# The Cosmic Baryon Partition between the IGM and CGM in the SIMBA Simulations

Ilya S. Khrykin,1,2,3 Daniele Sorini,4 Khee-Gan Lee, 1,2 and Romeel Davé5,6,7
1Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583,
Japan
2Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan
3Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla
4059, Valparaíso, Chile
4Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK
5Scottish Universities Physics Alliance (SUPA), Institute for Astronomy,
University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK
6University of the Western Cape, Department of Physics and Astronomy,
Bellville, Cape Town 7535, South Africa
7South African Astronomical Observatories, Observatory, Cape Town 7925, South
Africa E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use the Simba suite of cosmological hydrodynamical simulations to
investigate the importance of various stellar and AGN feedback mechanisms in
partitioning the cosmic baryons between the intergalactic (IGM) and
circumgalactic (CGM) media in the $z\leq 1$ Universe. We identify the AGN jets
as the most prominent mechanism for the redistribution of baryons between the
CGM and large-scale IGM. In contrast to the full feedback models, deactivating
AGN jets results in $\approx 15-20$ per cent drop in fraction of baryons that
reside in the IGM and a consequent increase of baryon fraction in the CGM by
$\approx 50$ per cent. Additionally, we find that stellar feedback mechanisms
modify the baryon partition between IGM and CGM on a $10$ per cent level. We
further examine the physical properties of simulated haloes in different mass
bins, and their response to various feedback models. On average, a sixfold
decrease in the CGM mass fraction due to the inclusion of feedback from AGN-
driven jets is detected in $10^{12}M_{\odot}\leq M_{\rm 200}\leq
10^{14}M_{\odot}$ haloes. Examination of the average radial gas density
profiles of $M_{200}>10^{12}M_{\odot}$ haloes reveals up to an order of
magnitude decrease in gas densities due to the AGN jet feedback. We compare
gas density profiles from Simba simulations to the predictions of the
analytical modified NFW model, and show that the latter provides a reasonable
approximation of the gas radial profiles within the virial radii of the full
range of halo masses when rescaled by the CGM fraction of the halo. The
relative partitioning of cosmic baryons and, subsequently, the feedback models
can be constrained observationally with fast radio bursts (FRBs) in upcoming
surveys.

###### keywords:

galaxies: formation — galaxies: haloes — intergalactic medium — large-scale
structure of Universe — methods: numerical

††pubyear: 2022††pagerange: The Cosmic Baryon Partition between the IGM and
CGM in the SIMBA Simulations–The Cosmic Baryon Partition between the IGM and
CGM in the SIMBA Simulations

##  1 Introduction

It is believed that the intergalactic medium (IGM) contains the bulk of cosmic
baryons, providing the basic building blocks for the formation and evolution
of galaxies and large-scale structures (see McQuinn, 2016, for a recent
review). Indeed, at $z\gtrsim 3$, observations of H i Ly$\alpha$ absorption in
the spectra of high-z quasars (e.g. O’Meara et al., 2015), coupled with
estimates of the extragalactic ultraviolet background (e.g. Faucher-Giguère et
al., 2008), provide solid evidence that more than $90$ per cent of cosmic
baryons reside in the IGM. This estimate is in good agreement with predictions
from Big Bang Nucleosynthesis theory, deuterium abundance calculations, and
cosmic microwave background experiments (Steigman, 2010; Cooke et al., 2018;
Planck Collaboration et al., 2020). At lower redshifts ($z\lesssim 1$),
however, observations have failed to meet the same predictions, with up to
$\approx 20$ per cent of the expected cosmic baryons unaccounted for (the so-
called “missing baryon problem"; e.g. Fukugita & Peebles, 2004; Shull et al.,
2012). Theoretical models posit that a substantial fraction of these baryons
might exist within or around galactic haloes in a warm-hot phase, eluding
detection (Tumlinson et al., 2017).

According to the hierarchical structure formation paradigm (Lacey & Cole,
1993), at $z\lesssim 2$, baryons residing in the diffuse IGM sink into the
potential wells of the galactic haloes created by gravitational collapse. As
they are funnelled to the centres of the newly formed haloes, part of these
baryons get shock-heated to extreme temperatures ($T_{\rm vir}>10^{6}~{}{\rm
K}$) and become a part of the diffuse hot circumgalactic medium (CGM; White &
Rees, 1978; Efstathiou & Silk, 1983; Cole et al., 2000). However, despite the
abundance of observational methods that have been employed to probe the low-z
IGM/CGM gas, both in emission (e.g. Yoshino et al., 2009; Lim et al., 2020;
Tanimura et al., 2020) and absorption (e.g. Prochaska et al., 2011; Tumlinson
et al., 2013; Werk et al., 2014; Mathews & Prochaska, 2017; Heckman et al.,
017b; Nicastro et al., 2018; de Graaff et al., 2019), currently, it has been
challenging to make a full accounting of the cosmic baryon budget in the late-
time Universe. This is, in part, because individual techniques can only probe
specific phases of the gas occupying relative small fractions of the overall
cosmic baryon budget, while requiring specific assumptions (e.g., gas
temperature and metalicity, photon ionising background, etc) in order to
interpret observations.

The situation is further exacerbated by different astrophysical feedback
mechanisms that might impact the distribution of baryons in the CGM vis-a-vis
the IGM at low-z (e.g. Heckman & Thompson, 017a; Tumlinson et al., 2017;
Angelinelli et al., 2022; Sorini et al., 2022; Angelinelli et al., 2023;
Ayromlou et al., 2023). Cosmological hydrodynamical simulations have shown
that feedback is highly effective in evacuating the large fraction of baryons
from the galactic haloes out into the IGM (e.g., Davé et al., 2010; Martizzi
et al., 2019), while modifying the intrinsic radial density profiles of the
haloes (e.g. Pillepich et al., 2018; Ayromlou et al., 2021). These works also
suggest that the importance of different feedback mechanisms is correlated
with the halo mass.

For instance, Sorini et al. (2022) showed that at $z\lesssim 2$, in $M_{\rm
halo}>10^{12}M_{\odot}$ haloes, the feedback from active galactic nuclei (AGN)
is the dominant mechanism that pushes the baryons out of CGM into the IGM (see
also Appleby et al., 2021). The energy injected by the AGN feedback
additionally heats up the CGM gas, quenching the star-formation and modulating
the galaxy evolution (Scannapieco et al., 2005; Terrazas et al., 2020;
Fielding et al., 2020). On the other hand, there is both numerical and
observational evidence that stellar feedback is more prominent in the low-mass
haloes, and also capable of evacuating the baryons to large distances from the
centres of the haloes (e.g., Stinson et al., 2006; Rubin et al., 2014; Sorini
et al., 2022; Ayromlou et al., 2023). Therefore, unravelling the impact of
different feedback processes on the gas in and out of galactic haloes is
crucial for establishing the evolution of the cosmic baryon distribution and
locating all unaccounted baryons.

Among the observational probes, the emerging field of fast radio bursts (FRBs;
see Cordes & Chatterjee, 2019, for a review) offers a unique opportunity to
obtain insights into the distribution of cosmic baryons, as well as to
identify the physical nature of the feedback machinery. One of the key
measurable properties of these extragalactic millisecond radio transients is
the so-called dispersion measure,

$\mathrm{DM}=\int n_{e}(l)dl,$ (1)

which is a measure of the free electron column density, $n_{e}$, along the
line of sight $l$. Under the assumption of a fully ionised IGM and CGM, which
is a nearly perfect approximation in the post-reionisation universe, the
electrons probed by DMs of FRBs directly trace the distribution of cosmic
baryons (McQuinn, 2014; Ravi, 2019; Prochaska & Zheng, 2019; Macquart et al.,
2020; Simha et al., 2020; Batten et al., 2022).

The vast majority of FRBs detected to date, mostly by the Canadian Hydrogen
Intensity Mapping Experiment (CHIME; CHIME/FRB Collaboration et al., 2021),
have not been localised, i.e. the positional uncertainty of the FRBs are too
large to identify their host galaxies. This means that their redshifts are
unknown, hence so are the limits on the integral of Equation (1). It has been
estimated that samples of $\gtrsim 10^{4}$ unlocalised FRBs would be required
to place constraints on the CGM baryons via cross-correlation with galaxy data
(Shirasaki et al., 2022; Wu & McQuinn, 2023). With samples of localised FRBs,
such as those by Commensal Real-Time ASKAP Fast Transient (CRAFT; Macquart et
al., 2010), MeerKAT TRAnsients and Pulsars (MeerTRAP; Sanidas et al., 2018),
and Deep Synoptic Array (DSA; Kocz et al., 2019), the host galaxy and hence
the redshift is known, significantly improving the constraining power of FRBs
dispersion measures. Even then, samples of $\sim 10^{3}$ localised FRBs would
be required to discern the effect of AGN feedback on the cosmic diffuse gas
(Batten et al., 2022), i.e. far more than the $\sim 60$ known at the time of
writing. However, Lee et al. (2022) recently argued that combining FRBs’
dispersion measures with spectroscopic observations of foreground galaxies
(“FRB foreground mapping”) will allow simultaneous constraints on the IGM and
CGM baryon distributions with far greater precision than feasible with
localised FRBs alone. Specifically, they forecast that a sample of just $\sim
100$ FRBs at $0.1<z<0.8$ should be able to constrain the overall fraction of
cosmic baryons residing in the diffuse IGM, to within
$\sigma(f_{\mathrm{igm}})/f_{\mathrm{igm}}=0.075$. They also argued that,
assuming a simple modified Navarro-Frenk-White profile (mNFW; Navarro et al.,
1997; Mathews & Prochaska, 2017; Prochaska & Zheng, 2019) to represent all
intervening haloes, the sample could simultaneously constrain the halo cut-off
radius, $r_{\mathrm{max}}$, and fraction of halo baryons residing in the CGM,
$f_{\mathrm{gas}}$, to within $\sigma(r_{\mathrm{max}})/r_{\mathrm{max}}=0.11$
and $\sigma(f_{\mathrm{gas}})/f_{\mathrm{gas}}=0.12$, respectively. One can
easily relate $f_{\rm gas}$ to the total fraction of baryons residing in all
haloes of a given mass range $\left[M_{1},M_{2}\right]$ as

