# Star formation efficiency across large-scale galactic environments

Laya Ghodsi1,2, Allison Man1, Darko Donevski3,4, Romeel Davé5,6,7, Seunghwan
Lim8, Christopher C. Lovell 9,10, Desika Narayanan 11,12

1 Department of Physics & Astronomy, University of British Columbia, 6224
Agricultural Road, Vancouver BC, V6T 1Z1, Canada
2 ESO, Karl Schwarzschild strasse 2, 85748 Garching, Germany
3 National Centre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland
4 SISSA, Via Bonomea 265, 34136 Trieste, Italy
5 Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh, EH9 3HJ, United Kingdom
6University of the Western Cape, Bellville, Cape Town 7535, South Africa
7South African Astronomical Observatories, Observatory, Cape Town 7925, South
Africa
8 Canadian Institute for Theoretical Astrophysics, University of Toronto, 60
St. George St., Toronto, ON M5S 3H4, Canada
9 Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby
Road, Portsmouth, PO1 3FX, United Kingdom
10Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, United
Kingdom
11Department of Astronomy, University of Florida, 211 Bryant Space Sciences
Center, Gainesville, FL 32611 USA
12Cosmic Dawn Center (DAWN), Niels Bohr Institute, University of Copenhagen,
Jagtvej 128, København N, DK-2200, Denmark E-mail<EMAIL_ADDRESS>
###### Abstract

Environmental effects on the formation and evolution of galaxies have been one
of the leading questions in galaxy studies during the past few decades. In
this work, we investigate the relationship between the star formation activity
of galaxies and their environmental matter density using the cosmological
hydrodynamic simulation Simba. The galactic star formation activity indicators
that we explore include the star formation efficiency (SFE), specific star
formation rate (sSFR) and molecular hydrogen mass fraction
($f^{*}_{\mathrm{H_{2}}}$) and the environment is considered as the large
scale environmental matter density, calculated based on the stellar mass of
nearby galaxies on a 1 Mpc/h grid using the cloud in cell (CIC) method. Our
sample includes galaxies with $9<\log\frac{M_{*}}{M_{\odot}}$ at $0<z<4$,
divided into three stellar mass bins to disentangle the effects of stellar
mass and environment on the star formation activity of galaxies. For low- to
intermediate-mass galaxies at low-redshifts ($z<1.5$), we find that the star
formation efficiency of those in high-density regions are $\sim 0.3$ dex lower
than those in low-density regions. However, there is no significant
environmental dependence of the star formation efficiency for massive galaxies
over all our redshift range, and low- to intermediate-mass galaxies at high
redshifts ($z>1.5$). We present a scaling relation for the depletion time of
cold molecular hydrogen (${\mathrm{t_{depl}}}=1/\mathrm{SFE}$) as a function
of galaxy parameters including environmental density. Our findings provide a
framework for quantifying the environmental effects on the star formation
activities of galaxies as a function of stellar mass and redshift. The most
significant environmental dependence is seen at later cosmic times ($z<1.5$)
and towards lower stellar masses ($9<\log\frac{M_{*}}{M_{\odot}}<10$). Future
large galaxy surveys can use this framework to look for the environmental
dependence of the star formation activity and examine our predictions.

###### keywords:

Galaxy formation and evolution, Star formation, Molecular gas, Depletion time,
Hydrodynamic simulation

††pagerange: Star formation efficiency across large-scale galactic
environments–A

##  1 Introduction

It is well established that the environment of galaxies affects their
properties including colour (Balogh et al., 2004; Bamford et al., 2009),
morphology (Dressler, 1980; Goto et al., 2003; Skibba et al., 2009) and star
formation rate (SFR; Kauffmann et al., 2004; Peng et al., 2010; Woo et al.,
2013; Old et al., 2020). These works show that in the low-redshift universe
($z<1$), galaxies in dense regions tend to be redder, more elliptical, and
less star-forming. In contrast to the low-redshift universe, there is no
consensus in the literature about the relationship between SFR and the
environment at high redshift ($z>1$). Some studies claim that at high
redshift, galaxies in dense regions have higher SFRs than galaxies in low-
density regions, on average (Elbaz et al., 2007; Cooper et al., 2008; Santos
et al., 2014, 2015; Shimakawa et al., 2018). This suggested change in the
relationship between SFR and the environment at roughly $z\sim 1$ is called
the reversal of the SFR-density relation. This trend partially arises from the
virialization process of galaxy clusters. The overdense regions in the low-
redshift universe are mostly virialized galaxy clusters subject to physical
processes including starvation and ram pressure stripping that can reduce the
star formation activity of the member galaxies. Whereas, the overdense regions
in the higher-redshift universe are mostly non-virialized protoclusters and
have large intergalactic gas reservoirs to fuel the star formation in
galaxies. However, other observational and theoretical studies find no
evidence for a relationship between SFR and the environment of galaxies
(Scoville et al., 2013; Darvish et al., 2016; Duivenvoorden et al., 2016;
Lovell et al., 2021) or find the same relationship as the low-redshift
universe (Patel et al., 2009). Observational studies on this topic suffer from
selection biases due to observational limits that restrict the sample redshift
and stellar mass ranges and different methods to define and measure
environments and galaxy properties (Muldrew et al., 2012, 2015; Lovell et al.,
2018). Exploring this research question using hydrodynamic simulations helps
in understanding the background physics of galaxy formation and evolution
producing the observed trends (Yajima et al., 2022). Although simulations also
suffer from known limitations, they have been able to predict many
observational trends. For instance, Tonnesen & Cen (2014) and Hwang et al.
(2019) have been able to reproduce the reversal of SFR-density relation using
hydrodynamic simulations up to redshifts $z=1$ and $z=2$, respectively.

Cold molecular gas, mainly consisting of molecular hydrogen, is the star
formation fuel in galaxies. Studying the effects of the environment on the
molecular hydrogen content is thus essential to dissect the role of the
environment on the star formation activity of galaxies. Observational studies
find different trends for the role of the environment on galactic molecular
gas content. At low-redshift, a few studies find no correlation between the
environment and the molecular gas content of galaxies (Kenney & Young, 1989;
Lavezzi & Dickey, 1998; Koyama et al., 2017). However, some works report that
molecular gas content increases with the environmental density (Mok et al.,
2016) while others report that molecular gas content decreases with
environmental density (Fumagalli et al., 2009; Scott et al., 2013; Boselli et
al., 2014). Molecular gas observations for high-redshift galaxies ($z>1.5$)
are currently quite limited in sample sizes and affected by selection biases.
Nevertheless, Some works at this redshift range find higher gas fractions for
cluster galaxies than the field galaxies (Noble et al., 2017; Hayashi et al.,
2018), while other works find no environmental dependence for gas fraction
(Lee et al., 2017; Tadaki et al., 2019).

Since SFR and molecular hydrogen content, two tracers for galactic star
formation activity, are correlated, it is useful to define another parameter
that relates these two parameters to trace the variations of SFR and molecular
hydrogen mass at the same time. Star formation efficiency (SFE) is typically
defined as the SFR per molecular hydrogen mass ($\mathrm{SFE\equiv
SFR/M_{H_{2}}}$; Young et al. 1996; Boselli et al. 2001). Similarly, depletion
time ($\mathrm{t_{depl}=1/SFE=M_{H_{2}}/SFR}$) is the timescale during which
the galaxy converts all of its molecular hydrogen into new stars at the
current star formation rate. A fall in SFR could be due to a lower molecular
hydrogen mass supply, and/or less efficient star formation. It is thus
important to monitor these three galactic star formation activity parameters
together to gain a comprehensive understanding of galaxy evolution (Scoville
et al., 2017; Lu et al., 2022).

Previous studies have tried to quantify the SFE and gas depletion time as
functions of other galaxy properties. Scoville et al. (2017) quantified the
gas depletion time as a function of redshift, star formation rate, and stellar
mass. Similarly, Tacconi et al. (2018) introduced a scaling relation for the
depletion time as a function of redshift, star formation rate, stellar mass,
and the radius of galaxies. These works, however, lack the environmental
effects on galactic star formation activity. Darvish et al. (2018b) includes a
number of environmental indicators to the Scoville et al. (2017) scaling
relation, but they do not find an environmental dependence of the gas
depletion time in their sample of galaxies with
$\log\frac{M_{*}}{M_{\odot}}>10$ at $0.5<z<3.5$. It is worth mentioning that
most of the known scaling relations are for star-forming galaxies in the
field, without explicit measurements of the environment they reside in
(Tacconi et al., 2018; Liu et al., 2019). Systematic investigations of the
star formation efficiency of galaxies versus the environment remain sparse to
date.

To derive more complete knowledge of galaxy gas properties, we must also
investigate whether galaxy large-scale environment regulates the link between
the gas and star formation. The environment of galaxies is defined in a
variety of ways based on the available data set and the goal of different
studies. On large scales, the matter content in the Universe is distributed in
a web-like structure, called the cosmic web. This structure consists of dense
massive nodes, long filaments connecting the nodes, vast thin walls, and huge
low-density regions called voids. Some works define the environment of
galaxies as the cosmic web component in which the galaxy resides (Hahn et al.,
2007; Moorman et al., 2016; Xu et al., 2020). However, the complexity of this
structure and the wide range of different physical definitions for the
components of the cosmic web makes it complicated to study the environmental
effects on galaxies using this environment tracer. In addition, membership in
a galaxy cluster or field is another definition for galactic environment
(Vulcani et al., 2013; Zavala et al., 2019; Lemaux et al., 2020). The result
of studies using this method is a strong function of the definition of
cluster/field galaxies for which there is not a consensus in the community. At
high redshift ($z>2$), this environment measure is even more tricky given the
lack of massive virialized galaxy clusters. Moreover, this dichotomy does not
cover the full dynamic range of different galactic environments. The
environmental density of galaxies is another measure of the environment. As
summarized in Muldrew et al. (2012) and Etherington & Thomas (2015), two
commonly used methods in the literature to measure environmental density
include: (i) the density within a volume around the galaxy determined by the
distance between the galaxy and its $\mathrm{N^{th}}$ nearest neighbour, where
N typically ranges from 5 to 10 (Bamford et al., 2009; Ellison et al., 2010;
Wu, 2020). (ii) the density within a cell, sphere, or cylinder around the
galaxy determined by a fixed aperture (Berrier et al., 2011; Smolčić et al.,
2017). These methods are computationally fast to perform. The results of these
methods might be functions of $\mathrm{N}$ for the first method and the fixed
aperture size for the second method. The nearest neighbour-based methods
probes the internal trends of massive halos better, while the fixed aperture-
based method is a better probe for scales larger than individual halos
(Muldrew et al., 2012). Variations of the last two methods use two/three-
dimensional measures and number/luminosity/mass density which makes the
comparison of different studies on this topic even more complicated and
tricky. Moreover, the peculiar motion of galaxies along the line of sight,
leading to the Kaiser effect (finger of god), makes it complicated to measure
the exact densities (Kaiser, 1987). The reader can refer to Haas et al.
(2012), Muldrew et al. (2012), and Darvish et al. (2015) for a more detailed
discussion about different environment indicators.

