# The physical origins of gas in the circumgalactic medium using
observationally-motivated TNG50 mocks

Simon Weng,1,2,3,4 Céline Péroux,1,5 Rahul Ramesh,6 Dylan Nelson,6 Elaine M.
Sadler,2,3,4 Martin Zwaan,1 Victoria Bollo,1 Benedetta Casavecchia,7
1 European Southern Observatory, Karl-Schwarzschildstrasse 2, D-85748 Garching
bei München, Germany
2 Sydney Institute for Astronomy, School of Physics A28, University of Sydney,
NSW 2006, Australia
3 ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D)
4 ATNF, CSIRO Space and Astronomy, PO Box 76, Epping, NSW 1710, Australia
5 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de
Marseille) UMR 7326, 13388, Marseille, France
6 Universität Heidelberg, Zentrum für Astronomie, Institut für theoretische
Astrophysik, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany
7 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, D-85748
Garching b. München, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The gas in the circumgalactic medium (CGM) of galaxies is being probed by
absorbers in the spectrum of background objects in observations, yet the
physical properties of the medium remain unconstrained. We use the
cosmological hydrodynamical simulation TNG50 to statistically trace the
origins of H i Ly-$\alpha$ absorbers around galaxies at $z=0.5$ with stellar
masses ranging from 108 to 1011 M⊙. By including all gas with a line of sight
velocity within $\pm 500$ $\text{km s}^{-1}$ of the central, we mimic
observational studies of the CGM to quantitatively assess the impact of other
galaxy haloes and overdense gas in the IGM that intersect sightlines. We find
that 75 per cent of H i absorbers with column densities
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>16.0$ trace the central galaxy
within $\pm 150$ (80) $\text{km s}^{-1}$ of $M_{*}=10^{10}(10^{8})$ M⊙ central
galaxies. The impact of satellites to the total absorber fraction is most
significant at impact parameters $0.5R_{\rm vir}<b<R_{\rm vir}$ and satellites
with masses below typical detection limits ($M_{*}<10^{8}$ M⊙) account for 10
(40) per cent of absorbers that intersect any satellite bound to $10^{10}$ and
$10^{11}$ $(10^{9})$ M⊙ centrals. After confirming outflows are more dominant
along the minor axis, we additionally show that at least 20 per cent of
absorbers are found without strong radial motions, highlighting that absorbers
can also trace quasi-static gas. The metallicity of absorbers also depends on
the azimuthal angle, but this signal is largely driven by enriched inflowing
and quasi-static gas. Our work shows that determining the stellar mass of
galaxies at $z_{\rm abs}$ is essential to constrain the physical origin of the
gas traced in absorption, which in turn is key to characterising the
kinematics and distribution of gas and metals in the CGM.

###### keywords:

quasars: absorption lines – galaxies: evolution – galaxies: kinematics and
dynamics – galaxies: haloes

††pubyear: 2023††pagerange: The physical origins of gas in the circumgalactic
medium using observationally-motivated TNG50 mocks–13

##  1 Introduction

The discovery of extragalactic absorption lines (e.g. Lynds & Stockton, 1966;
Burbidge et al., 1966; Stockton & Lynds, 1966) shortly followed the discovery
of the first quasar/quasi-stellar object (QSO) 3C 273 (Schmidt, 1963). At the
time, it was suggested that the absorption arose from discrete clouds of gas
found in galaxies located along the line of sight (LOS) towards the background
source (Bahcall & Salpeter, 1965). Almost six decades on, the study of
absorbers has evolved significantly but fundamental questions remain
unanswered. One such question concerns the origin of absorbers; while
absorption lines are commonly associated with galaxies near the line-of-sight
(Bahcall & Spitzer, 1969; Bergeron, 1986), what is their spatial distribution,
kinematic behaviour and origin of the gas being traced in and around galaxies?

The distribution of gas in and around galaxies remains in a constant state of
flux. Like a galactic bank account, reservoirs of gas are depleted by constant
withdrawals in the form of star formation. More sudden ejections of gas in the
form of stellar-wind-driven and AGN-driven outflows further reduce the amount
of gas available for star formation but also enrich the surrounding galaxy
halo (Veilleux et al., 2005). Deposits in the form of accreting gas from dark
matter filaments (Kereš et al., 2005, 2009; Nelson et al., 2013), condensed
material from galactic fountains (Fraternali & Binney, 2008; Marinacci et al.,
2010; Fraternali, 2017) and ‘clumpy’ accretion in the form of satellites
(Mihos & Hernquist, 1994; Hernquist & Mihos, 1995; Ramón-Fox & Aceves, 2020)
replenish the gas supply. These processes take place simultaneously in a
delicate balance and affect the stellar properties of galaxies. The study of
the circumgalactic medium (CGM), the region where inflows and outflows leave
their signatures, is of paramount importance to understanding how galaxies
form and evolve.

Intervening quasar absorbers have long been found at the same redshift of
galaxies and intersecting gaseous haloes around galaxies (Bahcall & Spitzer,
1969; Bergeron, 1986). The circumgalactic medium that extends from the
interstellar medium (ISM) to the intergalactic medium (IGM) can be traced by
absorption lines towards background QSOs. Some surveys study the CGM by
preselecting galaxies (using a property such as mass or luminosity) that are
located near quasar sightlines and then searching for and analysing absorption
lines in the QSO spectrum (e.g. Chen et al., 2010; Tumlinson et al., 2011;
Werk et al., 2016; Heckman et al., 2017; Johnson et al., 2017; Berg et al.,
2018; Muzahid et al., 2018; Pointon et al., 2019). Alternatively, one can
identify absorption lines towards a distant background source, typically of a
certain species such as H i or Mg ii, and then search for galaxies at the
absorber redshift (e.g. Steidel & Hamilton, 1992; Schroetter et al., 2016;
Hamanowicz et al., 2020; Lofthouse et al., 2020; Muzahid et al., 2020; Nielsen
et al., 2020; Péroux et al., 2022; Banerjee et al., 2023; Berg et al., 2023).
Finally, surveys can target fields with a UV-bright QSO without a pre-
selection on absorption lines or galaxies (e.g. Prochaska et al., 2011;
Prochaska et al., 2019; Burchett et al., 2019; Chen et al., 2020; Muzahid et
al., 2021). Despite the differing sample selections, surveys of absorption
lines and their associated galaxies have benefited from the advent of integral
field spectroscopy (IFS). The ability to simultaneously obtain imaging and
spectroscopy has led to a proliferation of galaxy–absorber pairs (e.g. $>100$
for Mg ii; Schroetter et al., 2019; Dutta et al., 2020).

It is then inevitable that absorption lines in QSO spectra will intersect gas
associated with processes such as inflows and outflows or structures like
satellites and filaments (Lehner et al., 2013; Lehner et al., 2019; Lehner et
al., 2022; Berg et al., 2023). However, disentangling the relationship between
absorber and galaxy and understanding the origin of the absorber becomes
difficult for many reasons. First and foremost, we are unable to directly
measure the distance between absorber and galaxy in the direction along the
line of sight. The only indirect measure of distance is the line of sight
velocity difference between the galaxy and absorber ($\Delta v_{\rm LOS}$).
However, this is determined by a combination of the Hubble flow and peculiar
velocity of the gas cloud. Several studies using simulations show that
selecting absorbers in velocity space can select for gas beyond several times
the virial radius (Rahmati et al., 2015; Ho et al., 2020, 2021). Identifying
the origin of an absorber becomes further complicated in studies where galaxy
overdensities are found associated with absorbers (e.g. Gauthier, 2013; Bielby
et al., 2017; Manuwal et al., 2019; Hamanowicz et al., 2020). Typically, the
galaxy found at closest impact parameter to the QSO sightline is selected as
the absorber host, but other methods have also been used (Berg et al., 2023).
Finally, the inherent sensitivity limit in observations means that a
population of low-mass and/or quiescent galaxies will remain undetected,
particularly at cosmological redshifts. We can infer the existence of these
galaxies from studies that do not detect any object near high column density
absorbers that are expected to arise from the galaxy disk (e.g. Fumagalli et
al., 2015; Weng et al., 2023a). The combination of these factors means that
assigning an origin to absorbers is challenging but few works have attempted
to quantify these effects and how they vary with properties such as species
and column density.

Intervening absorption lines are expected to probe gas flows into and out of
galaxies. However, distinguishing between outflows and inflows is challenging
because of the line of sight velocity degeneracy between the two gas flows.
Gas infalling onto a galaxy from behind and gas outflowing towards the
observer will both be blueshifted with respect to the systemic redshift of the
galaxy. Likewise, redshifted absorbers can originate from inflows in front of
the galaxy or outflows ejecting gas away from the observer. This degeneracy is
not present in down-the-barrel studies where inflows (outflows) are identified
by redshifted (blueshifted) absorption against the background stellar
continuum (e.g. Martin, 2005; Veilleux et al., 2005; Rubin et al., 2014;
Heckman et al., 2017). To overcome this uncertainty in transverse absorption-
line studies, one must rely on other assumptions.

One possible way to distinguish outflowing absorbers is to measure the
azimuthal angle ($\Phi$) between the absorber and a galaxy’s major axis. In
both observations and simulations of gas outflows that originate in the galaxy
centre, galactic winds commonly form an expanding biconical shape
perpendicular to the galaxy disk as this is the path of least resistance (e.g.
Bordoloi et al., 2011; Lan et al., 2014; Nelson et al., 2019). Hence, one
could assume that outflowing absorbers can be identiifed by their alignment
with the minor axis and inflowing absorbers with the major axis as the
accreting gas co-rotates with the galaxy (Rahmani et al., 2018b; Schroetter et
al., 2019; Zabl et al., 2021). Perhaps further evidence of this major-minor
axis dichotomy can be found in the bimodal distribution of azimuthal angles
for absorbers relative to their galaxy hosts (Bouché et al., 2012; Kacprzak et
al., 2012; Bordoloi et al., 2014; Schroetter et al., 2019). It should follow
then that metal-enriched gas will be preferentially found near the minor axis.
Indeed, various simulations predict an azimuthal angle dependence in
metallicity profiles (Péroux et al., 2020; van de Voort et al., 2021), as well
as density, temperature (Truong et al., 2021; Yang et al., 2023) and magnetic
field strengths (Ramesh et al., 2023a), in line with some observational
studies (Cameron et al., 2021; Zhang & Zaritsky, 2022; Heesen et al., 2023),
although other observations find little to no evidence for such angular
anisotropies (Péroux et al., 2016; Kacprzak et al., 2019; Pointon et al.,
2019; Huang et al., 2021; Wendt et al., 2021).

We also expect outflowing gas to be metal-enriched as heavy elements from the
ISM are ejected into the CGM and IGM via feedback processes (Oppenheimer &
Davé, 2006; Muratov et al., 2017; Nelson et al., 2019; Mitchell & Schaye,
2022). Studies of high-velocity clouds (HVCs) in the Milky Way use a
combination of kinematics and metallicity to distinguish between inflows and
outflows, and also calculate the mass rates of both processes (e.g. Wakker et
al., 1999; Fox et al., 2004, 2016, 2019; Ramesh et al., 2023c). Beyond the
local Universe, some studies of H i absorbers find a multimodal gas-phase
metallicity distribution (Lehner et al., 2013, 2022). The separate populations
have been proposed to trace outflows (metal-rich), inflows (metal-poor) and
gas in the IGM (pristine) (Berg et al., 2023). However, simulations do not
find such large metallicity contrasts between inflows and outflows,
particularly at lower redshift where galactic winds are more efficiently
recycled (Hafen et al., 2017).

In this work, we study the origins of absorbers using the cosmological
magnetohydrodynamical simulation TNG50. We explore how the line of sight
velocity difference between galaxy and absorber compares with the physical
distance. Then, we quantify the contributions of satellite galaxies and gas in
the intergalactic medium to the fraction of absorbers with varying column
densities. We also test the fidelity of azimuthal angle and metallicity
assumptions when identifying gas flows in the CGM. Finally, we compare our
results to studies of Ly-$\alpha$ absorbers at redshift $z\lesssim 1$ such as
the MUSE-ALMA Haloes survey (Péroux et al., 2022) and the COS CGM Compendium
(Lehner et al., 2018). We adopt a cosmology consistent with Planck
Collaboration et al. (2016) and halo masses, circular velocities and radii are
defined at 200 times the critical density of the Universe (e.g. $R_{\rm
vir}=R_{\rm 200c}$), consistent with TNG50.