$f_{\rm cgm}=\frac{\int^{M_{2}}_{M_{1}}\left[\int^{r_{\rm max}}_{0}f_{\rm
gas}\Omega_{b}\rho_{h}\left(M_{h},z,r\right)4\pi r^{2}{\rm
d}r\right]\phi(M_{h})\,{\rm
d}\ln\frac{M_{h}}{M_{\odot}}}{(\Omega_{b}/V)\int_{V}\bar{\rho}_{\rm
m}\left(z\right){\rm d}V}.$ (2)

In the denominator, the cosmic matter density at a given redshift,
$\bar{\rho}_{\rm m}\left(z\right)=\Omega_{\rm m}\left(z\right)\rho_{\rm
crit}\left(z\right)$, is averaged over a sufficiently large volume of the
Universe $V$ and multiplied by the cosmic baryon fraction, $\Omega_{b}$, to
yield the baryon mass density within the volume $V$. In the numerator, the
inner integral integrates over $\rho_{h}\left(z,r\right)$, the radial matter
density profile of collapsed haloes with mass $M_{h}$, scaled by the
aforementioned cosmic baryon and CGM mass fractions. The outer integral then
integrates CGM mass weighted by the halo mass function $\phi(M_{h})$, defined
as the number density of haloes per logarithmic mass bin, over the mass range
$\left[M_{1},M_{2}\right]$. One limitation of the Lee et al. (2022) study is
that their simplistic model assumed that a fixed set of
$\\{f_{\mathrm{gas}},r_{\mathrm{max}}\\}$ describes all haloes intersected by
their simulated FRB sight lines. Similar assumptions have also been adopted in
observational papers studying FRB foregrounds, (e.g. Simha et al., 2020, 2021,
2023; Lee et al., 2023). Recent results from hydrodynamical simulations
already hint that this simple assumption is unrealistic, in that
$f_{\mathrm{gas}}$ will vary as a function of halo mass (Schaller et al.,
2015; Wang et al., 2017; Davies et al., 2019; Tollet et al., 2019; Davies et
al., 2020; Angelinelli et al., 2022; Sorini et al., 2022; Angelinelli et al.,
2023; Ayromlou et al., 2023), and possibly other galaxy properties as well.

[caption = Main parameters of the Simba simulation runs that were used in this
worlk. From left to right, columns show: name of the simulation, size of the
simulation box, total number of particles, different feedback mechanisms that
were used/not used in a given simulation., width = center, doinside = ]X c c c
c c c Name $L_{\rm box}$ $N_{\rm p}$ Feedback prescription
$h^{-1}{\rm cMpc}$ Stellar AGN winds Jets X-ray heating
Simba-100 100 $2\times 1024^{3}$ ✓ ✓ ✓ ✓
Simba-50 50 $2\times 512^{3}$ ✓ ✓ ✓ ✓
No-X-ray 50 $2\times 512^{3}$ ✓ ✓ ✓ –
No-Jet 50 $2\times 512^{3}$ ✓ ✓ – –
No-AGN 50 $2\times 512^{3}$ ✓ – – –
No-feedback 50 $2\times 512^{3}$ – – – –

The main goal of this paper is to examine how the basic distribution of cosmic
baryons between the IGM and CGM changes under the influence of different
feedback mechanisms in the low-z Universe. In order to do that, we follow up
recent results of Simba cosmological hydrodynamical simulations performed by
Sorini et al. (2022), focusing on quantities in $z=0.0-1.0$ range that are
more closely related to those that might be directly constrained by FRBs. In
addition to stellar feedback prescription, Simba offers a unique
implementation of the AGN feedback, which is directly tied to the process of
the accretion onto the central black holes (BH) including a torque-limited
model for the cold gas accretion (Anglés-Alcázar et al., 2015; Anglés-Alcázar
et al., 2017) and Bondi accretion for the hot gas (Bondi, 1952). Due to high
adaptability of Simba, we can deactivate different feedback modules and
dissect the importance and impact of each one of them on the distribution of
gas in and outside galactic haloes. Finally, we also outline the application
of the FRB observations to probing different feedback prescriptions.

This paper is organized as follows. In Section 2, we summarize the main
properties of the cosmological hydrodynamical simulations in the core of the
analysis presented in this work. In Section 3, we discuss the effect of
various feedback mechanisms on partitioning of cosmic baryons. In Section 4,
we discuss our findings and describe how observational constraints on the
cosmic baryon distribution from Fast Radio Bursts can be used to infer the
feedback models. Finally, we summarize and conclude in Section 5.

##  2 Cosmological Simulations

In this work, we use the Simba suite of cosmological simulations (Davé et al.,
2019), developed based on the meshless finite mass implementation of the
hydrodynamical code Gizmo (Hopkins, 2015). Simba incorporates prescriptions
for star formation, black hole seeding and accretion, as well as stellar and
AGN feedback. It also accounts for radiative cooling, photoionisation heating,
metal cooling and out-of-equilibrium evolution of primordial elements via the
Grackle-3.1 library (Smith et al., 2017). Since Simba has been extensively
described in previous works, we only outline the main parameters of the
simulations, and refer the interested reader to Davé et al. (2019) for a more
detailed description.

To study the distribution of cosmic baryons, we utilize the suite of six Simba
runs considered by Sorini et al. (2022). The box size, number of resolution
elements, and feedback prescriptions implemented in each run are summarised in
Table 1. All simulations adopt cosmological parameters consistent with Planck
Collaboration et al. (2016) measurements ($\Omega_{\rm
m}=0.3,~{}\Omega_{\Lambda}=0.7,~{}\Omega_{\rm
b}=0.048,~{}h=0.68,\sigma_{8}=0.82,n_{\rm s}=0.97$, with the usual definition
of the symbols). The fiducial run Simba-100 has a box size of
$100~{}h^{-1}~{}{\rm cMpc}$ and contains $1024^{3}$ dark matter particles and
as many gas resolution elements. It includes the numerical implementation of
the following feedback mechanisms:

Figure 1:  Upper panel: Redshift evolution of the baryon mass fraction in
different locations within the Simba fiducial simulation with $L_{\rm
box}=100\,h^{-1}\,\rm cMpc$ and all feedback prescriptions included
(Simba-100). The $y$\- axis is normalized to the cosmic baryon mass fraction.
Middle: partitioning of various contributions to the mass fraction of the gas
inside the CGM phase, shown in the upper panel. Right: corresponding pie-
chart, illustrating the per cent contributions from different sources to the
mass fraction of the gas in the CGM phase at $z=0.1$. Figure 2:  Left: same as
in the upper panel of Fig. 1, but for variants of the Simba simulation with
different feedback prescriptions. All runs are characterized with a box size
of $L_{\rm box}=50~{}h^{-1}{\rm Mpc}$. Middle: corresponding partitioning of
various contributions to the mass fraction of the gas inside the CGM in each
of the runs illustrated in the left panels. Right: pie-charts illustrating the
relative contributions from different halo mass ranges to the mass fraction of
all gas in the CGM.

  * •

Stellar feedback incorporates the combined effect of Type II supernovae (SN)
winds, radiation pressure, and stellar winds. It is modelled via metal-loaded
winds, whose mass-loading factor and velocity follows the scaling in the FIRE
zoom-in simulations (Anglés-Alcázar et al., 2017; Muratov et al., 2015). The
temperature of $30\%$ of the ejected wind particles is set by the difference
between the energy produced by the SN explosions and the kinetic energy of the
wind itself, while the remaining wind particles are assigned the temperature
of $10^{3}~{}{\rm K}$.

  * •

AGN feedback is implemented in three main modes:

    1. 1.

AGN winds \- perfectly bipolar outflows, produced by BH accreting at $>0.2$
times the Eddington accretion rate. The winds are kinetically coupled to the
surrounding gas, and do not change its thermal state. The speed of the
radiative wind outflows depends logarithmically on the BH mass.

    2. 2.

AGN jets \- produced in the BH with $M_{\rm BH}>10^{7.5}M_{\sun}$, accreting
at $<0.2$ times the Eddington accretion rate. Similarly to the radiative
winds, the jets are modelled as purely bipolar outflows that are kinetically
coupled to the gas. However, they can achieve much higher velocities,
logarithmically proportional to the inverse of the accretion rate. The maximum
allowed speed is set to $7000~{}{\rm km\ s^{-1}}$ at $f_{\rm edd}\leq 0.02$.

    3. 3.

X-ray heating is added to the AGN jets mode if the gas fraction in the BH host
galaxy drops below 0.2. This mode only affects the gas within the BH kernel,
both kinetically and thermally. In the latter mode, the intensity decreases as
the inverse square of the distance of the gas particle from the BH.

Like the fiducial Simba-100 run, the Simba-50 version includes all above-
mentioned feedback prescriptions, but is done in a smaller $L_{\rm
box}=50~{}h^{-1}{\rm cMpc}$ box with the same resolution. Finally, we
considered four more variants of the Simba-50 run, turning off different
feedback modules, to test their importance for the evolution of cosmic baryons
distribution (see Table 1 for details). We note that all versions of the
$50~{}h^{-1}{\rm cMpc}$ runs have identical initial conditions, and they have
not been recalibrated to match observational properties.

