occupation relation would
probably under-estimate the mean halo occupation.

Based on the assumptions above, the lifetime of a given galaxy subsample is
defined as the average timescale that galaxies in the similarly massive DM
halos of the subsample spend in the phase of the corresponding extinction-
corrected rest-frame UV color111If average more or fewer galaxies reside in
the DM halos experience the phase of the UV color of the subsample, the
estimated lifetime will be correspondingly shorter or longer, respectively.,
such that,

$lifetime_{\rm sample}=\Delta t\frac{N_{\rm sample}}{N_{\rm halo}},$ (2)

where $\Delta t$ is the time interval corresponding to the redshift range of
the galaxy sample, $N_{\rm sample}$ is the source number of the galaxy sample,
and $N_{\rm halo}$ is the number of the DM halos that have a similar mass to
the halo mass of the sample. In equation 2, $N_{\rm halo}$ is not necessarily
equivalent to $N_{\rm sample}$. This is because although we assume that each
galaxy in the similar DM halos would undergo the respective phase of the
sample, galaxy color is not constant during its evolution. When we observe a
galaxy whose host halo has a similar mass to the measured halo mass of a given
galaxy sample, its color may have already changed (i.e., it may have already
passed the phase of that galaxy sample) or its color may have not yet reached
the color of the given galaxy sample (i.e., it may have not yet evolved into
that phase). Thus the lifetime here can be a rough estimate of the time that a
galaxy is alive in a respective phase of color.

To calculate $N_{\rm halo}$, we should integrate the halo mass function (HMF)
over the respective redshift range of the sample,

$N_{\rm halo}=\int_{V(z_{\rm max})}^{V(z_{\rm
min})}\int_{M_{-}}^{M_{+}}\frac{dn_{\rm halo}}{dM_{\rm h}}(M_{\rm h},z)dM_{\rm
h}dV,$ (3)

where $V(z_{\rm max})$ and $V(z_{\rm min})$ are the comoving volumes
corresponding to the maximum and minimum redshifts of the sample, $M_{+}$ and
$M_{-}$ correspond to the upper and lower limits of the DM halo mass of the
subsamples in Table 1. The HMF, $\frac{dn_{\rm halo}}{dM_{\rm h}}(M_{\rm
h},z)$, follows the functional form of

$\frac{dn_{\rm halo}}{dM_{\rm h}}(M_{\rm
h},z)=f(\sigma)\frac{\bar{\rho}_{m}}{M_{\rm h}}\left|\frac{d~{}{\rm
ln}~{}\sigma}{dM_{\rm h}}\right|,$ (4)

where $\sigma$ is the cosmology-dependent mass variance of the linear density
field smoothed on the scale of $R(M)=(3M/4\pi\cdot\bar{\rho}_{m})^{1/3}$,
$\bar{\rho}_{m}$ is the mean density of the universe, and $f(\sigma)$
represents the functional form that defines a particular HMF fit. In this
paper, we should explore how the form of HMF fit influences the lifetimes of
our galaxy subsamples (see Section 4.1).

Beside HMF, we also discuss multiple factors that can influence the resultant
lifetimes of our galaxy subsamples, including the choices of the width of
redshift bin, the growth of DM halo mass, and the stellar mass of the galaxy
sample.

##  4\. Discussion

In this section, we explore several factors that may affect lifetimes of
galaxy samples.

###  4.1. Influence of Halo Mass Functions

The lifetimes of galaxies can be affected by different forms of HMFs to some
degree. We investigate the influence of several forms of HMFs, including that
of Tinker et al. (2008), Sheth et al. (2001), Crocce et al. (2010), and Reed
et al. (2007), as well as two additional forms of HMFs derived respectively by
the friends-of-friends (FOF) and spherical overdensity (SO) halo-finding
methods in Watson et al. (2013). We list these HMFs in Table 2, while a
comparison of them is shown in Figure 2.

Figure 2.— Comparison of different HMFs at four redshifts within our concerned
redshift range.

In general, the Watson et al. (2013) FOF HMF shows larger numbers of halos
than other HMFs at all redshifts, so it leads to smaller lifetimes estimated
through Equation 2. At $z>1.5$, the Watson et al. (2013) SO HMF appears to be
the smallest among all HMFs over a large range of halo mass, but the
discrepancy becomes very small towards the most massive end.

Table 2Comparison of different HMFs Ref. | Fitting Function $f(\sigma)$
---|---
Tinker et al. (2008) | $f_{\rm T}(\sigma,z)=A((\frac{b}{\sigma})^{a}+1){\rm exp}\left[-\frac{c}{\sigma^{2}}\right],$
$A=0.186(1+z)^{-0.14},a=1.47(1+z)^{-0.06},b=2.57(1+z)^{-\alpha},c=1.19$,
$\alpha={\rm exp}\left[-(\frac{0.75}{{\rm ln}(\Delta_{\rm
vir}/75)})^{1.2}\right]$
Sheth et al. (2001) | $f_{\rm ST}(\sigma)=A\sqrt{\frac{2a}{\pi}}\left[1+(\frac{\sigma^{2}}{a\delta_{c}^{2}})^{p}\right]\frac{\delta_{c}}{\sigma}{\rm exp}\left[-\frac{a\delta_{c}^{2}}{2\sigma^{2}}\right]$
Crocce et al. (2010) | $f_{\rm Cr}(\sigma)=A(\sigma^{-a}+b){\rm exp}\left[-\frac{c}{\sigma^{2}}\right],$
$A=0.58(1+z)^{-0.13},a=1.37(1+z)^{-0.15},b=0.3(1+z)^{-0.084},c=1.036(1+z)^{-0.024}$
Watson et al. (2013) FOF | $f_{\rm W_{FOF}}(\sigma)=A((\frac{b}{\sigma})^{a}+1){\rm exp}\left[-\frac{c}{\sigma^{2}}\right],$
$A=0.282,a=1.406,b=2.163,c=1.21$
Watson et al. (2013) SO | $f_{\rm W_{SO}}(\sigma,z)=\Gamma(\Delta,\sigma,z)A((\frac{b}{\sigma})^{a}+1){\rm exp}\left[-\frac{c}{\sigma^{2}}\right],$
$(A,a,b,c)_{z=0}=(0.194,2.267,1.805,1.287),$
$(A,a,b,c)_{z>6}=(0.563,874,3.810,1.453),$
$(A,a,b,c)_{0<z<6}=\Omega_{M}(z)\times(1.907(1+z)^{-3.216}+0.074,3.136(1+z)^{-3.058}+2.349,5.907(1+z)^{-3.599}+2.344,1.318),$
$\Gamma(\Delta,\sigma,z)=C(\Delta)(\Delta/178)^{d(z)}{\rm
exp}\left[\frac{p(1-\frac{\Delta}{178})}{\sigma^{q}}\right],$
$C(\Delta)=0.947{\rm
exp}\left[0.023(\frac{\Delta}{178}-1)\right],d(z)=-0.456\Omega_{M}(z)-0.139,p=0.072,q=2.130$
Reed et al. (2007) | $f_{R}(\sigma)=A\sqrt{\frac{2a}{\pi}}\left[1+\left(\frac{\sigma^{2}}{a\delta_{c}^{2}}\right)^{p}+0.6G_{1}+0.4G_{2}\right]\times\frac{\delta_{c}}{\sigma}{\rm exp}\left[-\frac{ca\delta_{c}^{2}}{2\sigma^{2}}-\frac{0.03}{(n_{\rm eff}+3)^{2}}(\frac{\delta_{c}}{\sigma})^{0.6}\right],$
$A=0.310,a=0.75,ca=0.764,p=0.3,\delta_{c}=1.686,n_{\rm eff}=6\frac{d~{}{\rm
ln}~{}\sigma^{-1}}{d~{}{\rm ln}~{}M}-3,$
$G_{1}={\rm exp}\left[-\frac{({\rm
ln}~{}\sigma^{-1}-0.4)^{2}}{0.72}\right],G_{2}={\rm exp}\left[-\frac{({\rm
ln}~{}\sigma^{-1}-0.75)^{2}}{0.08}\right]$

Since we use a mass-selected sample, it may suffer some selection effects,
e.g., the detection completeness limit of the sources, photometric
uncertainties in the derivation of stellar mass using a set of stellar
population models, and the effect of clustering. In this paper, we adopt the
stellar mass functions (SMFs) computed by the $1/V_{\rm max}$ method as well
as their 1$\sigma$ uncertainties in the COSMOS/UltraVISTA field (Muzzin et al.
2013a) to account for the selection effects.

Here is a brief introduction of the computation of $1/V_{\rm max}$ Muzzin et
al. (2013a) SMFs in the COSMOS/UltraVISTA field. First, they determined the
maximum volume ($V_{\rm max}$) within which an object of the stellar mass of
$M_{\ast}$ could be detected above the stellar mass completeness level. Then
they counted the galaxies in bins of $M_{\ast}$, and corrected those bins with
$1/V_{\rm max}$. The 1$\sigma$ uncertainties in the $1/V_{\rm max}$ Muzzin et
al. (2013a) SMFs are made up of three parts: Poisson error when counting
galaxies within $V_{\rm max}$, errors from photometric uncertainties, and
cosmic variance due to galaxy clustering. To estimate the error from
photometric uncertainties, they performed 100 Monte Carlo (MC) realizations of
the catalog. Within each realization, they allowed the photometry in the
catalog to perturb according to the measured photometric uncertainties. Then
these 100 MC catalogs were used to recalculate SMFs, and they determined the
empirical uncertainties in the SMFs due to photometric uncertainties as the
range of these recalculated SMFs. For the estimation of cosmic variance, they
adopted the prescriptions of Moster et al. (2011).