##  2 Methods

###  2.1 The TNG50 Simulation

We present results from TNG50-1 (Nelson et al., 2019; Pillepich et al., 2019),
the highest resolution version in the IllustrisTNG (henceforth, TNG) suite of
cosmological magneto-hydrodynamical simulations (Marinacci et al., 2018;
Naiman et al., 2018; Nelson et al., 2018; Pillepich et al., 2018b; Springel et
al., 2018). Building on the original Illustris simulation (Genel et al., 2014;
Vogelsberger et al., 2014b, a; Sijacki et al., 2015), IllustrisTNG
incorporates magnetic fields (Pakmor et al., 2014) and amends the original
Illustris galaxy formation model (Vogelsberger et al., 2013; Torrey et al.,
2014) with updated feedback processes (Weinberger et al., 2017; Pillepich et
al., 2018a).

With a box size of $\sim$50 comoving Mpc (cMpc) and 21603 resolution elements,
the TNG50 simulation has a baryonic (dark matter) mass resolution of
$8.5\times 10^{4}$ M⊙ ($4.5\times 10^{5}$ M⊙). The relatively large volume
combined with the high spatial resolution are optimal for the study of
structures in the circumgalactic medium of galaxies.

In the TNG simulations, galaxy stellar mass growth is regulated by supernovae
(SN) and supermassive black holes (SMBH) feedback. The SN and SMBH feedback
models in TNG use an isotropic energy injection. Hence, any directionality is
a natural result of subsequent hydrodynamical and gravitational interactions.
The model for stellar feedback uses a kinetic wind approach where star-forming
gas is stochastically ejected from galaxies by Type II supernovae (Springel &
Hernquist, 2003). Galactic winds are isotropic and the ejected wind particles
are decoupled from surrounding gas in the star-forming environment until the
density or time reaches a threshold, at which point they hydrodynamically
recouple and deposit their mass, momentum, and energy (see Pillepich et al.,
2018a, for details). The TNG stellar feedback model produces high mass loading
galactic-scale outflows which are highly directional and metal-enriched
(Nelson et al., 2019; Péroux et al., 2020; Ramesh et al., 2023b).

In TNG, supermassive black holes are seeded in haloes exceeding a total mass
of $\sim 7\times 10^{10}M_{\odot}$. SMBHs then grow by merging with other
black holes and accreting gas at the Eddington-limited Bondi rate. The mode in
which the active galactic nucleus (AGN) ejects energy into its surroundings is
dictated by this accretion rate. At low-accretion rates relative to the
Eddington limit, kinetic energy is stochastically injected into neighbouring
cells after enough energy is accumulated. On the other hand, thermal energy
heats the surrounding gas at high-accretion rates proportional to the accreted
mass. The thermal feedback mode typically dominates for SMBH masses $\lesssim
10^{8}$ M⊙, and the kinetic mode for high-mass black holes (Weinberger et al.,
2017). Feedback from SMBHs in the TNG model is the physical mechanism of
galaxy quenching, ejecting gas from halo centers (Truong et al., 2020) out to
the scale of the closure radius (Ayromlou et al., 2022), while heating gaseous
haloes and thus preventing future cooling of the CGM (Zinger et al., 2020).

###  2.2 TNG50 Galaxy sample

We identify haloes and subhaloes in the simulation using the subfind algorithm
(Springel et al., 2001). The central subhalo is defined as the one found at
the gravitational potential minimum of a friends-of-friends halo (FoF; Davis
et al., 1985) and the associated baryonic component is termed the central
galaxy. Baryonic components of all other subhaloes associated with the FoF
halo are dubbed satellites.

We focus on galaxies at $z=0.5$ with stellar masses ($M_{*}$) ranging from
$\log(M_{*}/M_{\odot})=8.0$ to $11.0$ in order to match the properties of
galaxies from the MUSE-ALMA Haloes survey (Péroux et al., 2022; Karki et al.,
2023). The survey targets 32 high column density
($\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$) H i absorbers and finds 79
galaxies within $\pm 500$ $\text{km s}^{-1}$ of the absorbers at impact
parameters ranging from 5 to 250 kpc (Weng et al., 2023a).

We consider galaxies that are the central galaxies of their halo and measure
their stellar masses within twice the stellar half mass radius. For the four
stellar mass bins centred around $[10^{8.0},10^{9.0},10^{10.0},10^{11.0}]$
$\rm{M_{\odot}}$, we select [100, 100, 50, 20] galaxies at random within bins
of $\pm 0.3$ dex. We choose the number of central galaxies in each mass bin
such that the number of ‘sightlines’ passing within the virial radius of the
galaxies in each bin is approximately equal. The physical extent of the mocks
along the plane perpendicular to the projection is the minimum of [400 pkpc,
$2R_{\rm vir}$]. The limit of 400 pkpc is set by the physical scales covered
in a single VLT/MUSE (Bacon et al., 2010) pointing at $z\approx 0.5$.

###  2.3 Mock absorption columns

The TNG simulations track the global neutral hydrogen content of gas cells. In
order to split the gas between atomic or molecular hydrogen fractions
individually, we adopt the molecular hydrogen fraction model of Gnedin &
Kravtsov (2011) to estimate the H2 fraction (following Popping et al., 2019).
We then obtain the H i mass by subtracting the H2 gas mass from the total
neutral hydrogen content. The masses of all metallic ions including Mg ii, C
iv and O vi ions are computed in post-processing using v17.00 of the cloudy
(Ferland et al., 2017) code, assuming collisional and photoionization
equilibrium in the presence of the meta-galactic UV background (UVB) from the
2011 update of Faucher-Giguère et al. (2009) (following Nelson et al., 2020;
Nelson et al., 2021).

Our separation of galaxies into centrals versus satellites allows us to flag
each gas cell based on whether it is gravitationally bound to the central, a
satellite of the central or another FoF halo along the line of sight. Gas
cells not bound to any halo (largely gas in the intergalactic medium) are also
flagged. We henceforth refer to this family of flags as the set of
(gravitational) origin labels.

Additionally, we separately categorise each gas cell as outflowing, inflowing
or quasi-static using the cell’s radial velocity ($v_{r}$) with respect to the
central galaxy. Hence, to be classified by one of these three flags, it is a
pre-requisite that the gas cell is gravitationally bound to the central
subhalo (designated as ‘central’ in the origin labels). We calculate the
radial velocity of each gas cell using the scalar product of the gas cell
position relative to the galaxy centre with the velocity in the frame of
reference of the subhalo excluding the Hubble flow. A radial velocity cutoff
of $20$ $\text{km s}^{-1}$ is used to identify inflowing and outflowing cells,
that is, cells with $v_{r}>20$ $\text{km s}^{-1}$ are outflowing and cells
with $v_{r}<-20$ $\text{km s}^{-1}$ are inflowing. Cells found at velocities
$-20<v_{r}<20$ $\text{km s}^{-1}$ are considered to be in quasi-hydrostatic
equilibrium and encompasses gas that is static or rotationally-dominated. We
also include satellite galaxies in this set of flags so all gas cells within
the central subhalo are included. Henceforth, this set of flags are the gas
flow labels. We consider other choices of the radial velocity boundary and the
implications in section 5.

We show a cartoon depicting the two sets of labels (origin and gas flow) in
Figure 1. Within the central galaxy’s halo, they may trace gas accretion
(bluish purple), outflows (orange) or satellite galaxies (green). However,
absorbers may also intersect other galaxy haloes in front of or behind the
central galaxy and these haloes are depicted in yellow. Finally, there is the
chance that absorbers trace overdense gas in filamentary structures (purple).
The various, intermixed possibilities highlight that relating the gas observed
in absorption with its physical origin requires a statistical approach.

Figure 1: A cartoon view of intervening absorption lines towards background
quasars intersecting galaxy haloes. Absorbers can have various origins in the
circum- and intergalactic media. They trace gas flows such as outflows
(orange) and accretion (dark blue) as well as satellites (green). There is
also the possibility of the sightline intersecting other galaxy haloes along
the line of sight (yellow) or filaments in the intergalactic medium (purple).
The gas can be found at a large range of impact parameters and velocities with
respect to any given galaxy and the diversity of possible origins mean that
connecting absorbers with their host galaxies requires a statistical approach.
We use the TNG50 simulation to quantify the contribution of the various
physical origins presented, and how these contributions vary as a function of
observable including the impact parameter, line of sight velocity, metallicity
and azimuthal angle.

To determine the column density of the H i and other ions, we project the gas
along the line of sight around each galaxy onto a large grid with pixel size
$1\times 1$ pkpc2, through a line of sight depth of $\pm 500$ $\text{km
s}^{-1}$ relative to the systemic velocity of the galaxy. We test pixel sizes
with side length 0.5, 1 and 2 pkpc, finding only marginal differences in our
forthcoming results. This is consistent with Szakacs et al. (2022), where the
H2 and H i column density distribution functions in TNG100 are found to differ
only at column densities $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>22$ when
comparing pixel sizes with side length 150 pc and 1 kpc. These regions are
limited to the centre of galaxies that comprise a small gas covering fraction
when compared to the CGM. Hence, we adopt a 1 pkpc2 pixel to match the median
CGM resolution of $\approx 1$ pkpc in TNG50 at $R_{\rm vir}>0.1$ which
increases for larger radii.

The $\pm 500$ $\text{km s}^{-1}$ line of sight depth includes the contribution
from the Hubble flow and we choose $\pm 500$ $\text{km s}^{-1}$ to be
consistent with surveys of absorber-galaxy pairs (e.g. Dutta et al., 2020;
Weng et al., 2023a). We use the standard cubic-spline deposition method
(following Nelson et al., 2016) to spatially distribute the masses of each gas
component. Every pixel in this map corresponds to a single, observable
‘sightline’. For each pixel, we calculate the mass contribution from cells
belonging to the origin and gas flow flags mentioned earlier: [IGM, central,
satellite and other halo] (origin) and [inflow, quasi-static, outflow and
satellite] (gas flow). We assign a flag to each pixel identifying which
component contributes the most mass for that given sightline and then we
determine the gas column density using the chosen flag only. The orientations
of the central galaxies are random as we use three projections of a given halo
that correspond to the fixed $x$-, $y$\- and $z$-axes of the simulation
volume.

In addition to the column density, we calculate other quantities measurable in
observations such as the mass-weighted (using H i) metallicity of the gas,
line of sight velocity difference between absorber and central galaxy ($\Delta
v_{\rm LOS}$), two-dimensional projected distance from the galaxy centre
(impact parameter, $b$) and azimuthal angle between the galaxy’s major axis
and absorber ($\Phi$). We also include mass-weighted quantities such as the
gas temperature, distance along the projection axis ($d_{\rm LOS}$) and
distance from the galaxy centre (3D distance) which are not directly
measurable in observations. These calculations are determined using only the
cells that belong to a given flag. Ultimately, we generate a catalogue of
absorbers with the properties discussed above that are labelled by two sets of
flags that inform us of the origin and gas flow.

