#  ZFIRE \- The Gas Inflow Inequality for Satellite Galaxies in Cluster and
Field Halos at $z=2$

Anishya Harshan,1,2 Kim-Vy Tran,1,2 Anshu Gupta2,3 Glenn G. Kacprzak2,4
Themiya Nanayakkara2,4
1School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
2ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia
3International Centre for Radio Astronomy Research (ICRAR), Curtin University,
Bentley WA, Australia
4Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn, VIC 3122, Australia

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Gas inflow into galaxies should affect the star formation and hence the
evolution of galaxies across cosmic time. In this work, we use TNG100 of the
IllustrisTNG simulations to understand the role of environment on gas inflow
rates in massive galaxies at $z\geq 2$. We divide our galaxies ($\log({\rm
M}_{\star}/{\rm M}_{\odot})$$\geq 10.5$) into cluster
($\log\rm{M_{halo}/M_{\odot}}\geq 13$) and field
($\log\rm{M_{halo}/M_{\odot}}<13$) galaxies at $z=2$ and further divide into
centrals and satellites. We track their gas inflow rates from $z=6$ to 2 and
find that the total gas inflow rates of satellite galaxies rapidly decline
after their infall into cluster halos and as they reach the cluster center. At
$z=2$, the gas inflow rate of cluster satellite galaxies is correlated with
the cluster-centric radii and not the host halo mass. In contrast, the gas
inflow rate in centrals is strongly correlated with the host halo mass at
$z\geq 2$. Our study indicates that between redshifts 6 to 2, the gas that
normally is accreted by the satellite galaxies is redirected to the center of
the cluster halo as inflows to the cluster centrals and forming the intra-
cluster medium. Our analysis suggest that the inequality of gas accretion
between massive satellite and central galaxies is responsible for the
starvation of cluster satellite galaxies that evolve into the massive quenched
cluster galaxies observed at $z<0.5$.

###### keywords:

galaxies: evolution – galaxies: clusters: general – galaxies: high-redshift

††pubyear: 2022††pagerange: ZFIRE \- The Gas Inflow Inequality for Satellite
Galaxies in Cluster and Field Halos at $z=2$–A

##  1 Introduction

Understanding gas flows is quintessential to understanding the evolution of
star formation in a galaxy and hence galaxy growth. In the $\Lambda$CDM model
of the universe, pristine or metal-poor gas is accreted onto galaxies as fuel
for star formation over cosmic time. As star formation progresses, the gas
reservoir of the galaxy is depleted and energetically ejected in form of winds
and outflows driven by star formation or the active-galactic nuclei (AGN). The
ejected gas is metal-enriched and may either be accreted back onto the galaxy
or enrich the circumgalactic medium (CGM) and the inter-galactic medium (IGM).
The progression of the baryonic cycle across cosmic time describes the
properties of galaxies, and yet remains poorly understood.

Numerical simulations predict that gas accretion in a galaxy can proceed in
several ways. Gas accreted onto a massive galaxy ($\log M_{h}/M_{\odot}>11$)
from the surrounding IGM is shock heated and will eventually radiatively cool
(Rees & Ostriker, 1977; Silk, 1977) and settle onto the galactic disk. This is
called “hot-mode" accretion. Alternatively, pristine gas can flow into the
galaxy via filaments from the cosmic web and is called “cold-mode" accretion
(Kereš et al., 2005, 2009; Nelson et al., 2013; Stern et al., 2019; Wright et
al., 2021). Galaxies can also acquire gas from mergers with other galaxies
(Anglés-Alcázar et al., 2017; Sparre et al., 2022). The different modes of gas
accretion occur simultaneously and leave different imprints on the evolution
of galaxies and their star formation .

A series of paper from the ZFIRE survey suggests that, the environment of a
galaxy has been shown to correlate with star formation rates and interstellar
medium properties of galaxies as early as $z=2$ (Tran et al., 2010;
Kawinwanichakij et al., 2017; Forrest et al., 2017; Alcorn et al., 2019;
Watson et al., 2019; Harshan et al., 2020, 2021). Gravitational and
hydrodynamical interactions with other galaxies and the intra-cluster medium
(ICM) in dense environments preferentially affects the gas in satellite
galaxies (Gunn & Gott, 1972; Moore et al., 1996; Fumagalli et al., 2009;
Cortese et al., 2021). At lower redshifts $z<0.8$, studies find a deficiency
of atomic and molecular gas fractions in galaxies in high density environments
(Scott et al., 2012; Peng et al., 2015; Brown et al., 2017). In contrast,
studies of galaxies in proto-clusters at $z>1.5$ find higher gas fractions
(Noble et al., 2017; Rudnick et al., 2017; Hayashi et al., 2017; Aoyama et
al., 2022). However, measuring gas fractions in galaxies at $z>1$ remains
challenging due to sensitivity limits and access to key spectral lines.

Environment plays a crucial role in regulating the gas accretion history of
individual galaxies. Numerical simulations suggest that galaxies residing in
the filaments of the cosmic web acquire their gas mass through cold-mode
accretion (Birnboim & Dekel, 2003; Ocvirk et al., 2008; Kereš et al., 2009;
Liao & Gao, 2019). Using cosmological simulations, studies like Simha et al.
(2009) and van de Voort et al. (2017a) find that satellite galaxies in massive
halos have lower gas accretion compared to central galaxies. van de Voort et
al. (2017a) also find that the gas inflow rate in central galaxies is not
strongly affected by the environment of the galaxy. In a recent study, Chun et
al. (2020) show that at $z<5$, when the gas in the cosmic web is ionized,
accretion of gas is a function of surrounding gas density and the
position/proximity of the galaxy to the cosmic web. On the other hand,
galaxies that fall into massive halos lose their gas reservoir due to
interaction with the cluster potential, the ICM, or nearby galaxies.

Observational studies of gas inflows remain challenging at higher redshifts
and have relied on the detection of Ly$\alpha$ emission around quasars and
protoclusters, or quasar sightlines (Dijkstra & Loeb, 2009; Bouché et al.,
2013; Kacprzak et al., 2016; Zabl et al., 2019; Umehata et al., 2019; Fu et
al., 2021; Daddi et al., 2021, 2022). Additionally, interpretation of the
observational results of inflows is made complicated by the effect of
outflows, active galactic nuclei (AGN), and star formation. Studies using
Ly$\alpha$ emission line find a higher occurrence of Ly$\alpha$ emitting halos
around high density environments at $z>2$ indicating active gas accretion
(Umehata et al., 2019; Daddi et al., 2021, 2022), however, accretion in
isolated galaxies in the high redshift universe is not easily observable.