The 1$\sigma$ uncertainties of the Muzzin et al. (2013a) SMFs can give us a
rough estimation of selection effects, reflected by fluctuations in the number
counts of all galaxy subsamples. We integrate the upper and lower limits of
SMFs to obtain the 1$\sigma$ fluctuations of the galaxy amounts in each
redshift bin. Then we assume the over/under-estimated amount of sources to
share the same distributions of AGN fraction and extinction-corrected UV color
as that of normal galaxies in that redshift bin. Thus the over/under-estimated
amount of each galaxy subsample can be determined proportional to the total
over/under-estimated amount of sources.

Figure 3 shows the distributions of number counts of all galaxy subsamples
selected with $\log(M_{\ast}/M_{\odot})\geq 10.0$ as well as their 1$\sigma$
uncertainties caused by selection effect, with the maximum number at each
redshift normalized to 1. At $z<2$, the fluctuations in the number counts of
all galaxy subsamples caused by selection effect are generally around 8-9% of
the total subsample source amounts, while it increases to $\sim 15\%$ in the
highest redshift bin of $2.0\leq z\leq 2.5$.

Figure 3.— Number distribution of COSMOS/UltraVISTA galaxies. The number
counts of RS, GV and BC galaxies in different redshift bins are shown as the
red, green and blue open histograms respectively, with the maximum number
counts in each redshift bin normalized to 1. The line-filled parts in the
histograms show the fluctuations in number counts due to selection effect for
all galaxy subsamples.

We account for this source selection uncertainty in the calculation of
lifetime error bars for all galaxy subsamples. The error bar of each subsample
is computed as the quadrature sum of the uncertainty caused by selection
effect and the uncertainty caused by the measured DM halo mass. In general,
the uncertainty of selection effect is smaller than the uncertainty caused by
the measured DM halo mass for each galaxy subsample. For BC and GV galaxies,
the lifetime uncertainties caused by selection effect only contribute to
$\approx 1/3$ of the total uncertainties. While for RS galaxies, this fraction
is smaller, being about $1/4\sim 1/5$.

Figure 4.— Comparison of the lifetimes of UV-selected RS (red symbols in the
top row), GV (green symbols in the middle row), and BC galaxies (blue symbols
in the bottom row) in four redshift bins calculated using different HMFs. The
X-axis shows different HMFs where “T08” refers to Tinker et al. (2008), “S01”
refers to Sheth et al. (2001), “C10” refers to Crocce et al. (2010), “W13 FOF”
and “W13 SO” refers to FOF and SO halo-finding methods respectively in Watson
et al. (2013), and “R07” refers to Reed et al. (2007).

We show the lifetimes of our subsamples calculated with various HMFs in Figure
4, and tabulate them in Table 3. The variations of lifetime between different
HMFs are generally around ${\rm 0.2~{}dex}$ for all galaxy subsamples, with
that for the low-redshift subsamples ($z<1.5$) being slightly below ${\rm
0.2~{}dex}$ and that for the high-redshift subsamples ($z\geq 1.5$) being
slightly above ${\rm 0.2~{}dex}$, respectively. These variations are much
smaller than the intrinsic variations of lifetimes of galaxies with different
colors at different redshifts.

The RS galaxies generally exhibit longer lifetimes than the BC and GV galaxies
at all redshifts, which means that the RS galaxies are more likely in a stable
phase. For the BC galaxies, their lifetimes are approximately around ${\rm
1~{}Gyr}$ in all redshift bins, while for the GV galaxies, their lifetimes
show a difference between high and low redshift bins. At $z\geq 1.5$, the
lifetime of GV galaxies is generally around ${\rm 300~{}Myr}$, and it
increases significantly to above ${\rm 1~{}Gyr}$ when the redshift goes to
$z<1.5$. We make a more detailed discussion about the variations of lifetimes
among different subsamples in Section 5.

Table 3Lifetimes of galaxy subsamples derived from different HMFs in the COSMOS/UltraVISTA field Galaxy subsample | T08 | S01 | C10 | W13 FOF | W13 SO | R07 |
---|---|---|---|---|---|---|---
$0.5\leq z<1.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $13.7^{+5.9}_{-4.3}$ Gyr |  $13.8^{+6.2}_{-4.4}$ Gyr |  $12.6^{+6.2}_{-3.8}$ Gyr |  $9.9^{+4.5}_{-3.2}$ Gyr |  $9.9^{+3.7}_{-2.8}$ Gyr |  $12.8^{+5.3}_{-3.9}$ Gyr |
$\rm GV$ |  $3.0^{+0.9}_{-0.7}$ Gyr |  $2.9^{+0.9}_{-0.7}$ Gyr |  $3.0^{+0.8}_{-0.7}$ Gyr |  $2.1^{+0.6}_{-0.5}$ Gyr |  $2.5^{+0.6}_{-0.6}$ Gyr |  $2.9^{+0.8}_{-0.7}$ Gyr |
$\rm BC$ |  $1.2^{+0.3}_{-0.3}$ Gyr |  $1.1^{+0.4}_{-0.3}$ Gyr |  $1.2^{+0.4}_{-0.3}$ Gyr |  $0.8^{+0.2}_{-0.2}$ Gyr |  $1.0^{+0.3}_{-0.3}$ Gyr |  $1.2^{+0.3}_{-0.4}$ Gyr |
$1.0\leq z<1.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $2.2^{+1.2}_{-0.8}$ Gyr |  $2.1^{+1.2}_{-0.8}$ Gyr |  $1.9^{+1.1}_{-0.6}$ Gyr |  $1.5^{+0.9}_{-0.5}$ Gyr |  $2.1^{+1.0}_{-0.6}$ Gyr |  $2.0^{+1.0}_{-0.7}$ Gyr |
$\rm GV$ |  $1.6^{+0.6}_{-0.4}$ Gyr |  $1.5^{+0.6}_{-0.4}$ Gyr |  $1.4^{+0.5}_{-0.4}$ Gyr |  $1.1^{+0.4}_{-0.3}$ Gyr |  $1.5^{+0.4}_{-0.4}$ Gyr |  $1.4^{+0.5}_{-0.4}$ Gyr |
$\rm BC$ |  $1.3^{+0.4}_{-0.3}$ Gyr |  $1.3^{+0.3}_{-0.3}$ Gyr |  $1.2^{+0.4}_{-0.2}$ Gyr |  $0.9^{+0.3}_{-0.2}$ Gyr |  $1.5^{+0.4}_{-0.3}$ Gyr |  $1.2^{+0.4}_{-0.2}$ Gyr |
$1.5\leq z<2.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $1.8^{+1.2}_{-0.7}$ Gyr |  $1.7^{+1.1}_{-0.7}$ Gyr |  $1.5^{+0.8}_{-0.6}$ Gyr |  $1.2^{+0.9}_{-0.4}$ Gyr |  $1.8^{+1.0}_{-0.6}$ Gyr |  $1.5^{+0.9}_{-0.6}$ Gyr |
$\rm GV$ |  $390^{+140}_{-120}$ Myr |  $360^{+140}_{-110}$ Myr |  $320^{+120}_{-100}$ Myr |  $260^{+100}_{-70}$ Myr |  $440^{+140}_{-110}$ Myr |  $330^{+120}_{-100}$ Myr |
$\rm BC$ |  $1.4^{+0.4}_{-0.3}$ Gyr |  $1.3^{+0.4}_{-0.3}$ Gyr |  $1.2^{+0.3}_{-0.3}$ Gyr |  $0.9^{+0.3}_{-0.2}$ Gyr |  $1.6^{+0.4}_{-0.3}$ Gyr |  $1.2^{+0.3}_{-0.3}$ Gyr |
$2.0\leq z\leq 2.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $14.3^{+17.9}_{-7.8}$ Gyr |  $10.8^{+11.9}_{-5.5}$ Gyr |  $8.2^{+8.6}_{-4.2}$ Gyr |  $10.0^{+12.6}_{-5.5}$ Gyr |  $12.6^{+15.7}_{-6.7}$ Gyr |  $9.3^{+10.7}_{-4.8}$ Gyr |
$\rm GV$ |  $370^{+190}_{-130}$ Myr |  $330^{+180}_{-110}$ Myr |  $280^{+140}_{-90}$ Myr |  $240^{+140}_{-80}$ Myr |  $400^{+190}_{-130}$ Myr |  $290^{+150}_{-100}$ Myr |
$\rm BC$ |  $1.1^{+0.5}_{-0.3}$ Gyr |  $1.0^{+0.4}_{-0.3}$ Gyr |  $0.8^{+0.3}_{-0.2}$ Gyr |  $0.7^{+0.3}_{-0.2}$ Gyr |  $1.2^{+0.5}_{-0.3}$ Gyr |  $0.9^{+0.3}_{-0.3}$ Gyr |

###  4.2. Influence of Redshift Bin Width

The width of redshift bin can also affect the resultant lifetime of a galaxy
sample. This is because if we choose a very narrow redshift bin, our
assumption on the DM halo experiencing a specific phase of galaxy evolution
would no longer hold. Thus it is necessary to know, to which extent the width
of redshift bin can affect the lifetime estimation.