The haloes are identified using a three-dimensional friends-of-friends
algorithm from Gizmo, originally developed by V.Springel for the Gadget-3
code, with linking length equal to $0.2$ times the mean interparticle
separation. The outputs of the Simba simulations are post-processed with the
yt-based package Caesar111https://caesar.readthedocs.io/en/latest/ in order to
cross-match the positions of galaxies with identified haloes, and produce the
catalogues of their key quantities that are analysed in what follows.

##  3 Results

Table 1: Partition of cosmic baryons at $z=0.1$, in Simba simulation runs characterised by different feedback prescriptions. Run | Stars | ISM | HI | CGM | IGM
---|---|---|---|---|---
|  |  |  | Clusters | High-mass groups | Low-mass groups | Galaxies |
|  |  |  |  ($\geq 10^{14}M_{\odot}$) |  ($10^{13}M_{\odot}-10^{14}M_{\odot}$) |  ($10^{12}M_{\odot}-10^{13}M_{\odot}$) |  ($10^{10}M_{\odot}-10^{12}M_{\odot}$) |
Simba-100 | 3.70% | 0.82% | 0.79% | 3.15% | 2.10% | 0.76% | 1.82% | 86.78%
Simba-50 | 3.67% | 0.75% | 0.76% | 2.04% | 2.91% | 0.74% | 1.63% | 87.34%
No-X-ray | 5.33% | 0.99% | 0.93% | 2.02% | 2.79% | 0.93% | 1.75% | 85.06%
No-Jet | 10.65% | 1.66% | 1.15% | 2.58% | 6.51% | 4.17% | 2.48% | 70.55%
No-AGN | 12.20% | 1.30% | 1.05% | 2.64% | 6.60% | 3.84% | 2.64% | 69.51%
No-feedback | 21.27% | 21.64% | 0.46% | 2.01% | 4.89% | 3.34% | 2.67% | 58.82%

Hereafter, we describe the results of the Simba simulations and how different
feedback mechanisms affect the overall distribution of cosmic baryons.

###  3.1  Partitioning of cosmic baryons

We begin by inspecting the redshift evolution of the relative mass fraction of
cosmic baryons in our fiducial model Simba-100, which is illustrated in the
top panel of Fig. 1. In order to assign the baryons to one of four considered
phases, we inspect all simulation snapshots covering the redshift range
$0.0\leq z\leq 1.0$, and use the following criteria (see Appleby et al., 2021,
for details):

  * •

CGM: gas elements are considered to be part of the halo, if they are located
at distances $r\leq r_{200}$, where $r_{200}$ is the radial distance from the
minimum of the gravitational potential of the halo that encloses a total
matter density equal to $200$ times the critical density of the universe.
Hereafter, we will use $r_{200}$ as a proxy for the virial radius of a given
halo. The ‘CGM’ component excludes gas elements that are assigned to ISM and
Stars, described below;

  * •

ISM: gas particles located in the galactic haloes with total hydrogen number
density $n_{\rm H}>0.13~{}{\rm cm^{-3}}$ at temperatures
$\log_{10}\left(T/\mathrm{K}\right)<4.5+\log_{10}(n_{\rm H}/10~{}{\rm
cm^{-3}})$;

  * •

Stars: all star particles in the simulation box;

  * •

IGM: gas elements that are located outside the galactic haloes, at distances
$r>r_{200}$.

This categorization is simpler than the temperature-based categorization shown
in Sorini et al. (2022) for the simple reason that FRB measurements of the
free electron column are insensitive to the gas temperature. This will
potentially allow observers to deconstruct the IGM and CGM baryon
distributions with only the minimal assumption that the gas is ionised (e.g.,
Simha et al., 2020; Lee et al., 2022). In the following discussion, we shall
refer to the mass fraction of cosmic gas that resides in the IGM and CGM as
$f_{\mathrm{igm}}$ and $f_{\mathrm{cgm}}$, respectively. Also, when computing
the mass fraction of each of the gaseous phases described above, we will
remove the contribution of neutral hydrogen, so that ‘CGM’, ‘ISM’ and ‘IGM’
represent a partition of ionised gas only. As seen in Table 1, the global
baryon mass fraction of neutral hydrogen is unsurprisingly very low – at most
of the order $1$ per cent in all runs considered. Most cosmic neutral hydrogen
is known to reside within galaxies (Zwaan et al., 2003; Rhee et al., 2018) and
is not distributed co-spatially with the CGM and IGM. Therefore, to a very
good approximation, the diffuse ionised gas in the IGM and CGM track the
overall baryon distribution in those regions.

The top panel of Fig. 1 shows the mass fraction of baryons locked in the
aforementioned phases, with respect to the cosmic baryon mass in the
simulation volume, over the redshift range $0<z<1$. It is apparent that the
majority of cosmic baryons in Simba-100 reside inside the diffuse IGM gas
($f_{\rm igm}\simeq 87\%$), while only about $f_{\rm cgm}\simeq 8\%$ are
associated with the halo gas. This is consistent with the best observational
estimates, although there are large uncertainties on the latter (see Shull et
al., 2012; de Graaff et al., 2019, for an extensive review). Moreover, this
distribution hardly changes within the simulated redshift range, indicating
that the feedback from star-formation and AGN activity balances the
gravitational inflow of baryons from the IGM at $z<1$.

In the middle panel of Fig. 1, we focus on the CGM phase, further
distinguishing the contribution due to gas in haloes of different masses. We
adopt the following definitions for the halo mass bins considered:

  * •

Galaxies: $10^{10}M_{\odot}\leq M_{200}<10^{12}M_{\odot}$;

  * •

Low-mass Groups: $10^{12}M_{\odot}\leq M_{200}<10^{13}M_{\odot}$;

  * •

High-mass Groups: $10^{13}M_{\odot}\leq M_{200}<10^{14}M_{\odot}$;

  * •

Clusters: $M_{200}\geq 10^{14}M_{\odot}$.

In the above partition, $M_{200}$ is the total mass enclosed within $r_{200}$.
We only consider haloes that contain at least one galaxy substructure, as
identified by the Caesar package. This ensures that poorly resolved haloes at
the low-mass end are excluded from the analysis. The gaseous media within
haloes in the different mass categories are often given specific terms in the
literature, especially the Intra-Cluster Medium (ICM) when referring to
cluster gas or Intra-Group Medium (IGrM) for groups. For simplicity, in this
paper we refer to all circum-halo gas with the umbrella term ‘CGM’ regardless
of halo mass.

At lower redshift, the contribution from lower-mass haloes diminishes in
favour of the higher-mass haloes, especially clusters. At $z=1$, 70 per cent
of the CGM gas in the universe resides in galaxies and low-mass groups, and
only $\sim 10$ per cent in clusters. By contrast, at $z=0.1$, the CGM within
clusters accounts for $\sim 40$ per cent of the total, as shown in the pie
chart at the bottom of Fig. 1. Such dramatic evolution could be either a
consequence of the overall larger number of clusters at lower redshift, or of
the different impact of feedback processes on the baryonic content of haloes
with different masses.

In order to disentangle the role of different feedback mechanisms on the
partitioning of cosmic baryons, we thus examine the set of smaller-volume
($L_{\rm box}=50~{}h^{-1}{\rm cMpc}$) Simba simulations, listed in Table 1.
The left column of Fig. 2 illustrates the large-scale distribution of cosmic
baryons in these simulations that adopt different feedback prescriptions. In
addition, we indicate contributions to the mass fraction of the CGM gas from
different sources in the simulations in the middle and right columns of Fig.
2, using the same categorisation introduced in Fig. 1.

The top row of Fig. 2 illustrates the resulting distribution in the full-
feedback run Simba-50. It is apparent that the results in the left panel are
virtually indistinguishable from those of the Simba-100 simulation (Fig. 1,
top panel). However, we observe larger differences if we focus on the CGM
component only (middle panel in Fig. 1 and middle panel in the first row of
Fig. 2). The evolution of the mass fraction in the CGM of galaxies and low-
mass groups is essentially unchanged in the Simba-100 and Simba-50 runs. But
unlike the $100\,h^{-1}\,\rm cMpc$ fiducial run, the smaller-box variant does
not contain any clusters above $z\approx 0.75$. The poorer cluster statistics
in the smaller box systematically affects the results at lower redshift as
well. At $z=0.1$, about 28 per cent of the CGM gas resides in clusters in the
Simba-50 simulation, whereas this fraction is 40 per cent in the Simba-100
run. Therefore, the results on the overall classification of cosmic baryons
are robust to changes in the simulation volume, but the rarity of clusters in
the smaller volume affects the statistics of cosmic baryons within the CGM.
However, since the different feedback variants were run on the same box size
and initial conditions as Simba-50, we can still gain insight from the
relative changes in the gas fractions even in the most massive halos. Indeed,
the halo mass function varies only weakly across the different runs (Sorini et
al., 2022).

The second row of Fig. 2 shows the effect of turning off the X-ray heating
mode in the AGN feedback. Clearly, the absence of X-ray heating does not have
a significant effect on the overall distribution of baryons compared to the
Simba-50 run. By construction, the X-ray heating mode of the AGN feedback
affects matter only within the BH kernel and has very limited impact on
material outside of haloes (see Section 2). Therefore, the relative
$f_{\mathrm{igm}}\simeq 85\%$ and $f_{\mathrm{cgm}}\simeq 8\%$ fractions do
not change significantly with respect to the full feedback model. For the ISM
gas, half of the X-ray heating energy is converted to the kinetic energy by
imparting a radial outwards kick to the ISM particles (Sorini et al., 2022).
Turning off the X-ray heating allows modestly more baryons ($\sim 30-50\%$) to
accumulate in the ISM and stellar component compared to the full feedback run,
as seen in Table 1. These ISM and stellar contributions are however still very
subdominant compared to the CGM and IGM.