In Harshan et al. (2021), using IllustrisTNG we show that the massive galaxies
in the proto-cluster environment at $z\sim 2$ have been gas poor since $z\sim
5$ compared to their field counterparts. We hypothesize that lower gas
fractions in massive proto-cluster galaxies directly result in lower star
formation in cluster environment as measured using the star formation
histories from ZFOURGE \- ZFIRE observations (Straatman et al., 2016;
Nanayakkara et al., 2016; Tran et al., 2015). The lower gas fraction can arise
due to a lower inflow of gas from the surrounding medium either due to
feedback, (Zinger et al., 2020) or the position of the galaxy in the cosmic
web, gas poor mergers, and outflows. In this work, we test our hypothesis by
tracking how gas inflow onto individual galaxies varies with the environment.

This paper is organized as follows. In section 2, we describe the data from
cosmological simulations IllustrisTNG and methodology to describe gas inflows.
In sections 3 and 4 we state our results and discussion, and in section 5
summarise the results.

##  2 IllustrisTNG and Methodology

IllustrisTNG is a suite of magneto-hydro-dynamical cosmological simulations
based on the $\Lambda$CDM cosmology (Pillepich et al., 2018b; Nelson et al.,
2018; Springel et al., 2018; Marinacci et al., 2018; Naiman et al., 2018).
IllustrisTNG extends the Illustris framework with kinetic black hole feedback,
magneto-hydrodynamics, and a revised scheme for galactic winds, among other
changes (Weinberger et al., 2017; Pillepich et al., 2018a). It spans a range
of galaxy environments including halo masses as high as $\rm{M}_{halo}\sim
10^{14.6}\rm{M}_{\odot}$ at $z=0$.

In this work, we use TNG100 of the IllustrisTNGproject with
$\rm{L}_{box}=110.7$ cMpc, and a total volume $\sim 10^{6}~{}\rm{Mpc}^{3}$, to
map the gas inflow rate histories of the cluster and field galaxies at $z=2$.
The data of TNG100 is publicly available and described by Nelson et al.
(2019). The TNG100 simulation has a baryonic mass resolution of
$\rm{m}_{b}=1.4\times 10^{6}~{}\rm{M}_{\odot}$ which provides about 1000
stellar particles per galaxy for a galaxy with stellar mass $\log({\rm
M}_{\star}/{\rm M}_{\odot})$$=9$, and proportionally more stellar particles
for more massive galaxies.

Following the selection in Harshan et al. (2021), we select galaxies
associated with halos of $\rm{M}_{halo}\geq 10^{13}\rm{M}_{\odot}$ as cluster
galaxies and galaxies in halos $\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$ as field
galaxies at $z=2$.

In observational studies, field galaxies are defined as galaxies in relatively
under-dense environments including smaller group halos. We further divide the
cluster and field sample into central and satellite populations. The central
galaxy of each halo is defined as the most massive galaxy in the friends-of-
friends (FOF) group (halo) and satellite galaxies in each halo are all
galaxies except the central in the FOF group (halo).

In Harshan et al. (2021), we find that only the most massive galaxies show a
consistently low total gas fraction in the progenitors and hence, for this
analysis we are only considering massive galaxies at
$\log\rm{M}_{*}/\rm{M}_{\odot}\geq 10.5$ in both cluster and field halos. To
separate the effect of stellar mass and environment, we have mass matched the
field sample (both centrals and satellites) with the cluster satellite sample
(Table 1). Mass matching is done by selecting all galaxies (not one-to-one) in
the field sub-samples within $\pm 0.01$ dex range of each cluster satellite
galaxy. We excluded cluster satellites without a stellar mass match in the
field samples. Ultimately, we obtain a sample of 394 galaxies in the field
centrals, 69 field satellites and 44 cluster satellite galaxies. Due to the
stellar-to-halo mass relation (Moster et al., 2018; Behroozi et al., 2019;
Pillepich et al., 2018b), central galaxies of massive clusters are hosted by
more massive halos than central galaxies in the field. Thus the central
galaxies in cluster halos are significantly more massive than the rest of the
sample, hence we are unable to mass match the cluster central sample to the
cluster satellite sample meaningfully (Sample sets are summarised in Table 1).

Figure 1: The stellar mass distribution of field
($\log\rm{M_{halo}/M_{\odot}}<13$) centrals (solid blue line) and field
satellites (dashed yellow line) stellar mass matched with the cluster
($\log\rm{M_{halo}/M_{\odot}}\geq 13$) satellites (red dashed line) from
TNG100 of IllustrisTNG at $z=2$. The solid teal line shows the stellar mass
distribution of the cluster centrals. The dashed vertical lines indicate the
median value of each sample indicated by the respective color and the number
of galaxies in each sample is indicated in the label. The histograms are
normalised by the total number of counts and the bin width.

The stellar mass distribution of four samples is presented in Figure 1 with
median stellar mass for each sample indicated with vertical dashed lines. The
median stellar mass for field centrals ($\log({\rm M}_{\star}/{\rm
M}_{\odot})$$=10.6\pm 0.1$), field satellites ($\log({\rm M}_{\star}/{\rm
M}_{\odot})$$=10.6\pm 0.1$) and cluster satellites ($\log({\rm M}_{\star}/{\rm
M}_{\odot})$$=10.6\pm 0.1$) fall within $1\sigma$ deviation. The median
stellar mass for the cluster centrals is $\log({\rm M}_{\star}/{\rm
M}_{\odot})$$=11.2\pm 0.2$. We perform the Kolmogorov-Smirnov test to test the
equality of the mass matched sample (excluding the massive cluster central
population). We calculate the K-S test statistic of $0.1$ and p-value = $0.73$
between the field centrals and cluster satellites and 0.15 (K-S statistic) and
0.50 (p-value) between the field and cluster satellites. Thus the stellar mass
distribution of selected field centrals and satellites conforms to the cluster
satellites. Throughout the paper, field centrals are referred to as ‘F
centrals’, field satellites as ‘F satellites’, cluster centrals are referred
to as ‘C centrals’ and cluster satellites as ‘C satellites’ in the figure
legends.