In this study, we use the environmental density as an environment indicator
calculated considering the stellar mass of neighbour galaxies on a 1
$\mathrm{h^{-1}Mpc}$ grid. An advantage of using this method over using the
number density of galaxies as an environment measure is that the stellar mass
density in the vicinity of a galaxy can be directly linked to the luminosity
density of neighbour galaxies on the same scale which is measurable from
observations. Moreover, stellar mass density captures the underlying matter
distribution better than number density which is susceptible to stochastic
effects. Furthermore, in both observations and simulations, the uncertainties
in mass density are smaller than that in number density since small galaxies
that are difficult to detect in data or resolve in simulations are large in
number but small in mass (Tonnesen & Cen, 2014).

Although observational capabilities have significantly advanced recently,
their ability to detect faint galaxies is still limited by the sensitivity of
the instrument. Hence, observed galaxy catalogues, specifically at high
redshift, are not complete and suffer from selection biases. This limitation,
in addition to various approaches used to measure galactic properties from
observations, makes it difficult to capture bias-free trends in galaxy
properties. Using simulations is a fruitful method to model galaxies
theoretically, avoid observational selection biases, and explore the physical
processes that drive the observed trends. In this work, using a set of
simulated galaxies from the Simba hydrodynamic cosmological simulation (Davé
et al., 2019), we aim to shed new light on a question: Does the environment of
galaxies affect their star formation activity? We investigate the galactic
star formation activity parameters, including SFE, sSFR, molecular hydrogen
mass fraction, and depletion time, as functions of redshift, stellar mass, and
environmental density. The data set used spans a relatively wide redshift
range ($0<z<4$), wide stellar mass range
($9<\log{\frac{M_{*}}{M_{\odot}}}<13$), wide environmental density range
($-4<\log{[1+\delta_{\mathrm{gal}}^{*}]}<2$), and a large number of galaxies
($\sim$ 330,000 galaxies). The environmental density is calculated using the
stellar mass density of neighbour galaxies on a 1 Mpc/h grid, calculated using
the Cloud in Cell (CIC) method.

This paper is structured as follows: section 2 presents the properties of the
Simba simulation and the data we use. In section 3, we explain our methods to
calculate the environmental density and error propagation. In section 4 we
present our results. Section 5 discusses the results and possible
interpretations. Finally, section 6 summarises the main findings of the work.

##  2 The Simba simulation

In this work, we study the environmental dependence of the star formation
activity of galaxies in Simba, a state-of-art cosmological hydrodynamic
simulation. In this section, we briefly explain the main methods and physical
prescriptions used to derive the galaxy properties in this simulation.

Simba (Davé et al., 2019) is a suite of galaxy formation and evolution
simulations in a cosmological context built on the Mufasa project (Davé et
al., 2016) with updated physics for the modelling of black hole growth and
feedback. Simba uses the meshless finite mass version of the GIZMO code
(Hopkins, 2015), assuming the Planck Collaboration et al. (2016) cosmological
parameters of $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$, $\Omega_{b}=0.048$,
$H_{0}=68\ \mathrm{kms^{-1}\ Mpc^{-1}}$, $\sigma_{8}=0.82$, $n_{s}=0.97$, and
the star formation model of Schmidt (1959). This simulation explores the
evolution of galaxies, black holes, and galactic gas over a wide redshift
range. The molecular hydrogen content in Simba is calculated on the fly using
the prescription provided in Krumholz & Gnedin (2011) based on the local gas
metallicity and column density as follows:

$f_{H_{2}}=1-0.75\frac{s}{1+0.25s}$ (1)

Where $f_{H_{2}}$ is the molecular hydrogen mass fraction (mass of molecular
hydrogen over the total gas mass of each gas cell) for each gas cell and $s$
is defined as

$s=\frac{\ln{(1+0.6\chi+0.01\chi^{2})}}{0.0396Z(\Sigma/M_{\odot}pc^{-2})}$ (2)

Where $Z$ is the metallicity in solar units. The gas column density ($\Sigma$)
needs to be estimated using the Sobolev (Sobolev, 1960) approximation as
$\Sigma=\rho h$ where $\rho$ is the mass density of gas and $h$ is the local
density scale height in each gas cell in the simulation box, to be computed
using $h=\rho/|\nabla\rho|$. In this equation, $\chi$ is a function of
metallicity, defined as

$\chi\approx 3.1(\frac{1+3.1Z^{0.365}}{4.1})$ (3)

Having the molecular hydrogen mass fraction from the above equations, one can
calculate the number density of molecular hydrogen $n_{H_{2}}$ in each gas
cell.

Stars only form in gas cells with $n_{H}>0.13\ \mathrm{cm^{-3}}$. The star
formation rate (SFR) in each star-forming gas cell is calculated using the
molecular hydrogen mass fraction $f_{H_{2}}$, gas mass density $\rho$, and
dynamical timescale $t_{\mathrm{dyn}}$ with the Schmidt (1959) star formation
model as $SFR=\epsilon_{*}f_{H_{2}}\rho/t_{\mathrm{dyn}}$, assuming the star
formation efficiency per free-fall time to be $\epsilon_{*}=0.02$ (Robert C.
Kennicutt, 1998). Galactic SFR values are instantaneous and calculated by
summing over gas cells. These values are also consistent with the SFRs
calculated using young stellar particles, averaged over tens of Myr. Simba
finds galaxies by applying a 6-dimensional Friends-of-Friends (6DFoF) galaxy
finder on all stars and gas cells with $n_{H}>0.13\ \mathrm{cm^{-3}}$. Halos
are found using a 3DFoF method applied on dark matter particles with a linking
length set to 0.2 times the mean interparticle spacing.

Simba models the star formation-driven galactic winds as decoupled two-phase
winds with $30\%$ of the particles consisting of hot particles. The scaling
relation of the mass loading factor with stellar mass comes from Anglés-
Alcázar et al. (2017b) using the Fire simulation (Hopkins et al., 2014). One
of the main improvements of Simba compared to Mufasa is that black holes are
seeded and grown on the fly and the feedback from black hole accretion
contributes to galaxy quenching. Simba has two black hole accretion modes;
torque-limited accretion for cold gas within the black hole ($T<10^{5}K$)
using the model of Anglés-Alcázar et al. (2017a) based on Hopkins & Quataert
(2011), and Bondi accretion (Bondi, 1952) for hot gas ($T>10^{5}K$). The
active galactic nuclei (AGN) feedback in Simba includes X-ray energy feedback
based on the model of Choi et al. (2012) and a kinetic subgrid model
consisting of radiative and jet modes. The radiative mode is used for high
Eddington ratios
($f_{\mathrm{Edd}}=\dot{M}_{\mathrm{BH}}/\dot{M}_{\mathrm{Edd}}>0.2$) when
multi-phase winds of molecular and warm ionized gas flow from AGNs with a
speed of roughly $10^{3}\mathrm{\ km\ s^{-1}}$. The jet mode feedback begins
to be implemented at lower Eddington ratios ($f_{\mathrm{Edd}}<0.2$) and only
becomes dominant at $f_{\mathrm{Edd}}<0.02$. In this feedback mode, AGNs
produce collimated jets of hot gas with a speed of roughly $10^{4}\mathrm{\
km\ s^{-1}}$. Additionally, these jets only exist in early-type galaxies with
black hole mass of $\mathrm{M_{BH}>M_{BH,lim}=10^{7.5}M_{\odot}}$ to be
consistent with observations. The velocities and temperatures of the bipolar
kinetic feedback model are mostly taken from observation to reproduce the
observed energy release on larger scales of tens of kpc (Davé et al., 2019).
The jet mode AGN feedback has been shown to be the main mechanism responsible
for galaxy quenching in Simba. The X-ray mode feedback is the complementary
process to fully quench galaxies. However, the radiative mode does not
contribute to galaxy quenching (Davé et al., 2019).

Simba considers the galaxy stellar mass function (GSMF) as a test for its
feedback model since the GSMF is well measured over a reasonable redshift
range (Davé et al., 2019). Assuming the observed GSMF from Bernardi et al.
(2017), the combined CANDELS and zFOURGE data from Tomczak et al. (2014), and
CANDELS data from Song et al. (2016), Simba has a GSMF in good agreement with
these observations with the exception of a subtle overproduction of massive
galaxies at $z<2$. This inconsistency of the GSMF of massive galaxies between
Simba and observations could be due to a number of observational biases or
numeric uncertainties in Simba (Davé et al., 2019): first, the massive end of
GSMF could be underestimated in observations, which arises from missing
massive dusty galaxies in the considered rest-frame optical surveys. Second,
the most massive galaxies have bottom-heavy initial mass functions (IMF with a
larger ratio of low-mass stars with $M<M_{\odot}$ compared to Milky Way-like
galaxies; Conroy et al. 2013). This may cause an underestimation of the
stellar masses of most massive galaxies in observations. Third, the most
massive simulated galactic systems usually have extended envelopes of stars
and satellites around them that might be over-merged into the main central
part of the galaxy because of poor resolution. As a result, the stellar mass
of a fraction of these galaxies might be overestimated in simulations.
Moreover, the Simba run used in this work with a box length of $100\
\mathrm{h^{-1}Mpc}$ produces a halo with a virial mass of
$M_{\mathrm{halo}}=1.16\times 10^{15}M_{\odot}$ by $z=0$ as its largest halo.
However, this halo is $50\%$ more massive than expected for this volume. This
may contribute to the overproduction of massive galaxies in Simba. If galaxies
in Simba are found using a fixed aperture of 30 kpc (similar to the EAGLE
simulation; Schaye et al. 2015), the GSMF would be in significantly better
agreement with observations.

In this work, we use 22 snapshots of the Simba full run simulation (m100n1024)
over the redshift range of $0<z<4$. This simulation has been run in a box of
$100\ \mathrm{h^{-1}Mpc}$ in length with $1024^{3}$ gas particles and
$1024^{3}$ dark matter particles. The minimum gravitational softening length
is $0.5\ \mathrm{h^{-1}kpc}$. In this run, the gas particle mass, the dark
matter particle mass, and the minimum stellar mass of a resolved galaxy are
$1.82\times 10^{7}M_{\odot}$, $9.6\times 10^{7}M_{\odot}$, and $5.8\times
10^{8}M_{\odot}$ respectively. We apply a lower stellar mass limit of
$M_{*}>10^{9}M_{\odot}$ on our galaxy sample at all analyzed redshifts, which
is the same as the lower mass limit of the sample that Tacconi et al. (2018)
uses to find scaling relations between galaxy molecular gas masses, stellar
masses, and star formation rates. Assuming this limit, we can compare our
results with this scaling relation.

##  3 Methods

In this section, we explain the methods used to investigate the effect of the
environment on the star formation activity of galaxies. To achieve our goal,
we study a number of important parameters in the star formation process of
galaxies:

  * •

Specific star formation rate ($sSFR=SFR/M_{*}$, unit [$\mathrm{yr^{-1}}$]):
The rate of forming new stars per stellar mass.

  * •

Molecular hydrogen fraction
($f^{*}_{\mathrm{H_{2}}}=M_{\mathrm{H_{2}}}/M_{*}$): The mass fraction of the
molecular hydrogen compared to stellar mass.