In Figure 2, we display four randomly selected galaxies in the sample that
span the entire range of stellar masses from roughly 108.0 to 1011.0 M⊙. The
left column of plots are H i column density maps where the subhalo
identification, stellar mass and physical scale are also given. The central
and rightmost plots separately colour each pixel by the dominant mass
contribution for that sightline using the origin and gas flow flags. We keep
the colour schemes for the two sets of flags consistent throughout the
remainder of this work. The dashed white circle marks the virial radius of the
central galaxy. From the first and second columns, we find that the dense H i
gas typically arises from the centre of centrals, satellites or other haloes,
while lower $\log N(\mbox{\text{H\,{i}}})$sightlines trace the intergalactic
medium. Moreover, we see that satellites and other coincident galaxy haloes
can dominate the projected H i mass for a sightline even at small impact
parameters from the central galaxy (most prominently seen in the second and
third rows). Using the gas flow flags, we show that gas in the CGM has a rich
structure in radial velocity. For the central galaxies depicted at larger
inclinations (second and third rows), outflows appear to be preferentially
directed along the minor axes. Gas accretion can be seen directed along the
major axes but can arise from the central galaxy stripping gas from another
galaxy (second row) or the accretion of gas clouds that co-rotate with the
disk (bottom row). These velocity structures are washed out when the galaxy
appears more face-on (top row), where much of the gas is quasi-static.

Figure 2: A visualisation of $z=0.5$ galaxy haloes in the TNG50 sample of
galaxies. In the left column, we show the projected H i column density maps of
four galaxies belonging to different stellar mass bins from top to bottom. The
plots are also accompanied by the Subhalo index and a bar indicating the
physical scale of the image. The central and rightmost columns are the same
galaxy shown in the left column but pixels are now coloured by the flag that
dominates the H i mass in projection. The black pixels in the right column
signify that none of the gas flow labels meet the column density threshold of
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>13.0$ in those regions. The
colours used directly correspond to the labels used in Figure 1. We show the
virial radius of these haloes with the dashed white circle.

We note that there are subtleties in the categorisation of pixels using the
origin and gas flow depicted in Figures 1 and 2. First, we separate gas
inflows from satellites that may be accreting onto galaxies. Components of
accreting satellites that have been stripped and are infalling onto the
central galaxy will be considered ‘inflows’, whereas gas that is still more
tightly bound to the satellite will be labelled as ‘satellite’. Similarly,
large-scale streams from the IGM, as seen in the rightmost panels for $10^{8}$
and $10^{11}$ M⊙ central galaxies, will be considered inflows only if they are
gravitationally bound to the central galaxy. In the middle column, yellow and
purple dominates the regions beyond the virial radius because other haloes and
the intergalactic medium contribute the most H i for these sightlines. As we
do not consider the second halo term and IGM in the rightmost column, the
black pixels correspond to regions where none of the four gas flow labels
[inflow, quasi-static, outflow and satellite] meet a column density threshold
of $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>13.0$ (see Figure 2). It is
also for this reason that the central galaxy and satellites appear more
extended in the right column of that same figure (e.g. satellites in the top
left corner of the bottom two rows). Pixels previously assigned to the ‘IGM’
or ‘other halo’ origin labels may change into the most dominant of the four
gas flow labels, assuming the column density threshold is met.

##  3 Where is the gas along the line of sight?

In observational studies of absorber-galaxy systems, the impact parameter and
line of sight velocity are two reliably-determined measurables that inform us
on the true distance between absorbers and galaxies, and, by extension,
determine how absorbers are related to surrounding galaxies. While the impact
parameter is a measure of the physical distance in the plane of the sky, the
velocity difference ($\Delta v_{\rm LOS}$) does not directly correlate with
the distance along the line of sight ($d_{\rm LOS}$). An absorber’s line of
sight velocity is a combination of the gas flow being traced, viewing angle of
the galaxy and the Hubble flow. Hence, it is important to evaluate the
position of the absorbers with $\Delta v_{\rm LOS}$ within 300 to 1000
$\text{km s}^{-1}$ of the galaxy systemic redshift (typical values used in
recent surveys: Schroetter et al., 2016; Péroux et al., 2022; Galbiati et al.,
2023) may be beyond the virial radius. In this section, we examine the
relation between $\Delta v_{\rm LOS}$ and line of sight distance, and how the
relation evolves as a function of the gas origin and motion (e.g. absorbers
tracing the IGM, accretion and outflows). Additionally, we estimate the ranges
in $\Delta v_{\rm LOS}$ where the majority of the CGM gas mass lies.

In Figure 3, we show the relationship between the line of sight distance and
line of sight velocity for absorbers associated with a stacked sample of
central galaxies with stellar mass $\log(M_{*}/M_{\odot})=10.0$ at $z=0.5$.
This particular figure is limited to partial Lyman limit systems (pLLS;
$16.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<17.2$) associated with 1010
M⊙ central galaxies. Because this choice is arbitrary, we discuss the effects
of changing column density and stellar mass. We separate absorbers by their
respective origin or gas flow. The top-left panel includes all absorbers
associated to the central galaxy. The panels labelled as ‘Inflow’ and
‘Outflow’ in the top row are decompositions of the top-left plot (we have not
included ‘quasi-static’ gas with radial velocities $|v_{r}|<20$ $\text{km
s}^{-1}$). We display the properties of absorbers that trace gas in
satellites, the intergalactic medium or another halo along the line of sight
in the bottom row respectively from left to right. The colour of each hexbin
is the median impact parameter of all absorbers found in the bin.

Figure 3: A comparison of the line of sight distance with the line of sight
velocity for mock absorbers in TNG50 at $z=0.5$. Positive (negative) line of
sight distances are behind (in front of) the central galaxy away from
(towards) the observer. Similarly, positive (negative) line of sight
velocities are directed away from (towards) the observer. We consider H i
absorbers with column densities
$16.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<17.2$ (partial Lyman-limit
systems) associated with $\log(M_{*}/M_{\odot})\approx 10.0$ central galaxies.
Absorbers are separated by their origin or gas flow which are given in the top
right corner of each plot. Each hexbin is coloured by the median impact
parameter of all absorbers found within that hexbin. We plot the mean $d_{\rm
LOS}$ in 40 bins with width $\sim$10 $\text{km s}^{-1}$ for the ‘Central’,
‘Inflow’, ‘Outflow’ and ‘Satellite’ panels as a thick black line and the
shaded region represents the standard deviation. In general, close separations
in line of sight velocity can correspond to significant separations in line of
sight distance, even when an absorber is tracing the central galaxy. Using a
kernel density estimator, the white contours on the top row show where the
absorbers are concentrated with contour levels of [0.25, 0.50, 0.75] times the
total number of absorbers. We find that 75 per cent of the H i mass traced by
pLLS is enclosed within $|\Delta v|<150$ $\text{km s}^{-1}$ of the 1010 M⊙
central galaxies. This $\Delta v$ limit varies between 70 and 250 $\text{km
s}^{-1}$ for the stellar mass range in our sample but does not change
significantly with column density.

For gas associated with the central galaxy and its satellites, we find a wide
array of line of sight distances for any given line of sight velocity, with
values varying by up to $\sim$ $500$ pkpc at fixed $\Delta v_{\rm LOS}$. One
cause for this is the degeneracy between the physical location of the
absorbing gas and its kinematics. Redshifted absorbers can trace gas accreting
onto the galaxy from behind or gas ejected towards the observer. Similarly,
blueshifted absorption lines can probe outflowing gas expelled away from the
observer or gas infalling onto the galaxy from in front. This is evident from
the black horizontal bars that display the average $d_{\rm LOS}$ associated
with absorbers in 40 bins with width $\sim$ 10$\text{km s}^{-1}$in Figure 3.
Gas at negative line of sight velocities trace both inflowing (outflowing) gas
at positive (negative) distances down the sightline. For centrals, the black
line is almost horizontal at $d_{\rm LOS}=0$ kpc and there is roughly $50$ kpc
of scatter in $d_{\rm LOS}$ at all $\Delta v_{\rm LOS}$. In general, inflows
are indistinguishable from outflows by the velocity of the absorber relative
to the galaxy alone (Tumlinson et al., 2017) and we require other diagnostics
to break the degeneracy (see Section 5).

We also find absorbers associated with satellite galaxies, the intergalactic
medium and other galaxy haloes. In particular, Figure 3 shows that partial
Lyman-limit systems with $|\Delta v_{\rm LOS|}<500$ $\text{km s}^{-1}$ can
trace gas in the IGM or intersect other haloes that are located more than
several pMpc along the line of sight. While these absorbers are typically
found at larger impact parameters, we still find that gas in the IGM and other
haloes can mimic CGM gas of the central galaxy. We quantify the relative
frequency of these absorbers in Section 4. The black horizontal bars
indicating the median line of sight velocity for gas tracing satellites is
similar to absorbers tracing accretion, highlighting that satellites are
infalling onto the central and their average motion resembles gas accretion.
In contrast, the almost linear relationship between $d_{\rm LOS}$ and $\Delta
v_{\rm LOS}$ for gas in the IGM and other haloes is caused by the Hubble flow
dominating at larger scales. We note that the range in LOS distances of the
‘IGM’ and ‘Other Halo’ panels is not the same as the other four.

Nevertheless, it is clear from the inflow-outflow degeneracy combined with the
contribution of absorbers from satellites, the IGM and other haloes that the
line of sight velocity difference between absorber and galaxy is not a direct
indicator of actual distance. By extension, a small velocity separation (e.g.
$|\Delta v_{\rm LOS}|<500$ $\text{km s}^{-1}$) does not imply an absorber is
associated with a given galaxy. This result is in line with the findings of
previous works using the EAGLE simulations (Schaye et al., 2015; Crain et al.,
2015) to analyse H i around massive galaxies (M200 $\gtrsim$
$10^{12}~{}\rm{M_{\odot}}$) at $z\sim 2$$-$3 (Rahmati et al., 2015), and Mg ii
and O vi around stellar mass $10^{9}$ to 1011 M⊙ galaxies at $z\approx 0.3$
(Ho et al., 2020, 2021). More recently, the FOGGIE simluations, find that
structures of varying temperatures, metallicities and densities in the CGM
that span $>100$ kpc can be tightly constrained in velocity space (Peeples et
al., 2019). We similarly show here that the velocity restrictions we apply in
observations lead to a larger LOS path length compared to the virial radius of
galaxies. In addition, when multiple absorption components are separated in
velocity (from tens to hundreds of $\text{km s}^{-1}$) down a single sightline
in observations, our results emphasise that even these smaller velocity
differences can lead to different spatial locations. Studies find
discrepancies larger than 2 dex in metallicity for distinct absorption
components (e.g. Zahedy et al., 2019; Lehner et al., 2022) and we suggest that
these separate components may arise from different physical origins such as
the intergalactic medium.

We are able to give a simple prescription for an appropriate line of sight
velocity limit that encompasses most of the H i gas belonging to the CGM of
the central using the distribution of absorbers in velocity space. Using a
kernel density estimator with contour levels of [0.25, 0.50, 0.75] in Figure
3, we find that 75 per cent of LLS absorbers associated with the central are
found within $\pm 150$ $\text{km s}^{-1}$of galaxies with stellar mass 1010
M⊙. In addition, we find that our prescription of $\pm 150$ $\text{km s}^{-1}$
varies marginally for absorbers of H i column densities larger than 1017.2
atoms cm-2. However, this value is strongly dependent on the stellar (and more
importantly, the halo) mass and decreases to $\pm 70$ (120) $\text{km s}^{-1}$
for $M_{*}\approx 10^{8}$ ($10^{9}$) M⊙ galaxies and is $\pm 250$ $\text{km
s}^{-1}$ for the $M_{*}\approx 10^{11}$ M⊙ galaxies in our sample.
Fundamentally, the increasing virial velocities for more massive haloes drives
the variation in $v_{\rm LOS}$ values. However, this relation is not linear,
with $\Delta v_{\rm LOS}>V_{200}$ (halo circular velocity) for smaller haloes,
and vice versa, for larger haloes. We also find these values are roughly
consistent with the findings of Peeples et al. (2019), where the strongest
absorption is confined to be within $\pm 200$ $\text{km s}^{-1}$ of a
$M_{*}\approx 10^{10}$ M⊙ galaxy at $z=2$. We refrain from a more direct
comparison with FOGGIE because their cosmological zoom simulations are at a
different redshift and do not include contributions from the IGM or other
haloes. These $\Delta v_{\rm LOS}$ estimates serve as useful indicators of
whether an H i absorber is within the halo of a nearby galaxy.