Sample | Sample Size |  $\log({\rm M}_{\star}/{\rm M}_{\odot})$a | Gas Radiib |  $\rm{R/R}_{200}$c |  $\log D_{5}$d
---|---|---|---|---|---
F Satellites | 69 | $10.6\pm 0.1$ | $15.8\pm 0.8$ | $0.9\pm 0.1$ | $2.0\pm 0.1$
F Centrals | 394 | $10.6\pm 0.1$ | $24.5\pm 1.0$ | - | $2.2\pm 0.0$
C Satellites | 44 | $10.6\pm 0.1$ | $17.3\pm 1.3$ | $0.5\pm 0.2$ | $1.9\pm 0.0$
C Centrals e | 24 | $11.2\pm 0.2$ | $27.6\pm 2.4$ | - | $1.6\pm 0.1$

Table 1: Summary of selected centrals and satellites sample in field and
cluster environments

  * a

Median log Stellar Mass of the sample

  * b

Median Gas Radii is presented in kpc

  * c

Median Halo-centric Distance (R) normalised by the $R_{200}$ of the host halo

  * d

Median log of fifth nearest neighbour distance is presented in kpc

  * e

C Centrals are not stellar mass matched with C Satellites

###  2.1 Measuring Gas Inflows

We measure the gas radii of the galaxies by calculating the gas density
profile of each galaxy within 10 times the stellar half mass radii
($10R_{h}$), in order to minimize the contribution of gas from the halo
itself. We consider the radii where the gas density first drops to within 10%
of the average gas density at $10R_{h}$ as the gas radii of the galaxy.
Although the satellite galaxies show a clear flattening of the gas density
profiles, central galaxies do not. The gas in the halo is associated to the
central galaxy which would describe gas density profile and the higher gas
radii in the centrals compared to the stellar mass matched satellites. The
median gas radii of the four samples are reported in Table 1. The median gas
radii for the entire stellar mass matched sample (i.e., field centrals, field
satellites and cluster satellites) is $22.4\pm 0.8$ kpc.

Inside the gas disk, the gas turbulence is mainly driven by radial flows.
Whereas, gas accretion affects the outskirts of the gas disk (Forbes et al.,
2022). Thus, we calculate the inflowing gas at approximately the gas radii. We
calculate the inflowing gas at 20 kpc radii in a 5 kpc shell by calculating
the radial velocity of each gas cell in the prescribed shell and measure the
total gas mass inflow rate at a specific redshift as:

$\dot{\rm{M}}_{\rm{in,gas}}(\rm{z})=\sum_{i=0}^{N}{\rm{v_{rad}}_{i,z}\rm{M}_{i,z}/\Delta
r}$ (1)

where $\rm{v_{rad}}_{i,z}$ is the radial velocity of each gas cell of mass
$\rm{M}_{i,z}$ at radius $20<\rm{r_{i}}<25$ kpc from the center of the galaxy
at each redshift $z$. The radial velocity is calculated with respect to the
center of the galaxy taking into account the motion of the galaxy and the
local Hubble expansion. Additionally, we measure the gas inflow rates using
gas cells with conservative threshold of $\rm{v_{rad}}<-50\rm{km/s}$ in the
considered shell.

Figure 2: Gas column density (top) and radial velocity (bottom) distribution
of gas in two example galaxies at z = 2 (left: satellite; right: central)
projected face-on. The black circles (top panels) show twice the stellar half
mass radius of the galaxy and white shells depict the 20 kpc and 25 kpc radius
from the center of the galaxy. We calculate the rate of inflowing gas in the 5
kpc shell at 20 kpc radii from the center of the galaxy. The inflowing gas is
shown in green with darkness increasing with the increasing radial velocity
and the outflowing gas is depicted in red. The images are 100 kpc x 100 kpc at
$z=2$ created using the IllustrisTNG data visualisation tool (https://www.tng-
project.org/data/vis/; credit: TNG Collaboration).

Figure 2 shows an example of a central and a satellite galaxy. The top panel
shows the column density of total gas with density increasing from purple to
yellow. The bottom panel show the radial velocity of the gas cells associated
with the galaxy such that the inflowing gas is colored in green and the
outflowing gas is red. The images are overlayed with the spherical shell shown
in white in which the gas inflow rate is calculated.

We use data from the SUBLINK algorithm which tracks the merger histories of
the galaxies (Rodriguez-Gomez et al., 2015) to quantitatively follow the
evolution of galaxies and consider only the main progenitor branch
(considering the most massive progenitors at each snapshot) to calculate the
gas inflow rates and associated histories of the galaxies.

##  3 Results: History of Gas Inflow Rates

We measure the history of total gas inflow rates of satellite and central
galaxies in field ($\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$) and cluster
($\rm{M}_{halo}\geq 10^{13}\rm{M}_{\odot}$) halos at $z\geq 2$ as described in
section 2.1 and show in Figure 3. The stellar mass matched field satellites
(Yellow dashed line; Figure 3) and field centrals (Blue solid line ; Figure 3)
have a similar history of gas inflow rates. In contrast, the stellar mass
matched cluster satellite galaxies (Red dashed line; Figure 3) have higher gas
inflow rates compared to the field (centrals + satellites) sample at $z>4$,
after which the gas inflow rates in cluster satellites decreases ($\approx
0.5$ dex decline) by $z=2$. The decline in gas inflow rates in cluster
satellites indicates an effect of environment in the early universe (at
$6>z>2$). However, this effect is measured in satellites of cluster halos
only.

Cluster centrals (Teal solid line; Figure 3) have an increasing gas inflow
rate at $6>z>3$, after which the gas inflow rate is constant for the next 1.3
Gyrs. By $z=2$, cluster centrals have an $\approx 0.5$ dex higher gas inflow
rate compared to the field centrals (Table 2). However, cluster centrals are
not stellar mass matched with the other three sample sets, therefore it is
difficult to disentangle if their stellar mass and/or environment is driving
their higher gas mass inflow rate. Within the stellar mass matched sample, in
the appendix A we explore the effect of stellar mass of the galaxy on the gas
inflow rates. We find that at $z\leq 3$, the stellar mass of the galaxy has no
significant correlation with the gas inflow rates before and after infall.

Sample | $z=2$ | $z=3$ | $z=4$ | $z=6$
---|---|---|---|---
F Satellites | $10.8\pm 0.1$ | $11.1\pm 0.2$ | $11.0\pm 0.2$ | $10.2\pm 0.6$
F Centrals | $10.9\pm 0.01$ | $11.1\pm 0.2$ | $11.0\pm 0.3$ | $10.2\pm 0.6$
C Satellites | $10.7\pm 0.7$ | $11.1\pm 0.3$ | $11.2\pm 0.2$ | $10.4\pm 0.6$
C Centrals | $11.4\pm 0.3$ | $11.5\pm 0.3$ | $11.4\pm 0.5$ | $10.8\pm 0.6$
Table 2: Log of gas inflow rates ($\rm{M_{\odot}Gyr^{-1}}$) at z = 2, 3, 4 and
6 Figure 3:  Gas inflow rate histories of massive galaxies selected at $z=2$
depend on their position in the halo. The solid lines show the bootstrapped
median of gas inflow histories and the shaded region shows the $1\sigma$
error. The gas inflow rates of satellite galaxies in cluster halos (red) are
higher at $z>4$ compared to the field centrals (blue). The rate of gas inflow
of total gas decreases rapidly in the satellite galaxies associated with
larger cluster halos at $z\gtrsim 3$. The number of galaxies in each bin is
given in parenthesis in each label.