  * •

Star formation efficiency (SFE, unit [$\mathrm{yr^{-1}}$]): The efficiency of
producing new stars in a galaxy, also the inverse of the time a galaxy takes
to consume all of its molecular gas to form new stars ($\mathrm{t_{depl}}$):

$\mathrm{SFE=sSFR/f^{*}_{H_{2}}=SFR/M_{H_{2}}=1/t_{depl}}$ (4)

Having these quantities from the simulation, we need an environment indicator
to explore the relationship between the star formation process in galaxies and
their large-scale environment.

###  3.1 Stellar mass density as an environment indicator

Among the various existing methods of measuring the environment of galaxies,
we decided to investigate the relation between the stellar mass environmental
density of the neighbour galaxies and the star formation activity of each
galaxy. Stellar mass density is a good indicator of the large-scale
environment of galaxies and traces the underlying matter distribution in the
Universe, therefore giving more intrinsic environmental trends. (Mo & White,
1996; Wang et al., 2018). We also performed our analysis using the number
density of neighbour galaxies that resulted in the same trends, but weaker. In
this study, we calculate the stellar mass density by considering the stellar
mass of each galaxy and all of its neighbour galaxies in a $1\
h^{-1}\mathrm{Mpc}$ grid. We choose $1\ h^{-1}\mathrm{Mpc}$ for our grid size
because it represents the typical size of virialized galaxy clusters, so it is
large enough to capture the super-halo environmental trends (Kauffmann et al.,
2004; Muldrew et al., 2012). The general trends reported in this work remain
the same when changing this parameter by a few megaparsecs. Our mass
interpolation method is the Cloud In Cell (CIC) method that gives a fraction
of the weight of each particle to all vertices of the cell it occupies
(Birdsall & Fuss, 1969). This method reduces density measurement errors
compared to other mass assignment schemes. The dimensionless quantity
environmental density (also known as overdensity or density contrast) is
defined with the following equation:

$\delta_{\mathrm{gal}}^{*}(x)=(\rho(x)-\bar{\rho})\ /\bar{\rho}$ (5)

Where $\rho(x)$ is the stellar mass density of galaxies in 3-dimensional space
at point $x$ and $\bar{\rho}$ is the mean stellar mass density of the
simulation box.
Pylians3111https://pylians3.readthedocs.io/en/master/(Villaescusa-Navarro et
al., 2018) is our tool for this calculation which is a set of python
libraries, written in Python, Cython and C to facilitate the analysis of
numerical simulations. We use the defined environmental density
($\delta_{\mathrm{gal}}^{*}$) as an indicator of the environment in this work.
The results of this work remain the same if we consider dark matter mass
rather than stellar mass density.

###  3.2 Error and scatter estimation

We use the median as a statistical tool to investigate the overall evolution
of sSFR, $\mathrm{f^{*}_{H_{2}}}$, and SFE with galaxy properties including
stellar mass, redshift, and environmental density. The median is chosen as the
statistical indicator of our data sample because it is robust against outliers
and false measurements. In order to capture the entire behaviour of the data,
we need to look at the measurement error of our statistical indicator and
scatter of the data as well.

We use bootstrapping to measure the error of the median of sSFR,
$\mathrm{f^{*}_{H_{2}}}$, and SFE for our analysis. Bootstrapping is a
resampling method useful for estimating bias and variance. In order to
calculate a function in a data set with N data points using bootstrapping, the
first step is to generate B new data sets (B=10000 in our case) from the
original data set each of size N with replacement. Then one can calculate the
function on each new data set and keep track of the statistical distribution
of these values.

Furthermore, we need to quantify how the data points are scattered around the
median values to avoid extracting trends only from median values. We use
median absolute deviation (MAD) for measuring the scatter of data points
around the median values. The MAD is defined as the median of the absolute
deviations from the median of the data for a data set $X_{1},X_{2},...,X_{n}$:

$MAD=\mathrm{median}(|X_{i}-\mathrm{median}(X_{i})|)$ (6)

In our case, data points $X_{i}$ can be the sSFR, $\mathrm{f^{*}_{H_{2}}}$,
and SFE of each galaxy with index $i$.

##  4 Results

In the previous sections, we defined our galaxy sample, our methods, and the
important parameters to investigate including specific star formation rate
(sSFR), molecular hydrogen mass fraction ($\mathrm{f^{*}_{H_{2}}}$), star
formation efficiency (SFE) of galaxies, and environmental density
($\delta_{\mathrm{gal}}^{*}$). In the remainder of the paper, we present and
interpret our results based on galaxy evolution theories and other works on
this topic, focusing on our central question: "What is the effect of the Mpc-
scale environment on the star formation activity of galaxies?"

Figure 1: Top left: Median galactic SFE($\mathrm{=sSFR/f^{*}_{H_{2}}}$) as a
function of the environmental density, $\delta_{\mathrm{gal}}^{*}$, at
different redshifts, shown by the colour bar. Green colours show the present
day at $z=0$. In each environmental density bin, the median values and their
uncertainties (shaded areas) are obtained from bootstrapping. Top middle:
similar to the left panel for sSFR ($\mathrm{=SFR/M_{*}}$) versus
environmental density. Top right: Same as the top left panel for the molecular
hydrogen mass fraction $\mathrm{f^{*}_{H_{2}}(=M_{H_{2}}/M_{*})}$ versus
environmental density. Bottom row: Median Absolute Deviation (MAD) of galaxy
SFE, sSFR, and $\mathrm{f^{*}_{H_{2}}}$ versus environmental density at
different redshifts. The slope of the SFE-density curve is changing by
redshift, showing the reversal of SFE-density relation.

###  4.1 The redshift evolution of star formation efficiency across galactic
environments

In this subsection, we present the trends we detect between sSFR,
$\mathrm{f^{*}_{H_{2}}}$, SFE, and environmental density of galaxies in the
Simba simulation. Figure 1 presents the median star formation efficiency,
median specific star formation rate and the median molecular hydrogen mass
fraction of galaxies as functions of the environmental density (as defined in
section 3.1) in the redshift range of $0<z<4$ in the top row. We only show
eight redshift snapshots in this figure for visual clarity. The star formation
efficiency is shown in the top left panel of Figure 1, indicating that at high
redshifts ($z>1$) galaxies in denser regions form stars more efficiently than
galaxies in low-density regions, for instance by $\sim 0.2$ dex at $z=4$.
However, at lower redshifts ($z<1$) galaxies in denser regions form stars less
efficiently than galaxies in low-density regions, for instance by $\sim 0.25$
dex at $z=0$.

Investigating the environmental density dependence of the specific star
formation rate and the molecular hydrogen mass fraction is insightful to
interpret the visible trends for SFE. The top middle panel of Figure 1
presents the median sSFR values for galaxies as a function of the
environmental density for different redshift snapshots. At high redshifts
($z>1$), the sSFR-density curves are almost flat on average, with subtle
variations in the densest bins. The slope of the curves becomes negative
towards lower redshift ($z<1$), meaning that at these redshifts galaxies in
denser regions have lower sSFR values compared to galaxies in low-density
regions, for example by $\sim 0.9$ dex at $z=0$. Finally, the top right panel
of Figure 1 shows the environmental density dependence of the molecular
hydrogen mass fraction of galaxies at different redshift snapshots. This plot
shows that the molecular hydrogen mass fraction of galaxies decreases with
environmental density at all redshifts by $\sim 0.2$ for $z=4$ and $\sim 0.15$
for $z=0$.

Considering all of the trends shown in Figure 1, one can see that at high
redshift ($z>1$), the increase of SFE with environmental density arises from
the decrease of $\mathrm{f^{*}_{H_{2}}}$ with environmental density. However,
at low redshift ($z<1$), the decrease of sSFR with environmental density seems
to be stronger than the decrease of $\mathrm{f^{*}_{H_{2}}}$ with
environmental density, leading the $\mathrm{SFE=sSFR/f^{*}_{H_{2}}}$ to
decrease with environmental density as well. Overall, the slope of the
$\mathrm{SFE-\delta_{gal}^{*}}$ curves is negative at low redshifts ($z<1$)
and positive at high redshifts ($z>1$). We can call this change in the slope
of the $\mathrm{SFE-\delta_{\mathrm{gal}}^{*}}$ at $z\sim 1$ a "reversal" of
SFE-density relation (analogous to the reversal of SFR-density relation
introduced in Elbaz et al. 2007; Popesso et al. 2015).

The bottom row of Figure 1 presents the median absolute deviation (MAD) of
SFE, sSFR, and the molecular hydrogen mass fraction versus the environmental
density of galaxies with different colours indicating the different explored
redshifts. MAD is a measure of the scatter in the data. The scatter of galaxy
sSFR, plotted in the bottom middle panel of Figure 1, increases with
environmental density at all redshift snapshots. We think it results from high
galactic activity in dense regions, including merger and AGN feedback,
compared to lower-density regions. However, the scatters of SFE and
$\mathrm{f^{*}_{H_{2}}}$ do not directly depend on the environmental density.

Figure 2: SFE, sSFR, and molecular hydrogen mass fraction as functions of
redshift for different environmental densities, shown by colours. The reversal
point of the SFE-density and sSFR-density relations are at $z\sim 1$ and
$z\sim 2.5$, respectively.

In order to investigate the reversal of the SFE-density relation, we look at
the redshift evolution of the SFE-density relation directly in Figure 2. These
plots show SFE, sSFR, and $\mathrm{f^{*}_{H_{2}}}$ as functions of redshift
for different overdensities, shown by different colours. The redshift
evolution of SFE is plotted in the left panel of Figure 2, suggesting a
reversal of SFE-density relation around $z\sim 1$. At high redshift ($z=4$),
galaxies in dense regions form stars more efficiently than galaxies in low-
density regions by $\sim 0.3$ dex. While in the local universe ($z=0$),
galaxies in dense regions form stars less efficiently than those in low-
density regions by $\sim 0.3$ dex.

The top right panel of Figure 2 shows that in the high-redshift universe in
Simba ($z\sim 4$), galaxies in denser regions (darker curves) have slightly
higher median sSFR ($\sim 0.2$ dex) than galaxies in low-density regions
(lighter curves). However, this trend changes at lower redshifts in a way that
at $z\sim 0$, galaxies in high-density regions have much lower median sSFR
($\sim 0.9$ dex) than galaxies in low-density regions. Based on this panel,
one can argue that we have a reversal of the sSFR-density relation with a
turning point around $z\sim 2.5$. The drop in the sSFR-redshift curves occurs
earlier in more dense regions. Hence, this reversal in the sSFR-density
relation reflects that galaxy quenching happens earlier in high-density
regions compared to low-density regions. The bottom right panel of Figure 2
shows the redshift evolution of the molecular hydrogen mass fraction for
different environment densities. From this plot, it is clear that galaxies in
the two densest bins on average have larger molecular hydrogen mass fractions
at high redshift compared to the low-redshift snapshots. The same trend can be
seen for galaxies in low-density regions after a certain redshift.
Furthermore, galaxies in denser regions have a lower molecular hydrogen mass
fraction compared to those in low-density regions at all redshifts.
Consequently, the reversal of SFE-density relation mainly arises from the
redshift evolution of sSFR.