We resample each of our galaxy populations with finer redshift bin widths of
$\Delta z=0.1,0.2,0.3,0.4,0.5$, centering at $\bar{z}=0.75,1.25,1.75,2.25$.
Here, we only apply the Tinker et al. (2008) HMF and do not consider the
change of DM halo mass within the redshift bin since we have checked, in Lin
et al. (2019), that the halo mass does not show an obvious evolution within
each redshift bin. The results are shown in Figure 5 and tabulated in Table 4.

Figure 5.— The lifetimes (left panels) and the corresponding duty cycles (right panels) of the RS (red symbols in the top row), GV (green symbols in the middle row), and BC (blue symbols in the bottom row) galaxies as a function of the redshift bin width, centering at the median of each redshift bin annotated at the top of each column.  Table 4Lifetimes of galaxy subsamples derived by different redshift bin widths in the COSMOS/UltraVISTA field Galaxy subsample | $\Delta z=0.1$ | $\Delta z=0.2$ | $\Delta z=0.3$ | $\Delta z=0.4$ | $\Delta z=0.5$ |
---|---|---|---|---|---|---
$0.5\leq z<1.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $2.1^{+0.9}_{-0.6}$ Gyr |  $5.6^{+2.3}_{-1.7}$ Gyr |  $8.5^{+3.6}_{-2.5}$ Gyr |  $10.6^{+4.5}_{-3.2}$ Gyr |  $13.7^{+5.7}_{-4.2}$ Gyr |
$\rm GV$ |  $0.6^{+0.2}_{-0.1}$ Gyr |  $1.2^{+0.3}_{-0.3}$ Gyr |  $1.9^{+0.5}_{-0.5}$ Gyr |  $2.4^{+0.7}_{-0.6}$ Gyr |  $3.0^{+0.9}_{-0.7}$ Gyr |
$\rm BC$ |  $0.2^{+0.1}_{-0.1}$ Gyr |  $0.4^{+0.2}_{-0.1}$ Gyr |  $0.7^{+0.2}_{-0.2}$ Gyr |  $0.9^{+0.2}_{-0.3}$ Gyr |  $1.2^{+0.3}_{-0.4}$ Gyr |
$1.0\leq z<1.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $0.5^{+0.3}_{-0.2}$ Gyr |  $1.0^{+0.6}_{-0.3}$ Gyr |  $1.5^{+0.8}_{-0.6}$ Gyr |  $1.8^{+1.0}_{-0.7}$ Gyr |  $2.2^{+1.2}_{-0.8}$ Gyr |
$\rm GV$ |  $0.3^{+0.1}_{-0.1}$ Gyr |  $0.7^{+0.2}_{-0.2}$ Gyr |  $1.0^{+0.4}_{-0.3}$ Gyr |  $1.2^{+0.5}_{-0.3}$ Gyr |  $1.6^{+0.6}_{-0.4}$ Gyr |
$\rm BC$ |  $0.2^{+0.1}_{-0.1}$ Gyr |  $0.6^{+0.1}_{-0.1}$ Gyr |  $0.9^{+0.2}_{-0.2}$ Gyr |  $1.1^{+0.3}_{-0.3}$ Gyr |  $1.3^{+0.4}_{-0.3}$ Gyr |
$1.5\leq z<2.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $0.4^{+0.3}_{-0.1}$ Gyr |  $0.8^{+0.4}_{-0.3}$ Gyr |  $1.1^{+0.8}_{-0.4}$ Gyr |  $1.5^{+0.9}_{-0.6}$ Gyr |  $1.8^{+1.2}_{-0.7}$ Gyr |
$\rm GV$ |  $70^{+30}_{-20}$ Myr |  $150^{+50}_{-40}$ Myr |  $220^{+80}_{-60}$ Myr |  $300^{+110}_{-80}$ Myr |  $390^{+140}_{-110}$ Myr |
$\rm BC$ |  $0.3^{+0.1}_{-0.1}$ Gyr |  $0.5^{+0.2}_{-0.1}$ Gyr |  $0.8^{+0.3}_{-0.2}$ Gyr |  $1.1^{+0.3}_{-0.2}$ Gyr |  $1.4^{+0.4}_{-0.3}$ Gyr |
$2.0\leq z\leq 2.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $2.7^{+3.4}_{-1.4}$ Gyr |  $5.1^{+6.6}_{-2.7}$ Gyr |  $8.1^{+10.2}_{-4.3}$ Gyr |  $11.0^{+13.7}_{-5.9}$ Gyr |  $14.3^{+17.7}_{-7.6}$ Gyr |
$\rm GV$ |  $80^{+30}_{-30}$ Myr |  $150^{+70}_{-50}$ Myr |  $220^{+110}_{-70}$ Myr |  $290^{+150}_{-90}$ Myr |  $370^{+180}_{-120}$ Myr |
$\rm BC$ |  $0.2^{+0.1}_{-0.1}$ Gyr |  $0.4^{+0.2}_{-0.1}$ Gyr |  $0.6^{+0.3}_{-0.2}$ Gyr |  $0.9^{+0.3}_{-0.3}$ Gyr |  $1.1^{+0.4}_{-0.3}$ Gyr |

The lifetimes of all galaxy subsamples at each redshift generally increase
with the increasing redshift bin width, $\Delta z$. But the increasing trend
becomes flatter when the bin width is large enough, and the lifetime of each
subsample almost approaches a constant value when $\Delta z$ reaches 0.5.
Therefore, we conclude that $\Delta z=0.5$ is large enough to correctly
estimate the lifetime of each UV-selected galaxy subsample.

We note that although the lifetime of each subsample shows some dependence on
the choice of redshift bin width, its duty cycle remains roughly unchanged
with $\Delta z$, as is shown in the right panels of Figure 5. We define the
duty cycle of a subsample as the fraction of DM halos hosting the respective
type of galaxies within its redshift interval, following the definition in
e.g., Martini & Weinberg (2001),Haiman & Hui (2001),Hickox et al.
(2012),Eftekharzadeh et al. (2015),Toba et al. (2017), and He et al. (2017):

$f_{\rm duty}=\frac{n_{\rm sample}}{\int^{\infty}_{M_{\rm
h,min}}\frac{dn}{dM_{\rm h}}dM_{\rm h}},$ (5)

where $n_{\rm sample}$ is the number density of each subsample,
$\frac{dn}{dM_{\rm h}}$ is the HMF following the functional form of Tinker et
al. (2008), and $M_{\rm h,min}$ is adopted as the estimated lower limit of the
DM halo mass of the subsample, respectively.

It is clear that BC and GV galaxies generally have smaller duty cycles than RS
galaxies at the same redshifts. And for BC and GV galaxies, their duty cycles
show distinct behaviors as the redshift evolves. The duty cycle of BC galaxies
increases with the increasing redshift, from 4% at $z\sim 0.7$ to 13% at
$z\sim 2.2$; while that of GV galaxies shows an opposite trend, decreasing
from 11% at $z\sim 0.7$ to 6% at $z\sim 2.2$.

In addition, it is interesting to find that the duty cycles of BC and GV
galaxies are comparable to that of AGN at corresponding redshifts. Shankar et
al. (2009) constructed evolutionary models to estimate the properties of AGNs
and found that their duty cycle shows a declination from 7% at $z=3$ to 0.4%
at $z=1$, when the black hole mass is fixed to $10^{9}~{}M_{\odot}$. Some
other studies also utilized the same clustering estimator to evaluate the duty
cycles of AGNs (e.g., White et al. 2008; Gilli et al. 2009; Shankar et al.
2010), implying a result of $\sim 10\%$ on average. And it has also been
discovered that the duty cycle of AGNs is sensitively dependent on the
luminosity, as well as other cosmological parameters (e.g., $\sigma_{8}$) and
the choice of HMF (e.g., Adelberger et al. 2005; Shen et al. 2007;
Eftekharzadeh et al. 2015; He et al. 2017). More luminous AGNs usually tend to
have shorter lifetimes and smaller duty cycles.

###  4.3. Influence of DM Halo Growth

As stated in the Introduction, the activity of galaxies is usually accompanied
with the growth of their host DM halos, therefore it is of necessity to know
whether and to which extent the growth of DM halos would affect the lifetimes
of our galaxy subsamples. We assume the growth of DM halos following the
median mass growth rate of DM halos derived by Fakhouri et al. (2010) through
fitting the evolution of DM halos in the Millennium-II cosmology simulation.
The growth of DM halos will change the integral interval in Equation 3, since
now we need to redefine the DM halos with a mass similar to that of the galaxy
subsamples. With the growing DM halo mass, the upper and lower limits of the
halo mass of each galaxy subsample would also increase in time following the
growth function by Fakhouri et al. (2010), thus the integral interval of halo
mass is redshift-dependent within each redshift bin. The original integral
intervals as well as those considering the growth of DM halos are shown as the
dashed open and solid line-filled polygons in the left panel of Figure 6,
respectively.

