# The Effect of AGN Feedback on the Lyman-$\alpha$ Forest Signature of Galaxy
Protoclusters at $z\sim 2.3$

Chenze Dong,1,2 Khee-Gan Lee,1,2 Romeel Davé3,4 Weiguang Cui3,5 and Daniele
Sorini6
1Kavli Institute for the Physics and Mathematics of the Universe (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
3Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
4University of the Western Cape, Bellville, Cape Town 7535, South Africa
5Departamento de Física Teórica and CIAFF, Modulo 8 Universidad Autónoma de
Madrid, 28049 Madrid, Spain
6Institute for Computational Cosmology, Durham University, South Park Road,
DH1 3LE, United Kingdom

E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The intergalactic medium (IGM) in the vicinity of galaxy protoclusters are
interesting testbeds to study complex baryonic effects such as gravitational
shocks and feedback. Here, we utilize hydrodynamical simulations from the
SIMBA and The Three Hundred suites to study the mechanisms influencing large-
scale Lyman-$\alpha$ transmission in $2<z<2.5$ protoclusters observed in the
COSMOS field. We focus on the matter overdensity-Lyman-$\alpha$ transmission
relation $(\delta_{m}-\delta_{F})$ on Megaparsec-scales in these
protoclusters, which is hypothesized to be sensitive to the feedback
implementations. The lower-density regions represented by the SIMBA-100
cosmological volume trace the power-law $\delta_{m}-\delta_{F}$ relationship
often known as the fluctuating Gunn-Peterson approximation (FGPA). This trend
is continued into higher-density regions covered by the 300-GadgetMUSIC
simulations that implement stellar feedback only. The 300-GadgetX and
300-SIMBA simulations, with AGN thermal and AGN jet feedback respectively,
exhibit progressively more Lyman-$\alpha$ transmission at fixed overdensity.
Compared with the 7 protoclusters observed in the CLAMATO$\times$COSTCO data,
only 2 appear consistent with the FGPA. The others exhibit clear deviations: 4
follow the trend of AGN X-ray thermal feedback models while the COSTCO-I
protocluster appears to reflect intense jet feedback. The large discrepancy
with the stellar-feedback-only 300-GadgetMUSIC model disfavours large-scale
heating from gravitational collapse and/or stellar feedback. This indicates
that some form of AGN feedback is likely at play in the observed
protoclusters, and possibly long-ranged AGN jets in the case of COSTCO-I.
While more detailed and resolved simulations are required to move forward, our
findings open new avenues for probing AGN feedback at Cosmic Noon.

###### keywords:

galaxies: intergalactic medium – galaxies: clusters: general – methods:
numerical – cosmology: large-scale structure of Universe

††pubyear: 2024††pagerange: The Effect of AGN Feedback on the Lyman-$\alpha$
Forest Signature of Galaxy Protoclusters at $z\sim 2.3$–4

##  1 Introduction

Galaxy clusters did not form in a day; these giants were still in their
adolescence stage at redshift $z>2$, in structures commonly known as “galaxy
protoclusters". At this cosmic noon epoch, they were still in an active stage
of mass assembly, and their member galaxies may or may not end up in a
present-day cluster (Chiang et al., 2013). Protoclusters contribute a
significant fraction to the cosmic star formation rate density (Chiang et al.,
2017) at Cosmic Noon when cosmic star-formation activity was at its peak.
Galaxy protoclusters do not necessarily exhibit very conspicuous galaxy
overdensities, with most nascent protocluster cores accumulating masses of
$M\lesssim 10^{14}\,M_{\odot}$ by this epoch, and the mass growth of these
protoclusters is not necessarily monotonic (i.e. the most massive protocluster
cores at $z\sim 2$ will not always grow into the most massive $z\sim 0$
clusters, for example, Remus et al. 2023; Cui et al. 2020).