###  4.2 Redshift evolution of SFE-density relation:
Environment or mass dependency?

Figure 3: Top: Median SFE as a function of redshift for different stellar
masses and environment overdensities, calculated using bootstrapping as
described in section 3.2. Each column shows a stellar mass bin and colours
represent the environmental density bin. Error bars show the standard
deviation of bootstrapped median values. Middle: Same as the top row, but for
the specific star formation rate (sSFR). Bottom: Same as the top row for
molecular hydrogen mass fraction $\mathrm{f^{*}_{H_{2}}}$. The environmental
dependence of SFE is mostly prominent for low- and intermediate-mass galaxies
at low redshift.

We detect an environmental dependence for the redshift evolution of the star
formation activity of galaxies in Figure 2. However, environmental density may
just be one of the many factors affecting the star formation activity of
galaxies. According to observed scaling relations, other factors like the
stellar mass of galaxies may also affect their star formation efficiency (Peng
et al., 2010). Moreover, the correlation between massive galaxies and dense
environments may bias the interpretation of environmental trends (Bolzonella
et al., 2010; Darvish et al., 2015; Bahé et al., 2017). Furthermore, If an
environmental dependence persists after controlling for stellar mass, we can
conclude that the environment is indeed one factor leading to the trends
reported in the last section. In this section, we divide our galaxy sample
into three stellar mass bins in order to distinguish between the effect of
mass and the effect of the environment on the star formation activity of
galaxies.

Figure 3 shows the dependence of the SFE, sSFR, and the molecular hydrogen
mass fraction of galaxies on the environmental density for different redshifts
and galaxy stellar masses. Each column in this figure demonstrates galaxy
properties in a specific stellar mass bin, starting from galaxies with
$9<\log\frac{M_{*}}{M_{\odot}}<9.5$ in the left column, to galaxies with
$9.5<\log\frac{M_{*}}{M_{\odot}}<10$ in the middle column, and galaxies with
$10<\log\frac{M_{*}}{M_{\odot}}$ in the right column. We have 3 stellar mass
bins, 22 redshift bins, and 5 environmental density bins, generating 330 data
points in each row of Figure 3. The detailed information on the number of
galaxies in each bin is provided in Table 1.

The top row panels of Figure 3 explore the redshift evolution of SFE. In the
top left and top middle panels (low- and intermediate-mass bins) we can see
that for the galaxies in the local universe ($z\sim 0$), those in denser
regions have $\sim 0.3$ dex lower SFE values than those in low-density
regions. However, we do not see a significant dependence on the environmental
density for low- and intermediate-mass galaxies in the high redshift universe.
Furthermore, the effect of the environmental density on the SFE of massive
galaxies (top right panel) is much less prominent than in lower-mass galaxies.
In fact, only galaxies in the densest bin have $\sim 0.2$ dex higher SFE than
other galaxies at $z=0$. Comparing the top row of Figure 3 with the left panel
of Figure 2, we can see that at $z\sim 4$, the visible trend in Figure 2 is
not an effect of the environmental density, but it mainly arises from the
large range of SFE values in different mass bins. The overall trend in the
low-redshift universe ($z<1$) of Figure 2 is seen in the low and intermediate-
mass bins of Figure 3, but not in its most massive bin. Hence, the
environmental density has a small effect on the star formation efficiency of
low- and intermediate-mass galaxies in the low-redshift universe.

The middle row of Figure 3 presents the redshift evolution of sSFR. Similar to
the SFE plots in the top row, an environmental density dependence of sSFR in
the local universe ($z\sim 0$) is noticeable ($\sim 0.5$ dex) in the low- and
intermediate-stellar mass galaxies. However, these low- and intermediate-mass
galaxies do not show a density dependence of their sSFR in the high redshift
universe ($z\sim 4$). For the most massive galaxies, an environmental
dependence is only discernible between redshifts $z\sim 1$ and $z\sim 3$. At
this redshift range, galaxies in denser regions have lower sSFR values than
those in low-density regions. There is no obvious trend seen for these
galaxies in the local universe ($z\sim 0$) or the highest redshift bins
($z>3$). By comparing the middle row of Figure 3 with the top right panel of
Figure 2, one can argue that the overall trend at the local universe ($z\sim
0$) is primarily driven by lower-mass galaxies
($9<\log\frac{M_{*}}{M_{\odot}}<10$).

The redshift evolution of molecular hydrogen mass fraction is illustrated in
the bottom row of Figure 3 for galaxies at different stellar mass bins, and
environment overdensities. The most significant trend in these plots is that
all galaxies in the densest bins have the lowest hydrogen mass fraction at
almost all redshifts explored in this work ($0<z<4$). Moreover, more massive
galaxies have lower hydrogen fractions, on average, which is consistent with
the trend seen in Davé et al. (2020). The molecular hydrogen mass fraction
shows the most prominent dependence on environmental density, more than that
of the sSFR and the SFE (Figures 2 and 3).

In conclusion, our analysis shows a slight dependence of the star formation
efficiency of galaxies in Simba at low- to intermediate-mass bins
($9<\log\frac{M_{*}}{M_{\odot}}<10$) at low-redshifts ($z<1.5$). However, we
do not detect a reversal of SFE-density relation in Figure 3 in each mass bin,
meaning that this is a pure effect of the stellar mass of galaxies and not
their environment.

###  4.3 Scaling relations for gas depletion time

We fit scaling relations to the galaxy parameters to examine the statistical
significance of the trends observed in Figures 2 and 3. Tacconi et al. (2018)
fits the depletion time ($t_{\mathrm{depl}}$) as a function of redshift ($z$),
specific star formation rate ($sSFR$), stellar mass ($M_{*}$), and galaxy’s
half-light radius ($R_{\mathrm{h}}$) in the rest frame optical band 5000 Å
with the following equation which we refer to as the PHIBSS scaling relation
(Plateau de Bure High-z Blue Sequence Survey222https://www.iram.fr/
phibss2/Home.html) in this work:

$\begin{array}[]{l}\log
t_{\mathrm{depl}}(\mathrm{Gyr})=A_{t}+B_{t}\log(1+z)+C_{t}\log(\delta MS)+\\\
D_{t}(\log M_{*}-10.7)+E_{t}\log(\delta R_{\mathrm{h}})\end{array}$ (7)

Where $\mathrm{\delta MS=sSFR/sSFR_{(MS,z,M_{*})}}$ in which
$\rm{sSFR_{(MS,z,M_{*})}}$ is the specific star formation rate of the star-
forming main sequence galaxies, defined in Speagle et al. (2014) with the
following equations:

$\begin{array}[]{l}\log{\frac{sSFR_{(MS,z,M_{*})}}{\mathrm{Gyr^{-1}}})}=9-(6.51-0.11\times\frac{t_{c}}{\mathrm{Gyr}})+\\\
(-0.16-0.026\times\frac{t_{c}}{\mathrm{Gyr}})\times(\log{\frac{M_{*}}{M_{\odot}}}+0.025)\end{array}$
(8)

Where

$\begin{array}[]{l}\log{\frac{t_{c}}{\mathrm{Gyr}}}=1.143-1.026\times\log{(1+z)}-0.599\times\log^{2}{(1+z)}+\\\
0.528\times\log^{3}{(1+z)}\end{array}$ (9)

Furthermore, $\rm{\delta
R_{\mathrm{h}}=R_{\mathrm{h}}/R_{\mathrm{e0}}(z,M_{*})}$, where
$\rm{R_{\mathrm{e0}}(z,M_{*})}$ is the average half-light radius of the star-
forming population, defined in van der Wel et al. (2014) as
$\rm{R_{\mathrm{e0}}=8.9\mathrm{kpc}(1+z)^{-0.75}(M_{*}/5\times
10^{10}M_{\odot})^{0.23}}$.

Tacconi et al. (2018) fits Equation 7 to the PHIBBS survey, consisting of a
large sample of 1444 star-forming galaxies in the redshift range of $0<z<4$, a
stellar mass range of $9<\log{\frac{M_{*}}{M_{\odot}}}<11.8$, and the star
formation rate relative to the main sequence in range of $\rm{10^{1.3}<\delta
MS<10^{2.2}}$. The overall best-fit parameters are:

$A_{t}=+0.09,\quad B_{t}=-0.62,\quad C_{t}=-0.44,\quad\\\ D_{t}=+0.09,\quad
E_{t}=+0.11$

Krumholz & Dekel (2012); Hunt et al. (2016) argue that metallicity affects the
star formation activity of galaxies as well. Following this idea, we include
this parameter in our study on Simba. In this work, we modified Equation 7 to
include two additional terms to fit for the environmental density and the
galaxy mass-weighted metallicity:

$\begin{array}[]{l}\log
t_{{}_{\mathrm{depl}}}(\mathrm{Gyr})=A+B\log(1+z)+C\log sSFR+\\\ D\log
M_{*}+E\log(R_{\mathrm{h}})+F\log(Z)+G\log(1+\delta_{\mathrm{gal}}^{*})\end{array}$
(10)

We evaluate the significance of each term by performing linear regression to
ten slightly different variations of Equation 10. As summarized in Table 2,
the first six models exclude either the environment factor or gas metallicity.
Models 7 to 10 include both of them, and some use $\delta MS$ instead of sSFR
and some use $\delta R_{\mathrm{h}}$ instead of $R_{\mathrm{h}}$. The
$\mathrm{R^{2}}$ value, the so-called "coefficient of determination", ranges
between 0 to 1 with higher values showing statistically better fits. The
coefficients, their standard errors, their $95\%$ confidence intervals, and
their p-values are also presented for each model. The p-value shows the
significance of the parameter in the fit. A p-value of less than 0.05 is
considered to be statistically significant. The highest $\mathrm{R^{2}}$ value
(0.866192) belong to the 8th/10th model where we have redshift ($1+z$),
specific star formation rate ($sSFR$), stellar mass ($M_{*}$), radius
($R_{\mathrm{h}}$) or normalized size ($\delta
R_{\mathrm{h}}=R_{\mathrm{h}}/R_{\mathrm{e0\ (z,M_{*})}}$), metallicity ($Z$),
environmental density ($1+\delta_{\mathrm{gal}}^{*}$) included in the model.
This evaluation shows that the environment is an important term for
determining the gas depletion time of galaxies. Interestingly, using sSFR
gives a better fit than using $\mathrm{\delta MS}$ based on their
$\mathrm{R^{2}}$, but using radius ($R_{\mathrm{h}}$) or the normalized radius
($\delta R_{\mathrm{h}}=R_{\mathrm{h}}/R_{\mathrm{e0\ (z,M_{*})}}$) does not
make a different. Excluding the environmental density or the metallicity leads
to a (slightly) poorer fit. This is consistent with the recent observational
studies, showing the environmental dependences of the gas-phase metallicity of
galaxies (Chartab et al., 2021; Calabrò et al., 2022).