The jet mode of the AGN feedback plays a significantly more important role in
regulating the distribution of cosmic baryons. Indeed, as shown in the left
panel of the third row of Fig. 2, additionally turning AGN jets off increases
the $f_{\rm cgm}$ by $\simeq 50$ per cent at all considered redshifts when
compared to the full feedback model. Turning the AGN jets off drastically
reduces the supply of kinetic energy that was pushing the gas out of the
haloes, allowing more gas particles to stay within the CGM. Moreover, there is
a clear redshift evolution of the baryon fractions in the No-jet case. The
lack of AGN jets allows an increasingly larger amount of baryons to sink
towards the CGM and centres of the haloes. Additional gas is then consumed by
star formation, progressively increasing the fraction of baryons in stars
towards lower redshifts, reaching $\approx 10$ per cent by $z=0$ (see Table
1). It is also apparent from the middle and the right-hand side panels of the
No-jet row of Fig. 2 that turning AGN jets off result in larger relative
contribution of low-mass galaxy groups to the budget of baryons residing in
the CGM. Instead, the input from individual galaxies and clusters is reduced
by $\approx 10$ per cent when compared to the full-feedback simulation run.

On the other hand, as illustrated in the left panel of the fourth from the top
row of Fig. 2, additionally shutting down AGN winds does not alter the baryon
distribution substantially when compared to the No-jet version of the
simulations. Sorini et al. (2022) showed that the impact of the AGN winds is
sub-dominant effect with respect to AGN jets in evacuating baryons from haloes
across all mass ranges, and does not significantly change the $f_{\rm igm}$
and $f_{\rm cgm}$ fractions. A similar trend is present in the middle and
right columns on the No-AGN row of Fig. 2, where the relative contributions to
the CGM fraction from different sources hardly changes compared to the No-jet
run. This is quantified in Table 1, where the fractions of baryons in various
phases in these two versions of Simba are almost identical. This illustrates
the role of supernova feedback, which is still active in the No-AGN run, in
regulating the cosmic baryon distribution.

Finally, we explore the No-feedback model, in which all AGN-related and
stellar feedback processes are turned off. The corresponding scenario is
illustrated in the bottom row of Fig. 2. Overall, the evolution of the baryon
distribution is qualitatively similar to the No-jet and No-AGN cases, albeit
with a several per cent decrease in the total amount of gas in favour of
stars. However, the fraction of baryons residing in the IGM at $z\simeq 0.0$
drops by $\approx 10$ per cent compared to the No-jet or No-AGN models (see
Table 1). Therefore, the SN-driven winds do play a significant role in
depleting the haloes of gas and pushing it to the IGM at low redshift (see
also Sorini et al., 2022). Turning them off results in $f_{\rm igm}$ not
exceeding $\simeq 60$ per cent, and $f_{\rm cgm}\approx 10\%$ at $z\simeq
0.0$. Consequently, the formation of stars is more effective in such model
because of the lower gas temperatures in the inner part of haloes, hence the
fraction of baryons in stars reaches $\approx 20$ per cent by $z=0.0$ (see
Table 1 for details).

It is clear that different feedback prescriptions result in a different
partition of baryons, which, in principle, can be constrained by observations.
Having established the role of different feedback prescriptions in shaping the
distribution of cosmic baryons across multiple phases and redshifts, we now
focus our attention on the physical properties of the galactic haloes and
their response to various feedback models.

Table 2: Gas mass fraction $f_{\rm gas}$ within haloes at $z=0.1$, in Simba simulation runs characterised by different feedback prescriptions. The values correspond to the median and $16^{\rm th}-84^{th}$ percentiles of the distributions, presented in Fig. 3 for the halo mass bins defined in Section 3.1 Halo Category | Mass range | Simba-100 | Simba-50 | No-X-ray | No-Jet | No-AGN | No-feedback
---|---|---|---|---|---|---|---
Galaxies | $10^{10}M_{\odot}\leq M_{200}<10^{12}M_{\odot}$ | $0.30^{+0.19}_{-0.19}$ | $0.29^{+0.19}_{-0.20}$ | $0.28^{+0.20}_{-0.16}$ | $0.37^{+0.18}_{-0.14}$ | $0.40^{+0.17}_{-0.15}$ | $0.28^{+0.11}_{-0.13}$
Low-mass Groups | $10^{12}M_{\odot}\leq M_{200}<10^{13}M_{\odot}$ | $0.06^{+0.14}_{-0.04}$ | $0.07^{+0.11}_{-0.05}$ | $0.10^{+0.16}_{-0.06}$ | $0.54^{+0.12}_{-0.12}$ | $0.53^{+0.11}_{-0.11}$ | $0.35^{+0.06}_{-0.05}$
High-mass Groups | $10^{13}M_{\odot}\leq M_{200}<10^{14}M_{\odot}$ | $0.20^{+0.19}_{-0.09}$ | $0.21^{+0.25}_{-0.09}$ | $0.22^{+0.17}_{-0.10}$ | $0.65^{+0.04}_{-0.05}$ | $0.64^{+0.05}_{-0.04}$ | $0.47^{+0.05}_{-0.04}$
Clusters | $M_{200}\geq 10^{14}M_{\odot}$ | $0.69^{+0.15}_{-0.12}$ | $0.73^{+0.05}_{-0.05}$ | $0.71^{+0.05}_{-0.05}$ | $0.73^{+0.02}_{-0.05}$ | $0.71^{+0.03}_{-0.03}$ | $0.59^{+0.00}_{-0.03}$

###  3.2 Gas mass fraction in the CGM

We begin by investigating how the baryon fraction enclosed within
circumgalactic haloes (i.e. outside the galactic stellar and ISM components)
varies with respect to the overall halo mass in the simulations with different
feedback mechanisms (see Table 1).

For a given simulation run, we estimate the median and $16^{\rm th}$-$84^{\rm
th}$ percentiles of the distribution of the CGM gas mass fraction across all
haloes, normalized by $f_{\rm b}M_{200}$, where $f_{\rm b}=\Omega_{\rm
b}/\Omega_{\rm m}$ is the cosmic baryon mass fraction. We then plot the
resulting $f_{\rm gas}$ calculated at $z=0.1$ in Fig. 3. In addition, in Table
2, we list the values of $f_{\rm gas}$ for specific halo mass ranges defined
in Section 3.1.

It is immediately noticeable that halos within the Simba runs with at least
stellar feedback agree reasonably well with each other for the least massive
($10^{10.5}~{}M_{\odot}\leq\mathrm{M}_{\rm 200}\leq
10^{11}~{}\mathrm{M}_{\odot}$) and the most massive ($M_{\rm 200}\geq
10^{14}~{}\mathrm{M}_{\odot}$) ranges. Thus, in these halo mass intervals, the
AGN feedback processes have little effect on the amount of gas within the CGM.
Because of the deeper potential wells and larger virial radii, cluster-sized
haloes still retain a similar CGM mass fraction with or without AGN-driven
jets, even though jets do contribute significantly to increasing the
temperature of the CGM gas (see Sorini et al., 2022). On the other end of the
mass range, the BHs residing in the central galaxies of the simulated haloes
are typically below the mass threshold that can trigger AGN-driven jets in the
Simba model (Thomas et al., 2019). Therefore, stellar feedback is the dominant
mechanism ejecting gas from the shallow gravitational potentials and thus
decreasing the CGM gas fraction.

Similar to the discussion in Section 3.1, there is no apparent difference in
the results obtained in the fiducial Simba-100 and Simba-50 runs across the
full mass range considered. This supports the conclusion that the results in
Fig. 3 are converged with respect to the simulation volume. Additionally,
$f_{\rm gas}$ evolution in the No-X-ray version of the simulation is
effectively indistinguishable from the full feedback runs. As shown in Section
3.1, the X-ray heating is only effective in the very inner region of the
haloes within the BH kernel and does not significantly change the CGM mass
fraction inside (and, likewise, outside) of the haloes.

It is apparent from Fig. 3 that, consistent with the discussion in Section
3.1, AGN jets are the dominant feedback mechanism that strongly suppresses the
CGM mass fraction, $f_{\rm gas}$, in the halo mass range
$10^{12}\,\mathrm{M_{\odot}}<M_{200}<10^{14}\,\mathrm{M_{\odot}}$, namely the
mass range occupied by galaxy groups. In the absence of the AGN jets, the
median gas mass fraction inside the haloes increases sixfold in comparison to
the fiducial Simba runs with the full feedback model. Moreover, as mentioned
previously, the effect of the AGN winds is sub-dominant compared to that of
AGN jets, and does not by itself change significantly the partitioning of
cosmic baryons between the IGM and CGM. Therefore, the gas fraction inside the
CGM of haloes should not change in the absence of the AGN winds. Indeed, as it
is clear from Fig. 3, the evolution of $f_{\rm gas}$ in No-AGN model is
extremely similar to the track of the No-jet run.

Finally, the evolution of the gas mass fraction within the CGM of haloes in
the No-feedback shows larger values in the intermediate mass range
$10^{12}\,\mathrm{M_{\odot}}\lesssim M_{200}\lesssim
10^{14}\,\mathrm{M_{\odot}}$ compared with models including all feedback
modes. Similar to the discussion in Section 3.1, due to the lack of feedback
mechanisms, there is more gas present in the CGM, compared to the
Simba-50/Simba-100 runs. On the other hand, tuning off stellar feedback
results in increased star formation in the No-feedback run. This leaves less
gas within the CGM when compared to the No-jet/No-AGN models, in which star
formation is suppressed.

Figure 3: Gas mass fraction within haloes, as a function of $M_{200}$, for
runs of the Simba simulation with different feedback variants. The plot refers
to redshift $z=0.1$. The solid lines are colour coded according to the run
considered (see legend), and indicate the median gas mass fraction within each
$M_{200}$ bin, while the shaded areas show the $16^{\rm th}-84{\rm th}$
percentiles of the distribution.