Aside from their important role in star formation and massive galaxy assembly,
protoclusters also show interesting interactions with the intergalactic medium
(IGM), (for example Kikuta et al., 2019; Kooistra et al., 2022a; Emonts et
al., 2023; Cai et al., 2017). After the end of cosmic reionization around
$z\sim 6$, the free streaming photons and the expansion of the universe
quickly erase the thermal memory of IGM towards reionization. Equilibrium
between adiabatic expansion and photo-ionisation heating leads to a power-law
relation between the temperature and density in the IGM,
$T=T_{0}(1+\delta)^{\gamma-1}$ (Hui & Gnedin, 1997), where
$\delta=\rho/\bar{\rho}-1$ is the overdensity, $T_{0}$ is the typical
temperature for IGM with an average density. The temperature-density equation-
of-state index, $\gamma\approx 1.7$ (Lee et al., 2015; Hiss et al., 2018;
Rorai et al., 2018), is a parameter arising from the balance between
recombination and photo-ionisation. This leads to a relation between hydrogen
Lyman-alpha (Ly$\alpha$) forest optical depth and IGM state, often known as
the Fluctuating Gunn-Peterson Approximation (FGPA; Croft et al., 1998, 2002):

$\tau_{\mathrm{Ly}\alpha}\propto\frac{T^{-0.7}}{\Gamma_{UV}}(1+\delta)^{2}\propto(1+\delta)^{\beta},$
(1)

where $\Gamma_{UV}$ corresponds to the background ultraviolet (UV) photo-
ionisation rate, and $\beta=2-0.7(\gamma-1)$ is another positive power-law
index. By observing Ly$\alpha$ absorption, this relation facilitates the
probing of overdensity $\delta$ along lines of sight to background objects,
for example, CLAMATO (Lee et al. 2014, Horowitz et al. 2022, Lee et al. 2018)
and LATIS (Newman et al. 2020). Protoclusters are usually the sites with the
highest overdensities and strongest Ly$\alpha$ absorption, so this relation
further enables the detection of protoclusters at Cosmic Noon (Stark et al.,
2015; Lee et al., 2016; Cai et al., 2016; Miller et al., 2021; Qezlou et al.,
2022).

Baryonic processes associated with galaxy formation in and around the
protoclusters add great complexity to the connection between the IGM and
galaxy protoclusters. Stellar and AGN feedback can inject energy into the IGM
as they regulate the star formation, potentially causing deviations from the
FGPA due to collisional heating and ionization. Furthermore, even shock
heating purely driven by gravitational collapse can raise the temperature of
the gas within high-mass halos (Kereš et al., 2005), eventually driving the
evolution into the complex and “missing” phases of the IGM at late times (Davé
et al., 2001; Cen & Ostriker, 2006). Early works (Theuns et al., 2001) hinted
at this more complex connection via inspection of individual quasar spectra.
On the simulation side, various studies assessed the impact of feedback
processes both on the summary statistics of the Ly$\alpha$ forest (for example
Kollmeier et al., 2003; Christiansen et al., 2020; Nagamine et al., 2021;
Burkhart et al., 2022; Tillman et al., 2023) and on Ly$\alpha$ absorption in
the circumgalactic medium of galaxies and quasars (Kollmeier et al., 2006;
Meiksin et al., 2015, 2017; Turner et al., 2017; Sorini et al., 2018, 2020;
Appleby et al., 2021). Kooistra et al. (2022a) also predicted an elevation of
IGM temperature by the manual phenomenological implementation of AGN feedback.