Figure 4 presents a comparison between the actual star formation efficiency of
Simba galaxies, the SFE predicted by our tenth model, and the SFE predicted by
the Tacconi et al. (2018) scaling relation. As expected, the environmental
dependence of galactic SFE seen in Simba at different redshifts, stellar mass,
and environmental density bins is captured in the predicted SFEs from our
scaling relation. Interestingly, the SFE values predicted by the Tacconi et
al. (2018) relation show an environmental dependence even though environmental
density is not an explicit parameter in their scaling relation. This might be
due to the environmental dependence of other fit parameters, for instance,
sSFR and size or simply because denser regions tend to contain more massive
galaxies. Figure 5 shows a relative comparison between the actual Simba SFE
values and the SFE predicted by our tenth scaling relation. The model is
predicting the SFE values with accuracy within $30\%$ of the actual values.
The predicted values follow the general trends of our SFE-redshift curves in
most of the bins.

Figure 4: Cosmic evolution of SFE for galaxies with different stellar masses.
Top: The actual SFE inferred from Simba. Middle: The SFE predicted by the best
scaling relation from this work ($10^{\mathrm{th}}$ model in Table 2). Bottom:
The SFE computed from the observational scaling relation by Tacconi et al.
(2018) following the procedures described in Sec 4.3. Colours show
environmental density. The environmental dependence of SFE (seen in Figure 3)
is reproduced using our scaling relation.  Figure 5: The relative difference
of the predicted SFE using the second model (SFE as a function of z, $M_{*}$,
sSFE, $\delta_{\mathrm{gal}}^{*}$) and the actual SFE values as a function of
redshift for different stellar mass bins and environmental density bins. The
model predictions are close to the data.

##  5 Discussion

Mass and environment are the two determinants suggested in the literature for
the evolution of galaxies (Peng et al., 2010, 2012; Darvish et al., 2016). The
effects of these factors on star formation activity have been studied from
different viewpoints and using different methods. In this section, we discuss
our results in the context of the literature findings.

###  5.1 Impact of the stellar mass on star formation

The stellar mass of a galaxy is an important indicator of its evolutionary
path, including its past and ongoing star formation activity. For instance,
more massive galaxies in Simba have more massive black holes consistent with
observations (Davé et al., 2019). According to the black hole growth model of
Simba, the accretion rate of black holes increases with their mass (Davé et
al., 2019). Therefore, galaxies with larger stellar masses and corresponding
higher black hole masses have higher black hole accretion rates. Moreover,
Thomas et al. (2019) show that SFR increases with the black hole accretion
rate for star-forming ($\mathrm{SFR>1M_{\odot}yr^{-1}}$) galaxies in Simba at
redshift $0<z<5$. They suggest this arises from the common gas reservoir used
for star formation and torque-limited black hole growth mode.

The AGN jet feedback in Simba is the main quenching responsible for galaxies
with black hole mass $\mathrm{M_{BH}>10^{7.5}M_{\odot}}$ which corresponds to
stellar mass $\mathrm{M_{*}>10^{10}M_{\odot}}$ (Thomas et al., 2019).
Consequently, the most massive galaxies we investigate in this work are
significantly affected by the AGN jet feedback which might be one of the
reasons their SFE and sSFR drop more rapidly than lower-mass galaxies (Figure
3). Since AGN jet particles in Simba are decoupled until they reach outside
the galaxy (tens of kpc), they cannot directly remove gas from the
interstellar medium (ISM) of the galaxy (Davé et al., 2019). However, the AGN
jets in Simba are implemented to heat up the gas in the circumgalactic medium
(CGM) of galaxies which truncates the accretion of cold gas from the CGM into
the galaxy (Appleby et al., 2021). We do not investigate the impacts of AGN in
this work, however, a potential trend could be as follows: since more massive
galaxies tend to reside in more dense regions (Bahé et al., 2017), their
strong AGN jet feedback can disturb the cooling and accretion of the CGM gas,
resulting in less fuel for star formation and a drop in the SFE and sSFR of
these galaxies compared to those in lower-density regions (the right column of
Figure 3). The faster drop of SFE and sSFR of massive galaxies compared to
lower mass galaxies, can partially cause the reversal of SFE-density and sSFR-
density relations we show in Figure 2.

It is commonly known that mass quenching is more effective on more massive
galaxies at higher redshifts while environment quenching is mostly effective
on lower mass galaxies at lower redshifts (Peng et al., 2010; Darvish et al.,
2018a). Recent observational studies find quenched massive galaxies (with mass
completeness limit of $\log{M_{*}/M_{\odot}}>10-11$) in high redshift
protoclusters ($1.4<z<3.3$) (Zavala et al., 2019; Alberts et al., 2022;
McConachie et al., 2022; Ito et al., 2023; Mei et al., 2023). Since galaxy
protoclusters are not fully virialized in this phase, the effect of
environment quenching might not be as significant as the effect of mass
quenching. Assuming that these recently reported protoclusters are overdense
regions at the observed redshifts, they could be in line with the trends seen
in Figure 3, showing that the most massive galaxies of Simba in dense regions
have lower sSFR compared to those in low-density regions at $0.5<z<3$.

Zavala et al. (2019) attributed the observed quenched massive galaxies in
protoclusters to the early phases of environment quenching. They argue that
this process is due to the environment because of the high fraction of
quenched galaxies in overdense regions. They use ALMA Band 6 observations to
investigate the SFE and gas fractions (defined as the ratio of the ISM gas to
the total gas and stellar mass) in 68 members of two massive protoclusters at
$2<z<2.5$. Their sample spans a stellar mass range of
$9<\log{\frac{M_{*}}{M_{\odot}}}<11.6$. While they find, on average, similar
scaling relations between the most massive galaxies
($\log{\frac{M_{*}}{M_{\odot}}}>10$) in protoclusters and those from the
field, they argue that a discernible fraction of the lower-mass galaxies might
have enhanced efficiencies. Their observed trend for massive galaxies is
consistent with our detected trends presented in Figure 3, showing no
environmental dependence for SFE for massive galaxies. However, their low-mass
galaxy sample ($\log{\frac{M_{*}}{M_{\odot}}}<10$) is incomplete and cannot be
used to extract any trends.

###  5.2 Impact of the environment on star formation

Our analysis shows that at low redshift ($z<1.5$), lower-mass galaxies with
$9<\log{\frac{M_{*}}{M_{\odot}}}<10$ in the Simba simulation that reside in
denser regions have, on average, lower SFEs and lower sSFRs compared to
galaxies in low-density regions (Figure 3). This trend has been seen in other
observational and simulation works as well and usually is attributed to
environmental quenching caused by different environmental processes.

More mergers in dense regions could be a reason for environmental quenching.
Rodríguez Montero et al. (2019) studies mergers and the resulting starburst
and quenching at $0<z<2.5$ in Simba. They find in merger events the SFR can
increase up to 2-3 times the SFR of normal star-forming galaxies. They show
this jump in SFR for low-mass galaxies
($\mathrm{\log{\frac{M_{*}}{M_{\odot}}}<10.5}$) is due to the more molecular
gas content in the final galaxy. While, for more massive galaxies
($\mathrm{\log{\frac{M_{*}}{M_{\odot}}}>10.5}$), this jump arises from higher
SFE because of more dense molecular gas in the galaxy after the merger. Since
merger events are more abundant in denser regions, this SFR jump for mergers
could result in an enhancement of SFR of galaxies in denser regions, in
contradiction to the trends seen in our work. However, the SFR jump caused by
mergers in Rodríguez Montero et al. (2019) makes up a very little part (a
couple of percent) of the overall cosmic SFR of Simba and its effect cannot be
significantly detected. Consequently, their results are not in direct
contradiction to our results.

Darvish et al. (2016) finds that the environment and mass quenching depend on
each other in a way that environment quenching happens more efficiently for
massive galaxies than lower mass galaxies and mass quenching occurs more
efficiently in dense regions. They suggest mergers as the main cause of this
finding. In this work, they investigate the relationship between the galactic
environment, stellar mass, and star formation activity for 73481 galaxies in
the COSMOS field in the redshift range of $0.1<z<3.1$. Their mass-complete
galaxy sample has a $\mathrm{K_{s}}$-band magnitude limit of
$\mathrm{K_{s}}<24$ and a stellar mass range of
$9.14<\log{\frac{M_{*}}{M_{\odot}}}<11.5$. They measure the SFRs and stellar
masses using a spectral energy distribution (SED) template fitting to the
available UV, optical, and mid-IR data. The limiting stellar mass of this data
set grows with redshift, such that in the redshift range of $0.1<z<0.5$ the
data set is complete for galaxies with stellar mass
$9.14<\log{\frac{M_{*}}{M_{\odot}}}$, while this limit grows to $9.97$ at
$1.5<z<3.1$. Darvish et al. (2016) uses the Voronoi tessellation method to
calculate the environmental density of galaxies. This work shows that at $z<1$
the median SFR and median sSFR decrease with increasing density, while they
become independent of redshift at $z>1$. They argue that environmental
quenching is the dominant quenching process at $z<1$. At $z>1$, mass quenching
is the dominant process that likely arises from stellar feedback.

In general, a direct and precise comparison of observations and simulations is
not feasible due to the different nature of these data. Furthermore, the
methods used to measure the galactic parameters, including SFR, molecular
hydrogen, and environment in our work are different from Darvish et al.
(2016). Nonetheless, it is still instructive to perform a qualitative
comparison. Keeping these points in mind, we look at SFR and sSFR as functions
of density for different redshifts in our work and Darvish et al. (2016) in
Figure 6 for a qualitative comparison. We apply the same mass completeness
limits on our galaxies as their work (Table 1 of Darvish et al. 2016) for a
fair comparison. Figure 6 shows that the density dynamic range of the Darvish
et al. (2016) data is smaller compared to our data (by $\sim 4$ dex), but the
SFR and sSFR dynamic ranges of their galaxies are larger than our galaxies (by
$1-2$ dex each). Hence, Darvish et al. (2016) reports stronger trends than our
work. The first row of Figure 6, showing SFRs from both works, suggests that
both data sets follow qualitatively the same trends. An exception is that the
median SFR of Simba increases towards $z>3$ and the highest density bins
($\log{(1+\delta_{\mathrm{gal}}^{*})}>1$), a parameter space beyond that
probed by Darvish et al. (2016). Based on Table 1 from Darvish et al. (2016),
the mass completeness limit for their galaxies with redshifts higher than 1.1
is larger than $10^{9.93}\mathrm{M_{\odot}}$, so their sample at high redshift
($z>1$) just covers the most massive galaxies from our sample and is
incomplete in detecting what we classify as lower mass galaxies in this work.
Although there are noticeable differences in the dynamic range in median SFR
between our work and Darvish et al. (2016), we find qualitatively similar
trends for star formation as a function of the environment for all of our
galaxies, on average.

Figure 6: Comparison of the environmental dependence of star formation
activity in Simba and the observational study based on the COSMOS survey
presented in Darvish et al. (2016). Both data sets have the same stellar mass
lower limits. Left: SFR (top) and sSFR(bottom) as functions of the
environmental density at different redshifts, shown by the colour bar for
Simba. Right: Same as the left column for Darvish et al. (2016) data (the
right column is a reproduction of Figure 1 of Darvish et al. 2016). For visual
purposes, the x-axes of the plots in the right column are more zoomed-in than
the left column. Simba is qualitatively reproducing the observed trends in the
COSMOS field, albeit Simba shows a smaller dynamic range in SFR and sSFR
overall..