Figure 6.— The left panel shows the integral intervals of the HMF in Equation
3 as a function of redshift for each of the UV-selected galaxy subsample with
$\log(M_{\ast}/M_{\odot})\geq 10.0$. The original integral intervals are shown
as the dashed open rectangles, while the integral intervals considering the
growth of DM halo mass are shown as the solid line-filled polygons. The filled
circles show the halo masses of these galaxy subsamples at their respective
median redshifts. The right panel shows the lifetime offset as a function of
the extinction-corrected UV color.

For each of our galaxy subsample selected by $\log(M_{\ast}/M_{\odot})\geq
10.0$, we evaluate the offset of the lifetime accounting for the growth of DM
halos. The offset is defined as

$lifetime~{}offset=\frac{lifetime_{\rm halo~{}growth}-lifetime}{lifetime},$
(6)

where $lifetime_{\rm halo~{}growth}$ is the lifetime considering the growth of
DM halos. We apply the Tinker et al. (2008) HMF during this computation.

The result is shown in the right panel of Figure 6. For all of our subsamples,
the lifetime offsets are no more than 5%, considering the median growth rate
of DM halos by Fakhouri et al. (2010). Thus we conclude that, the growth of DM
halo would have a negligible effect on the lifetimes of our UV-selected
galaxies in these redshift bins.

###  4.4. Influence of Stellar Mass

We also compare the lifetimes of galaxy subsamples selected by different
stellar-mass limits $\log(M_{\ast}/M_{\odot})\geq 10.5$ as tabulated in Table
1. For all subsamples selected with a higher stellar-mass threshold, we
calculate their lifetimes using the Tinker et al. (2008) HMF and show the
results in Figure 7. We have also accounted for source bias and selection
effect similarly to what we have done in the lifetime calculation for the
subsamples selected with $\log(M_{\ast}/M_{\odot})\geq 10.0$.

Figure 7.— Comparison of the lifetimes of galaxy subsamples selected by
different stellar-mass limits in the COSMOS/UltraVISTA field. The filled
circles, open rectangles, and open triangles represent the lifetimes of galaxy
subsamples selected by ${\log(M_{\ast}/M_{\odot})\geq 10.0}$, 10.5, and 11.0,
respectively. For galaxy subsamples with the same colors and in the same
redshift range, the derived lifetime increases with the decreasing stellar-
mass limit; however, this difference diminishes with the increasing redshift.

Generally speaking, for a given color of galaxies, the subsample selected with
a larger stellar-mass cut would have a shorter lifetime at a given redshift.
In the low-redshift bins this trend is obvious, but it seems to diminish when
the redshift goes higher.

The lifetimes of different galaxy populations, which characterize the time
when a certain type of galaxies stay in this phase, can also reflect the
typical quenching timescale of SFGs when we consider the quenching scenario.
In this scenario, due to mechanisms such as AGN feedback (e.g., Bower et al.
2006; Cicone et al. 2014; Schawinski et al. 2014) or merger-driven disk
instability (e.g., Martig et al. 2009; Leauthaud et al. 2012; Zolotov et al.
2015), the high-redshift SFGs would consume their gas and transform into QGs.
Multiple studies have revealed that this quenching process is much more
efficient in more massive galaxies (e.g., Springel et al. 2005b; Zolotov et
al. 2015; Contini et al. 2020). But at higher redshifts, the difference
between the quenching timescales of galaxies with different stellar-mass
limits seems to be diluted, as it is evident from some studies on the galaxy
mass function (e.g., Ilbert et al. 2013), that over the entire stellar mass
range, the number density of high-redshift ($z>1$) QGs increases smoothly. It
is also supported by some other studies on the emergence of QGs that the
quenching timescale of high-redshift compact SFGs is very short, which can be
as short as several hundred million years (e.g., Barro et al. 2013, 2016;
Kocevski et al. 2017). And this quenching timescale seems to be even shorter
with the increasing redshift, which was proposed by some studies on the
luminosity function (e.g., Cucciati et al. 2012) and the stellar-mass function
(e.g., Rowlands et al. 2018) of SFGs. Thus it is difficult to distinguish the
differences of the SFR transition timescales between galaxies with different
stellar masses at high redshifts.

##  5\. Comparison with the SED Modeling

In this section, we compare the galaxy lifetimes with those derived from the
SED modeling. In section 5.1, we introduce the method of our SED modeling and
discuss about the evolutionary track degeneracy due to the choice of different
sets of parameters. In section 5.2, we make a comparison between the SED
modeled lifetimes and the aforementioned clustering-based lifetimes. We also
compare the lifetimes between the populations of different colors and their
evolutions as a function of redshift. In section 5.3, we discuss the possible
reasons that cause the short lifetime of GV galaxies at $z\geq 1.5$. In
section 5.4, we compare the lifetimes of the subsamples with their typical gas
depletion timescales. In section 5.5, we try to reproduce the photometric
redshift distribution of all UV-selected galaxies in the COSMOS/UltraVISTA
catalog (Muzzin et al. 2013b) based on the derived lifetimes of this paper and
a preset HMF, and we also compare the best parameters in reproducing the
galaxy redshift distributions with our lifetime results.

###  5.1. The SED modeling

We utilize a simple stellar population generated from the Bruzual & Charlot
(2003) stellar population synthesis library assuming a Chabrier (2003) initial
mass function and solar metallicity. To construct the SED templates, we
further assume exponentially-declining star-formation histories with the form
of ${{\rm SFR}\propto\exp(-t/\tau)}$, where $t$ is the time since the onset of
star formation, and $\tau$ is the e-folding timescale, respectively. We do not
take dust extinction into account in this SED modeling, since we have divided
our galaxy sample using the dust-corrected rest-frame UV colors. We generate
galaxy SEDs formed at different redshifts between $z=0-10$ and the e-folding
timescale $\tau$ is allowed to vary between $10^{7}~{}\rm{yr}$ and
$10^{10}~{}\rm{yr}$. The dust-corrected UV colors of a modeled SED at
different redshifts are computed through the convolution between the
redshifted galaxy SED and the UV filters. The evolution track of each galaxy
subsample is determined by finding the right one whose dust-corrected UV color
($\rm{UV_{rest,ext-cor}}$), stellar mass and e-folding timescale $\tau$ match
those respective median SED-fitting values of each galaxy sample at its median
redshift in the Muzzin et al. (2013b) catalog. The modeled evolution tracks of
color-stellar mass relation for all UV-selected galaxy subsamples with
${\log(M_{\ast}/M_{\odot})\geq 10.0}$ are shown as solid color lines in Figure
8.

We define the modeled lifetime as the period when the dust-corrected UV color
and stellar mass of the modeled evolution track of each galaxy subsample in
the color-mass diagram satisfy the separation criteria of the respective
galaxy population in Equation 1, shown as the solid portions of the lines in
Figure 8. Specifically, we define the modeled lifetime of the BC, GV, or RS
galaxies as the period when the respective color-mass relation computed from
the modeled SED agrees with that of BC, GV, or RS galaxies since the formation
redshift $z_{f}$. In addition, for the modeled lifetime of each RS galaxy
population, we restrict the redshift to be no smaller than the observed lower
redshift limit of each subsample during the evolution. But if we consider the
red sequence as a stable phase in the quenching scenario and neglect other
factors such as rejuvenation for a small amount of red dead galaxies (e.g.,
Reipurth et al. 1997; Portegies Zwart 2000; Perets 2009), the expected
lifetime of the RS galaxies can be much longer than what we model. The modeled
lifetimes of all galaxy populations together with those inferred through
clustering analyses are all tabulated in Table 5.

Figure 8.— The color-stellar mass diagram for the UV-selected galaxies with
${\log(M_{\ast}/M_{\odot})\geq 10.0}$ and $0.5\leq z\leq 2.5$ (contours).
Black lines show the rough galaxy separation criteria fixing the redshift in
Equation 1 to the median of each redshift bin. The modeled evolution tracks on
the color-mass diagram for different galaxy populations are shown as the lines
of different colors with the filled circles representing the median stellar
masses and dust-corrected UV colors of different subsamples. The solid portion
of each line shows the modeled lifetime of each sample, which is defined as
the period of the track on the color-mass diagram that satisfies the
separation criteria of Equation 1 for each galaxy population.

Given that we use a single initial mass function with a fixed metallicity and
assume a uniform star-formation history, the evolutionary tracks in the color-
mass diagram (Figure 8) are subject to track degeneracy. The most prominent
degeneracy comes from the choices of formation redshift $z_{f}$ and the
exponential declination timescale $\tau$ of the modeled star-formation
history, which can reproduce almost the same track across the median
$M_{\ast}$ and $(U-V)_{\rm rest}-0.47A_{v}$ for each observed galaxy
subsample. Thus we study the choices of $z_{f}$ and $\tau$ that can produce
tracks across each pair of $M_{\ast}$ and $(U-V)_{\rm rest}-0.47A_{v}$ in the
color-mass diagram. We do not consider the variations of initial mass function
and metallicity in this paper. We show our results in Figure 9.