###  3.1 Gas Inflow Rates: Before and After Infall

In Figure 3, we find that the cluster satellites undergo a drastic change and
the gas inflow rates decrease by $\approx 0.5$ dex in $\sim 1.5$ Gyrs (at
redshifts 4 to 2), but the same is not observed in field satellites. In this
section, we investigate how the gas inflow rates of satellite galaxies in
cluster ($\rm{M}_{halo}\geq 10^{13}\rm{M}_{\odot}$) and field
($\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$) halos change before and after infall
into their host halo at $z=2$.

We measure the infall time of satellite galaxies at $z=2$ in the following
manner. We assume that the merger tree of the most massive galaxy in a FOF
halo also tracks the merger history of the FOF halo (Gupta et al., 2018). We
reconstruct the main progenitor branch history by identifying the host halo of
the progenitor of the most massive galaxy at each previous snapshot between
$2\leq z\leq 6$. We calculate the infall times of each satellite galaxy as the
snapshot when the galaxy becomes bound to the progenitor halo. Figure 4 shows
the gas inflow rates of the cluster satellites (left panel) and field
satellites (right panel) before and after their infall into the host halo at
$z=2$.

Figure 4:  The gas inflow rates in cluster satellites rapidly decline after
infall, in contrast, the stellar mass matched field satellites have comparable
history of gas inflow rates with the field centrals before and after infall.
The red (left) and yellow (right) lines show the history of gas inflow rates
of cluster (left) and field (right) satellites before (solid) and after
(dashed) infall into the host halo at $z=2$ respectively. The solid blue lines
in both panels show the history of gas inflow rates for field centrals. Shaded
regions show $1\sigma$ of the distribution.

Before infall into the host halo at $z=2$, cluster satellite sample have
higher gas inflow rates compared to the stellar mass matched field centrals at
$z\gtrsim 3$. At $z\lesssim 3$, the gas inflow rates of cluster satellites
sample closely follows that of the field centrals. However, Gas inflow rates
of field satellites closely follow that of the field centrals throughout their
history between $6>z>2$ (yellow solid line; Figure 4 right panel).

After infall into the host halo at $z=2$, cluster satellites (red dashed line;
Figure 4 left panel) have rapidly declining gas inflow rates. The gas inflow
rates of galaxies decline by $\approx 0.6-0.4$ dex in $\sim 1.5$ Gyrs once
they become bound to the massive cluster halo. In contrast, after infall into
their respective host halo at $z=2$, field satellites continue to have
comparable gas inflow rates with the field centrals (yellow dashed line;
Figure 4 right panel). The lack of decline in gas inflow rates in the field
satellites suggests that the environment affects the gas inflow rate of
galaxies only in massive cluster halos (median
$\log\rm{M_{halo}/M_{\odot}}=13.3\pm 0.03$).

###  3.2 Gas Inflow Rates vs Host Halo Mass

In Figure 3, we show that the cluster centrals ($\rm{M}_{halo}\geq
10^{13}\rm{M}_{\odot}$) have higher ($\approx 0.5$ dex) gas inflow rates
compared to the field centrals ($\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$) at
$z=2$. In contrast, the satellite sample show an inverse relation at $z=2$
with the field satellites having higher gas inflow rates than the cluster
satellites. In this section, we explore the correlation between gas inflow
rates of galaxies with their host halo mass at different redshifts.

Figure 5 shows the history of gas inflow rates of field centrals and cluster
centrals selected at $z=2$. The gas inflow rates are presented at $z=6,3$ and
2 against the host halo mass of the galaxies at respective redshifts. We
calculate the Spearman correlation coefficient (r) to quantify any correlation
between the gas inflow rates of the galaxies and their host halo masses (Table
3). Gas inflow rates in cluster and field centrals show a very strong
correlation with the host halo mass in the early universe until $z=3$, after
which the strength of the correlation is moderate by $z=2$ (Table 3).

Throughout the history ($6>z>2$) of central galaxies, the cluster centrals
occupy higher halo masses (median $\log\rm{M_{halo}/M_{\odot}}=13.2\pm 0.03$
at $z=2$) and have higher correlation between gas inflow rates and their host
halo masses compared to the field centrals (median
$\log\rm{M_{halo}/M_{\odot}}=12.3\pm 0.02$ at $z=2$). The high value of r
indicates that the gas inflow rates of centrals are dependent on their host
halo mass at $6>z>2$, and the strength of the correlation at each redshift
increases with increasing halo mass (Table 3).

Figure 5 shows the history of gas inflow rates of stellar mass matched field
satellites and cluster satellites at redshifts 6, 3 and 2. In the early
universe at $z=6$, we find a strong positive correlation between the gas
inflow rates in galaxies and their respective host halo masses in both cluster
satellites and field satellites (Table 3). At $z=6$, the cluster and field
satellites are associated with halos of similar halo masses and have
comparable gas inflow rates. By $z=3$, the gas inflow rates in field
satellites and their host halo mass show moderate correlation, in contrast,
the cluster satellites show no significant correlation (Table 3). By $z=2$,
both field and cluster satellites show no significant correlation (Table 3)
between the gas inflow rates and their host halo mass.

We show in Figure 5 and Section 3.1, that a population of cluster satellites
have lower gas inflow rates compared to the field centrals at $z=2$. The lack
of correlation between the gas inflow rate in the cluster satellites with
their host halo mass indicate that the lower gas inflow rates in this
population of cluster satellites is not driven by the host halo mass. We
investigate the cause of low gas inflow rate in this population in Section
3.3.

Figure 5:  History of gas inflow rates of field and cluster satellite galaxies
(top panel: shown in yellow and red contours respectively) and central
galaxies (lower panel: shown in blue and teal contours respectively) elected
at $z=2$ vs parent halo mass of the galaxy at redshifts 6, 3 and 2 (left to
right). The gas inflow rates of both cluster and field satellites and centrals
show a significant positive correlation with the halo mass of the host halo at
$z=6$. However by $z<=3$, gas inflow rates of satellite galaxies have no
significant dependence on the host halo mass of the galaxy, whereas centrals
continue to have a positive correlation with the halo mass. The crosshairs
help visually compare the gas inflow rates - host halo mass plane relation for
the different populations at each redshift snapshot.  Table 3: Correlation
Coefficient between Gas inflow rates and Galaxy Properties

Sample |  $\rm{M_{*}}$ (z=2) |  $\rm{M_{*}}$ (z=3) |  $\rm{M_{*}}$ (z=6) |  $\rm{M_{halo}}$ (z=2) |  $\rm{M_{halo}}$ (z=3) |  $\rm{M_{halo}}$ (z=6) |  $\rm{R/R}_{200}$ (z=2) |  $\log D_{5}$ a (z=2)
---|---|---|---|---|---|---|---|---
F Satellites | 0.10 | 0.36 | 0.10 | 0.11 | 0.46 | 0.78 | 0.36 | 0.0
F Centrals | 0.11 | 0.13 | 0.11 | 0.39 | 0.45 | 0.88 | - | $-0.40$
C Satellites | 0.16 | $-0.17$ | 0.16 | 0.01 | 0.19 | 0.92 | 0.54 | 0.27
C Centrals | - | - | - | 0.55 | 0.72 | 0.95 | - | $-0.67$

  * a

$\log D_{5}$ is inversely proportional to local galaxy overdensity, hence, a
negative value of correlation coefficient (r) implies a positive correlation
of gas inflow rate with the local galaxy overdensity.