##  4 The physical origin of absorbers

While the limits in $\Delta v_{\rm LOS}$ provided in the previous section are
practically useful for observational studies of absorber-galaxy systems, it is
also important to quantify the likelihood of absorbers originating from the
central galaxy as opposed to satellites, other galaxy haloes along the line of
sight or the intergalactic medium. Here, we explore how the fraction of
sightlines that trace gas outside the central subhalo changes as a function of
impact parameter and H i column density, two observables that are readily
available. In addition, we specifically study the contribution of lower mass
satellites that are below the current sensitivity limit of many observations
($M_{*}\lesssim 10^{8}$ M⊙).

###  4.1 What fraction of H i arise from the central halo?

We have shown that the combination of gas peculiar velocities and the Hubble
flow lead to absorbers of various origins masquerading as belonging to the
central galaxy. Here, we quantify the fractional contribution of the various
gas origins as a function of the impact parameter. We use the impact parameter
in particular because typically, the galaxy at lowest impact parameter is
considered to be associated with the absorber in observational surveys (e.g.
Langan et al., 2023; Weng et al., 2023a). Likewise, we consider all gas within
$\pm 500$ $\text{km s}^{-1}$ of the central galaxy to mimic a typical line of
sight velocity cut in observations (Dutta et al., 2020, 2021; Hamanowicz et
al., 2020; Galbiati et al., 2023; Weng et al., 2023a).

Figure 4: The fraction of H i absorbers that arise from gas in the IGM,
central, satellites or other haloes as a function of the impact parameter
(normalized by $R_{\rm vir}$ in the top axis) at $z=0.5$. A vertical dashed
black line indicates the virial radius. Here, we consider four stellar mass
bins ranging from $\log(M_{*}/M_{\odot})=8.0$ to $11.0$ in one dex intervals
and three column density ranges:
$17.2<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$ (LLS),
$19.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<20.3$ (sub-DLAs) and bottom;
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>20.3$ (DLAs). The central galaxy
dominates at impact parameters up to $\approx 0.5$ $R_{\rm vir}$, though this
limit decreases for higher column densities. At roughly $0.5R_{\rm
vir}<b<R_{\rm vir}$, satellite galaxies dominate the fraction of absorbers and
beyond the virial radius, other haloes along the line of sight contribute most
to the absorber count. There is only a marginal contribution from the
intergalactic medium at these H i column densities. These results highlight
that knowledge of the stellar mass of galaxies associated with quasar
absorbers is key to relate the gas probed in absorption with its origin.

In Figure 4, we show the fraction of absorbers that trace gas in the central
(blue), a satellite of the central (green), another galaxy halo along the line
of sight (yellow) and gas in the intergalactic medium (purple). We express the
fraction as a function of impact parameter (normalized by the virial radius in
the top $x$-axis) for each of the four origins. The twelve panels span four
stellar mass bins in ascending order from left to right and three H i column
density ranges: LLS (top;
$17.2<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$), sub-DLAs (centre;
$19.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<20.3$) and DLAs (bottom;
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>20.3$). We use bins of size 10
(20) kpc to calculate the fractions for the LLS and sub-DLA (DLA) column
densities.

We find that these fractions are strongly dependent on the stellar mass of the
central galaxy. Absorbers associated with more massive central galaxies
dominate out to larger physical impact parameters, but this is caused by more
massive galaxies having a larger virial radius (see normalized $b/R_{\rm vir}$
on the top $x$-axis). Indeed, the slopes of the central curves when viewed as
a function of the normalized impact parameter are almost identical across our
four stellar mass bins. Moreover, for stellar masses
$\log(M_{*}/M_{\odot})\leq 10.0$, we often find a peak in the contribution
from satellites before the ‘other halo’ term begins dominating. This is caused
by the fact that satellites are limited to being gravitationally bound to the
central. It is only for the 1011 M⊙ central galaxies where the virial radius
is larger than the mocks (as seen in Figure 2) that the contributions from
satellite galaxies have not peaked. We note again that these size restrictions
for the most massive galaxies in the plane perpendicular to projection are
physically motivated by the scales covered by the field of view of VLT/MUSE at
$z\sim 0.5$ which is the median redshift of the MUSE-ALMA Haloes survey
(Péroux et al., 2022). Ultimately, the general trend we find, which is roughly
independent of the mass and column density, is that the central galaxy
dominates the covering fraction up to $b\approx 0.5R_{\rm vir}$, followed by
satellite galaxies until the virial radius and then other galaxy haloes beyond
$R_{\rm vir}$.

There are also differences between the H i column density bins (increasing
column density from top to bottom rows). Figure 4 reveals a steeper decline in
absorbers associated with centrals for higher H i column densities, as these
densities drop rapidly in the CGM towards larger impact parameters (e.g Ramesh
et al., 2023b). This is accompanied by a greater fractional contribution of
satellites and other haloes at these impact parameters, as a result of
sightlines intersecting the dense ISM of these objects. The curves are more
irregular because strong absorbers are rarer, even with larger bin sizes. We
also see that the contribution from gas in the intergalactic medium (purple)
is minor for column densities $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]\geq
19.0$, but increases in prevalence at lower N${}_{\text{H\,{i}}}$ and larger
impact parameters, as expected since this gas is predominantly not H i as a
result of ionization by the UV background.

This becomes clearer in Figure 5 where we plot the trend of H i column density
as a function of the impact parameter for the four stellar mass bins. The
solid coloured line in the primary panel is the median $N_{\text{H\,{i}}}$
value at a given impact parameter for $M_{*}=10^{11}$ M⊙ central galaxies. The
faded grey curves represent the $N_{\text{H\,{i}}}$-$b$ for the lower stellar
mass bins which are shown and coloured in the secondary panels. Grey vertical
lines mark the impact parameters where the median column density drops below
the DLA, sub-DLA, LLS and pLLS thresholds. The coloured area is proportional
to the fractional contribution (right $y$-axis) of the four origin flags for
each range in column density. For instance, DLAs found towards our
$M_{*}=10^{11}$ M⊙ central galaxies (primary panel) can be attributed 75 per
cent of the time to the central, 15 per cent to satellites and the remainder
to other galaxy haloes. This calculation is made using DLAs found across all
impact parameters, as opposed to Figure 4 where the fractions are given as a
function of $b$. The black dashed vertical lines mark 0.15, 0.5 and 1 times
the virial radius to mark the inner and outer boundaries of the CGM.

Figure 5: The median H i column density of absorbers versus given impact
parameter for galaxies with stellar masses covering
$\log(M_{*}/M_{\odot})=8.0$ to $11.0$ at $z=0.5$. We show the
$N_{\text{H\,{i}}}$-$b$ relation in the primary panel for $M_{*}=10^{11}$ M⊙
central galaxies with the coloured line. The faded grey lines represent the
same relation but for the lower stellar mass bins found in the secondary
panels. The vertical dashed grey lines mark the impact parameters where the
median $\log N(\mbox{\text{H\,{i}}})$ transitions from damped Ly-$\alpha$
absorbers to partial Lyman-limit systems for $\log(M_{*}/M_{\odot})=11.0$
galaxies. The last impact parameter bin covers column densities
$13.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$ (Ly-$\alpha$ forest).
The coloured area in each bin is proportional to the fraction of absorbers
(with $\log N(\mbox{\text{H\,{i}}})$consistent with the respective bin) that
belong to the categories of IGM, central, satellite and other halo. For
example, the first bin that covers DLAs shows that most
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>20.3$ absorbers will arise from
the $M_{*}=10^{11}$ M⊙ central galaxy, with small contributions from
satellites and other haloes along the line of sight. At lower column
densities, the fraction of sightlines that begin to trace gas in the
intergalactic medium increases and then the IGM dominates for absorbers at
Ly-$\alpha$ forest column densities. Similar to Figure 4, we find that the
contribution of satellites, other haloes and the IGM increases with the impact
parameter. Additionally, we notice that higher mass central galaxies account
for a larger fraction of dense H i absorbers
($\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>17.2$) but at lower
$N_{\text{H\,{i}}}$, the central fractions remain consistent across the four
stellar mass bins. Black dashed vertical lines mark impact parameters that are
0.15, 0.5 and 1 times the virial radius. Despite the various potential
contributions from satellites, interloping haloes or the intergalactic medium,
the trend of decreasing H i column density with impact parameter remains
clear.

In all panels of Figure 5, we find an increase in the incidence of absorbers
tracing the intergalactic medium as we move to larger impact parameters (and
thus lower median H i column densities). This highlights that weaker absorbers
have a high likelihood of originating from the IGM, with more than 30 per cent
of $\log N(\mbox{\text{H\,{i}}})$< 1016 atoms cm-2 absorbers tracing gas
outside of haloes. We also see a larger fraction of absorbers belong to the
central across all column densities for the more massive haloes. Crucially, we
note that this signal is caused by our restrictions in the mock size
perpendicular to the plane of projection which is intended to mimic
observational surveys that have fixed observed size. The less massive central
galaxies will have a larger portion of their maps outside the virial radius
and hence, have a higher incidence of absorbers arising from other haloes or
the intergalactic medium.

We also see in Figure 5 that the median H i column density is larger at all
impact parameters $\gtrsim 10$ kpc for the more massive central galaxy. At
smaller impact parameters, this trend does not hold because of the
supermassive black hole at the centre of the most massive galaxies ejecting
gas via the kinetic feedback mode (Zinger et al., 2020). In addition, we find
a scatter in $N_{\text{H\,{i}}}$ up to 3 dex across the range of impact
parameters. This scatter can be attributed to the intrinsic inhomogeneity of
gas in the circumgalactic medium, but our results also emphasise that we may
be intersecting gas clouds outside galaxy haloes in the IGM. Despite the
various potential contributions from satellites, interloping haloes or the
intergalactic medium, the trend of decreasing H i column density with impact
parameter remains clear (Lehner et al., 2013; Nelson et al., 2020; van de
Voort et al., 2021; Berg et al., 2023; Weng et al., 2023a).

###  4.2 What fraction of absorbers intersect low-mass satellites?

The Milky Way halo contains tens of satellite galaxies (McConnachie, 2012) and
we expect absorption lines to occasionally intersect these smaller haloes
found in the CGM of more massive galaxies. The probability of sightlines
intersecting satellites, particularly faint ones, is of interest to observers
as strong absorbers occasionally do not have galaxy counterparts down to some
magnitude ($R\approx 25$ mag) and/or star-formation rate (SFR) limit
($SFR\approx 0.1$) (e.g. Fumagalli et al., 2015; Berg et al., 2023; Weng et
al., 2023a). Here, we estimate the gas mass contribution from low-mass
satellite galaxies in the TNG50 simulation. This is motivated by the shallower
completeness expected for galaxies below some stellar mass limit in
observational surveys.

For a given TNG50 mock with satellites and other haloes, each gas cell is
assigned a subhalo ID. We calculate the subhalo ID that dominates the HI mass
for each pixel in our 2D projection down the line of sight of the mock. By
crossmatching this list of IDs with sightlines that have been assigned a
‘satellite’ or ’other halo’ flag, we then compute the fractional contribution
of galaxies with varying stellar masses to the total number of sightlines that
are dominated by gas in satellites or a secondary halo along the sightline.
This is depicted in Figure 6 where the four colours correspond to the four
stellar mass bins in our sample. We show the cumulative contribution from
satellites (left) and other haloes (right) with stellar masses ($M_{\rm
satellite}$) that begin at 10-4 times the mass of the central galaxy ($M_{\rm
central}$). The shaded region reflects the scatter when considering different
column density bins, from
$13.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$ to
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>20.3$ systems.