We have shown that the AGN feedback, and more specifically, AGN jets are the
dominant mechanism evacuating CGM gas from haloes with $M_{200}\gtrsim
10^{12}\,M_{\odot}$. However, while other feedback mechanisms considered in
this work do not impact the partitioning of baryons between IGM/CGM as
strongly, they could in principle affect the spatial distribution of baryons
within haloes more significantly. In order to assess this, we will look into
the radial density profiles of the haloes in the next Section.

\begin{overpic}[width=433.62pt]{Figures/dprof_ratios_145.pdf}
\put(10.0,66.5){(a)} \put(32.0,66.5){(b)} \put(55.0,66.5){(c)}
\put(80.5,66.5){(d)} \put(10.0,45.0){(e)} \put(32.0,45.0){(f)}
\put(55.0,45.0){(g)} \put(80.0,45.0){(h)} \put(10.0,23.5){(i)}
\put(32.0,23.5){(j)} \put(55.0,23.5){(k)} \put(80.5,23.5){(l)} \end{overpic}
Figure 4: Upper panels: Radial profiles of the average comoving gas density
around haloes within different classes of haloes (see main text for details),
as indicated above every panel. Lines of different colours correspond to the
results from Simba runs with different feedback prescriptions, as reported in
the legend. The error bars show the extent of the $16^{\rm th}-84^{\rm th}$
percentiles of the distribution. To aid readability, lower error bars have
been omitted if the $16^{\rm th}$ percentile lies below the lower bound of the
$y$-axis. In all panels, the $x$-axis has been normalised to the $r_{200}$
radius. All results refer to redshift $z=0.1$. Middle panels: Ratio of the
radial profiles of the average gas density shown in the corresponding upper
panels and the mean dark matter density profile obtained from the No-feedback
run, rescaled by the cosmic baryon mass fraction. Lower panels: As in the
middle panels, except that the ratio is taken with respect to the fiducial
mNFW profile with $f_{\rm gas}=0.75$ in each halo mass bin. We also with the
dashed-horizontal line show the ratio $f_{\rm gas,simba100}/0.75$ (Table 2),
and their $16^{\rm th}-84^{\rm th}$ percentile range in the shaded area.

###  3.3  Radial gas density profile

In this section, we study the gas density distribution within haloes, and how
this relates to the underlying dark matter profile, in the various feedback
runs. We will also test how analytical models of the CGM gas distribution
within haloes, such as the mNFW profile, compare against the simulations
considered.

We first select all haloes in the $z=0.1$ snapshot of each Simba simulation
volume, and consider all gas elements (excluding the ISM component) that are
located at distances $0.01r_{200}\leq r\leq 5r_{200}$. The elements are then
divided into $20$ equal-length logarithmic bins of distance and split between
the CGM phases as defined in Section 3.1. Additionally, we divide identified
haloes into $4$ bins according to their mass, corresponding to the mass ranges
defined in the discussion in Section 3.1 and in Table 1. We estimate the mean
and $16^{\rm th}-84^{\rm th}$ percentiles of the gas co-moving density
distribution as a function of radial distance $r$ in each halo mass bin. The
resulting gas density profiles estimated for each feedback prescription model
are illustrated in the top row of Fig. 4.

We then obtain the dark matter density profile within haloes in the No-
feedback run, $\rho_{\rm DM}^{\rm No-fb}$, with the same procedure utilised
for obtaining the gas density. This quantity, multiplied by the cosmic baryon
mass fraction, $f_{b}\equiv\Omega_{b}/\Omega_{m}$ represents the radial
profile expected if halo baryons trace their dark matter profiles perfectly.
The ratios of the gas density profiles for all runs with respect to
$f_{b}\,\rho_{\rm DM}^{\rm No-fb}$ are reported in the middle row of Fig. 4.

Finally, in the bottom row of Fig. 4 we make a comparison with the mNFW halo
profile. In the semi-analytic mNFW model, the NFW radial profile is modified
to more accurately capture the hydrodynamics of halo gas:

$\rho_{b}\left(r\right)=f_{\rm
gas}\left(\frac{\Omega_{b}}{\Omega_{m}}\right)\frac{\rho_{0}\left(M_{\rm
200}\right)}{y^{1-\alpha}\left(y_{0}+y\right)^{2+\alpha}},$ (3)

where $\rho_{0}\left(M_{200}\right)$ is the central density (see Prochaska &
Zheng 2019), $y\equiv c\left(r/r_{200}\right)$, $c$ is the concentration
parameter, $y_{0}$ and $\alpha$ are the profile parameters. We adopt the
fiducial values $c=7.67$, $y_{0}=2.0$, $\alpha=2.0$ and $f_{\rm gas}=0.75$, as
in Prochaska & Zheng (2019). In principle, $\rho_{0}\left(M_{\rm 200}\right)$
is normalized such that the volume integral over the mNFW profile yields a
total CGM mass that is a fraction $f_{\mathrm{gas}}$ relative to the total
halo baryonic mass, i.e. $M_{\rm cgm}\equiv
f_{\mathrm{gas}}(\Omega_{b}/\Omega_{m})M_{200}$. However, note that there is
no a priori guarantee that $f_{\rm gas}$ used in Equation 3 is necessarily
consistent with that measured in the simulations through Equation 2 — this is
a hypothesis we will examine. We plot the ratio between the mean gas density
profiles measured from the simulations and the estimated mNFW profile in the
bottom row of Fig. 4.

It is evident from the Galaxies column ($10^{10}M_{\odot}\leq
M_{200}<10^{12}M_{\odot}$) of Fig. 4 that the radial gas density profiles of
individual galaxies in all feedback models are generally similar to each
other. As discussed in Section 3.2, AGN feedback has little effect on the
radial distribution of baryons in haloes with masses below
$M_{200}<10^{12}M_{\odot}$ that are unlikely to host massive BHs. Similarly,
Fig. 4e shows that models with AGN jets and winds turned off have only
slightly higher ($\approx 1$ per cent) gas to dark matter density ratio than
the full feedback runs. The No-feedback model displays a deprivation of CGM
baryons below $r\sim 0.3r_{200}$, where the gas has condensed almost entirely
into the ISM phase. While stellar feedback is not the most effective mechanism
to expel gas outside more massive haloes (see discussion in Section 3.1), in
low-mass haloes it nevertheless ejects a significant amount of baryons and
reshuffles the distribution within the virial radius. Once this feedback
mechanism is turned off, the CGM gas at $r\lesssim r_{200}$ is more
effectively accreted and converted into stars. Fig. 4i shows that the
simulated gas density profiles in all feedback models disagree within a factor
of $\sim 3$ with the theoretical prediction of the mNFW model at
$0.2r_{200}\lesssim r\lesssim r_{200}$. This difference is likely caused by
the choice of the scaling factor, $f_{\rm gas}$, in the mNFW profile. While we
set the fiducial value $f_{\rm gas}=0.75$, from Fig. 3 it appears that $f_{\rm
gas}$ should be a factor of $\approx 2-7$ smaller depending on the exact
feedback prescription as shown in Fig. 3 (see also Table 2). To visualise this
point, in the bottom panels of Fig. 4 we plot as a horizontal dashed black
line the ratio $f_{\rm gas,simba100}/0.75$, which indicates the mNFW profile
if the Simba-100 $f_{\mathrm{gas}}$ values in Table 2 were adopted. We
additionally show the $16^{\rm th}-84^{\rm th}$ spread of this ratio as a grey
shaded area.

For the Low-mass groups with $10^{12}M_{\odot}\leq M_{200}<10^{13}M_{\odot}$
that are capable of sustaining massive BHs, one sees from Fig. 4 that AGN
feedback plays a significantly more important role in redistributing baryons
across the CGM and beyond. Indeed, at $r\lesssim r_{200}$, the halo gas
densities found in No-jets and No-AGN models are up to $\sim 6-7$ times higher
compared to models with the full feedback prescription (Fig. 4b).
Interestingly, the density profile of the No-feedback model occupies an
intermediate regime between the full feedback models and models without AGN
feedback. As for the Galaxies, this is due to the fact that the lack of any
feedback allows gas to accrete into stars, not only in the central regions,
but across the full extent of the haloes. Once the stellar feedback is turned
on (the No-AGN run), it suppresses the star formation and partially ejects gas
out of the member galaxies into the halo to retain more baryons in the CGM
relative to the No-feedback run. On the other hand, as we illustrated
previously, AGN winds are not effective at expelling gas out of haloes,
whereas AGN jets are. Therefore, in Simba runs with at least AGN jets turned
on, one sees a decrease in gas density throughout the halo.

Fig. 4f provides additional evidence of the AGN feedback efficacy in expelling
gas particles to the outskirts of the higher-mass haloes. It is apparent that
within the halo extent ($r\leq r_{200}$) the full feedback models contain
$\approx 3-4$ less gas than dark matter compared to the No-jet/No-AGN runs.
The gas density profiles in the Low-mass groups also show significantly more
deviation from the predictions of the mNFW model, as illustrated in Fig. 4j.
While the No-jet/No-AGN models show good agreement with the fiducial mNFW
model ($f_{\rm gas}=0.75$) at $0.2r_{200}\lesssim r\lesssim 2r_{200}$, the gas
densities within full feedback Simba runs, on the other hand, are $\approx
4-5$ times lower than the fiducial mNFW model forecasts in the same regime.
However, while the full feedback models retain less gas, they remain
relatively flat with respect to the mNFW at $0.2r_{200}\lesssim r\lesssim
2r_{200}$. The mNFW profile with a rescaled $f_{\mathrm{gas}}$ (Fig. 3 and
Table 2) should thus be a reasonable approximation in this regime, as
indicated by the dashed line in Fig. 4j.