Figure 9.—  The degeneracy of evolutionary track in the color-mass diagram due
to multiple choices of parameter pairs of $z_{f}$ and $\tau$. The left panel
shows the degeneracy of $z_{f}$ and $\tau$ by fixing the normalized stellar
mass to the median $M_{\ast}$ and allowing $(U-V)_{\rm rest}-0.47A_{v}$ to
vary in the step of 0.2 (RS: $[1.2,2.0]$, GV: $[0.8,1.6]$, BC: $[0.0,0.8]$).
Each point of the parameter pair of $z_{f}$ and $\tau$ in the figure
corresponds to an evolutionary track running through the median $M_{\ast}$ and
$(U-V)_{\rm rest}-0.47A_{v}$ at the respective redshift of each subsample. The
points are color coded by their modeled lifetimes according to the color bars
on the right. Error bars show the $1\sigma$ distributions of $\tau$ and
$(U-V)_{\rm rest}-0.47A_{v}$ around their medians for each subsample. The
modeled lifetime of each subsample and the $1\sigma$ uncertainties (see the
text for detail) are labeled as color numbers in each sub-panel. The right
panel shows the degeneracy of $z_{f}$ and $\tau$ by fixing the extinction-
corrected rest-frame UV color to the median $(U-V)_{\rm rest}-0.47A_{v}$ of
each subsample and allowing the stellar mass to vary between
$\log(M_{\ast}/M_{\odot})=10.0$ and $\log(M_{\ast}/M_{\odot})=10.8$. We have
also labeled the parameter pair of $z_{f}$ and $\tau$ corresponding to the
evolutionary track running through the median $M_{\ast}$ and $(U-V)_{\rm
rest}-0.47A_{v}$ of each subsample in each sub-panel, as well as their
$1\sigma$ distributions. Color numbers in each sub-panel show the modeled
lifetimes and their $1\sigma$ uncertainties.

In the left part of Figure 9, we study track degeneracy by fixing the stellar
mass to the median $M_{\ast}$ of each observed subsample selected with
$\log(M_{\ast}/M_{\odot})\geq 10.0$ and allowing rest-frame extinction-
corrected UV color to vary in the step of 0.2. For the modeling of RS
galaxies, the $(U-V)_{\rm rest}-0.47A_{v}$ values vary from 1.2 to 2.0 shown
as five different red curves in each sub-panel of the left part of Figure 9.
For the modeling of GV and BC galaxies, the $(U-V)_{\rm rest}-0.47A_{v}$
values are set to vary within $[0.8,1.6]$ and $[0.0,0.8]$ respectively, also
shown as color-coded curves in the sub-panels. Each point on the color-coded
curves represents a pair of choices of formation redshift $z_{f}$ and the
exponential star-formation declination timescale $\tau$, corresponding to one
of the modeled evolutionary tracks in the color-mass diagram. The points along
the curves are coded with the color gradients according to the color bars of
the SED modeled lifetimes on the right hand side. In each sub-panel, we have
also added the modeled parameter pair of $z_{f}$ and $\tau$ corresponding to
the evolutionary track running across the median $M_{\ast}$ and $(U-V)_{\rm
rest}-0.47A_{v}$ of each subsample, shown as a filled dot with its color coded
as its corresponding lifetime. The horizontal and vertical error bars in each
sub-panel denote the $1\sigma$ uncertainties of the exponential star-formation
declination timescale $\tau$ and extinction-corrected rest-frame UV color
around the median values of each subsample. The numbers annotated in the sub-
panels show the exact modeled lifetimes of subsamples as well as their
uncertainties with respect to the maximum and minimum values of the modeled
lifetimes in the $1\sigma$ parameter box of $\tau$ and $(U-V)_{\rm
rest}-0.47A_{v}$.

Although there are many parameter pairs that may produce almost the same
evolutionary track crossing the median $M_{\ast}$ and $(U-V)_{\rm
rest}-0.47A_{v}$ of each subsample in the color-mass diagram, the estimated
lifetime does not seem to exhibit a large variation within the $1\sigma$ range
of $\tau$ and $(U-V)_{\rm rest}-0.47A_{v}$. The SED modeled lifetimes of RS
galaxies are generally larger than that of BC and GV galaxies at all
redshifts. For GV galaxies, they have similar modeled lifetimes to that of BC
galaxies at higher redshifts ($z>1$). But at lower redshifts ($z\leq 1$), the
modeled lifetimes of GV galaxies increase rapidly, while that of BC galaxies
remain at a low level.

We have also studied track degeneracy by fixing the rest-frame extinction-
corrected UV color to the median $(U-V)_{\rm rest}-0.47A_{v}$ of each
subsample, and allowing the median normalized stellar mass to vary in the step
of 0.2 between $\log(M_{\ast}/M_{\odot})=10.0$ and
$\log(M_{\ast}/M_{\odot})=10.8$. But this time we do not find a clear division
of parameter pairs of $z_{f}$ and $\tau$ between different normalized stellar
masses, as can be seen in the right part of Figure 9. This is because when we
model the color evolution of a galaxy SED, the stellar mass is a flexible
parameter which is independent of the color evolution. The normalization of
stellar mass only depends on the exact value of the median $M_{\ast}$ of each
subsample at the respective redshift. We also label the modeled lifetimes of
each subsample together with their $1\sigma$ uncertainties corresponding to
the maximum and minimum modeled lifetimes within the $1\sigma$ distributions
of $\tau$ and normalized $M_{\ast}$.

Table 5Comparison of the clustering-based Lifetimes with those from the SED
modeling

Galaxy subsample | $\rm{lifetime~{}(clustering)^{a}}$ | $\rm{lifetime~{}(SED~{}modeling)^{b}}$ | $\rm{formation~{}redshift~{}(z_{f})^{c}}$ | $\rm{lifetime~{}(simulation~{}peak)^{d}}$ |  |
---|---|---|---|---|---|---
$0.5\leq z<1.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $12.1^{+5.0}_{-3.6}$ Gyr | 2.6 Gyr | 1.91 | 2.8 Gyr |  |
$\rm GV$ |  $2.7^{+0.8}_{-0.6}$ Gyr | 1.6 Gyr | 1.63 | 1.9 Gyr |  |
$\rm BC$ |  $1.1^{+0.3}_{-0.3}$ Gyr | 0.7 Gyr | 0.95 | 1.1 Gyr |  |
$1.0\leq z<1.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $2.0^{+1.0}_{-0.7}$ Gyr | 1.7 Gyr | 2.71 | 1.8 Gyr |  |
$\rm GV$ |  $1.4^{+0.5}_{-0.4}$ Gyr | 1.1 Gyr | 1.97 | 1.2 Gyr |  |
$\rm BC$ |  $1.3^{+0.3}_{-0.3}$ Gyr | 1.5 Gyr | 1.76 | 1.4 Gyr |  |
$1.5\leq z<2.0$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $1.6^{+1.0}_{-0.6}$ Gyr | 1.4 Gyr | 3.48 | 1.3 Gyr |  |
$\rm GV$ |  $350^{+120}_{-90}$ Myr | 780 Myr | 2.62 | 970 Myr |  |
$\rm BC$ |  $1.3^{+0.3}_{-0.3}$ Gyr | 0.6 Gyr | 1.95 | 0.8 Gyr |  |
$2.0\leq z\leq 2.5$ | $\log(M_{\ast}/M_{\odot})\geq 10$ |
$\rm RS$ |  $10.9^{+12.7}_{-5.7}$ Gyr | 0.9 Gyr | 3.88 | 0.9 Gyr |  |
$\rm GV$ |  $320^{+150}_{-100}$ Myr | 620Myr | 2.94 | 610 Myr |  |
$\rm BC$ |  $1.0^{+0.3}_{-0.3}$ Gyr | 0.7 Gyr | 2.60 | 0.7 Gyr |  |

a The lifetimes of galaxy subsamples with $\rm{log(M_{\ast}/M_{\odot})\geq
10.0}$ estimated through the clustering results and a Tinker et al. (2008)
HMF.
---
b The lifetimes of galaxy subsamples estimated through SED modeling by setting
the key parameters (redshift, stellar mass, dust-corrected rest-frame UV
colors and e-folding star-formation decline timescale $\tau$) to that of the
observed subsamples, and leaving formation redshift $z_{f}$ as a free
parameter.
c The formation redshift of the subsample in SED modeling.
d The peak of lifetime for each galaxy subsample in the Monte-Carlo
simulation.

To make our analyses more systematically, we also perform a Monte-Carlo
simulation to show the lifetime distribution of the SED modeling in comparison
with the clustering-based lifetime. We randomly generate 10,000 data points in
representation of the parameters of 10,000 random galaxies, ensuring that they
have the same redshift ($z$), stellar mass (${M_{\ast}}$), rest-frame dust-
corrected UV color ($\rm{UV_{rest,ext-cor}}$), and exponential star-formation
declination timescale ($\tau$) distributions as each observed galaxy subsample
selected with ${\log~{}(M_{\ast}/M_{\odot})\geq 10.0}$ in the
COSMOS/UltraVISTA field. The distributions of all these parameters for the
random galaxy samples (same as the observed subsamples) are shown in Figure
10. For each random galaxy (having a set of four parameters, $z$,
${M_{\ast}}$, $\rm{UV_{rest,ext-cor}}$, $\tau$), we calculate its evolutional
track in the color-mass diagram (Figure 8) similarly to what we have done
above for each galaxy subsample. The lifetime of each random galaxy is
computed through the track. The peak of the simulated lifetime distribution is
very close to the result of our SED modeling for each galaxy subsample, which
is also tabulated in Table 5. The $1\sigma$ distributions of the lifetimes of
all random galaxies are plotted as the envelops surrounding their peaks in
Figure 11.