Figure 6: The contribution of satellites (left) and other haloes (right) with
varying masses to the fraction of sightlines that intersect satellites. The
four colours correspond to the four stellar mass bins for the central galaxy.
We performed this calculation using five column density bins, from Ly-$\alpha$
forest column densities
($13.0\leq\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]\leq 16.0$) to DLAs. The
shaded region between the solid lines for each stellar mass bin reflects the
maximum discrepancy (typically $\sim$0.1) in the fraction found between the
five column density bins. We use stellar mass bins of size
$\log(M_{*}/M_{\odot})=0.2$. On the left, we find that galaxies with masses
$M_{*}<10^{8}$ M⊙, which are possibly missed in current observational surveys,
comprise 10 per cent of sightlines that intersect satellites for central
galaxies with stellar masses 1010 and 1011 M⊙. This value increases to 40 per
cent for 109 M⊙ central galaxies. On the right, secondary haloes belonging to
galaxies with $M_{*}<10^{8}$ M⊙ contribute to 25 per cent of sightlines that
intersect another halo across all central masses. The dashed vertical and
horizontal lines help guide the eye to the numbers given above. We emphasise
here that low-mass galaxies account for a significant portion of the H i
covering fraction.

For $M_{*}=10^{10}$ and $10^{11}$ M⊙ central galaxies, $M_{*}<10^{8}$ M⊙
satellite galaxies only contribute roughly ten per cent to the total number of
absorbers that intersect any satellite (see dashed vertical and horizontal
lines on the left plot). This fraction increases to $\approx$40 per cent of
sightlines for a central galaxy mass of $10^{9}$ M⊙. For other haloes along
the line of sight, the contribution from $M_{*}<10^{8}$ M⊙ galaxies remains
roughly constant at 25 per cent (right). There is a larger scatter of roughly
0.2 between the different column density bins, but the upper and lower bounds
are not driven by any particular column density. This effect highlights that
even low-mass galaxies in the CGM of massive galaxies and along the line of
sight contribute to absorption and might remain undetected in observational
surveys of absorbers, particularly at high redshift.

There are occasional cases in observational studies of galaxy counterparts to
absorbers where no galaxy is detected near the QSO sightline down to some
limit in stellar mass or star-formation rate (Fumagalli et al., 2015; Berg et
al., 2023; Weng et al., 2023a). At $z>2$ in surveys of Ly-$\alpha$ emitters
(LAEs) associated with H i absorbers, there are also systems where LAEs are
found at impact parameters $>50$ kpc from DLAs (Mackenzie et al., 2019;
Lofthouse et al., 2023). Given that the H i is not expected to extend out to
such large distances from simulations (Stern et al., 2021), the more plausible
explanation is that we are tracing low-mass satellite galaxies near the QSO
sightline. Figure 4 suggests that satellite galaxies at $z=0.5$ are
responsible for up to 50 per cent of absorbers at impact parameters within the
virial radius of the central but particularly at $0.5<R_{\rm vir}<1.0$. We
then see in Figure 6 that $M_{*}\leq 10^{8}$ M⊙ satellites make up $\approx$10
(40) per cent of the total number of sightlines that trace satellites
gravitationally bound to central galaxies with stellar masses 1010 and 1011
(109) M⊙. Hence, strong H i absorbers without a nearby galaxy counterpart may
simply be associated with objects below the sensitivity limit (typically
$10^{8}$ M⊙ at $z=0.5$ for a one hour MUSE exposure). Such objects require
deeper observations or larger telescopes such as the Extremely Large Telescope
(ELT), Giant Magellan Telescope (GMT) or the Thirty Meter Telescope (TMT).

##  5 Gas flows in the CGM

As depicted in Figure 1 and 2, disentangling inflowing from outflowing gas in
observations of the circumgalactic medium is not straightforward. Two commonly
used indicators to observationally infer whether the gas is inflowing or
outflowing are the azimuthal angle ($\Phi$) and metallicity ($Z$). In this
section, we characterise how the inflowing to outflowing gas fraction changes
as a function of $\Phi$ and study the origins of the metallicity anistropy
between the major and minor axes of galaxies.

###  5.1 Using the azimuthal angle to identify gas flows

In transverse absorption-line studies, the 2D projected azimuthal angle is
often advocated as an indicator to distinguish outflows from inflows (e.g.
Bouché et al., 2012; Kacprzak et al., 2012; Schroetter et al., 2016, 2019;
Weng et al., 2023b) where accretion is assumed to align with the major axis
($\Phi=0^{\circ}$) and outflows with the minor axis ($\Phi=90^{\circ}$). To
mimic observations, we create 2D images of galaxies in TNG50 using the stellar
density. These images are fully idealised mocks that neglect observational
effects such as noise, instrumental response and seeing and the effects of
dust attenuation and scattering (Bottrell et al., 2023). We run statmorph
(Rodriguez-Gomez et al., 2019), an algorithm utilised to determine the
morphology of sources, on the mock images to model the galaxy profile and
return the position angle (PA) and axis ratio, $b/a$. From here, we generate
the azimuthal angles for each pixel in the galaxy projection, excluding the
pixels within 5 kpc of the galaxy centre as $\Phi$ values are unreliable at
small distances. For the figures in this section, we select only galaxies with
$b/a$ values below the median of the sample. This is to exclude galaxies that
may be at lower inclinations (more face-on) where the measured position angle
is no longer reliable. We adopt this rather simple approach to measure the PA
and $b/a$ because our goal is not to test the accuracy of azimuthal angle
measurements in observations but rather, to provide a first-order
approximation for what observers might measure.

Figure 7: The fraction of absorbers intersecting gas originating from one of
the four gas flow flags as a function of the azimuthal angle. We use a
consistent radial velocity cutoff of $\pm 20$ $\text{km s}^{-1}$ to label
inflows and outflows and consider only central galaxies with a major-to-minor
axis ratio less than the median value for the sample. Using bins in $\Phi$ of
10∘, we colour each bin by the fraction of absorbers belonging to categories
seen in the colour bar (left $y$-axis). We find a significant contribution
from gas that is in quasi-hydrostatic equilibrium (purple) at all azimuthal
angles. The solid white line represents the fraction of outflowing to
inflowing absorbers (right $y$-axis) and the horizontal dashed line depicts an
equal ratio. There is an increase in this ratio as we move from the major axis
to the minor axis. This is most prominent in the primary panel where we
consider LLS absorbers found at an impact parameter of 50 to 100 kpc from
central galaxies of stellar mass $\sim$109 M⊙. However, accounting only for
absorbers at 0 to 50 kpc (bottom left) or absorbers around $\sim$1010 M⊙
central galaxies (bottom right) produces a minor difference of $\approx$0.2 in
the outflow to inflow ratio between the major and minor axes. While there is a
clear increase in the outflow to inflow ratio towards larger $\Phi$ at lower H
i column densities (bottom middle), this ratio never exceeds unity.

We show the fraction of absorbers that belong to one of the four gas flow
flags as a function of the azimuthal angle in Figure 7. The shaded background
conveys the fraction of absorbers belonging to the four categories using bins
of ten degrees in azimuthal angle. The solid white line depicts the fraction
of outflows to inflows, also as a function of the azimuthal angle where the
horizontal dashed line marks an equal ratio. In the primary panel, we consider
Lyman-limit systems found at an impact parameter of 50 to 100 kpc from 109 M⊙
central galaxies. This cut in column density enables us to maximise the signal
as higher $N_{\text{H\,{i}}}$ systems are much rarer tracers of gas flows. In
the bottom panels, we consider smaller impact parameters (left), lower column
densities (middle) and larger stellar masses (right).

We see a clear trend in the primary plot where the outflow to inflow ratio
increases towards larger azimuthal angles. While similar signals have been
detected before in simulations (Nelson et al., 2019; Mitchell et al., 2020),
previous studies calculate the azimuthal angle of pixels in a frame where the
galaxy is viewed directly edge on. The method used in this paper, based on
observational practices, only loosely constrains the inclination of the galaxy
by considering axial ratios that are below the median value. This is a
reassuring confirmation of observational practices where one uses $\Phi$ to
distinguish gas flows (Schroetter et al., 2016; Zabl et al., 2019).
Additionally, we also account here for the contribution from satellite
galaxies and gas in quasi-hydrostatic equilibrium. Still, we confirm that the
ratio of outflows to inflows is $\approx$0.5 near the major axis, rising to a
peak of $\approx$2 at the minor axis in the primary panel.

In the main panel, we have considered a particular subset of absorbers (LLS at
impact parameters 50 to 100 kpc) associated with $M_{*}=10^{9}$ M⊙ central
galaxies. When varying the impact parameter, H i column density and central
stellar mass, we obtain more varied results. Looking at the bottom left panel
of Figure 7 where we consider only absorbers within the inner 50 kpc of the
galaxy, we still find an increasing outflow to inflow ratio, but the
proportion of inflows dominates at all azimuthal angles. A similar signal is
found when considering typical Ly-$\alpha$ forest column densities
($13.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$, bottom middle) and
stellar mass (bottom right), namely, inflows dominate in number even near the
minor axis. The influence of impact parameter and stellar mass are
particularly significant, with only half the gas flows at the minor axis
attributed to outflows. In particular, we find that accretion dominates
sightlines passing through central galaxies belonging to the $M_{*}\approx
10^{10}$ and 1011 M⊙ bins. Current observational studies, which are more
sensitive to massive galaxies, find signatures of outflows more common than
inflows (Martin et al., 2012; Rubin et al., 2012; Péroux et al., 2017; Rahmani
et al., 2018a; Zabl et al., 2019; Weng et al., 2023b) and we discuss the
possible reasons for this in Section 7.

Looking specifically at the fraction of gas in quasi-hydrostatic equilibrium,
we find little evolution with azimuthal angle. The bottom left panel of Figure
7 shows that the fraction does not vary at lower impact parameters and only
mildly increases for lower column densities (bottom middle). Due to an
increasing fraction of inflowing absorbers, there is much less gas in quasi-
hydrostatic equilibrium for $M_{*}=10^{10}$ M⊙ central galaxies (bottom
right). Curiously, we find a tentative signal of an increasing satellite LLS
fraction at $50<b<100$ kpc as we move towards the minor axis for both
$M_{*}=10^{9}$ and $10^{10}$ M⊙ central galaxies. For these stellar masses
where outflows are typically starburst-driven, the gas density is expected to
be greater along the minor axis. Hence, there should be stronger ram pressure
stripping, leading to more quiescent and less gas-rich satellites near
$\Phi=90^{\circ}$ in the CGM. We find instead that satellites along the minor
axis are more gas-rich, reminiscent of the anisotropic galaxy quenching found
in galaxy clusters (Martín-Navarro et al., 2021; Ando et al., 2023). A
possible cause for this is if satellite galaxies along the major axis have
been accreted at earlier times and hence, more likely to be quenched (Karp et
al., 2023). At this stage, the origins of this anistropic LLS fraction from
satellites remain ambiguous.

Figure 8: The metallicity with respect to solar of absorbers belonging to the
four gas flow categories as a function of the azimuthal angle are plotted as
solid lines. The median metallicity of all absorbers is plotted in white. We
use a radial velocity cutoff of $\pm 20$ $\text{km s}^{-1}$ and only consider
central galaxies with a major-to-minor axis ratio less than the median value
for the sample. The shaded regions reflects the $0.5\sigma$ uncertainty in the
metallicity for only the inflowing and outflowing absorbers. Absorbers
associated with quasi-static gas and satellites do not have their errors
plotted for clarity. In the primary panel, we adopt the same impact parameter
(50 to 100 kpc), column density (LLS) and stellar mass bins ($M_{\rm
central}\approx 10^{9}$ M⊙) as in Figure 8. For the secondary panels, we also
make the same variations from the fiducial parameters. The positive gradient
of the white dashed line in all panels shows that the median metallicity of
absorbers increases towards the minor axis regardless of physical origin,
although the effect diminishes for more massive central galaxies. When we
separate the absorbers into the four gas flow flags and plot their
metallicities (coloured lines), we discover that the inflowing and quasi-
static gas also increase in $Z$ at larger $\Phi$. These absorbers then drive
the trend in median metallicity as they are typically more abundant than
outflowing absorbers.