Overall, similar trends hold for the High-mass groups ($10^{13}M_{\odot}\leq
M_{200}<10^{14}M_{\odot}$), illustrated in the third column of Fig. 4. In the
absence of the AGN feedback, the CGM retains up to $\approx 10$ times more gas
(at $r\lesssim r_{200}$) than in the full feedback runs. However, strong AGN
feedback in the high-mass groups expels more gas beyond their virial radii.
Therefore, the gas density profiles of the full feedback models show an excess
at $r\gtrsim r_{200}$, compared to runs with AGN feedback turned off. This is
further illustrated in Fig. 4g panel, where full feedback models contain up to
$\approx 10$ times less gas than dark matter at $r\sim 0.1-0.2r_{200}$
compared to the No-jet/No-AGN runs. The latter, similar to the Low-mass
groups, display an almost flat ratio of gas and dark matter densities at
$r\gtrsim 0.5r_{200}$. It is apparent from Fig. 4k, that analogous to the Low-
mass groups case, the gas density profiles of High-mass groups in the No-
jet/No-AGN and No-feedback runs have a steeper radial fall-off than the mNFW
prediction in the centres of the haloes, at $r\lesssim 0.2r_{200}$. However,
they show mNFW-like density profiles with $f_{\rm gas}=0.75$ inside the CGM at
$0.2r_{200}\lesssim r\lesssim 1.0r_{200}$. On the contrary, the fiducial mNFW
model predicts densities up to $\approx 3-5$ times higher inside the CGM at
$0.1r_{200}\lesssim r\lesssim 1.0r_{200}$ compared to the full feedback
models. However, as indicated by the dashed line in Fig. 4k, rescaling the
mNFW model with the $f_{\rm gas}$ value inferred from Simba-100 run (see Table
2) instead of using the fiducial $f_{\rm gas}=0.75$, helps to mitigate the
discrepancy.

Finally, Fig. 4d shows that in the high-mass end of the haloes’ mass range the
only noticeable difference between various feedback models is retained in the
very centres of the haloes ($r\lesssim 0.5r_{200}$), where the AGN feedback
mechanisms are effective in evacuating baryons from the inner regions of
haloes. However, at the halo outskirts ($r\gtrsim 0.5r_{200}$) the CGM gas
density profiles of the highest-mass haloes once again become almost
indistinguishable. The strong gravitational potential wells of the clusters do
not allow gas ejection beyond the edge of the haloes into the IGM, resulting
in a very similar gas density profiles among all feedback models. This is
further evident from Fig. 4h, where the density of gas approaches that of dark
matter at $r\gtrsim 0.5r_{200}$ in all feedback models. Finally, as
illustrated in the Fig. 4l panel, the gas density profiles of the high-mass
haloes are in good agreement with the predictions of the mNFW model with
$f_{\rm gas}=0.75$ at $r\gtrsim 0.3r_{200}$.

##  4 Discussion

###  4.1 Prospects of using FRBs to infer feedback mechanisms

In Section 3.1 we showed that various feedback prescriptions adopted in Simba
simulation runs result in distinctly different partitions of the cosmic
baryons across the redshift range studied in this work. It is immediately
apparent from Table 1 that the fraction of cosmic baryons that reside in the
diffuse IGM is by itself a strong indicator of the relevant feedback
mechanisms. For instance, the difference between $f_{\rm igm}$ found in the
full-feedback models and no-feedback model is $\approx 30$ per cent, while the
differences between other adopted feedback models lie in range $\Delta f_{\rm
igm}\simeq 2-20\%$. At the same time, the different feedback models yield
strikingly different predictions for the fraction of baryons,
$f_{\mathrm{gas}}$, that reside within halo CGM of galaxies and groups at
different halo masses (Fig. 3). Thus, it becomes feasible to distinguish
between various models through observational methods that can constrain
$f_{\mathrm{igm}}$ and $f_{\mathrm{gas}}$.

For instance, the “FRB foreground mapping” technique (Lee et al., 2022) that
combines FRB dispersion measurements, spectroscopic survey of the foreground
galaxies, and Bayesian algorithms for foreground density reconstruction (Ata
et al., 2015; Ata et al., 2017), offers a unique opportunity to measure the
cosmic baryon partition in the low-redshift Universe. Lee et al. (2022) argued
that a sample of $N=30$ localised FRBs will be enough to constrain the $f_{\rm
igm}$ to $\approx 10$ per cent precision. Comparing this forecast to the
results presented in Section 3.1, the full-feedback and no-feedback Simba
models would be distinguished at a $\approx 2.5\sigma$ level using
$f_{\mathrm{igm}}$ alone. Further constraints would come from the CGM gas
fractions, $f_{\mathrm{gas}}$, of the galaxies and groups with directly
intersected by the FRB sightlines. Unlike X-ray emission and the Sunyaev-
Zel’dovich effects, however, $f_{\mathrm{gas}}$ constraints using FRBs have no
explicit dependency on the halo baryon density. Therefore, $f_{\mathrm{gas}}$
can be constrained for haloes down to $M<10^{11}\,M_{\odot}$ so long as they
can be clearly identified as foreground galaxies. However, a more quantitative
forecast combining $f_{\mathrm{igm}}$ with the $f_{\mathrm{gas}}$ from various
halo masses would require incorporating their covariances, which we defer to
future work. Nevertheless, the ongoing FRB Line-of-sight Ionization
Measurement From Lightcone AAOmega Mapping survey (FLIMFLAM; Lee et al., 2022,
Khrykin et al. in prep) is gathering foreground spectroscopic data on $N\sim
20$ FRB fields with the goal of delivering preliminary constraints on
$f_{\mathrm{igm}}$ and $f_{\mathrm{gas}}$.

On the other hand, 10 per cent precision on $f_{\mathrm{igm}}$ is not enough
to differentiate between different feedback mechanisms (see Table 1). However,
upcoming generations of multi-fiber spectroscopic facilities, such as the
ongoing Dark Energy Spectroscopic Instrument (DESI; Levi et al., 2013) and
William Hershel Telescope Enhanced Area Velocity Explorer (WEAVE; Dalton et
al., 2012), coupled with increased FRB localisation capabilities of CHIME/FRB
and DSA in the Northern Hemisphere will increase the number of FRBs suitable
for FRB foreground mapping to several hundreds. As forecast in Lee et al.
(2022), $N\simeq 100$ FRBs would be enough to achieve a $\approx 5$ per cent
precision on $f_{\rm igm}$. This level of precision would enable the
discrimination amongst the predictions of most Simba feedback models (see
Table 1), especially in conjunction with the $f_{\mathrm{gas}}$ constraints
based on intervening foreground galaxies or groups with accurately-estimated
halo masses (Hahn et al., in prep.).

###  4.2 Implications for the models of halo density profiles

The NFW profile (Navarro et al., 1997) has been widely used to describe the
density distribution in the collapsed dark matter haloes in the cosmological
framework. It is however not expected to be an adequate representation of the
baryonic density profiles, which are needed for modelling the dispersion
measure contributed by the intervening haloes’ CGM to the observed total
dispersion measure of a given FRB. In the context of FRB DM analysis and the
hydrodynamic effects that would alter the baryonic radial profiles, Prochaska
& Zheng (2019, see also , ) introduced a modified version of the NFW profile
(see Equation 3). According to the discussion in Section 3.3 and bottom row of
Fig. 4, the deviation of the inferred CGM gas density profiles from the
analytical mNFW model does not only depend on the considered halo mass, but
additionally on the exact feedback prescription, and distance from the centre
of the halo.

In comparison with a scaled version of the dark matter radial profile (middle
row of Fig. 4), the mNFW profile yields a better match for the slope of the
halo gas profiles, particularly in the range $0.1\lesssim r/r_{200}\lesssim
1.0$. By adjusting the $f_{\mathrm{gas}}$ scaling parameter in the profile to
the CGM gas fraction seen in the simulated haloes of various mass ranges, mNFW
can approximate the halo gas profiles at $0.1\lesssim r/r_{200}\lesssim 1.0$
to within a factor of $\sim 30-50\%$ across the mass range for most of the
feedback models. The agreement is best for galaxy clusters, presumably because
the hydrostatic equilibrium of the well-virialized halos is the most amenable
to the semi-analytic approximation of mNFW.

However, it seems clear that the choice of $f_{\mathrm{gas}}$ should vary as a
function of halo mass in order to match the simulated profiles, regardless of
the feedback model (see Table 2). Several previous works that utilised the
mNFW model to describe the density distribution of the galactic haloes have
used a fixed fiducial value of $f_{\rm gas}=0.75$ (Simha et al., 2020, 2021,
2023), similarly adopted in this work, as the baseline value for comparison.
Based on the results of our Simba runs, it is an oversimplification to adopt a
single value of $f_{\rm gas}$ across all halo masses (see the bottom row of
Fig. 4). The fiducial value of $f_{\rm gas}=0.75$, which was originally
proposed by Prochaska & Zheng (2019) based on somewhat hand-waving arguments,
can only be safely applied to describe the cluster-level massive haloes with
$M_{200}\gtrsim 10^{14}M_{\odot}$, but not lower-mass haloes including
individual galaxies. Recently, Lee et al. (2023) discovered two foreground
galaxy clusters that are intersected by the well-known FRB 20190520B
sightline. They adopted $f_{\mathrm{gas}}=0.9$ for the gas fraction of these
clusters; based on our findings, this is likely to lead to a slight
overestimation of the dispersion measure contributed by these clusters, if the
Simba simulation is a good representation of the real universe.

Our conclusions on the accuracy of the mNFW profile are only valid for the
simulation suite analysed here: the mNFW profile may or may not be a
reasonable fit to CGM profiles in other simulations with different underlying
physical models for feedback processes are considered (e.g. Schaye et al.,
2015; Pillepich et al., 2018; Pakmor et al., 2023; Schaye et al., 2023). This
remains to be verified in future works.