###  3.3 Gas Inflow Rates vs Halo-centric Radius

We show in section 3.1 and Figure 4 that after infall, the gas inflow rates
decrease rapidly in satellites of cluster halos only. However, the gas inflow
rates for satellite galaxies at $z=2$ do not depend on the host halo mass. To
resolve the seemingly contradictory results, we consider the correlation of
gas inflow rates and the halo-centric distance.

Figure 6:  The gas inflow rate of cluster satellites (red contours; right
panel) show a positive moderate correlation with the halo-centric radii
normalized by $\rm{R}_{200}$ of the host halo at $z=2$. In contrast, the gas
inflow rate of field satellites (yellow contours; left panel) shows a weak
correlation with the halo-centric distance. The contours show the distribution
of stellar mass matched sample of field satellite and cluster satellite
galaxies on gas mass inflow rates - halo-centric radii plane respectively. The
Correlation coefficient (r) for each sample is shown in respective panels.

Figure 6 shows gas inflow rates of the stellar mass matched field satellites
and the cluster satellites as a function of their halo-centric distance at
$z=2$. We calculate the proper distance (R) of galaxies from the halo center
(defined by the position of the particle with minimum gravitational potential
energy) and normalize it by $\rm{R}_{200}$ (radius of a sphere centered at the
halo center where mean density is 200 times the critical density of the
Universe) at $z=2$. We calculate the Spearman correlation coefficient (r)
between the gas inflow rates of satellite galaxies and their halo centric
distance $\rm{R/R}_{200}$ and report in Table (3).

The gas inflow rate is weakly correlated to $\rm{R/}R_{200}$ for field
satellites and moderately correlated for cluster satellites (Table 3). The
stellar mass matched cluster satellites occupy the halo closer to the cluster
center (median $\rm{R/R}_{200}=0.5\pm 0.2$) and have lower gas inflow rates
compared to the field satellites (median $\rm{R/R}_{200}=0.9\pm 0.1$). This
indicates that after galaxies fall into the massive cluster halos, their gas
inflow rates decrease as they approach the halo center (Table 2).

###  3.4 Gas Inflow Rates vs Local Galaxy Overdensity

In this section, we investigate the effect of local galaxy overdensity on the
gas inflow rates of galaxies. We calculate the local galaxy overdensity by
measuring the proper distance of the fifth nearest neighbor ($\rm{D_{5}}$;
neighbor $\log({\rm M}_{\star}/{\rm M}_{\odot})$$\geq 8$) for each galaxy at
$z=2$. Figure 7 shows gas inflow rates of the four sample sets as a function
of the fifth nearest neighbor distance.

We calculate the Spearman correlation coefficient (r) between the gas inflow
rates and the fifth nearest neighbor distance ($\rm{D_{5}}$). The gas inflow
rates in cluster and field centrals are positively correlated with the local
galaxy overdensity at $z=2$ (Table 3). The cluster centrals are in higher
density environments (median $\log(\rm{D}_{5}/\rm{kpc})=1.6\pm 0.1$ ) and have
higher gas inflow rates compared to field centrals (median
$\log(\rm{D}_{5}/\rm{kpc})=2.2\pm 0.02$ ; Table 2). This indicates an effect
of local galaxy overdensity on the gas inflow rates of centrals.

Figure 7:  Gas inflow rate vs local galaxy overdensity (quantified as 5th
nearest neighbor distance; $\rm{D_{5}}$) for stellar mass matched cluster
satellites (red contours; left top), field satellites (yellow contours; right
top) and field centrals (blue contours; right bottom) at $z=2$. The teal
contours show the gas inflow rates of cluster centrals (not stellar mass
matched) at $z=2$. The correlation coefficient (r) for each sample is shown in
respective panels. The gas inflow rates in cluster and field centrals show a
positive correlation with the local galaxy overdensity. The cluster satellites
show a weak negative correlation with the local galaxy overdensity. In
contrast, field satellites show no correlation with the local galaxy
overdensity.

The cluster satellite galaxies show a weak negative correlation between gas
inflow rates and the local galaxy overdensity. However, gas inflow rates in
field satellites show no correlation with the local galaxy overdensity. The
cluster satellite galaxies are in higher density environments (median
$\log(\rm{D}_{5}/\rm{kpc})=1.9\pm 0.04$) and have lower gas inflow rates
compared to field satellites (median $\log(\rm{D}_{5}/\rm{kpc})=2.0\pm 0.1$).
Contrary to the effect on centrals, the local galaxy overdensity has a
negative effect on the gas inflow rates of satellite galaxies.

##  4 Discussion

Gas inflow in galaxies is expected to depend on the galaxy environment. We use
IllustrisTNG cosmological simulations to study the effect of environment on
the gas inflow rates in massive galaxies ($\log({\rm M}_{\star}/{\rm
M}_{\odot})$$\geq 10.5$) at $z\geq 2$. We select central and satellites galaxy
samples in low mass (field; $\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$) and high
mass (cluster; $\rm{M}_{halo}\geq 10^{13}\rm{M}_{\odot}$) halos at $z=2$.
Furthermore, we have stellar mass matched the field satellite and centrals
with the cluster satellite sample. The same can not be done for the cluster
centrals because they are $\approx 4$ times more massive (median $\log({\rm
M}_{\star}/{\rm M}_{\odot})$$=11.2\pm 0.2$) compared to the cluster satellite
galaxies (median $\log({\rm M}_{\star}/{\rm M}_{\odot})$$=10.6\pm 0.1$). Our
analysis supports a picture where gas inflow rates vary as a function of
environment as characterized by mass of the host halo, halo-centric radius,
and local galaxy overdensity.