In Figure 7, we also find that 40 to 60 per cent of gas at all azimuthal
angles arises from gas that is in quasi-hydrostatic equilibrium or in
satellites. The contribution from satellites is most prominent in the primary
panel which is consistent with our findings in Figure 4 where satellites
dominate around $0.5$-$1R_{\rm vir}$. At smaller impact parameters and lower
column densities, their contribution diminishes. The contribution from quasi-
static gas is significant for impact parameters up to 200 kpc, all central
galaxy stellar masses and H i column densities tested in this study. While the
ratio of outflowing to inflowing sightlines increases with larger azimuthal
angles, this value ignores the contribution from gas moving at slower radial
speeds. These results show that a fraction of absorbers in galaxy haloes may
be associated with gas that is static or rotating. The latter has been
observed in 60 per cent of Ly-$\alpha$ absorbers at $z\lesssim 0.03$ (French &
Wakker, 2020) and recent simulations show that rotational support is
significant in the CGM (Lochhaas et al., 2023). In this work, we have set an
arbitrary restriction on the radial velocity ($\pm 20$ $\text{km s}^{-1}$) to
distinguish between inflows, quasi-static gas and outflows. Hence, the
fraction of gas that is rotating without loss of angular momentum (quasi-
static) compared to co-rotating inflows depends on the radial velocity
boundary. Any increase in magnitude of the $v_{r}$ cutoff used naturally leads
to an increase in the fraction of gas in hydro-static equilibrium. Likewise,
setting all negative (positive) radial velocities to be counted as inflows
(outflows) leads to no absorbers being labelled as quasi-static. However, we
find that unless the $v_{r}$ cutoff is $>150$ $\text{km s}^{-1}$, the ratio of
inflows to outflows remains similar; there is just a proportional increase in
the quasi-static absorber fraction. At high radial velocities, we only expect
to find outflows and hence, the fraction of inflows to outflows decreases
dramatically. Therefore, we caution that interpreting H i absorbers as
inflowing or outflowing using their projected azimuthal angle relative to the
major axis should include further consideration of other properties such as
$b$, $\log N(\mbox{\text{H\,{i}}})$ and $M_{*}$.

###  5.2 The metallicity of gas flows

Another absorber property used to differentiate outflows from inflows is the
gas phase metallicity (Fox et al., 2016, 2019; Ramesh et al., 2023c). In
Figure 8, we plot the metallicity as a function of the azimuthal angle. The
four coloured lines correspond to the four gas flow origin flags and the
dashed white line is the median metallicity of all absorbers in a given
azimuthal angle bin with size 10∘. The shaded regions indicate the 0.5$\sigma$
uncertainty in the metallicity for only outflows and inflows. We adopt the
same fiducial parameters (LLS found 50 to 100 kpc from $M_{*}=10^{9}$ M⊙
central galaxies) in the primary panel as Figure 7 and the same changes in the
secondary panels.

For almost all azimuthal angles and choices of $b$, $N_{\text{H\,{i}}}$ and
central galaxy stellar mass, the metallicity (with respect to solar) of
outflows is consistently 0.2 to 0.5 dex larger than inflows. The few
exceptions occur for absorbers at azimuthal angles $\Phi<30^{\circ}$ found
towards $M_{*}=10^{10}$ M⊙ central galaxies (bottom right) where the accreting
gas is roughly equal in metallicity to the outflowing gas, perhaps due to
recycling. We also find that absorbers are more metal-rich around more massive
central galaxies, while the impact parameter and H i column density change the
normalisation by $<0.1$ dex. It is for this same reason that the metallicity
of satellites is typically lower than the median value; as satellites are
usually less massive, their average metallicities are also going to be lower.

Similar to the findings of Péroux et al. (2020); van de Voort et al. (2021),
we find a positive gradient in the median metallicity of absorbers as a
function of azimuthal angle. The magnitude of the difference in $Z$ between
the minor and major axes is typically of order $0.2$ dex, but does diminish
with increasing central stellar mass. More curiously, we find that the driver
of this metallicity difference is only partly caused by an increase in the
fraction of absorbers tracing metal-enriched outflows. In general, gas that is
inflowing or in quasi-hydrostatic equilibrium also increases in metallicity
towards the minor axis, hinting that the pollution of metals into the
circumgalactic medium near the minor axis also results in metal-enriched
accretion in the form of recycled gas (Fraternali, 2017; Muratov et al.,
2017). As the fraction of quasi-static and inflowing gas typically dominates
the outflow fraction (bottom panels in Figure 7), the increase in median
metallicity of these types of absorbers contribute more significantly to the
median metallicity of the CGM. We discuss the metallicity distribution of
inflowing and outflowing gas in more detail in Section 6.

##  6 Observational Comparisons

###  6.1 Determining the host galaxy of absorbers

With the proliferation of IFS surveys targeting absorbers, there are an
increasing number of cases where multiple galaxies are found within some
velocity cutoff with respect to the absorber (e.g. Dutta et al., 2020;
Hamanowicz et al., 2020; Berg et al., 2023; Weng et al., 2023a). This points
to a more complex relationship between galaxies and absorbers, and one that
may only be captured by larger statistical studies. At the same time, it is
also critical to relate the properties of the CGM with galaxy properties in
individual cases to understand how galaxies evolve (e.g. Tumlinson et al.,
2013; Burchett et al., 2016; Werk et al., 2016; Langan et al., 2023). However,
the fidelity of our current methods used to associate galaxies with absorbers
remains ambiguous. We have shown already in Figures 3 to 6 the importance of
considering other haloes along the line of sight and $M_{*}<10^{8}$ M⊙
satellites. The predominant method of assuming the galaxy at lowest impact
parameter to the QSO sightline is the host neglects both these considerations.
Here, we analyse the potential host galaxy of absorbers using the results of
these simulations and prescribe a more reliable method for associating
absorbers to galaxies.

Figure 9: In the top row, we show the fraction of
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$ absorbers belonging to the
central galaxy given its impact parameter ($x$-axis) and line of sight
velocity difference ($y$-axis). In the bottom row, we show the regions where
satellites, other haloes and the IGM dominate (note the differing axis
ranges). Hexbins are shaded by the colour corresponding to the most likely
origin in the bottom row; we emphasise that the origin is probabilistic which
we demonstrate in the top row. We use four likelihood bins [$p>0.75$,
$0.5<p\leq 0.75$, $0.25<p\leq 0.5$, $0<p\leq 0.25$] to colour each hexbin by
the corresponding probability ($p$) that the central galaxy hosts the absorber
at some impact parameter and velocity. There are three plots for the stellar
mass bins: $10^{9}$ (left) $10^{10}$ (centre) and $10^{11}$ (right) M⊙. It is
clear that the central galaxy’s stellar mass drives the coverage of H i in
$\Delta v$-$b$ space. This figure represents a step forward from assuming the
galaxy at lowest impact parameter is the host of the absorber by taking into
account both $b$ and $\Delta v$ and can be used to guide observers when
selecting counterparts to absorbers. At the same time, we emphasise that the
contamination from origins such as satellites and other haloes are
significant. For current and future surveys of H i absorbers, we consider
different $\log N(\mbox{\text{H\,{i}}})$bins in Figure 13.

In Figure 9, we show the most likely origin flag of an absorber found at some
impact parameter and line of sight velocity difference to the central galaxy.
The likelihood changes significantly as a function of stellar mass and the
three panels represent stellar masses
$M_{*}=[10^{9},10^{10},10^{11}]\rm{M}_{\odot}$ from left to right. We also
impose a H i column density limit of $\log N(\mbox{\text{H\,{i}}})>18.0$; both
the stellar mass and column density restrictions are intended to match the
MUSE-ALMA Haloes sample (black stars). At lower column densities, there is an
increasing contribution from the intergalactic medium, replacing the
contribution from other haloes and satellites. However, the extent of the
central galaxy (blue) remains similar as a function of $\Delta v$ and $b$,
even when considering all absorbers with
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>13.0$. If we look at the
distribution of absorbers around galaxies from the COS-Halos survey (Tumlinson
et al., 2013; Werk et al., 2013; Werk et al., 2014, 2016; Peeples et al.,
2014), we find that the centroids of the total H i absorption profile are
predominantly within the region dominated by the central galaxy after matching
stellar masses. However, the centroids of individual H i components are
occasionally found at large velocities from the galaxy systemic redshift and
are more likely associated with other haloes or the IGM (if at lower column
density). The line of sight velocity difference between absorber and galaxy
remains an important consideration and we suggest that when associating
galaxies with absorbers, both $\Delta v$ and $b$ are considered (e.g. Berg et
al., 2023).

While we have coloured each hexbin by a given flag, we emphasise that this is
inherently probabilistic and there is much uncertainty, particularly at the
boundaries between the central and satellites or other haloes. Though Figure 9
is a useful observational diagnostic for determining the potential origins of
H i absorbers by moving beyond just using the impact parameter, it underscores
that absorbers may not have a single well-defined origin and the need for
statistical studies involving larger samples. This point is implicitly made by
works studying absorbers arising from the intragroup medium where absorbers
are not associated with a single galaxy (Gauthier, 2013; Nielsen et al., 2018;
Klitsch et al., 2018; Péroux et al., 2019; Dutta et al., 2021, 2023; Qu et
al., 2023).

Thus far, we have adopted a galaxy-centric approach, that is, we study how the
properties of H i absorbers vary as we move from a selected central galaxy.
Many of the current IFS absorber studies are centred on a background source
which is arbitrarily positioned in the sky with respect to foreground galaxies
(e.g. Schroetter et al., 2016; Chen et al., 2020; Hamanowicz et al., 2020;
Lofthouse et al., 2020; Muzahid et al., 2020; Nielsen et al., 2020; Muzahid et
al., 2021; Berg et al., 2023), including the MUSE-ALMA Haloes survey (Péroux
et al., 2022). To produce results that can be directly used to better
understand observations of this type, we select a random pixel within a 200
kpc square centred on the central that has a column density
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$. This becomes the location
of our absorber and we calculate the impact parameter of the remaining pixels
in the field with respect to the absorber. We can then determine the fraction
of absorbers belonging to the four origin flags in a similar manner to Figure
4. For each mock, we repeat this process five times for each projection
direction, leading to 15 iterations in total for each central subhalo.

The results are shown in Figure 10. Using a randomly-selected absorber, we
plot how the fraction of pixels tracing the central, satellites, other haloes
and the IGM surrounding the absorber changes as a function of impact
parameter. The dashed lines corresponding to the four origins show this
variation with $b$ and the coloured background elucidates the relative
fractions within each 10 kpc impact parameter bin. To make an appropriate
comparison with the MUSE-ALMA Haloes survey, we re-analyzed this observational
data and mimicked the definitions of centrals, satellites and other haloes
used in TNG50. We derive the halo mass and virial radius using the stellar-to-
halo mass relation (Rodríguez-Puebla et al., 2017). We define the central
galaxy as the most massive galaxy in the MUSE field of view and hence, other
galaxies associated with the absorber are described as satellites (other
haloes) if they are inside (outside) $R_{\rm vir}$ of the central (Karki et
al., 2023). While satellites can still be bound beyond $R_{\rm vir}$, we
choose to adopt this simpler definition for this exercise. Then, we apply a
column density cutoff of $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$ and
only consider $M_{*}=10^{9}$ to $10^{11}$ M⊙ central galaxies to match the
MUSE-ALMA Haloes survey. The impact parameter of these various galaxies in the
data are plotted using stars with a filled colour corresponding to their
origin. For clarity, we plot central galaxies in the top row, followed by
satellites and other haloes as we move down.

Figure 10: The fractional pixel origins ($y$-axis) given its impact parameter
($x$-axis) from a randomly-selected
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$ absorber. There are three
plots for the stellar mass bins: $10^{9}$ (left) $10^{10}$ (centre) and
$10^{11}$ (right) M⊙. The dashed lines show how the fraction of pixels tracing
the central, satellites, other haloes and the IGM vary as a function of impact
parameter. Using bins of 10 kpc in $b$, we also highlight their relative
fractions using the coloured background. Data points from the MUSE-ALMA Haloes
survey are plotted as stars and diamonds (for the galaxy nearest the absorber)
using the publicly available catalogue from Weng et al. (2023a). We find a
rough agreement between the abundances of central, satellite and other halo
galaxies and the predictions from TNG50. The excess of satellite galaxies at
low impact parameters for the $10^{11}$ M⊙ (right) plot arises from a large
galaxy group consisting of $>10$ galaxies found towards Q1130$-$1449 (Péroux
et al., 2019; Chen et al., 2019).