Figure 10.— The distributions of the redshift $z$, stellar mass $M_{\ast}$,
rest-frame extinction-corrected UV color $\rm{UV_{rest,ext-cor}}$, and
exponential star-formation decline timescale $\tau$ for galaxies in the
COSMOS/UltraVISTA field. Random galaxies in the Monte-Carlo simulation are
generated based on the same distributions of these galaxy parameters. Figure
11.— The lifetimes estimated through clustering are compared with those from
the SED modeling. The filled circles show the clustering-based lifetimes of
the RS, BC, and GV galaxies with ${\log~{}(M_{\ast}/M_{\odot})\geq 10.0}$ in
different redshift bins, which are the average lifetimes computed with the 6
different HMFs in Section 4.1. Each plus shows the lifetime estimated through
the SED modeling by fixing the key parameters (i.e., redshift, stellar mass,
extinction-corrected rest-frame UV color, and exponential star-formation
decline timescale) to the median values of each respective observed galaxy
subsamples. The color envelops show the $1\sigma$ lifetime distributions from
the Monte-Carlo simulation for all galaxy subsamples, the peaks of which are
very close to the pluses. The black dashed line shows the age of the universe
as a function of redshift.

###  5.2. Comparison of lifetime for different galaxy populations

In general, the clustering-based lifetimes are very close to the results from
the SED modeling as demonstrated in Figure 11. For the RS galaxies, lifetimes
computed through the SED modeling are slightly smaller than that from
clustering. This is because we define the lifetime of RS galaxies as the time
interval since the galaxy crosses the GV region till the lower limit of the
redshift bin of the respective observed subsample. But if we consider red
sequence as a stable phase, the time that the galaxy stays in it can be much
longer than expected if we do not consider other minor physical processes like
rejuvenation.

RS galaxies generally have larger modeled lifetimes than BC and GV galaxies at
all redshifts. The SED modeled lifetimes of GV and BC galaxies are similar at
$z>1$. But at $z\leq 1$ the lifetime of GV galaxies grows rapidly to $\rm\sim
1.6~{}Gyr$, while the lifetime of BC galaxies remains to be $\rm\sim
700~{}Myr$. At $z\geq 1.5$, the clustering-based lifetimes of GV galaxies are
shorter than BC galaxies at the same redshifts, but we do not observe this
trend for their SED modeled lifetimes. This may be due to the heavy degeneracy
of stellar population parameters when constructing their color evolution.
Besides, the intrinsic uncertainties of the parameters also add to the
difficulties in distinguishing the deviation of SED modeled lifetimes between
the two galaxy populations.

If we consider the quenching scenario of star-forming galaxies, the lifetimes
of BC/GV galaxies can be a rough estimate of their typical quenching
timescales. The clustering-based lifetimes in combination with the lifetimes
estimated by the SED modeling have depicted an image of galaxy quenching for
us. In each redshift bin, RS galaxies are a stable already quenched
population, and they have the longest lifetimes. The distinct levels of
lifetime of GV galaxies at $z\leq 1$ and $z>1$ may reflect that the quenching
timescale of this transition galaxy population is different at higher and
lower redshifts. This is consistent with the result in Rowlands et al. (2018),
where GV galaxies are found to follow a distinct quenching route at $z<1$ and
$z\geq 1$, which is defined as the “fast and slow quenching route”. At $z\geq
1$, the GV galaxies quench very fast with a visible timescale of only a few
hundred million years; while at $z<1$, the typical quenching timescale can be
as long as several giga years.

As a comparison, the BC galaxies seem to follow a more constant quenching
timescale independent of redshift, shown by their clustering-based lifetime in
Figure 11. And it seems that the lifetime of BC galaxies modeled with a
composite SED of exponentially declining star formation is slightly below
their clustering-based lifetime at each redshift. This is because we do not
consider the newly born BC galaxies at each redshift bin when modeling the
color evolution of BC galaxies using a single redshifted composite SED. These
newly born BC galaxies can delay the decrease of star formation of the entire
BC population and extend their true lifetimes.

###  5.3. What makes the lifetimes of GV galaxies short at $z\geq 1.5$?

At $z\geq 1.5$, the shortest clustering-based lifetimes of GV galaxies among
all galaxy populations may suggest that some additional physical processes are
likely to join accelerating their quenching, since the SED modeling alone
(only accounting for the effect of star formation) cannot clearly distinguish
their shorter lifetimes from the larger lifetimes of BC galaxies. One of the
possibilities of accelerating quenching in GV galaxies is through the feedback
of their central AGNs. It has been found that AGN feedback has a great
influence on regulating the star formation in galaxies (e.g., Kormendy et al.
2009; Feruglio et al. 2010; Scholtz et al. 2018). Since our subsamples have
removed all the AGNs in the COSMOS/UltraVISTA catalog, it is diffcult to
identify the remaining sources with potential AGN like features.

Nevertheless, some studies argue that morphological compaction may be
necessary to trigger the central engine of AGN feedback (Sales et al. 2010;
Dubois et al. 2013; Diamond-Stanic et al. 2012). If this is the case, a
fraction of the transition galaxies between the star-forming and quiescent
populations should have compact structures, which is indeed discovered by a
number of studies (e.g., Barro et al. 2013, 2016; Kocevski et al. 2017). Some
recent relevant studies also found that quenching in the GV galaxies is likely
to happen more rapidly in compact galaxies than the normal control group
(e.g., Powell et al. 2017; Nogueira-Cavalcante et al. 2019).

We have checked the morphology of our galaxy subsamples. We obtain the
morphology information of our galaxy subsamples by cross-matching the
COSMOS/UltraVISTA galaxies with the 3d-HST morphology catalog provided by
Huertas-Company et al. (2015), who identified H-band morphologies of $\sim
50,000$ galaxies in the five CANDELS fields through deep learning. Then
following the definition in Barro et al. (2013), we classify sources with
$M_{\ast}/r_{e}^{1.5}\geq 10.3$ as “compact” galaxies, where $r_{e}$ is the
half-light effective radius of each galaxy. This separation criterion was
proven to be valid at $1.4<z<3.0$ in Barro et al. (2013). Although the our
subsamples extend to lower redshift bins, it can still be a rough estimate of
the compact fraction of our subsamples. We find that in all redshift bins, the
fractions of compact galaxies are higher in GV galaxies than in BC galaxies.
Of the successfully matched galaxies, the fractions of GV galaxies with
compact morphologies are 1.7%, 15%, 28% at $z\sim 0.8$, $z\sim 1.2$ and $z\sim
1.7$, which are $\sim 3$ times the fractions of compact BC galaxies at these
redshifts respectively. At $z<1.5$, we may have under-estimated the compact
fractions, since lower-redshift galaxies are less compact than they are at
higher redshifts, and the morphology compactness classification criterion
should also be correspondingly scaled down. In the highest redshift bin of
$2.0\leq z\leq 2.5$, the discrepancy between the compact fractions of the two
populations becomes smaller, with 19% being compact GV galaxies and 13% being
compact BC galaxies.

We note that in some other studies AGN affecting star formation can happen in
host galaxies of all morphological types (Zubovas et al. 2013). Some external
processes such as major mergers (e.g., Springel et al. 2005a; Hopkins et al.
2009; Kormendy & Ho 2013) or violent disk fragmentation (e.g., Hopkins et al.
2008; Martig et al. 2009; Behrendt et al. 2016; Bieri et al. 2016) in star-
forming galaxies can also trigger AGN feedback. There are also studies about
the infrared properties of galaxies of different structures, e.g., Chen et al.
(2020), revealing that galaxies of different compactness share similar SFRs
and dust temperatures, and that compact star-forming galaxies tending to be
more quiescent than normal star-forming galaxies can turn out to be a
“selection effect”. Thus, whether these galaxies had undergone a stage of
morphological compaction in galactic scales before AGN feedback is still a
topic under debate.

###  5.4. Comparison between lifetime and typical gas depletion timescale for
different galaxy populations

Figure 12.—  Left: the distribution of sSFR of COSMOS/UltraVISTA galaxies as a
function of redshift. Colors from red to blue depict the density gradient of
galaxies distributed in the parameter plane. The SFRs are calculated as a sum
of contributions from the rest-frame UV and IR fluxes. Dashed curves are the
empirical sSFRs of the main-sequence galaxies derived using the prescription
of Speagle et al. (2014) (Equation 8 in this paper) by fixing the stellar mass
in the equation to the median $M_{\ast}$ of RS, GV and BC galaxies at all
redshifts. Solid lines show the power-law fitting of the sSFRs of our RS, GV
and BC galaxies as a function of redshift. The empirical $\rm sSFR-z$
relations of Speagle et al. (2014) generally lie between the fitting results
of our BC and GV galaxy subsamples at $0.5\leq z\leq 3$. Right: the comparison
between the lifetimes of different galaxy subsamples selected with
$\log(M/M_{\ast})\geq 10.0$ and the typical gas depletion timescales. The red,
green, and blue dashed curves show respectively the typical gas depletion
timescales for the RS, GV, and BC galaxies derived using the script of Tacconi
et al. (2018), with $\rm sSFR(MS,z,M_{\ast})$ set to the empirical relation of
Speagle et al. (2014) (see Section 5.4 for details). The colored shaded/line-
filled regions show their $1\sigma$ ranges adopted as the maximums and
minimums of $t_{\rm depl}$ when sSFRs and $M_{\ast}$ of the subsamples vary
within $1\sigma$.