###  4.1 Cluster vs Field

Satellite galaxies in cluster halos selected at $z=2$ have high gas inflow
rates compared to the field centrals before infall into the host halo at $z=2$
(solid red line in Figure 4). After the infall into massive cluster halos, gas
inflow rates in cluster satellites rapidly decline (dashed red line in Figure
4). However, the same trend is not observed in the field satellite galaxies
selected at $z=2$, that show comparable gas inflow rates with the field
centrals before and after infall (yellow solid and dashed lines in Figure 4).

Similar to our result with IllustrisTNG, studies using other simulations like
Simha et al. (2009) find that gas accretion rates in satellite galaxies at
$z=0$ have declined to zero within 0.5-1 Gyrs after infall. At $z=2$, we find
that decline in gas inflow rates in cluster satellite galaxies after infall is
$\approx 0.5$ dex in 1 Gyr. In agreement with our results, Simha et al. (2009)
and van de Voort et al. (2017b) also find a divergence in gas inflow rates of
satellite sample from the stellar mass matched "control" central sample only
when the satellite sample are residing in more massive halos.

Before infall into the host halo at $z=2$, the massive satellite galaxies
could be the centrals of their host halos. At $z=6$-4, cluster satellites have
high gas inflow rates (Figures 3 and 4) and are in more massive host halos
compared to the field satellites (Figure 5). As the gas inflow rates of
centrals at redshifts 6 to 2 is driven by their halo mass, we conclude that
the higher gas inflow rates in the cluster satellites at redshifts 6 to 4 is
also driven by their higher halo mass.

In section 3.3, we show the dependence of gas inflow rates of cluster and
field satellite galaxies on the halo centric distance. Both cluster and field
satellites (stellar mass matched) have lower gas inflow rates as they approach
the halo center. The trend is stronger in cluster satellites compared to the
field satellites as they are also in more overdense environment (Figure 7). We
also find that the massive field satellites are in the outskirts (median
$\rm{R}/{R}_{200}=0.9\pm 0.1$) of the halo, whereas the massive cluster
satellites are closer (median $\rm{R/R}_{200}=0.5\pm 0.2$) to the halo center
(Figure 6; Table 1). The cluster satellites are nearly a factor of two closer
to their central galaxy compared to the field satellites. Thus it is likely
that the cluster satellites have been part of their parent halo for longer
compared to the field satellites. This could drive the lower gas inflow rates
in cluster satellites compared to the field satellites.

###  4.2 Centrals vs Satellites

The gas inflow rates in satellite galaxies decreases with decreasing halo-
centric distance (Figure 6). Cluster satellites within $\sim R_{200}$ radii
have lower gas inflow rates and drive the decline in the median gas inflow
rates of the entire cluster satellite sample at $z=2$ (median
$\rm{R/R}_{200}=0.5\pm 0.2$, Figure 4). Similar decline in gas inflow rates is
not measured in the field satellite sample because they are in the outskirts
of the halo (median $\rm{R}/{R}_{200}=0.9\pm 0.1$; Figure 6). Gas inflow rates
in both cluster and field satellites show a significant correlation with their
halo and stellar mass at $z>3$ (Figures 5 and 8). At $z>3$, our satellite
sample has not fallen into the cluster potentials they are associated to at
$z=2$. Thus before infall, these massive galaxies, could be the centrals of
their parent halos at $z>3$ and their gas inflow rates are driven by their
host halo mass.

Contrary to the satellites, gas inflow rates in central galaxies of both
cluster and field halos have a strong dependence on the host halo mass (Figure
5) and local galaxy overdensity (Figure 7). van de Voort et al. (2017a) and
Simha et al. (2009) also find comparable results. The opposite correlation of
the gas inflow rate with galaxy overdensity between cluster centrals and
satellites could also indicate two different modes of gas accretion. While
centrals are being fed by the filaments in the cluster center increasing gas
accretion, the gas around the satellite galaxies in higher overdensities are
unable to cool efficiently and hence hinder gas accretion (van de Voort &
Schaye, 2012; Chartab et al., 2021).

Gas inflow rates of the field centrals are correlated with their stellar mass
at redshifts 6 to 3, but not after $z\leq 3$ (appendix A). However, we find
that the gas inflow rates in the field centrals are correlated with their halo
mass for longer (at redshifts 6 to 2) (Figure 5). Thus we infer that the gas
inflow rates in the centrals is driven mainly by their host halo mass and not
their stellar mass.

The emerging scenario from our results is that after infall, satellites
galaxies in massive cluster halos compete for gas with the cluster halo and
the central galaxy (closer to the lowest point in halo potential and higher
local galaxy overdensity). Therefore, cluster satellites are not able to
accrete gas as they approach the halo center in massive halos. The gas inflow
rates in satellite galaxies in field halos (that tend to be in the outskirts
of the halo) are not affected by the environment (Figure 7; Table 1).

###  4.3 Comparing Gas Fractions and Gas Inflow Rates from Observations to
IllustrisTNG

In Harshan et al. (2021), we find that the total gas fraction is lower only in
the most massive protocluster galaxies compared to the field galaxies. In this
work, we show that the total gas inflow rates (primordial +mergers) in the
cluster satellite galaxies is lower than the field counterparts at $z=2$. We
also find that the cluster satellites have a higher gas inflow rates compared
to their field counterparts until $z\gtrsim 4$ before their infall into the
cluster potential (driven by their halo mass, Figure 5). This indicates that
in the formative stages of the clusters, satellite galaxies have higher gas
inflow rates than field galaxies. This is consistent with observational
studies that find a higher molecular gas fraction in galaxies in the
protocluster environment (Cucciati et al., 2014; Noble et al., 2017; Rudnick
et al., 2017) at $z\sim 2$.

Direct detection of gas inflow rates continues to be elusive in observations
at high redshifts and relies heavily on $\rm{Ly}\alpha$ emission, CGM
absorption studies with quasar sightlines and mass-metallicity relation. In
the past few decades an increasing number of proto-cluster/ filament galaxies
at $z>2$ with a higher frequency of $\rm{Ly}\alpha$ emission line detections
have been made (Cantalupo et al., 2014; Hennawi et al., 2015; Umehata et al.,
2019; Daddi et al., 2021). Although $\rm{Ly}\alpha$ emission is used to
identify gravitationally driven gas inflows, the interpretation of
$\rm{Ly}\alpha$ emission is not without degeneracies. Often, $\rm{Ly}\alpha$
emission can be explained by outflows and photo-ionization by star formation
and quasar. Within the interpretation of $\rm{Ly}\alpha$, studies find that at
$2<z<3.3$ galaxies in filaments, galaxy groups or protoclusters (Daddi et al.,
2021, $12.8<\log\rm{M_{halo}/M_{\odot}}<14.2$),(Christopher Martin et al.,
2014; Umehata et al., 2019, $\log\rm{M_{halo}/M_{\odot}}\sim 15.4$) have
active cold gas inflows. In context of our results from IllustrisTNG, higher
gas inflow rates in protocluster and groups indicate early stage of cluster
formation.