The relative abundances of central, satellite and other halo galaxies as a
function of impact parameter from the MUSE-ALMA Haloes survey is broadly
consistent with the predictions from TNG50. This highlights that the most
massive halo is not necessarily the host galaxy of the absorber and that
satellites and other haloes can be significant contributors to the H i gas
down any line of sight. A larger sample size, particularly for the $10^{9}$ M⊙
central galaxies, is required to make more quantitative comparisons. We note
that the excess of satellites at low impact parameters for the $10^{11}$ M⊙
central galaxies is caused by the DLA towards Q1130$-$1449 being associated
with more than 10 galaxies (Péroux et al., 2019; Chen et al., 2019). The right
plot of Figure 10 also highlights that gas ejected from AGN that is no longer
gravitationally bound may contribute significantly to the covering fraction
around massive haloes. While this signal may be enhanced by the kinetic
feedback mode of AGN in the IllustrisTNG model pushing large quantities of
cool gas into the CGM, it highlights the potential for absorbers to be unbound
to any galaxy which is ignored in most observational studies (see Berg et al.,
2023, for an exception).

###  6.2 Metallicity distribution of H i absorbers

We now move to statistical studies on absorber metallicities and their
possible origins in and around galaxies. The distribution of pLLS and LLS
absorber metallicities at $z\lesssim 1$ appears to be bimodal from
observations (Lehner et al., 2013; Lehner et al., 2019; Wotta et al., 2016,
2019; Berg et al., 2023). In Lehner et al. (2013), the authors find tentative
evidence for a metallicity bimodality using absorbers with column densities
$16.2\lesssim\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]\lesssim 18.5$ at
$z\lesssim 1$. In a larger sample, Wotta et al. (2016, 2019) find that the
bimodality is driven by pLLS absorbers where the metallicity peaks at [X/H]
$=-1.7$ and $-0.4$. This bimodality is purported to arise from low-metallicity
absorbers tracing gas accretion or overdense regions of the Universe (Berg et
al., 2023), while more metal-rich absorbers are associated with the CGM of
galaxies. However, a study of $L^{*}$ galaxies from the COS-Halos Survey finds
a unimodal metallicity distribution centred on [X/H] $\approx-0.5$ (Prochaska
et al., 2017) and argues that low-metallicity absorbers are not necessarily
freshly accreted gas. As our TNG50 sightlines are centred on central galaxies
with specific halo masses, we do not attempt to reproduce the number counts in
these observations (see Hafen et al., 2017; Rahmati & Oppenheimer, 2018), but
rather investigate whether a bimodality exists in the normalized sense using
the various absorber origins.

Figure 11: The normalized distribution of absorber gas phase metallicities. We
adopt H i column densities
$16.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$ to match observational
samples and a central stellar mass of $M_{*}=10^{9}$ M⊙ as the fiducial
parameters (left panels). The vertical dashed lines indicate the peaks of the
bimodal metallicity distribution found in observations. In the top (bottom)
panels, we separate sightlines probing inflows (the central) from outflows
(the IGM and other haloes) and display the distribution of metallicities for
the separate populations. The histograms have been normalized such that the
peaks of each distribution are at unity. An exception is for ‘other halo’
galaxies in the bottom row which have been normalized by the peak of the
central galaxy metallicity distribution for clarity. There is a $\approx$0.2
dex difference in metallicity between inflows and outflows for the fiducial
parameters. At higher stellar masses (lower column densities), this difference
diminishes (increases) in magnitude. We find gas in the IGM spans a broad
range of metallicities, from metal-poor to metal-rich, due to significant
environmental enrichment from nearby galaxies ejecting metals beyond their
haloes. Similarly, gas belonging to other haloes along the line of sight will
be of varying stellar masses and hence, have a significant range of
metallicites.

In Figure 11, we display the metallicity distributions of absorbers separated
by their origin in TNG50. We first separate absorbers by whether they trace
inflows or outflows in the top panels and then, absorbers tracing the IGM from
absorbers tracing the central galaxy in the bottom panels. All distributions
are normalized to have equal peak heights and do not reflect the actual
incidence of absorbers. While outflowing gas is typically more metal-rich than
inflowing gas, we find only a $\sim$0.2 dex different in their metallicity
distribution peaks for the fiducial parameters of
$16.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$ and $M_{\rm
cen}=10^{9}$ M⊙. This difference increases to $\sim$0.4 dex if we only
consider $13.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$ absorbers (top
right), but almost disappears when considering $M_{*}=10^{11}$ central
galaxies (top centre). Regardless, any difference we find is minor when
compared to observations where the difference in $[\rm{X/H}]$ is 1.3 dex. The
strong overlap in metallicities is in part caused by efficient wind recycling,
where previous metal-enriched outflows later become inflows (Anglés-Alcázar et
al., 2017). We choose to not include absorbers that are in quasi-hydrostatic
equilibrium or associated with satellites in this sample for clarity but note
that their presence further dilutes any signal of bimodality (as seen in the
unimodal ‘central’ distributions in the bottom panels). These results are
consistent with previous simulated metallicity distributions (Hafen et al.,
2017; Rahmati & Oppenheimer, 2018) and in tension with some observations
(Fumagalli et al., 2016; Wotta et al., 2019).

In the bottom panels of Figure 11, we compare the metallicity distributions of
gas tracing the intergalactic medium with gas bound to the central galaxy.
Curiously, we find absorbers with
$16.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$ and metallicities
$[\rm{X/H}]>0.1$ that trace gas in the IGM. When we compare these values with
the gas bound to a $M_{*}=10^{9}$ M⊙ central galaxy (bottom left), we find the
IGM gas is metal-enriched with respect to the galaxy. These metal-enriched IGM
absorbers arise from the expelled material of more massive haloes found along
the sightline, as feedback processes drive metal enriched gas to large
distances from the central regions of galaxies (e.g. Madau et al., 2001;
Scannapieco et al., 2002; Cen & Chisari, 2011; Maio et al., 2011; Ayromlou et
al., 2022). This is why we see an alignment in metallicity distributions
between the peak IGM component and the $M_{*}=10^{11}$ M⊙ central galaxy
sample in the bottom middle panel. When we select for H i column densities
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$, we see a 3 dex discrepancy
between the distribution peaks, with IGM absorbers found at lower $Z$ as
expected.

Whether the metallicity distribution of partial Lyman-limit systems is
unimodal or bimodal at $z=0.5$ is tied directly to the incidence of absorbers
belonging to the various origins (e.g. IGM and central). We reiterate that any
direct comparison with observational studies is difficult as our sightlines
are inherently found near galaxies. However, we do find that in the TNG50
simulation, the metallicity is not a perfect discriminator of gas that is
accreting or outflowing (consistent with the findings of Hafen et al., 2017;
Rahmati & Oppenheimer, 2018). Moreover, the peak of the metallicity
distribution for absorbers associated with $10^{10}$ and $10^{11}$ M⊙ central
galaxies is consistent with both the metal-rich peak from the bimodal
distribution of Wotta et al. (2016) ([X/H] $\approx-0.4$) and the peak of the
unimodal distribution ([X/H] $\approx-0.5$) in Prochaska et al. (2017). While
the peaks align, Prochaska et al. (2017) finds a spread in metallicities
around $L^{*}$ galaxies that is roughly twice as large as predicted in TNG50.
In Figure 11, we observe that the increased spread towards lower [X/H] may
arise from less-massive satellite or secondary halo galaxies that are
typically lower metallicity from the mass-metallicity relation (Lequeux et
al., 1979). We note that satellite galaxies have not been shown for clarity,
but their metallicity distribution also extends to smaller [X/H] values. The
lower incidence of super-solar metallicities ([X/H] $>0.5$) may be caused by
the limited CGM resolution in TNG50 ($\sim$ 1 pkpc beyond $0.1R_{\rm vir}$),
although we note that Wotta et al. (2016, 2019) find an upper limit in [X/H]
of roughly 0.5. Ultimately, our results suggest that inferring the origin of
gas from metallicity requires caution. For random sightlines towards UV-bright
QSOs (e.g. Lehner et al., 2018), we find that gas expelled from massive haloes
into the intergalactic medium are a source of metal-enriched absorbers.
Likewise, the large spread in [X/H] values around $L^{*}$ galaxies may imply
that absorbers are perhaps associated with lower-mass satellites that have
lower metallicity.

##  7 Discussion

###  7.1 The CGM of star-forming and quiescent galaxies

The analysis in this work focuses on the CGM properties of galaxies with
varying stellar masses at $z=0.5$, coinciding with the $M_{*}$ range and
redshift of galaxies in MUSE-ALMA Haloes. At any given stellar mass bin, there
are galaxies above, below and on the SFR-$M_{*}$ main sequence. Here, we
discuss how the CGM properties of galaxies with similar stellar mass depend on
the star-formation rate.

The central galaxies in this work are randomly selected from TNG50 and
enforced to have stellar masses within 0.3 dex of
$M_{*}=[10^{8.0},10^{9.0},10^{10.0},10^{11.0}]$ $\rm{M_{\odot}}$. In total,
there are [100, 100, 50, 20] galaxies within each bin, and we further
categorise the galaxies as star-forming or quiescent. The distinction is made
somewhat arbitrarily; we consider galaxies with SFRs in the lowest quartile of
each bin as quiescent, and the remainder are star-forming. We also tested
separating star-forming and quiescent galaxies using the median SFR but found
little variation in the results presented below.

We find marginal differences in the results presented in this paper after
separating star-forming and quiescent galaxies. The fraction of sightlines
with H i as a function of impact parameter for the various origins (Figure 4)
are within the uncertainties for both groups. Likewise, there are marginal
differences in the fraction of inflows, quasi-static gas, outflows and
satellites as a function of azimuthal angle between star-forming and quiescent
galaxies (Figure 7).

These results highlight that the stellar mass (because it is tied to halo
mass) is the key driver of the amount, distribution and extent of the H i in
the circumgalactic medium. Additionally, it is unclear in observations whether
the cool gas content in the CGM of star-forming and quiescent galaxies differ.
From studies of luminous red galaxies, the cool gas mass in the CGM of passive
galaxies appears comparable to star-forming galaxies (Chen et al., 2018;
Zahedy et al., 2019). As H i is almost ubiquitous in galaxy haloes (e.g.
Tumlinson et al., 2013; Werk et al., 2013; Werk et al., 2014), Ly-$\alpha$ is
not the ideal absorption line to trace differences between the circumgalactic
media of star-forming and passive galaxies. Instead, it appears that the O vi
covering fraction is significantly larger around star-forming galaxies (Werk
et al., 2013; Tchernyshyov et al., 2023). To understand the different CGM
properties of star-forming and quiescent galaxies in simulations, we require a
more systematic study across all stellar masses rather than isolated bins in
this work. The simplistic method of separating star-forming and passive
galaxies used might weaken any potential differences in their CGM properties.
However, we note that our results are consistent with observations and
highlight that stellar (halo) mass drives the H i amount, extent and
distribution around galaxies.

###  7.2 The incidence of inflows in observations and simulations

The incidence of inflowing gas observed in both tranverse absorption line and
down-the-barrel observational studies is inferred to be $\approx$10 per cent
(Kacprzak et al., 2010; Martin et al., 2012; Rubin et al., 2012; Ho et al.,
2017; Zabl et al., 2019; Weng et al., 2023b). Comparatively, our TNG50 results
indicate a minimum inflow fraction of $\approx$20 per cent which rapidly
increases for higher stellar masses (Figure 7). We emphasise that
observational studies of gas flows using absorption-lines towards background
sources provide only fragmentary information about the physical origin of the
observed gas and perhaps our assumptions on whether the gas traced is
inflowing or outflowing require reassessment (see Section 5). Moreover, when
we collapse the mock into a 2D projection, there is almost always both
inflowing and outflowing gas in each sightline. This conflicts with
assumptions made in most transverse absorption-line studies, where an entire
absorption system is typically assigned to a single gas flow (e.g. Schroetter
et al., 2019; Zabl et al., 2019; Weng et al., 2023b). In reality, the
absorption could be produced by a combination of the two processes (see
Nielsen et al., 2022, for an example) and current observational methods do not
capture this adequately. This may be one cause of the disparity in inflow
incidences between observations and simulations. Despite these observational
and modelling challenges, the incidence of sightlines tracing inflowing gas
remains an important and potentially constraining quantity to measure.