A bump is noticeable in the SFE and molecular hydrogen mass fraction of our
lower-mass galaxies ($\log{\frac{M_{*}}{M_{\odot}}}<10$) at $z<1.5$ that
reside in lower-density regions in Figure 3. A previously quenched galaxy can
"rejuvenate" if it experiences a merger event or accretes a lot of cold gas
from an external gas reservoir. In this case, the gas content and star
formation activity of the galaxy would increase after a drop caused by
quenching. However, based on Lorenzon et al., in prep., the rejuvenation event
among the quenched and post-starburst Simba galaxies is less than $10\%$ at
each redshift and is independent of environments. Therefore, the rejuvenation
events cannot fully explain the detected trends in our work.

The "starvation" of satellite galaxies happens when they fall into galaxy
clusters with hot intracluster medium (ICM). As a result, they are prevented
from accreting enough cold gas from their surrounding which leads them to
quench after finishing their gas content. This theory has been tested by van
de Voort et al. (2017) using the Eagle cosmological simulation. They study
galaxies with $\log{\frac{M_{*}}{M_{\odot}}}>8$ at redshifts $0<z<2$ with the
environment defined as the local 3-dimensional number density of galaxies up
to the $\mathrm{10^{th}}$ nearest neighbour. They find a strong suppression of
gas accretion rate in dense environments, most effective on satellites at low
redshifts. This phenomenon could partially explain the low star formation
activity of galaxies in dense regions compared to low-density regions for low-
to-intermediate mass galaxies in Figure 3.

"Ram pressure stripping" is another environment-related phenomenon. When a
low-mass galaxy gets bound to the gravitational field of a galaxy cluster and
moves quickly in the ICM of the cluster, the pressure exerted from the hot ICM
strips the gas content of the galaxy from it (Boselli et al., 2022). Darvish
et al. (2018b) studies the relationship between the local environment and gas
content for 708 galaxies with stellar mass $\log{\frac{M_{*}}{M_{\odot}}}>10$
at $0.3<z<4.5$. The environment is parameterized as the environmental density
calculated using the adaptive kernel smoothing method (Scoville et al., 2013)
and the projected comoving distance to the $10^{\mathrm{th}}$ nearest
neighbour to each galaxy for high and low redshift, respectively. This work
finds no environmental dependence for gas mass fraction (defined as the ratio
of the interstellar medium gas to the total gas and stellar mass) and the
depletion timescale for their massive galaxies. Previous studies show that
environmental-related processes like ram-pressure stripping can strip the gas
content of galaxies in dense regions (Boselli & Gavazzi, 2014). However,
Darvish et al. (2018b) shows that ram pressure stripping is only strongly
effective in galaxies with stellar mass $\log{\frac{M_{*}}{M_{\odot}}}<9$ with
weak gravitational potential well (Fillingham et al., 2016). As a result, ram-
pressure stripping is ineffective in removing the gas content of massive
galaxies, including those studied in the work of Darvish et al. (2018b).
Furthermore, molecular hydrogen (main fuel for star formation) is much denser,
more bound to galaxies, and consequently less vulnerable than atomic hydrogen
to be stripped from galaxies. Kenney & Young (1989) and Koyama et al. (2017)
find no environmental dependence for molecular hydrogen while Fumagalli et al.
(2009) finds a deficiency in molecular gas in denser environments compared to
low-density environments. The stellar mass range of the galaxy sample used in
Darvish et al. (2018b) is the same as the most massive galaxies we explore in
Simba (the third row of Figure 3). We are not able to directly compare the
results of these works because of the fundamental differences between the
data, measurement methods, and the different definitions of galaxy properties.
However, we do see a lower gas mass fraction in the two densest regions
compared to lower-density regions and a small difference in SFE between the
densest region and the rest of the regions. This finding is not in direct
contradiction with Darvish et al. (2018b) because these dense regions of Simba
are not explored in the observations of Darvish et al. (2018b).

Lemaux et al. (2020) investigates the reversal of the star formation-density
relation using the VIMOS Ultra Deep Survey (VUDS). This work uses observations
of 6730 star-forming galaxies with a stellar mass range
$8<\log{\frac{M_{*}}{M_{\odot}}}<12$ and spectroscopic redshifts of
$2<z_{\mathrm{spec}}<5$ in three extragalactic fields of COSMOS, ECDFS, and
CFHTLS-D1 to explore the relationship between the star formation rate (SFR) of
galaxies and their environmental density ($\mathrm{\delta_{gal}}^{*}$). They
use the Voronoi Monte Carlo (VMC) mapping to estimate the projected density of
galaxies and the environmental density contrast. In contrast to the low-
redshift universe where SFR and $\mathrm{\delta_{gal}^{*}}$ are anti-
correlated, they find a positive correlation between these two quantities in
their high-redshift sample ($2<z<5$). They find that this trend is mainly
driven by the high fraction of massive galaxies in dense regions that are
forming stars more rapidly than lower-mass galaxies. A weak but statistically
significant positive correlation between SFR and $\mathrm{\delta_{gal}^{*}}$
is still visible after controlling for mass and redshift dependence in this
work. Although their galaxy sample and their density estimation method are
different from ours, our results in the redshift range of $2<z<4$ are in
qualitative agreement with their findings.

Wang et al. (2018) performs a similar analysis to our work on a re-simulated
data set from the three hundred
project333https://www.nottingham.ac.uk/astronomy/The300/index.php (Cui et al.,
2018). This data set includes 324 re-simulated clusters and 4 field regions
from the MultiDark Planck simulation (MDPL2; Klypin et al. 2016). Wang et al.
(2018) re-simulates these clusters and fields using hydrodynamical codes
GADGET-X and GADGET-MUSIC. They divide their galaxies into three categories:
(i) cluster galaxies are galaxies closer than $2R_{200}$ to the cluster center
which is the position of the most massive dark matter halo of galaxies in each
re-simulated cluster; (ii) cluster vicinity galaxies which have a distance to
the cluster center between $2R_{200}$ and the fixed comoving radius of
$10\mathrm{h^{-1}Mpc}$ ; (iii) field galaxies within a fixed radius of
$38\mathrm{h^{-1}Mpc}$ from the center of the re-simulated field region,
defined as the center of the region at $z=0$ and fixed for all redshifts. They
define the SFR of each galaxy as the sum of SFRs of all gas cells of the
galaxy. The SFR of each gas cell is derived from the Springel et al. (2005)
prescription. They define the environment of galaxies based on the density of
all matter (dark matter, stars, and gas) calculated in a $1\mathrm{h^{-1}Mpc}$
sphere around each galaxy and the density contrast ($\delta_{1}$) is
calculated compared to the average density of the universe. Their sSFR
threshold to define star-forming galaxies is $\mathrm{sSFR}>0.3/t_{H(z)}$.
Wang et al. (2018) find that for $z=0$ star forming galaxies, sSFR declines
when $\delta_{1}$ increases in all of their environment categories. However,
they claim this trend is driven by the high abundance of massive galaxies in
dense regions. Since massive galaxies have lower sSFRs compared to low-mass
galaxies, on average, they reduce the overall sSFR in dense regions.
Controlling for this effect in the data, they find that the environment does
not affect galaxies in cluster and cluster vicinity, but for galaxies in the
field, sSFR slightly decreases when environmental density increases. We cannot
directly compare these trends to our trends for Simba, because at $z=0$ many
galaxies in Simba are not star-forming based on the definition of star-forming
galaxies in Wang et al. (2018). They also show that the median sSFR for all
galaxies at the redshift range of $0<z<2.5$ falls when $\delta_{1}$ increases
and argue that this trend is also due to the high abundance of massive
galaxies in dense regions (Figure 6 of Wang et al. 2018). The difference
between the sSFR-$\delta_{1}$ curves for their three environment categories
(Cluster, vicinity, field) at low redshifts ($z=0$) is much larger than those
at high redshifts ($z>1$), showing that the environment matters more at lower
redshifts. Although the simulation configuration and measurement methods of
this work are different from our work on Simba, a qualitative comparison
between the trends shows that these results are broadly consistent. However,
this work has not investigated the molecular gas content and SFE, so we cannot
discuss these parameters in their simulation.

Lovell et al. (2021) introduces the First Light And Reionisation Epoch
Simulations (FLARES) zoom simulations and investigates the effects of the
environment on the galactic properties at the epoch of reionization ($5<z<10$)
in this simulation. Their results show a strong density dependence for the
galaxy stellar mass functions and star formation rate distribution function.
The galaxy star-forming main sequence, though, does not show any environmental
dependence. They re-simulate spherical regions taken from a parent simulation
with a box length of $3.2\ \mathrm{comoving\ Gpc}$ (Barnes et al., 2017).
Using the nearest grid point mass assignment scheme, the environmental density
is calculated on a $\sim 2.67$ comoving Mpc cubic grid. Then they find the
environmental density on large scales by convolving this grid with a $14\
\mathrm{h^{-1}\ cMpc}$ top-hat filter and define the environmental density as
$\delta(x)=\rho(x)/\bar{\rho}-1$ where $\rho(x)$ is the dark matter density on
grid points and $\bar{\rho}$ is the mean density in the box. They choose 40
regions in a way that the sample includes a number of the highest
overdensities (that include the first massive galaxies) and a range of
different overdensities to examine the environmental dependence of galaxy
formation. The redshift ranges of the FLARES simulations investigated in
Lovell et al. (2021) and Simba studied in our work do not overlap. Moreover,
the galaxy sample used in Lovell et al. (2021) contains galaxies with stellar
mass in the range of $\mathrm{7.5<\log{\frac{M_{*}}{M_{\odot}}}}<11.3$ and
$\mathrm{7.5<\log{\frac{M_{*}}{M_{\odot}}}<10.2}$ at redshifts $\mathrm{z=5}$
and $\mathrm{z=10}$, respectively. At higher redshifts, there are fewer
massive galaxies due to the hierarchical assembly of galaxies. On average,
they are looking at more low-mass galaxies than our work. Furthermore, FLARES
includes a larger volume than Simba with more abundant rare and extreme
environments. Despite all these differences between these two works, it is
still insightful to look at the physics behind the significant environmental
trends detected in FLARES at high redshifts in contrast to Simba.

###  5.3 Caveats and future work

We are aware that several biases and limitations exist in this study. For
example, hydrodynamic simulations use different theoretical prescriptions to
assign galaxy properties including SFR that may not completely capture the
complex physics behind the formation of galaxies. Furthermore, even the
periodic simulation boxes lack a sufficient number of the largest structures
in the real universe. On the other hand, the spatial resolution of simulations
are limited, making it hard to capture the subgrid physics of galaxies. As a
result, simulations and observations need to be used together to understand
the problem in a broader view. Our work also suffers from specific caveats. We
mainly explore median galaxy properties which may not represent the total
distribution and scatter of all galaxies and it is not sensitive to outliers,
so we might have missed some subtle variations in the distribution of
galaxies. Further work is needed to check the reported trends in our work
using other hydrodynamic simulations as well and confirm the physical origin
of these trends.