Similar to Prochaska & Zheng (2019), in this study, we have fixed the
concentration parameter of the gas inside the haloes ($c=7.67$) to the value
adopted in the original NFW model as characteristic for the Milky Way.
However, several numerical studies based on different N-body or hydrodynamical
simulations have shown that the concentration parameter is not, in fact,
constant, but instead evolves with halo mass and redshift, with lower-mass
haloes more concentrated than higher-mass haloes at $z=0$ (e.g., Macciò et al.
2007; Ishiyama et al. 2013; Dutton & Macciò 2014; Klypin et al. 2016; Schaller
et al. 2015; Rodríguez-Puebla et al. 2016; López-Cano et al. 2022; Shao et al.
2023; Sorini et al., in prep.). In particular, the average concentration of
haloes with $M_{200}<10^{12}\,M_{\odot}$ is typically larger than the $c=7.67$
value adopted throughout the mass bins considered in this work (see, e.g.
Schaller et al., 2015). The low concentration parameter and unrealistically
high $f_{\rm gas}$ adopted in the mNFW profile for the low-mass groups may
explain why the match with the numerical results is particularly poor in this
halo mass range. A mass-dependent concentration parameter may therefore yield
a better match between the simulated profiles mNFW profiles, and we defer such
investigations to future work.

Nevertheless, in the upcoming FLIMFLAM analyses that will aim to constrain
$f_{\mathrm{igm}}$ and $f_{\mathrm{gas}}$ as free parameters, most haloes
directly intersected by FRB sightlines are likely to have impact parameters at
$0.1\,r_{200}\lesssim r\lesssim 1\,r_{200}$. This implies that mNFW is likely
to be an adequate model for the radial CGM profiles, at least for the first
generation of analyses. As the observational constraints improve in future
data sets, a more precise model than the current version of mNFW would likely
be required.

##  5 Conclusions

In this work, we followed up on the results of Simba cosmological
hydrodynamical simulations from Sorini et al. (2022), and analysed how various
feedback mechanisms affect the partition of cosmic baryons between the diffuse
IGM and CGM of haloes at $z\leq 1.0$. We also examined halo properties under
different feedback prescriptions. The main conclusions of our work are as
follows:

  1. 1.

The jets mode of the AGN feedback plays the most important role in reshuffling
the baryons between the IGM and CGM at $z=0.0-1.0$ (see Fig. 2). In contrast
to the full-feedback Simba run ($f_{\rm igm}\simeq 87\%$, at $z=0.0$),
deactivating AGN jets results in a $\approx 20$ per cent drop in the IGM
baryon fraction ($f_{\rm igm}\simeq 70\%$, at $z=0.0$), and a subsequent
increase of the global CGM fraction from $f_{\rm cgm}\simeq 5\%$ to $f_{\rm
cgm}\simeq 13\%$, respectively.

  2. 2.

Stellar feedback, on the other hand, is found to have a $\approx 10$ per cent
effect on the amount of baryons in the IGM. Simba without stellar feedback
produces $\approx 20$ per cent more stars owing to decreased gas temperatures
and a more effective star-formation process.

  3. 3.

The evolution of the CGM gas mass fraction $f_{\rm gas}$ depends strongly on
the halo mass, as well as the exact feedback prescription (see Fig. 3). AGN
jets significantly expel gas from haloes, resulting on average in almost
sixfold drop of $f_{\rm gas}$ compared to the models without AGN feedback
within haloes in $10^{12}M_{\odot}\lesssim M_{200}\lesssim 10^{14}M_{\odot}$
mass range.

  4. 4.

Examination of the halo radial density profiles (see Fig. 4) additionally
indicates that the AGN feedback is the most effective mechanism for
redistributing the baryons between the CGM and IGM in the
$10^{12}M_{\odot}\lesssim M_{200}\lesssim 10^{14}M_{\odot}$ mass range. In the
absence of the AGN jets, such haloes retain on average $\approx 10$ times more
gas within their respective virial radius. On the contrary, we find that the
distribution of baryons within $M_{200}\lesssim 10^{12}M{\odot}$ and
$M_{200}\gtrsim 10^{14}M_{\odot}$ haloes is only weakly sensitive to the
different feedback prescriptions studied in this work.

  5. 5.

Comparison of the mNFW analytical model with the results of the full
hydrodynamical simulations indicates that the former is an adequate
description for the radial distribution of the gas within virial radii of the
$M_{200}\lesssim 10^{14}M_{\odot}$ haloes. However, we note that given the
dependence of the $f_{\rm gas}$ on the halo mass (see Fig. 3), future
applications of the mNFW model should avoid using a fixed value of $f_{\rm
gas}$ and instead either treat it as a free parameter or use the values shown
in Table 2 (see discussion in Section 4.2).

While the qualitative trends of the cosmic baryon distribution in this paper
are consistent with previous analyses (e.g., Ayromlou et al., 2023; Sorini et
al., 2022) that studied different feedback models, the parametrization in this
paper is explicitly intended to provide interpretation for the first
generation of FRB constraints on the IGM and CGM baryonic gas fractions. The
relative baryon partition of the IGM and CGM of various halo masses revealed
by FRBs will shed light on modes of galaxy or AGN feedback that otherwise
might have less noticeable effects on the stellar population of galaxies.

The large-scale distribution of baryons also has important cosmological
implications, especially the so-called ‘S8 tension’ in which the matter
density fluctuations measured by galaxy weak-lensing measurements at low-
redshift are smoother than those predicted by the primordial anisotropies
measured in the cosmic microwave background (CMB) (e.g., Heymans et al., 2021;
Abbott et al., 2022; Li et al., 2023; Dalal et al., 2023). One possible
solution lies in the so-called ‘baryonic effects’, i.e. the cosmic baryons
(and concomitantly the overall matter distribution) might have been altered
beyond predictions of simple N-body gravitational evolution, likely by
galaxy/AGN feedback (e.g., Chisari et al., 2019; van Daalen et al., 2020).
While recent work has focused on the role of X-ray and SZ measurements of
galaxy clusters and groups as proxies for baryonic feedback (Schneider et al.,
2022), direct measurements of the large-scale baryon distribution such as
those presented in this paper will provide complementary constraints. This
will be investigated in future papers.

## Acknowledgements

We thank Sunil Simha, Xavier Prochaska, and Hideki Tanimura for useful
discussions. Kavli IPMU is supported by World Premier International Research
Center Initiative (WPI), MEXT, Japan. ISK would like to acknowledge the
support received by the ESO-Chile joint committee grant. This work used the
DiRAC@Durham facility managed by the Institute for Computational Cosmology on
behalf of the STFC DiRAC HPC Facility. The equipment was funded by BEIS
capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and
ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC
is part of the National e-Infrastructure. This work made extensive use of the
NASA Astrophysics Data System and of the astro-ph preprint archive at
arXiv.org. For the purpose of open access, the author has applied a Creative
Commons Attribution (CC BY) licence to any Author Accepted Manuscript version
arising from this submission.

## Data Availability

The simulation data underlying this article are publicly
available222http://simba.roe.ac.uk/. The derived data will be shared upon
reasonable request to the corresponding author.