###  7.3 The number of clouds along each sightline

For the characterization of different origins, and measurements of absorber
properties per sightline, we assume that each sightline is dominated by a
single cloud of gas belonging to one of the flags. In reality, there is the
chance of intersecting multiple clouds which we commonly find in absorption
line observations. If there are multiple clouds along the line of sight found
at different velocities, then the measurement of properties such as the line
of sight distance or metallicity may be averaged out. However, Ramesh & Nelson
(2023) shows that the typical number of clouds intersected with
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>16.0$ along a sightline is
approximately two near the centre of Milky Way-like galaxies at $z=0$,
decreasing to a single cloud towards the virial radius for TNG50. Hence, at
this resolution in the CGM, it is likely we are only intersecting a single
cloud. In the event there are multiple clouds, we typically find that a single
cloud dominates the H i mass.

###  7.4 Extension to different gas phases

Thus far, we have analysed H i absorbers in this study. Other common metal
absorption lines used in studies of absorber counterparts include Mg ii, C iv
and O vi. In the top row of Figure 12, we show column density maps of each ion
for the same $M_{*}\approx 10^{10}$ M⊙ galaxy displayed in Figure 2. While the
H i and Mg ii absorbers are densest at the centre of galaxies, the C iv and O
vi absorbers that trace hotter gas phases are more diffuse and extended (Ho et
al., 2021; Nelson et al., 2023). Following the previous analysis for H i, we
designate each pixel by the identical set of origin and gas flow flags using
the aggregate Mg ii, C iv or O vi mass along each sightline. Looking at the
second row of Figure 12, we find that the more massive central galaxy
dominates most sightlines over satellite and other halo galaxies for C iv and
O vi. This is caused by smaller haloes not reaching the virial temperatures
($\approx 10^{5.5}$ K) required to form O vi. The structure of outflows in the
bottom row also differs between the ions. A visual inspection shows that ions
tracing the hotter phases of gas have a larger proportion of sightlines
dominated by outflows. This is in line with the earlier discussion of the high
incidence of H i inflows; because the outflowing gas is hotter, it is more
likely to reach the temperatures required to produce C iv and O vi. We present
here only some preliminary insights into how different absorber species have
varying origins. A complete analysis will be released in an upcoming work.

Figure 12: A visualisation of a $10^{10.2}$ M⊙ central galaxy (subhalo ID
455488) in TNG50. In the top row, we show the H i, Mg ii, C iv and O vi column
density maps respectively from left to right. The middle and bottom rows
colour the pixels by the flag that dominates the H i or ion mass in
projection. The dashed white circle outlines the virial radius of the galaxy.
We see that the extent, distribution and origin of absorbers depend
significantly on the tracer used. The C iv and O vi maps are dominated by the
massive central galaxy, while we find more contributions from satellites and
other haloes in the H i and Mg ii maps. More of the gas is found to be
outflowing in the higher-ionization tracers than in the H i or Mg ii, where
the latter is particularly marginal in extent.

##  8 Conclusions

We analyse the physical origins of H i absorbers in and around central
galaxies with stellar masses 108 to 1011 M⊙ at $z=0.5$ using the TNG50
simulation. We consider all gas cells within $\pm$ 500 $\text{km s}^{-1}$ of
the central galaxy and categorise all cells based on whether they are
gravitationally bound to the central galaxy, a satellite of the central or
another halo. Additionally, we also consider gas that is unbound and in the
intergalactic medium. These four flags [IGM, central, satellite, other halo]
form the origin labels. We also derive a second set of gas flow labels:
[inflows, quasi-hydrostatic, satellites and outflows]. We then connect
absorber properties such as impact parameter, line of sight velocity
difference, line of sight distance, metallicity, azimuthal angle and column
density with their origin or gas flow. Our major findings are summarised here.

  * •

The line of sight velocity difference is a poor indicator of the physical
distance between the absorber and galaxy, particularly if the absorber traces
gas in the intergalactic medium or another halo near the sightline (Figure 3).
Hence, studies that find large metallicity, temperature or density
discrepancies between individual components (Zahedy et al., 2019; Nielsen et
al., 2022; Sameer et al., 2022) may be tracing gas arising from different
origins or separated by large physical distances.
$\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<19.0$ H i absorbers with velocity
separations $|\Delta v_{\rm LOS}|<20$ $\text{km s}^{-1}$ can trace gas that is
several Mpc away. However, we find that 75 per cent of H i absorbers with
column densities $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>16.0$ can be
found within $\pm 150$ $\text{km s}^{-1}$ of $M_{*}=10^{10}$ M⊙ central
galaxies. This value decreases to 70 and 120 $\text{km s}^{-1}$ for $10^{8}$
and $10^{9}$ M⊙ galaxies, respectively and increases to $250$ $\text{km
s}^{-1}$ for $10^{11}$ M⊙ galaxies.

  * •

The fraction of absorbers that are associated with the central galaxy
decreases with impact parameter (Figure 4). This decline is steepest for
higher H i column densities and lower stellar masses. From impact parameters
$b>0.5R_{\rm vir}$, satellite and other halo galaxies begin to dominate the
fraction of absorbers. Contributions from the intergalactic medium are
marginal, particularly for higher $M_{*}$ central galaxies and larger
$N_{\text{H\,{i}}}$. However, we see in Figure 5 that $>80$ per cent of
absorbers with column densities
$13.0<\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]<16.0$ trace gas in the IGM.

  * •

Satellite galaxies with stellar mass $M_{*}<10^{8}$ M⊙ can contribute $\sim$40
per cent of the total sightlines that intersect satellites of a $10^{9}$ M⊙
central galaxy. This fraction reduces to $\sim$10 per cent for $10^{10}$ and
$10^{11}$ M⊙ centrals. For secondary haloes down the line of sight, 25 per
cent of absorbers are attributed to $M_{*}<10^{8}$ M⊙ galaxies. These findings
are a possible explanation of H i absorbers that do not have detected galaxy
counterparts near the background source in observational studies.

  * •

After modelling the azimuthal angle of absorbers in a similar manner to
observers, we find that the relative incidence of outflows compared to inflows
increases as we move towards the minor axis. This signal is strongly dependent
on the impact parameter and central galaxy stellar mass; at smaller $b$ and
larger $M_{*}$, inflows begin to dominate at all azimuthal angles. The larger
incidence in simulations may be attributed to the singular classification of
absorbers as either outflowing or inflowing; in TNG50, there is typically both
inflowing and outflowing gas found along each sightline.

  * •

The median metallicity of absorbers increases towards the minor axis,
consistent with the findings of previous theoretical studies (Péroux et al.,
2020; van de Voort et al., 2021) and observations (Wendt et al., 2021). When
decomposing the signal into individual gas flows, we find that the increasing
incidence of outflows does not drive the increasing metallicity, but rather
both inflows and outflows increase in $Z$ towards larger azimuthal angles.

  * •

We find that analysing absorber-galaxy systems from the MUSE-ALMA haloes
survey using the position of absorbers with respect to galaxies in $\Delta
v$-$b$ space is useful to statistically associate the gas with its surrounding
galaxies. Instead of assuming the galaxy nearest the absorber is the host
galaxy of the gas, we suggest these $\Delta v$-$b$ diagrams (Figure 9) are a
useful diagnostic that takes into account both the line of sight velocity
difference and impact parameter.

  * •

In Figure 10, we show a direct comparison between the distribution of galaxies
around absorbers in MUSE-ALMA Haloes and TNG50. The results are broadly
consistent and highlight that observers may need to consider absorbers not
bound to any galaxy as a plausible source of gas being probed, particularly
around larger haloes.

  * •

There is a marginal ($\lesssim 0.5$ dex) difference in the peaks of the
inflowing and outflowing gas metallicity distributions. The difference
decreases for larger central galaxy stellar masses and increases for lower H i
column densities but never reaches the large differences seen in observational
studies. In line with previous simulations (Hafen et al., 2017; Rahmati &
Oppenheimer, 2018), we find that the observed bimodal metallicity distribution
is not seen as strongly even after accounting for the various physical origins
of the gas.

We expect to expand this study to other common ions in absorption-line studies
such as Mg ii, C iv and O vi and to higher redshifts where more recent surveys
are searching for galaxy counterparts to absorbers (e.g. Díaz et al., 2021;
Bordoloi et al., 2023). Upcoming surveys with instruments such as the Dark
Energy Spectroscopic Instrument (DESI; DESI Collaboration et al., 2016a, b),
WEAVE (Dalton et al., 2016) and 4MOST (de Jong et al., 2012, 2019; Peroux et
al., 2023) will produce millions of absorber-galaxy pairs and their results
require a statistical interpretation. With the rapid proliferation of studies
of galaxies surrounding absorbers of varying species and redshifts, it becomes
more important than ever to interpret the results of these observational
studies using simulations.

## Acknowledgements

This research was supported by the International Space Science Institute
(ISSI) in Bern, through ISSI International Team project #564 (The Cosmic
Baryon Cycle from Space). This research is supported by an Australian
Government Research Training Program (RTP) Scholarship. EMS and SW acknowledge
the financial support of the Australian Research Council through grant
CE170100013 (ASTRO3D). DN and RR acknowledge funding from the Deutsche
Forschungsgemeinschaft (DFG) through an Emmy Noether Research Group (grant
number NE 2441/1-1). RR is a Fellow of the International Max Planck Research
School for Astronomy and Cosmic Physics at the University of Heidelberg
(IMPRS-HD). The TNG50 simulation was run with compute time awarded by the
Gauss Centre for Supercomputing (GCS) under GCS Large-Scale Project GCS-DWAR
on the Hazel Hen supercomputer at the High Performance Computing Center
Stuttgart (HLRS). Additional computations were carried out on the Vera machine
of the Max Planck Institute for Astronomy (MPIA) operated by the Max Planck
Computational Data Facility (MPCDF).

## Data Availability

The IllustrisTNG simulations, including TNG50, are publicly available and
accessible at www.tng-project.org/data, as described in Nelson et al. (2019).
Data directly related to this publication is available on request from the
corresponding author.

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##  Appendix A Probability that the central galaxy hosts an absorber

Current H i Ly$-\alpha$ surveys span a large variety of column densities and
redshifts. In Sections 3 and 4, we highlight the challenges when attributing
absorbers to a single galaxy. The commonly-adopted approach of assuming that
the galaxy at lowest impact parameter hosts the absorber is far too
simplistic. In Figure 9, we showed how the probability that an absorber
belongs to a central galaxy varies with $b$ and $\Delta v$. We have only
considered central galaxies with stellar masses $M_{*}=10^{9}$, $10^{10}$ and
$10^{11}$ M⊙ and $\log[N(\mbox{\text{H\,{i}}})/\rm{cm}^{-2}]>18.0$ absorbers
at $z=0.5$ which are most relevant for the MUSE-ALMA Haloes survey. Here, we
extend the plots to more specific bins in H i column density (pLLS, LLS, sub-
DLA and DLA) and also include $M_{*}=10^{8}$ M⊙ central galaxies. We will
expand the scope of this work to larger redshifts and Mg ii, C iv and O vi
absorbers in a forthcoming paper.

Figure 13: An extension of the bottom row of Figure 9 to include $10^{8}$ M⊙
central galaxies and more specific column density bins. Each hexbin is shaded
by the probability, corresponding to the four ranges in the colourbar, that an
absorber of given H i column density (rows) is associated with a central
galaxy of given stellar mass (columns). We only include absorbers at $z=0.5$,
although we plan an extension to higher redshifts and different ions in a
future work.