In this work, we study the star formation activity of a large set of galaxies
in a broad range of stellar mass, redshift, and environmental density. The
results of this study can guide the observers to look at the galaxy sets, with
specific properties, in which the environmental effects can be detected in
future observational surveys. We report the most significant environmental
dependences for intermediate-mass galaxies
($\mathrm{\log{\frac{M_{*}}{M_{\odot}}}}<10$) at $0<z<2$. Hence, observational
tests of our method need an estimate of the stellar mass of galaxies up to
$z=2$ that would be feasible using the large galaxy surveys to be carried out
with the James Webb Space Telescope (JWST) and the Euclid mission. It is a
synergy of Euclid and JWST that will allow studying the environmental impact
on physical and structural galaxy parameters introduced in Equation 10 up to
$z\sim 2$. Using these parameters, observers can probe the predictions of this
work.

##  6 Summary and Conclusion

In this work, we study the relationship between the star formation activity of
$\sim 330,000$ simulated galaxies with stellar mass
$\log{\frac{M_{*}}{M_{\odot}}}>9$ and their large-scale environment from the
cosmological simulation Simba in a redshift range of $0<z<4$. The environment
of galaxies is defined as the environmental density of nearby galaxies,
calculated on a 1 Mpc/h grid using the Cloud In Cell (CIC) method. We explore
the star formation efficiency (SFE), specific star formation rate (sSFR), and
molecular hydrogen mass fraction of galaxies in different redshift, stellar
mass, and environmental density bins. Additionally, we fit a scaling relation
for the molecular hydrogen depletion time scale as a function of redshift,
specific star formation rate, stellar mass, radius, metallicity, and
environmental density. Our most important findings include:

  * •

Across the entire stellar mass range considered in this work, galaxies
residing in denser regions at high redshift ($z>1$), tend to have higher SFE
than those in low-density regions. The difference is found to be $\sim 0.3$
dex at $z=4$. This trend reverses around $z\sim 1$: at later cosmic times
galaxies in dense regions have lower SFE than those in underdense regions, on
average ($\sim 0.3$ dex at $z=0$).

  * •

With a similar trend to SFE, our galaxy sample across the entire considered
stellar mass range shows a weak reversal of sSFR-density relation with a
turning point around $z\sim 2.5$.

  * •

Controlling for stellar mass variation, at low redshift ($z<1$), the low to
intermediate mass galaxies ($9<\log{\frac{M_{*}}{M_{\odot}}}<10$) in dense
regions have lower SFE and sSFR than galaxies in underdense regions. The
molecular hydrogen mass fraction is lower in dense regions compared to low-
density regions for all galaxies regardless of their stellar mass and
redshift.

  * •

We provide a scaling relation to determine depletion timescales (and therefore
SFE) in galaxies at $0<z<4$ with $\log{\frac{M_{*}}{M_{\odot}}}>9$. The
proposed model takes into account the environmental density parameter, which
is found to be a statistically important term in the fit.

## Acknowledgements

We warmly acknowledge Behnam Darvish for providing us with their data and Sara
Ellison for her thoughtful comments and discussions. DD. acknowledges support
from the National Science Center (NCN) starting grant SONATA
(UMO-2020/39/D/ST9/00720). LG. and AWSM. acknowledge the support of the
Natural Sciences and Engineering Research Council of Canada (NSERC) through
grant reference number RGPIN-2021-03046.

## Data Availability

The galaxy catalogues used in this work are available at the Simba project
repository444http://simba.roe.ac.uk/. The full tables including the
environment measurements are available upon request.

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##  Appendix A Tables

Table 1 shows the binning of galaxies based on stellar mass, redshift, and
overdensity, used in Figure 3. Table 2 shows the properties of the models we
fitted for the molecular gas depletion time of galaxies in Section 4.3.