We also compare the resulting lifetimes with the typical gas depletion
timescales of the RS, GV and BC galaxies calculated with the script of Tacconi
et al. (2018), who presented a fit for the galaxy gas depletion timescale,
which is dependent on redshift, stellar mass, SFR, and effective radius. The
gas depletion timescale was derived as total molecular gas mass divided by
SFR. In their paper, they combined three methods in the measurement of
molecular gas mass, i.e., the integrated CO line luminosity, FIR/submillimeter
dust emission SED, and a single-band measurement of the dust emission at 1mm.
Then given an appropriate assumption of the conversion factor, the molecular
gas mass can be obtained. The SFRs in their paper were estimated based on FIR
emission, MIR emission, and UV-to-NIR SED fitting in the order of decreasing
preference. We adopt their best error-weighted multi-parameter regression gas
depletion timescale, which is dependent on redshift, SFR and stellar mass:

$\begin{split}{\leftline{\hbox{$\log t_{\rm depl}(\rm
Gyr)=0.002-0.37\times\log(1+z)-0.40\times$}}}\\\
{\rm\log(sSFR/sSFR(MS,z,M_{\ast}))+0.17\times(\log~{}M_{\ast}-10.7)},\end{split}$
(7)

where $\rm sSFR(MS,z,M_{\ast})$ is the specific SFR (sSFR) of the main-
sequence galaxies as a function of redshift and stellar mass. Following
Tacconi et al. (2018), we adopt the prescription of the star-formation main
sequence proposed by Speagle et al. (2014) in this study,

$\begin{split}{\rm\leftline{\hbox{$\log(sSFR(MS,z,M_{\ast}))=(-0.16-0.026t_{c})\times$}}}\\\
(\log M_{\ast}+0.025)-(6.51-0.11t_{c})+9,\\\ \&\log
t_{c}=1.143-1.026\log(1+z)-0.599\log^{2}(1+z)\\\
+0.528\log^{3}(1+z).\end{split}$ (8)

In this equation, $t_{c}$ is the cosmic time of a flat $\Lambda$CDM universe
in units of Gyr.

Since the SFRs of COSMOS/UltraVISTA galaxies were derived with independent
data in contrast to the statistical data used in Speagle et al. (2014), it is
of necessity to check whether estimation of $\rm sSFR(MS,z,M_{\ast})$ through
Equation 8 is still valid for our galaxy subsamples. In the left panel of
Figure 12, we compare the $\rm sSFR(MS,z,M_{\ast})$ of Speagle et al. (2014)
prescription with the sSFRs of our COSMOS/UltraVISTA galaxies.

For each RS, GV, or BC galaxy subsample, we calculate its median values of the
stellar mass and SFR respectively. Then power-law fittings are provided to
derive the stellar mass and SFR as a function of redshift for galaxies of a
specific color (i.e., red, green or blue). The fittings of sSFRs as a function
of redshift for three galaxy populations are shown as solid lines with
respective colors in the left panel of Figure 12. The total SFR of each galaxy
is determined via $\rm{SFR_{tot}=SFR_{UV,uncorr}+SFR_{IR}}$, which is
presented in the Muzzin et al. (2013b) catalog, where the ${\rm SFR_{IR}}$ is
derived with the $\rm{24~{}\mu m}$ luminosity using the script of Kennicutt
(1998). Dashed curves in the left panel of Figure 12 show the $\rm
sSFR(MS,z,M_{\ast})$ of Speagle et al. (2014) prescription by fixing the
stellar mass in the equation to the median of our RS/GV/BC subsamples at each
redshift. It is clear that the $\rm sSFR(MS,z,M_{\ast})$ values of Speagle et
al. (2014) prescription generally lie between the median sSFRs of our BC and
GV galaxies at $0.5\leq z\leq 3.0$. Within the stellar mass range our
subsamples covered, $M_{\ast}$ only has minor influence on the resulting $\rm
sSFR(MS)~{}vs.~{}z$ relation for the Speagle et al. (2014) prescription. Thus
we can expect the prescription of Speagle et al. (2014) to be a good estimate
of the sSFRs of the main-sequence galaxies in the COSMOS/UltraVISTA field, and
is reliable in the computation of gas depletion timescales through Equation 7.

In the right panel of Figure 12, we compare the lifetimes of our galaxy
subsamples with their typical gas depletion timescales. The typical gas
depletion timescales of the RS, GV, and BC galaxies as a function of redshift
are shown as the color-coded dashed curves with their $1\sigma$ uncertainties
shown in shaded/line-filled regions. The boundary of the $1\sigma$ region of
galaxies of a specific color is set as the maximum and minimum of the $t_{\rm
depl}$ at each redshift when their sSFR and $M_{\ast}$ vary within $1\sigma$.

The gas depletion timescales of the RS and BC galaxies overlap with the
regions where the lifetimes (no matter clustering based or SED modeled ones)
of different populations lie in. The stratification of gas depletion timescale
with galaxy color is also in agreement with the separation of lifetimes
between quiescent (red) and star-forming (blue/green) population. It means
that gas consumption by pure star formation is very possible to be the major
cause of the transformation of galaxy color.

But we note that for the GV galaxies, the redshift evolution of the gas
depletion timescale could not reflect the distinct behavior of their lifetime
at higher and lower redshifts, as the gas depletion timescale of the GV
galaxies computed through the script of Tacconi et al. (2018) always lies
between that of the RS and BC galaxies at whatever redshifts. This incongruity
can be tuned if additional physical processes other than pure star formation
would help accelerate the gas depletion of the GV galaxies, making their
typical quenching timescale shorter than that of typical BC galaxies at high
redshifts ($z>1.5$). One natural possibility is AGN feedback. The jets
launched near the central supermassive black hole may quickly blow away the
cold gas within the GV galaxies and stop the star formation in a rather short
timescale (Bower et al. 2006; Cicone et al. 2014; Schawinski et al. 2014;
Powell et al. 2017; Nogueira-Cavalcante et al. 2019).

###  5.5. Reproducing the redshift distribution of UV-selected galaxies

Figure 13.— Left: The redshift distributions of the RS, BC, and GV galaxies in
the COSMOS/UltraVISTA field. The histograms show the redshift distributions of
the observed RS, BC, and GV galaxies, whose redshifts are directly taken from
the photometric-redshift catalog of Muzzin et al. (2013b). Upper panels show
the redshift distributions of RS, GV and BC galaxies in the COSMOS/UltraVISTA
field selected with $\log(M_{\ast}/M_{\odot})\geq 10.0$. Lower panels are the
same, but for the galaxies selected with $\log(M_{\ast}/M_{\odot})\geq 10.5$.
The curves in all panels show the estimated redshift distributions of the RS,
BC, and GV galaxies through Equation 9 assuming the Tinker et al. (2008) HMF.
For the solid curves, we use a power-law to fit the lifetimes of the RS, BC,
and GV samples, respectively, to derive $t_{\rm sample}$; while for the dashed
curves, we adopt the typical gas depletion timescales from Tacconi et al.
(2018) as $t_{\rm sample}$ (i.e., the color dashed curves in the right part of
Figure 12 and also shown in the right part of this figure). Right: The optimal
parameters of $t_{\rm sample}$ in Equation 9 to reproduce the redshift
distributions of RS, BC and GV galaxies (colored solid curves), and their
comparison with the clustering-based lifetimes (filled dots with error bars)
and the typical gas depletion timescales from the script of Tacconi et al.
(2018) (colored dashed curves with $1\sigma$ uncertainties).

Finally, we try to reproduce the photometric-redshift distributions of the RS,
BC, and GV galaxies in the COSMOS/UltraVISTA field (Muzzin et al. 2013b) based
on the HMF and the estimated lifetime of each subsample. In the Hickox et al.
(2009) scenario, the activities of active galaxies and QSOs are related to the
host DM halos. When the DM halo grows and reaches a certain mass ${M_{\rm
halo}=M_{\rm thresh}}$, the activity of the host galaxy will be triggered (see
Figure 16 of Hickox et al. 2009 for a schematic illustration of this picture).
Thus, the differential number of halos crossing the mass threshold ${M_{\rm
thresh}}$ as a function of redshift can be modeled as

$\frac{dN_{\rm thresh}}{dz}\propto\frac{dn_{\rm halo}}{dM_{\rm h}}(M_{\rm
thresh},z)\dot{M}_{\rm halo}(M_{\rm thresh},z)t_{\rm sample}\frac{dV}{dz},$
(9)

where $\frac{dn_{\rm halo}}{dM_{\rm h}}$ is the number density of DM halos per
unit halo mass computed using the Tinker et al. (2008) HMF, $\dot{M}_{\rm
halo}$ is the typical growth rate of DM halos (Fakhouri et al. 2010), $t_{\rm
sample}$ is the lifetime of the RS, GV, or BC galaxies as a function of
redshift and $\frac{dV}{dz}$ is the differential comoving volume over the
survey area, respectively. We derive $t_{\rm sample}$ by fitting a single
power-law to the lifetimes of RS, GV, and BC galaxies computed with the Tinker
et al. (2008) HMF in four redshift bins, respectively. Data points as well as
parts of the fitting results exceeding the age of the universe at a given
redshift (above the black dashed line in Figure 11) will be scaled down to the
age of the universe at the respective redshift. If a specific phase (RS, GV,
or BC) is triggered in a galaxy once the halo of the galaxy grows and reaches
the halo mass of the respective phase at a given redshift, then the observed
number density of galaxies in that phase will be expected to be proportional
to $\frac{dN_{\rm thresh}}{dz}$. Using appropriate normalizations, we can
reproduce the redshift distributions of the RS, BC, and GV galaxies and
compare with the observations. Hickox et al. (2012) also utilized this method
to estimate the redshift distribution of sub-millimeter galaxies in their
Section 4.5. We present our results in the left part of Figure 13.