Previous observational studies also investigate the effect of environment on
gas accretion using the mass-metallicity relations and find conflicting
results. At $z\sim 2$ studies like Kacprzak et al. (2015); Alcorn et al.
(2019) find no significant difference between the metallicities of
protocluster ($\log\rm{M_{halo}/M_{\odot}}\sim 14.4$) and field galaxies,
Shimakawa et al. (2015) find higher metallicities in protocluster
($\log\rm{M_{halo}/M_{\odot}}\sim 14$) and hypothesise that the enhancement of
metallicity in protocluster galaxies is due to the gas stripping. Chartab et
al. (2021) find relative metallicity of galaxies in overdense regions reverses
between redshifts 1.5 and 2. At $z\sim 2$ galaxies in high density
($1+\delta>1.77$) environment are relatively metal poor, potentially due to
the effective cooling and/or higher gas inflow rates in the early phases of
cluster assembly. However, by $z\sim 1.5$ galaxies in high density environment
($1+\delta>1.69$) becomes metal-rich. Our results suggest that as galaxies
fall into the higher density environment, their gas inflow rates decline hence
increasing their metallicities. Hence the contradicting observational results
could be explained by different stages of protocluster assembly or selection
of central and satellite galaxies. We also note that metallicity indicators
using nebular emission lines trace the ionized ISM and hence may not be robust
means of interpreting primordial gas inflows.

###  4.4 Gas Inflow and Galaxy Quenching

In Harshan et al. (2021), by constructing star formation histories using SED
fitting of multi-wavelength data (0.1 - 1 microns), we find that the massive
galaxies in the proto-cluster environment have formed $45\pm 8\%$ of their
total stellar mass by $z=3$ compared to field galaxies that have formed $31\pm
2\%$ of their total stellar mass. Using IllustrisTNG, we find that the
difference in star formation histories could be driven by the lower total gas
fraction in the progenitors of massive cluster galaxies compared to the field
counterparts. This indicates that the selected massive protocluster galaxies
had lower gas inflows (from primordial inflow + mergers) at $z\lesssim 5$. In
context of the results presented in this paper, we can confirm that the lower
gas fraction of proto-cluster galaxies is in part due to lower gas inflow
rates in cluster satellite galaxies. Thus our results suggest that the
observed onset of decline in star formation rates in $z=2$ massive
protoclusters galaxies is at least in part driven by gas starvation. This
early onset of environmental quenching in proto-cluster galaxies at $z=2$
could contribute to the large population of massive quenched galaxies in
clusters at $z<0.5$.

The gas inflow rate of the cluster satellites increases at redshifts 6 to 4
but starts declining after their infall into the cluster halo at $z~{}4$. On
the other hand, the gas inflow rate in the cluster central increases until
$z~{}2$. This indicates that as massive cluster satellites are falling into
the cluster potential, the gas that was available for accretion at redshifts 6
to 4 prior to infall, is now being redirected to the cluster potential and the
cluster centrals, increasing their gas inflow rates. This diversion of gas
could be due to the higher gravitational potential of the cluster centrals and
the cluster halo. Thus cluster centrals have a higher constant gas inflow
rates at redshifts 3 to 2, compared to the gas inflow rates in cluster
satellites. The gas that is accreted onto the cluster halo would eventually
build the hot ICM that is observed in $z>1$ clusters (Rabitz et al., 2017;
Stanford et al., 2001) and further impact the gas accretion in satellite
galaxies in cluster halos. The decreased gas inflow rates after infall in
addition to the interaction with the ICM will further decrease the star
formation in the satellite galaxies building up the quenched cluster
population observed at $z<0.5$. We will extend our work and investigate the
cause of constant gas inflow rates in centrals (a redshifts 4 - 2) and the
effect of the ICM on gas inflow rates and star formation in a future study.

##  5 Summary

In Harshan et al. (2021), we find that the lower star formation in massive
protocluster galaxies compared to the field galaxies at $z=2$ is driven by
lower total gas fractions since $z\sim 5$. To investigate the cause of lower
total gas fractions in protocluster galaxies, we study gas inflow rates. In
this work, we use IllustrisTNG cosmological simulations to understand the role
of environment in driving the gas inflow rates in the massive ($\log({\rm
M}_{\star}/{\rm M}_{\odot})$$>10.5$) galaxies. We divide our sample into
centrals and satellites in cluster ($\rm{M}_{halo}\geq 10^{13}\rm{M}_{\odot}$)
and field ($\rm{M}_{halo}<10^{13}\rm{M}_{\odot}$) halos. Our main results are
summarised as follows:

1) History of gas inflow rates at $6>z>2$: Until $z=2$, gas inflow rates in
cluster centrals is higher compared to the field centrals. At $z=2$, gas
inflow rates in cluster centrals ($\log\dot{\rm{M}}_{\rm{in,gas}}$ $=11.4\pm
0.3$ $\rm{M_{\odot}Gyr^{-1}}$) are $\approx 0.5$ dex higher compared to field
centrals which are not stellar mass matched ($\log\dot{\rm{M}}_{\rm{in,gas}}$
$=10.9\pm 0.01$ $\rm{M_{\odot}Gyr^{-1}}$). The cluster satellites also have
higher gas inflow rates compared to the stellar mass matched field centrals at
$6>z>\sim 4$ (Figure 3). At $z<\sim 3$, the gas inflow rates in cluster
satellites rapidly declines (by $\approx$ 0.4-0.6 dex) as they fall into the
cluster potential. However, the history of gas inflow rates in field satellite
galaxies do not show this trend and closely follow that of the field centrals
(Figure 4).

2) Gas inflow rates as a function of halo mass: At $z>4$, we find a strong
correlation between the gas inflow rates of both cluster and field satellites,
which is not observed at $z<4$. By $z=2$, the gas inflow rates of the cluster
satellites decline compared to the field satellites (Figure 5). On the other
hand, gas inflow rates of the central galaxies are strongly correlated with
the host halo mass until $z=2$ (Figure 5). This indicates that satellites are
competing for gas accretion with the centrals that have an advantageous
position in the halo.

3) Gas inflow rates as a function of halo-centric distance: Gas inflow rates
of cluster satellites show a moderate correlation with the halo-centric radius
such that the inflow rates decrease as galaxies approach the cluster center. A
weak correlation between gas inflow rate and halo-centric distance is seen in
the field satellites. Massive field satellites are also on average in the
outskirts of the field halo, unlike the cluster satellites. This indicates
that the field satellites have not been associated with the field halo for
long unlike the cluster satellites and could explain the lack of decline in
gas inflow rates in field satellites.