Table 1: The number of galaxies in each bin specified by stellar mass, redshift, and the stellar mass overdensity.  $9<\log{M*/M_{\odot}}<9.5$ | $9.5<\log{M*/M_{\odot}}<10$ | $10<\log{M*/M_{\odot}}$
---|---|---
$z$ | $\log{1+\delta}$ | N | $z$ | $\log{1+\delta}$ | N | $z$ | $\log{1+\delta}$ | N | $z$ | $\log{1+\delta}$ | N | $z$ | $\log{1+\delta}$ | N | $z$ | $\log{1+\delta}$ | N
0.0 | -1.62 | 794 | 1.9 | -1.62 | 817 | 0.0 | -1.62 | 667 | 1.9 | -1.62 | 223 | 0.0 | -1.62 | 163 | 1.9 | -1.62 | 62
| -0.69 | 2511 |  | -0.69 | 1906 |  | -0.69 | 2677 |  | -0.69 | 780 |  | -0.69 | 919 |  | -0.69 | 363
| 0.23 | 4025 |  | 0.23 | 2047 |  | 0.23 | 5572 |  | 0.23 | 1304 |  | 0.23 | 3248 |  | 0.23 | 1140
| 1.16 | 2118 |  | 1.16 | 1239 |  | 1.16 | 2640 |  | 1.16 | 709 |  | 1.16 | 2528 |  | 1.16 | 1161
| 2.09 | 60 |  | 2.09 | 276 |  | 2.09 | 64 |  | 2.09 | 156 |  | 2.09 | 73 |  | 2.09 | 290
0.1 | -1.62 | 794 | 2.0 | -1.62 | 756 | 0.1 | -1.62 | 640 | 2.0 | -1.62 | 191 | 0.1 | -1.62 | 170 | 2.0 | -1.62 | 43
| -0.69 | 2356 |  | -0.69 | 1874 |  | -0.69 | 2528 |  | -0.69 | 685 |  | -0.69 | 982 |  | -0.69 | 299
| 0.23 | 3738 |  | 0.23 | 2036 |  | 0.23 | 5158 |  | 0.23 | 1177 |  | 0.23 | 3437 |  | 0.23 | 1018
| 1.16 | 2120 |  | 1.16 | 1210 |  | 1.16 | 2512 |  | 1.16 | 657 |  | 1.16 | 2728 |  | 1.16 | 1035
| 2.09 | 57 |  | 2.09 | 268 |  | 2.09 | 63 |  | 2.09 | 145 |  | 2.09 | 72 |  | 2.09 | 290
0.3 | -1.62 | 752 | 2.2 | -1.62 | 713 | 0.3 | -1.62 | 546 | 2.2 | -1.62 | 133 | 0.3 | -1.62 | 191 | 2.2 | -1.62 | 47
| -0.69 | 2043 |  | -0.69 | 1808 |  | -0.69 | 2234 |  | -0.69 | 608 |  | -0.69 | 1082 |  | -0.69 | 246
| 0.23 | 3336 |  | 0.23 | 2047 |  | 0.23 | 4385 |  | 0.23 | 1084 |  | 0.23 | 3734 |  | 0.23 | 864
| 1.16 | 1881 |  | 1.16 | 1141 |  | 1.16 | 2208 |  | 1.16 | 577 |  | 1.16 | 2762 |  | 1.16 | 886
| 2.09 | 87 |  | 2.09 | 282 |  | 2.09 | 94 |  | 2.09 | 170 |  | 2.09 | 135 |  | 2.09 | 264
0.5 | -1.62 | 817 | 2.4 | -1.62 | 637 | 0.5 | -1.62 | 505 | 2.4 | -1.62 | 122 | 0.5 | -1.62 | 240 | 2.4 | -1.62 | 34
| -0.69 | 1944 |  | -0.69 | 1676 |  | -0.69 | 1937 |  | -0.69 | 452 |  | -0.69 | 1121 |  | -0.69 | 167
| 0.23 | 3041 |  | 0.23 | 2088 |  | 0.23 | 3546 |  | 0.23 | 1074 |  | 0.23 | 3683 |  | 0.23 | 635
| 1.16 | 1820 |  | 1.16 | 1040 |  | 1.16 | 2001 |  | 1.16 | 503 |  | 1.16 | 2859 |  | 1.16 | 713
| 2.09 | 81 |  | 2.09 | 271 |  | 2.09 | 90 |  | 2.09 | 163 |  | 2.09 | 170 |  | 2.09 | 224
0.7 | -1.62 | 755 | 2.5 | -1.62 | 583 | 0.7 | -1.62 | 446 | 2.5 | -1.62 | 100 | 0.7 | -1.62 | 203 | 2.5 | -1.62 | 21
| -0.69 | 1850 |  | -0.69 | 1640 |  | -0.69 | 1638 |  | -0.69 | 399 |  | -0.69 | 1099 |  | -0.69 | 153
| 0.23 | 2883 |  | 0.23 | 2017 |  | 0.23 | 2940 |  | 0.23 | 1034 |  | 0.23 | 3445 |  | 0.23 | 538
| 1.16 | 1725 |  | 1.16 | 988 |  | 1.16 | 1631 |  | 1.16 | 494 |  | 1.16 | 2789 |  | 1.16 | 606
| 2.09 | 107 |  | 2.09 | 286 |  | 2.09 | 104 |  | 2.09 | 154 |  | 2.09 | 187 |  | 2.09 | 231
0.8 | -1.62 | 741 | 2.7 | -1.62 | 494 | 0.8 | -1.62 | 421 | 2.7 | -1.62 | 91 | 0.8 | -1.62 | 201 | 2.7 | -1.62 | 8
| -0.69 | 1876 |  | -0.69 | 1532 |  | -0.69 | 1499 |  | -0.69 | 361 |  | -0.69 | 1022 |  | -0.69 | 115
| 0.23 | 2773 |  | 0.23 | 2033 |  | 0.23 | 2679 |  | 0.23 | 917 |  | 0.23 | 3324 |  | 0.23 | 424
| 1.16 | 1666 |  | 1.16 | 945 |  | 1.16 | 1459 |  | 1.16 | 456 |  | 1.16 | 2637 |  | 1.16 | 535
| 2.09 | 143 |  | 2.09 | 287 |  | 2.09 | 120 |  | 2.09 | 137 |  | 2.09 | 246 |  | 2.09 | 194
1.0 | -1.62 | 839 | 3.0 | -1.62 | 465 | 1.0 | -1.62 | 385 | 3.0 | -1.62 | 87 | 1.0 | -1.62 | 219 | 3.0 | -1.62 | 11
| -0.69 | 1859 |  | -0.69 | 1385 |  | -0.69 | 1285 |  | -0.69 | 284 |  | -0.69 | 880 |  | -0.69 | 80
| 0.23 | 2562 |  | 0.23 | 2077 |  | 0.23 | 2161 |  | 0.23 | 739 |  | 0.23 | 2756 |  | 0.23 | 300
| 1.16 | 1751 |  | 1.16 | 961 |  | 1.16 | 1369 |  | 1.16 | 453 |  | 1.16 | 2476 |  | 1.16 | 444
| 2.09 | 148 |  | 2.09 | 275 |  | 2.09 | 118 |  | 2.09 | 121 |  | 2.09 | 267 |  | 2.09 | 161
1.2 | -1.62 | 857 | 3.2 | -1.62 | 432 | 1.2 | -1.62 | 327 | 3.2 | -1.62 | 65 | 1.2 | -1.62 | 186 | 3.2 | -1.62 | 14
| -0.69 | 1925 |  | -0.69 | 1263 |  | -0.69 | 1132 |  | -0.69 | 251 |  | -0.69 | 770 |  | -0.69 | 56
| 0.23 | 2378 |  | 0.23 | 2145 |  | 0.23 | 1856 |  | 0.23 | 660 |  | 0.23 | 2344 |  | 0.23 | 237
| 1.16 | 1626 |  | 1.16 | 958 |  | 1.16 | 1171 |  | 1.16 | 422 |  | 1.16 | 2022 |  | 1.16 | 344
| 2.09 | 200 |  | 2.09 | 259 |  | 2.09 | 130 |  | 2.09 | 106 |  | 2.09 | 273 |  | 2.09 | 147
1.4 | -1.62 | 754 | 3.5 | -1.62 | 366 | 1.4 | -1.62 | 309 | 3.5 | -1.62 | 43 | 1.4 | -1.62 | 141 | 3.5 | -1.62 | 13
| -0.69 | 1987 |  | -0.69 | 1127 |  | -0.69 | 1052 |  | -0.69 | 186 |  | -0.69 | 667 |  | -0.69 | 32
| 0.23 | 2275 |  | 0.23 | 2102 |  | 0.23 | 1647 |  | 0.23 | 547 |  | 0.23 | 1972 |  | 0.23 | 147
| 1.16 | 1531 |  | 1.16 | 986 |  | 1.16 | 1010 |  | 1.16 | 443 |  | 1.16 | 1838 |  | 1.16 | 294
| 2.09 | 261 |  | 2.09 | 230 |  | 2.09 | 135 |  | 2.09 | 94 |  | 2.09 | 313 |  | 2.09 | 118
1.5 | -1.62 | 782 | 3.7 | -1.62 | 312 | 1.5 | -1.62 | 278 | 3.7 | -1.62 | 36 | 1.5 | -1.62 | 111 | 3.7 | -1.62 | 11
| -0.69 | 1965 |  | -0.69 | 1037 |  | -0.69 | 975 |  | -0.69 | 165 |  | -0.69 | 597 |  | -0.69 | 27
| 0.23 | 2201 |  | 0.23 | 2033 |  | 0.23 | 1588 |  | 0.23 | 479 |  | 0.23 | 1719 |  | 0.23 | 107
| 1.16 | 1461 |  | 1.16 | 971 |  | 1.16 | 921 |  | 1.16 | 424 |  | 1.16 | 1658 |  | 1.16 | 248
| 2.09 | 273 |  | 2.09 | 217 |  | 2.09 | 151 |  | 2.09 | 91 |  | 2.09 | 353 |  | 2.09 | 114
1.7 | -1.62 | 765 | 4.0 | -1.62 | 255 | 1.7 | -1.62 | 258 | 4.0 | -1.62 | 25 | 1.7 | -1.62 | 88 | 4.0 | -1.62 | 6
| -0.69 | 2010 |  | -0.69 | 849 |  | -0.69 | 883 |  | -0.69 | 126 |  | -0.69 | 466 |  | -0.69 | 25
| 0.23 | 2160 |  | 0.23 | 1794 |  | 0.23 | 1414 |  | 0.23 | 388 |  | 0.23 | 1338 |  | 0.23 | 71
| 1.16 | 1308 |  | 1.16 | 945 |  | 1.16 | 821 |  | 1.16 | 404 |  | 1.16 | 1372 |  | 1.16 | 174
| 2.09 | 281 |  | 2.09 | 235 |  | 2.09 | 166 |  | 2.09 | 92 |  | 2.09 | 315 |  | 2.09 | 99
Table 2: Different models fitted to the depletion time of galaxies as functions of redshift, stellar mass, specific star formation rate, stellar mass overdensity, metallicity, and galaxy size with this equation: $\log t_{{}_{\mathrm{depl}}}(Gyr)=A+B\log(1+z)+C\log sSFR+D\log M_{*}+E\log(R)+F\log(Z)+G\log(1+\delta)$ Model | Variable | Coefficient | STD | Conf.int | P-value | Model | Variable | Coefficient | STD | Conf.int | P-value
---|---|---|---|---|---|---|---|---|---|---|---
Model 1 | Const. | 0.672 | 0.005 | [0.662, 0.683] | 0.0 | Model 6 | Const. | 0.576 | 0.013 | [0.552, 0.601] | 0.0
$R^{2}=$ | 1+z | -2.11 | 0.002 | [-2.114, -2.106] | 0.0 | $R^{2}=$ | 1+z | -2.113 | 0.002 | [-2.118, -2.109] | 0.0
0.833259 |  $\delta$MS | -0.433 | 0.0 | [-0.434, -0.432] | 0.0 | 0.833295 |  $\delta$MS | -0.437 | 0.001 | [-0.438, -0.435] | 0.0
| $M_{*}$ | -0.04 | 0.001 | [-0.041, -0.039] | 0.0 |  | $M_{*}$ | -0.033 | 0.001 | [-0.035, -0.032] | 0.0
|  $\mathrm{R_{h}}$/Re0 | 0.82 | 0.001 | [0.818, 0.823] | 0.0 |  |  $\mathrm{R_{h}}$/Re0 | 0.808 | 0.002 | [0.804, 0.812] | 0.0
|  |  |  |  |  |  | Z | -0.015 | 0.002 | [-0.019, -0.012] | $2.5\times 10^{-17}$
Model 2 | Const. | 0.692 | 0.006 | [0.681, 0.703] | 0.0 | Model 7 | Const. | 0.586 | 0.013 | [0.561, 0.611] | 0.0
$R^{2}=$ | 1+z | -2.112 | 0.002 | [-2.117, -2.108] | 0.0 | $R^{2}=$ | 1+z | -2.116 | 0.002 | [-2.12, -2.112] | 0.0
0.833369 |  $\delta$MS | -0.432 | 0.0 | [-0.433, -0.432] | 0.0 | 0.833413 |  $\delta$MS | -0.437 | 0.001 | [-0.438, -0.435] | 0.0
| $M_{*}$ | -0.042 | 0.001 | [-0.043, -0.041] | 0.0 |  | $M_{*}$ | -0.035 | 0.001 | [-0.037, -0.033] | 0.0
|  $\mathrm{R_{h}}$/Re0 | 0.821 | 0.001 | [0.818, 0.824] | 0.0 |  |  $\mathrm{R_{h}}$/Re0 | 0.808 | 0.002 | [0.804, 0.812] | 0.0
| $1+\delta$ | 0.004 | 0.0 | [0.004, 0.005] | 0.0 |  | Z | -0.017 | 0.002 | [-0.021, -0.013] | $2.1\times 10^{-27}$
|  |  |  |  |  |  | $1+\delta$ | 0.005 | 0.000 | [0.004, 0.005] | 0.0
Model 3 | Const. | -2.346 | 0.005 | [-2.356, -2.335] | 0.0 | Model 8 | Const. | -2.954 | 0.014 | [-2.981, -2.927] | 0.0
$R^{2}=$ | 1+z | -1.036 | 0.002 | [-1.04, -1.031] | 0.0 | $R^{2}=$ | 1+z | -1.009 | 0.002 | [-1.013, -1.004] | 0.0
0.865279 | sSFR | -0.433 | 0.0 | [-0.433, -0.432] | 0.0 | 0.866192 | sSFR | -0.450 | 0.001 | [-0.451, -0.449] | 0.0
| $M_{*}$ | -0.168 | 0.001 | [-0.169, -0.167] | 0.0 |  | $M_{*}$ | -0.141 | 0.001 | [-0.143, -0.14] | 0.0
|  $\mathrm{R_{h}}$/Re0 | 0.837 | 0.001 | [0.834, 0.839] | 0.0 |  |  $\mathrm{R_{h}}$/Re0 | 0.775 | 0.002 | [0.772, 0.779] | 0.0
| $1+\delta$ | 0.002 | 0.0 | [0.001, 0.002] | $4.3\times 10^{-13}$ |  | Z | -0.078 | 0.002 | [-0.081, -0.075] | 0.0
|  |  |  |  |  |  | $1+\delta$ | 0.002 | 0.000 | [0.002, 0.003] | $2.8\times 10^{-24}$
Model 4 | Const. | 1.933 | 0.005 | [1.922, 1.944] | 0.0 | Model 9 | Const. | 1.807 | 0.015 | [1.778, 1.835] | 0.0
$R^{2}=$ | 1+z | -1.496 | 0.002 | [-1.499, -1.493] | 0.0 | $R^{2}=$ | 1+z | -1.510 | 0.002 | [-1.515, -1.506] | 0.0
0.833369 |  $\delta$MS | -0.432 | 0.0 | [-0.433, -0.432] | 0.0 | 0.833413 |  $\delta$MS | -0.437 | 0.001 | [-0.438, -0.435] | 0.0
| $M_{*}$ | -0.231 | 0.001 | [-0.232, -0.229] | 0.0 |  | $M_{*}$ | -0.220 | 0.001 | [-0.223, -0.218] | 0.0
| $\mathrm{R_{h}}$ | 0.821 | 0.001 | [0.818, 0.824] | 0.0 |  | $\mathrm{R_{h}}$ | 0.808 | 0.002 | [0.804, 0.812] | 0.0
| $1+\delta$ | 0.004 | 0.0 | [0.004, 0.005] | 0.0 |  | Z | -0.017 | 0.002 | [-0.021, -0.013] | $2.1\times 10^{-27}$
|  |  |  |  |  |  | $1+\delta$ | 0.005 | 0.000 | [0.004, 0.005] | 0.0
Model 5 | Const. | -1.081 | 0.005 | [-1.092, -1.07] | 0.0 | Model10 | Const. | -1.782 | 0.016 | [-1.813, -1.751] | 0.0
$R^{2}=$ | 1+z | -0.408 | 0.002 | [-0.412, -0.405] | 0.0 | $R^{2}=$ | 1+z | -0.427 | 0.002 | [-0.43, -0.424] | 0.0
0.865279 | sSFR | -0.433 | 0.0 | [-0.433, -0.432] | 0.0 | 0.866192 | sSFR | -0.450 | 0.001 | [-0.451, -0.449] | 0.0
| $M_{*}$ | -0.36 | 0.001 | [-0.361, -0.359] | 0.0 |  | $M_{*}$ | -0.319 | 0.001 | [-0.321, -0.318] | 0.0
| $\mathrm{R_{h}}$ | 0.837 | 0.001 | [0.834, 0.839] | 0.0 |  | $\mathrm{R_{h}}$ | 0.775 | 0.002 | [0.772, 0.779] | 0.0
| $1+\delta$ | 0.002 | 0.0 | [0.001, 0.002] | $4.3\times 10^{-13}$ |  | Z | -0.078 | 0.002 | [-0.081, -0.075] | 0.0
|  |  |  |  |  |  | $1+\delta$ | 0.002 | 0.000 | [0.002, 0.003] | $2.8\times 10^{-24}$