To make a comparison, we also substitute the lifetimes of the galaxy subsample
$t_{\rm sample}$ by the typical gas depletion timescales of the RS, BC, and GV
galaxies from Tacconi et al. (2018) (the color dashed curves in Figure 11 and
the right part of Figure 13). For the RS and BC galaxies, applying both the
fitted lifetime and gas depletion timescale in Equation 9 can reproduce their
redshift distributions well. But for the GV galaxies, only applying the
lifetime as $t_{\rm sample}$ can have a good reproduction of the redshift
distributions. This is especially obvious in the stellar-mass bin of
${\log~{}(M_{\ast}/M_{\odot})\geq 10.0}$, where we have a larger amount of
galaxies. The substitution of the lifetime of GV galaxies with their typical
gas depletion timescale is very likely to overestimate the number of the GV
galaxies at $z\sim 2$.

For a better visual identification of the difference between the clustering-
based lifetimes and typical gas depletion timescales and to evaluate their
performance on reproducing the redshift distributions for each galaxy
population, we display the optimal parameters of $t_{\rm sample}$ (colored
solid curves) in the right part of Figure 13, corresponding to the parameters
reproducing the exact redshift distributions (histograms in the left part of
Figure 13) of COSMOS/UltraVISTA galaxies in the Muzzin et al. (2013b) catalog.
The reproduced redshift distributions for each galaxy population are properly
normalized, such that the tracks of the parameters of $t_{\rm sample}$ can be
scaled up/down appropriately to best-fit their measured clustering-based
lifetimes.

It is clear that the optimal parameters reproducing the redshift histograms
are very close to the clustering-based lifetimes for each galaxy population.
Both of them show distinctive redshift evolution compared with the typical gas
depletion timescales (dashed curves in the right part of Figure 13). For RS
and BC galaxies, the evolutionary differences between the optimal parameters
and the typical gas depletion timescales are insignificant over the redshift
range we consider. But for GV galaxies, the optimal parameters reproducing
redshift histograms show great variations as redshift evolves, being well
consistent with the measured clustering-based lifetimes, but completely
different from the nearly constant gas depletion timescales computed through
the script of Tacconi et al. (2018).

Interestingly, the optimal parameters reproducing the redshift distribution
histogram of GV galaxies show minimums at the redshift of $z\sim 1-2$, which
is considered to be the peak of the global cosmic star formation and the peak
of the global central black hole accretion (Madau & Dickinson 2014; Heckman &
Best 2014; Delvecchio et al. 2014; Sijacki et al. 2015; Ma & Yan 2015;
Zakamska et al. 2016). This may be an implication that AGN feedback activities
help accelerate quenching in these galaxies, making their lifetimes shorter
than expected from the gas consumption by pure star formation. The scenario
that black hole feedback suppressing galaxy star formation at high masses is
supported by a number of semi-analytic models and numerical hydrodynamic
simulations (e.g., Dubois et al. 2014; Vogelsberger et al. 2014; Somerville &
Davé 2015; Schaye et al. 2015; Khandai et al. 2015). Since most of these
simulations are implemented with phenomenological models, further observations
are needed to confirm this black hole negative feedback effect on the star
formation of host galaxies. Mini simulations on multi scales with precise
physical assumptions could also help bridge the gap between stellar and
cosmological scales (McCarthy et al. 2010; Faucher-Giguère & Quataert 2012;
Torrey et al. 2014; Lacey et al. 2016; Weinberger et al. 2017; Bower et al.
2017). Observationally, debates about whether gas outflows driven by central
black hole trigger or suppress galaxy star formation still exist (Page et al.
2012; Walch et al. 2012; Silk 2013; Zubovas et al. 2013; Maiolino et al.
2017). Thus, it is currently difficult to make a firm conclusion that BH
feedback would inevitably result in the suppression of galaxy star formation.

Nevertheless, if this scenario is indeed the case, the inconsistency between
using clustering-based lifetime and typical gas depletion timescale to
reproduce galaxy redshift distribution can be fully reconciled, since pure
star formation is thus not the only way to consume the gas content within a GV
galaxy at $z\sim 2$. Other additional physical processes should also be
considered. By effectively shortening the gas depletion timescales in these
galaxies at $z\sim 2$ through shocks or gas outflows driven by their central
black holes, it is expected that the redshift distribution of the GV galaxies
can be better reproduced.

##  6\. Summary

In this paper, we estimate the lifetimes of UV-selected galaxies in the
COSMOS/UltraVISTA field at $0.5\leq z\leq 2.5$. We first separate the RS, BC,
and GV galaxies according to the color-separation criteria proposed by Wang et
al. (2017). Then based on the clustering results presented in Lin et al.
(2019), we estimate the lifetimes of these three galaxy populations at
different redshifts.

We explore multiple factors that can influence the resultant lifetimes,
including the choices of different HMFs, the width of redshift bins, the
growth of DM halo, and the stellar mass of the sample. We conclude that
applying different HMFs can slightly influence the lifetimes by $\rm{\approx
0.2~{}dex}$, $\Delta z=0.5$ is large enough to estimate the lifetimes, and
considering the growth of DM halo within each redshift bin has little effect
on lifetime estimation. We also find that the lifetimes of all galaxy
subsamples generally decrease with the increasing stellar mass, but this trend
becomes weaker as the redshift increases.

The lifetimes of galaxy samples agree well with those from the SED modeling.
The RS galaxies have the longest lifetime compared to those of the BC and GV
galaxies at all redshifts, indicating that the red sequence is likely a stable
phase. The BC galaxies have a nearly constant lifetime at different redshifts,
which is around $\rm{1~{}Gyr}$. And the lifetime of the GV galaxies seems to
be strongly dependent on redshift. At lower redshifts, the GV galaxies have
longer lifetime than that of the BC galaxies, but this trend goes to the
opposite when we focus on higher redshift bins, where the lifetime of the GV
galaxies can be as short as several hundred million years. We also compare our
result with the gas depletion timescale inferred from Tacconi et al. (2018).
For the RS and BC galaxies, the lifetimes generally agree well with the
respective gas depletion timescales, but for the GV galaxies, the gas
depletion purely by star formation cannot fully explain the huge difference of
their lifetimes at higher and lower redshifts. We conclude that the study on
the lifetime reflects the quenching scenario of SFGs. At high redshifts,
additional processes, such as AGN feedback, are required to accelerate the
quenching of the GV galaxies.

Finally, we try to reproduce the redshift distributions of the UV-selected
galaxies in the COSMOS/UltraVISTA field. Based on the Tinker et al. (2008)
HMF, the typical DM halo growth rate by Fakhouri et al. (2010), and the
clustering-based lifetimes inferred in this paper, we can well reproduce the
redshift distributions of the RS, BC, and GV galaxies. We also try to
reproduce the redshift distributions of galaxies by substituting the
clustering-based lifetimes with the SFR-dependent typical gas depletion
timescales derived from Tacconi et al. (2018). Again, we find that we can well
reproduce the redshift distributions of the RS and BC galaxies, but for the GV
galaxies, the number density at $z\sim 2$ seems to be overestimated. For GV
galaxies, the optimal parameters reproducing redshift distributions are very
close to the measured clustering-based lifetimes, but different from the
nearly constant gas depletion timescales over redshifts. And we stress that,
this disagreement can be reconciled if other additional physical processes
rather than star formation, such as AGN feedback, aid depleting gas in the GV
galaxies.

##  7\. acknowledgments

We acknowledge the constructive work by Adam Muzzin presenting the catalog in
the COSMOS/UltraVISTA field. This work is supported by the National Natural
Science Foundation of China (Nos. 11673004, 23002601, 1320101002, 11433005,
12025303, 11890693, and 11421303). X.Z.L. and Y.Q.X. acknowledge support from
the CAS Frontier Science Key Research Program (QYZDJ-SSW-SLH006) and the K.C.
Wong Education Foundation. G.W.F. acknowledges the support from Yunnan young
and middle-aged academic and technical leaders reserve talent program (No.
201905C160039), Yunnan ten thousand talent program-young top-notch talent and
Yunnan Applied Basic Research Projects (2019FB007).

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