## References

  * Abbott et al. (2022) Abbott T. M. C., et al., 2022, Phys. Rev. D, 105, 023520
  * Angelinelli et al. (2022) Angelinelli M., Ettori S., Dolag K., Vazza F., Ragagnin A., 2022, A&A, 663, L6
  * Angelinelli et al. (2023) Angelinelli M., Ettori S., Dolag K., Vazza F., Ragagnin A., 2023, A&A, 675, A188
  * Anglés-Alcázar et al. (2015) Anglés-Alcázar D., Özel F., Davé R., Katz N., Kollmeier J. A., Oppenheimer B. D., 2015, ApJ, 800, 127
  * Anglés-Alcázar et al. (2017) Anglés-Alcázar D., Faucher-Giguère C.-A., Kereš D., Hopkins P. F., Quataert E., Murray N., 2017, MNRAS, 470, 4698
  * Appleby et al. (2021) Appleby S., Davé R., Sorini D., Storey-Fisher K., Smith B., 2021, MNRAS, 507, 2383
  * Ata et al. (2015) Ata M., Kitaura F.-S., Müller V., 2015, MNRAS, 446, 4250
  * Ata et al. (2017) Ata M., et al., 2017, MNRAS, 467, 3993
  * Ayromlou et al. (2021) Ayromlou M., Nelson D., Yates R. M., Kauffmann G., Renneby M., White S. D. M., 2021, MNRAS, 502, 1051
  * Ayromlou et al. (2023) Ayromlou M., Nelson D., Pillepich A., 2023, MNRAS, 524, 5391
  * Batten et al. (2022) Batten A. J., Duffy A. R., Flynn C., Gupta V., Ryan-Weber E., Wijers N., 2022, MNRAS, 512, L49
  * Bondi (1952) Bondi H., 1952, MNRAS, 112, 195
  * CHIME/FRB Collaboration et al. (2021) CHIME/FRB Collaboration et al., 2021, ApJS, 257, 59
  * Chisari et al. (2019) Chisari N. E., et al., 2019, The Open Journal of Astrophysics, 2, 4
  * Cole et al. (2000) Cole S., Benson A., Baugh C., Lacey C., Frenk C., 2000, in Mazure A., Le Fèvre O., Le Brun V., eds, Astronomical Society of the Pacific Conference Series Vol. 200, Clustering at High Redshift. p. 109 (arXiv:astro-ph/9910233), doi:10.48550/arXiv.astro-ph/9910233
  * Cooke et al. (2018) Cooke R. J., Pettini M., Steidel C. C., 2018, ApJ, 855, 102
  * Cordes & Chatterjee (2019) Cordes J. M., Chatterjee S., 2019, ARA&A, 57, 417
  * Dalal et al. (2023) Dalal R., et al., 2023, arXiv e-prints, p. arXiv:2304.00701
  * Dalton et al. (2012) Dalton G., et al., 2012, in McLean I. S., Ramsay S. K., Takami H., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV. p. 84460P, doi:10.1117/12.925950
  * Davé et al. (2010) Davé R., Oppenheimer B. D., Katz N., Kollmeier J. A., Weinberg D. H., 2010, MNRAS, 408, 2051
  * Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
  * Davies et al. (2019) Davies J. J., Crain R. A., McCarthy I. G., Oppenheimer B. D., Schaye J., Schaller M., McAlpine S., 2019, MNRAS, 485, 3783
  * Davies et al. (2020) Davies J. J., Crain R. A., Oppenheimer B. D., Schaye J., 2020, MNRAS, 491, 4462
  * Dutton & Macciò (2014) Dutton A. A., Macciò A. V., 2014, MNRAS, 441, 3359
  * Efstathiou & Silk (1983) Efstathiou G., Silk J., 1983, Fundamentals Cosmic Phys., 9, 1
  * Faucher-Giguère et al. (2008) Faucher-Giguère C.-A., Prochaska J. X., Lidz A., Hernquist L., Zaldarriaga M., 2008, ApJ, 681, 831
  * Fielding et al. (2020) Fielding D. B., et al., 2020, ApJ, 903, 32
  * Fukugita & Peebles (2004) Fukugita M., Peebles P. J. E., 2004, ApJ, 616, 643
  * Heckman & Thompson (017a) Heckman T. M., Thompson T. A., 2017a, arXiv e-prints, p. arXiv:1701.09062
  * Heckman et al. (017b) Heckman T., Borthakur S., Wild V., Schiminovich D., Bordoloi R., 2017b, ApJ, 846, 151
  * Heymans et al. (2021) Heymans C., et al., 2021, A&A, 646, A140
  * Hopkins (2015) Hopkins P. F., 2015, MNRAS, 450, 53
  * Ishiyama et al. (2013) Ishiyama T., et al., 2013, ApJ, 767, 146
  * Klypin et al. (2016) Klypin A., Yepes G., Gottlöber S., Prada F., Heß S., 2016, MNRAS, 457, 4340
  * Kocz et al. (2019) Kocz J., et al., 2019, MNRAS, 489, 919
  * Lacey & Cole (1993) Lacey C., Cole S., 1993, MNRAS, 262, 627
  * Lee et al. (2022) Lee K.-G., Ata M., Khrykin I. S., Huang Y., Prochaska J. X., Cooke J., Zhang J., Batten A., 2022, ApJ, 928, 9
  * Lee et al. (2023) Lee K.-G., et al., 2023, arXiv e-prints, p. arXiv:2306.05403
  * Levi et al. (2013) Levi M., et al., 2013, arXiv e-prints, p. arXiv:1308.0847
  * Li et al. (2023) Li X., et al., 2023, arXiv e-prints, p. arXiv:2304.00702
  * Lim et al. (2020) Lim S. H., Mo H. J., Wang H., Yang X., 2020, ApJ, 889, 48
  * López-Cano et al. (2022) López-Cano D., Angulo R. E., Ludlow A. D., Zennaro M., Contreras S., Chaves-Montero J., Aricò G., 2022, MNRAS, 517, 2000
  * Macciò et al. (2007) Macciò A. V., Dutton A. A., van den Bosch F. C., Moore B., Potter D., Stadel J., 2007, MNRAS, 378, 55
  * Macquart et al. (2010) Macquart J.-P., et al., 2010, Publ. Astron. Soc. Australia, 27, 272
  * Macquart et al. (2020) Macquart J. P., et al., 2020, Nature, 581, 391
  * Martizzi et al. (2019) Martizzi D., et al., 2019, MNRAS, 486, 3766
  * Mathews & Prochaska (2017) Mathews W. G., Prochaska J. X., 2017, ApJ, 846, L24
  * McQuinn (2014) McQuinn M., 2014, ApJ, 780, L33
  * McQuinn (2016) McQuinn M., 2016, ARA&A, 54, 313
  * Muratov et al. (2015) Muratov A. L., Kereš D., Faucher-Giguère C.-A., Hopkins P. F., Quataert E., Murray N., 2015, MNRAS, 454, 2691
  * Navarro et al. (1997) Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493
  * Nicastro et al. (2018) Nicastro F., et al., 2018, Nature, 558, 406
  * O’Meara et al. (2015) O’Meara J. M., et al., 2015, AJ, 150, 111
  * Pakmor et al. (2023) Pakmor R., et al., 2023, MNRAS, 524, 2539
  * Pillepich et al. (2018) Pillepich A., et al., 2018, MNRAS, 475, 648
  * Planck Collaboration et al. (2016) Planck Collaboration et al., 2016, A&A, 594, A13
  * Planck Collaboration et al. (2020) Planck Collaboration et al., 2020, A&A, 641, A6
  * Prochaska & Zheng (2019) Prochaska J. X., Zheng Y., 2019, MNRAS, 485, 648
  * Prochaska et al. (2011) Prochaska J. X., Weiner B., Chen H. W., Mulchaey J., Cooksey K., 2011, ApJ, 740, 91
  * Ravi (2019) Ravi V., 2019, ApJ, 872, 88
  * Rhee et al. (2018) Rhee J., Lah P., Briggs F. H., Chengalur J. N., Colless M., Willner S. P., Ashby M. L. N., Le Fèvre O., 2018, MNRAS, 473, 1879
  * Rodríguez-Puebla et al. (2016) Rodríguez-Puebla A., Behroozi P., Primack J., Klypin A., Lee C., Hellinger D., 2016, MNRAS, 462, 893
  * Rubin et al. (2014) Rubin K. H. R., Prochaska J. X., Koo D. C., Phillips A. C., Martin C. L., Winstrom L. O., 2014, ApJ, 794, 156
  * Sanidas et al. (2018) Sanidas S., Caleb M., Driessen L., Morello V., Rajwade K., Stappers B. W., 2018, in Weltevrede P., Perera B. B. P., Preston L. L., Sanidas S., eds, Vol. 337, Pulsar Astrophysics the Next Fifty Years. pp 406–407, doi:10.1017/S1743921317009310
  * Scannapieco et al. (2005) Scannapieco E., Silk J., Bouwens R., 2005, ApJ, 635, L13
  * Schaller et al. (2015) Schaller M., et al., 2015, MNRAS, 451, 1247
  * Schaye et al. (2015) Schaye J., et al., 2015, MNRAS, 446, 521
  * Schaye et al. (2023) Schaye J., et al., 2023, MNRAS,
  * Schneider et al. (2022) Schneider A., Giri S. K., Amodeo S., Refregier A., 2022, Monthly Notices of the Royal Astronomical Society, 514, 3802
  * Shao et al. (2023) Shao M. J., Anbajagane D., Chang C., 2023, MNRAS, 523, 3258
  * Shirasaki et al. (2022) Shirasaki M., Takahashi R., Osato K., Ioka K., 2022, MNRAS, 512, 1730
  * Shull et al. (2012) Shull J. M., Smith B. D., Danforth C. W., 2012, ApJ, 759, 23
  * Simha et al. (2020) Simha S., et al., 2020, ApJ, 901, 134
  * Simha et al. (2021) Simha S., et al., 2021, ApJ, 921, 134
  * Simha et al. (2023) Simha S., et al., 2023, arXiv e-prints, p. arXiv:2303.07387
  * Smith et al. (2017) Smith B. D., et al., 2017, MNRAS, 466, 2217
  * Sorini et al. (2022) Sorini D., Davé R., Cui W., Appleby S., 2022, MNRAS, 516, 883
  * Steigman (2010) Steigman G., 2010, arXiv e-prints, p. arXiv:1008.4765
  * Stinson et al. (2006) Stinson G., Seth A., Katz N., Wadsley J., Governato F., Quinn T., 2006, MNRAS, 373, 1074
  * Tanimura et al. (2020) Tanimura H., Aghanim N., Kolodzig A., Douspis M., Malavasi N., 2020, A&A, 643, L2
  * Terrazas et al. (2020) Terrazas B. A., et al., 2020, MNRAS, 493, 1888
  * Thomas et al. (2019) Thomas N., Davé R., Anglés-Alcázar D., Jarvis M., 2019, MNRAS, 487, 5764
  * Tollet et al. (2019) Tollet É., Cattaneo A., Macciò A. V., Dutton A. A., Kang X., 2019, MNRAS, 485, 2511
  * Tumlinson et al. (2013) Tumlinson J., et al., 2013, ApJ, 777, 59
  * Tumlinson et al. (2017) Tumlinson J., Peeples M. S., Werk J. K., 2017, ARA&A, 55, 389
  * Wang et al. (2017) Wang L., Dutton A. A., Stinson G. S., Macciò A. V., Gutcke T., Kang X., 2017, MNRAS, 466, 4858
  * Werk et al. (2014) Werk J. K., et al., 2014, ApJ, 792, 8
  * White & Rees (1978) White S. D. M., Rees M. J., 1978, MNRAS, 183, 341
  * Wu & McQuinn (2023) Wu X., McQuinn M., 2023, ApJ, 945, 87
  * Yoshino et al. (2009) Yoshino T., et al., 2009, PASJ, 61, 805
  * Zwaan et al. (2003) Zwaan M. A., et al., 2003, AJ, 125, 2842
  * de Graaff et al. (2019) de Graaff A., Cai Y.-C., Heymans C., Peacock J. A., 2019, A&A, 624, A48
  * van Daalen et al. (2020) van Daalen M. P., McCarthy I. G., Schaye J., 2020, MNRAS, 491, 2424