4) Gas inflow rates as a function of local galaxy overdensity: Gas inflow
rates of central galaxies in both field and cluster halos are positively
correlated with the local galaxy overdensity. Cluster centrals that live in
higher mass halos and higher local galaxy overdensity have significantly
higher gas inflow rates compared to the field centrals. Thus for centrals, gas
inflow rates are primarily governed by the host halo mass and the local galaxy
overdensity at $z=2$. However, the gas inflow rates in cluster satellites only
show a weak negative correlation with the local galaxy overdensity and the
field satellites show no correlation with the local galaxy overdensity. This
indicates that for satellite galaxies the local galaxy overdensity have
minimal or no effect on the gas inflow rates.

We show that the effect of environment on the evolution of galaxies becomes
significant in the very early universe (at $z\sim$ 3-4). Unlike the effects of
environment in the local universe which are driven by galaxy-galaxy
interactions or hydrodynamical interactions with the intracluster medium, in
the early universe, the effect of environment on galaxies is driven by the
position of the galaxies in the cosmic web (e.g., central vs satellite and
high vs low host halo mass). While the gas inflow rates in centrals are driven
mainly by the host halo mass and the local galaxy overdensity, gas inflow
rates for satellite galaxies is driven by the halo-centric distance and the
time since infall.

The rapid decline in gas inflow in cluster satellites and the high gas inflow
in cluster centrals indicate that the gas is redirected from the satellites to
the cluster halo. This redirected gas should build up (starting as early as
$z\sim 3$) the ICM and is also accreted onto the cluster centrals. Whereas the
cluster satellites with declining gas inflow rates become gas starved. Thus
the quenching of the massive cluster satellites driven by starvation as shown
in this work would lead to the build up of the massive quenched cluster
populations that is observed in the local universe.

Further investigation of the inflowing gas by distinguishing the multi-phase
gas and studying the evolution to $z=0$ is required to understand the build up
of ICM and quenching in cluster galaxies driven by gas starvation. While the
observational signature for gas accretion remains elusive, future multi-
wavelength observational campaigns producing deep spatially resolved
spectroscopy of the high redshift galaxies will be able to test the
predictions made in cosmological simulations.

## Acknowledgements

The authors would like to thank Dr. Annalisa Pillepich for insightful feedback
for the paper. The authors acknowledge the support of the Australian Research
Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO
3D), through project number CE170100013.

## Data Availability

The TNG100 simulations are publicly available from the IllustrisTNG
repository: https://www.tng-project.org. The processed data of this article
will be shared on reasonable request to the corresponding author.

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##  Appendix A Appendix - Gas Inflow Rates vs Stellar Mass of a Galaxy

Figure 8: Before infall, the gas inflow rates of stellar mass matched field
(yellow contours; top row) and cluster satellite galaxies (red contours;
bottom row) are comparable at redshifts 6,4 and 2.6 (left to right). The blue
contours in both top and bottom rows show the comparison with the field
centrals at the respective redshifts. Spearman correlation coefficient (r)
between gas inflow rate and stellar mass of the galaxy is presented in the
respective color in each panel. The crosshairs help visually compare the gas
inflow rates for the different populations at each redshift snapshot. The
positive correlation of gas inflow rates with the stellar mass of field and
cluster satellites at $z>4$ vanishes at $z\leq 4$. Figure 9: After infall, the
gas inflow rates of cluster satellites (red contours; bottom row) decline
compared to the stellar mass matched field satellite sample (yellow contours;
top row). The blue contours in both top and bottom rows show the comparison
with the field centrals at the respective redshifts. Spearman correlation
coefficient (r) between gas inflow rates and stellar mass of the galaxy is
presented in the respective color in each panel. The crosshairs help visually
compare the gas inflow rates for the different populations at each redshift
snapshot. After infall into the host halo, gas inflow rates of both cluster
and field satellites show no significant correlation with the stellar mass of
the galaxy.

In Figure 3 we show that at $z=2$, the massive cluster centrals (median
$\log({\rm M}_{\star}/{\rm M}_{\odot})$$=11.2\pm 0.2$) have higher (by $\sim
0.5$ dex ) gas inflow rates compared to the field centrals and the satellite
sample (median $\log({\rm M}_{\star}/{\rm M}_{\odot})$$=10.6\pm 0.1$). We test
if the high gas inflow rate of cluster centrals are driven by their stellar
mass by investigating if the gas inflow rates of a galaxy depends on their
stellar mass at different redshifts and environments.

Figure 8 shows the gas inflow rates as a function of the stellar mass of the
galaxy before infall for field satellites (yellow contours, top row) and
cluster satellite samples (red contours, bottom row) selected at $z=2$. The
blue contours in both top and bottom rows show the comparison with gas inflow
rates of the stellar mass matched field centrals selected at $z=2$. We measure
the correlation between the stellar mass of the galaxy and the gas inflow
rates at redshifts 6,4 and 2.6 by performing the Spearman correlation test for
each sample set (summarised in Table 3).

In the early universe (at $z=6$), the Spearman correlation coefficient (r) of
the gas inflow rates with the stellar mass of the field centrals, field
satellites and cluster satellites are r $\geq 0.7$ indicating a strong
positive correlation (Table 3). The strength of correlation between gas inflow
rates and stellar mass of the galaxy decreases with decreasing redshift. At
$z=4$, the correlation between gas inflow rates and stellar mass is moderate
for field centrals and weak for the satellite sample (both field and cluster).
By $z=2.6$, there is no significant correlation between the gas inflow rates
with the stellar mass of the field satellites, field centrals and cluster
satellites ($r<0.2$). The correlation coefficient for field centrals ($r<0.2$;
Table 3) at redshifts 3 to 2 also indicates no significant correlation between
gas inflow rates and the stellar mass of the galaxy at $z\leq 3$ (Figure 9).

After infall in the host halo, field and cluster satellites show no
significant correlation ($r<0.2$; Table 3) between the gas inflow rates and
the stellar mass of the galaxy at $3\geq z\geq 2$ (Figure 9). However, at
$z=2$ a population of cluster satellites have lower gas inflow rates compared
to the field centrals and field satellites. We show in sections 3.3 and 3.4,
that the lower gas inflow rates in the cluster satellites is correlated to
their halo-centric radii and local galaxy overdensity.

We show Figure 3 that the gas inflow rates of cluster and field centrals have
similar shape, albeit the gas inflow rates in the cluster centrals is
significantly higher. The lack of correlation of gas inflow rates of field
centrals with their stellar mass at at redshifts 3 to 2, indicates that the
gas inflow rates of the cluster centrals would not be correlated to their
stellar mass at $z=2$, instead is correlated with the halo mass and the local
galaxy overdensity (Figures 5 and 7).