Recently, Dong et al. (2023) found a particularly interesting case, the
COSTCO-I protocluster, which potentially holds strong implications for the
interplay between heating processes on the Ly$\alpha$ forest in protoclusters.
This protocluster was first identified in density reconstructions of the
COSMOS spectroscopic catalogue (Ata et al., 2021) and resulting constrained
simulations (COSTCO project; Ata et al. 2022) as the progenitor of a modern-
day $M\approx(4.6\pm 2.2)\times 10^{14}\,M_{\odot}$ galaxy cluster. However, a
comparison with Wiener-filtered tomographic IGM map from the CLAMATO survey
(Horowitz et al., 2022) indicated that COSTCO-I is anomalously transparent to
Lyman-$\alpha$ absorption, contrary to the significant absorption expected
through FGPA given its galaxy overdensity. It is a special case among
protoclusters residing in the overlapping volume of CLAMATO and COSTCO
(hereafter CLAMATO$\times$COSTCO protoclusters), as none of the other
protoclusters have such transparent Ly$\alpha$ absorption at first glance.
Moreover, the unexpectedly high transmission is still significant after
averaging with a $r=15$ cMpc/h tophat kernel, indicating that this phenomenon
spans multiple physical Megaparsecs. Recent work by Miller et al. (2022)
suggests that the UV radiation from protocluster AGN seems unlikely to be the
cause of extended Ly-$\alpha$ transmission to the extent that we see unless an
ultra-luminous quasar is present (Schmidt et al., 2018). Therefore, according
to Equation 1, such a Ly$\alpha$ transparency could be attributed to a high
IGM temperature across several physical Mpc, given the uniform UV background
and overdensity confirmed by COSTCO. We hypothesise that the extended gas
around COSTCO-I is a possible site of large-scale heating processes other than
the UV background ionisation, which breaks the power-law relation between the
temperature and density of IGM. However, the mechanism that caused the heating
still remains unclear, as it could be some combination of AGN and/or stellar
feedback along with gravitational shock heating.

In this paper, we use the SIMBA cosmological galaxy formation simulation and
The Three Hundred Cluster zoom simulation suite to investigate the impact of
gravitational shock heating and various feedback processes on the Ly$\alpha$
forest within and around large halos at $z=2.5$. In Section 2, we introduce
the data adopted in this study and describe how we derive the quantities of
interest. In Section 3, we present our main findings based on the measurement
from both observations and simulations. In Section 4, we discuss the results
and summarize the study. Throughout this paper, we adopt the cosmology of
$H_{0}=100h\ \mathrm{km\,s^{-1}\,Mpc^{-1}}=70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$,
$\Omega_{m}=0.315$, and $\Omega_{\Lambda}=0.685$ for the observational data.

##  2 Data

In order to quantify the heating of IGM around the massive protoclusters in
simulations and compare with the observed case with large-scale gas heating,
we focus on the following quantities: overdensity $\delta_{m}$and Lyman-alpha
transmission $\delta_{F}$. The definition and the measurement methodology of
these quantities for the CLAMATO$\times$COSTCO observational data are
described in Section 2.1. The simulation suites used in this study are
introduced in Section 2.2, and in Section 2.3 we applied the same measurement
to the simulations.

###  2.1 CLAMATO and COSTCO Observations

This paper was motivated by the observational discovery of the COSTCO-I galaxy
protocluster at $z=2.30$ (Lee et al., 2016; Ata et al., 2022; Dong et al.,
2023). The unusual nature of this protocluster was first hinted at by Lee et
al. (2016), but the official discovery was through constrained N-body
simulations (Ata et al., 2022) based on the reconstructed matter density field
of the COSMOS galaxy spectroscopic sample in the redshift range $2.0<z<2.5$
(Ata et al., 2021). To briefly summarise, the initial conditions (ICs) of the
N-body simulations were first sampled using the COSMIC-BIRTH (Kitaura et al.,
2020) algorithm, conditional to the observed spectroscopic galaxy redshift
distribution at $2.00<z_{\mathrm{obs}}<2.52$ in the COSMOS field. Ata et al.
(2022) then ran PKDGRAV3 N-body code (Potter et al., 2017) on the initial
conditions derived from this analysis, with a total of 57 posterior
realisations and produced simulation snapshots at $z=2.3$ as well as at $z=0$.
This allows us to compute various quantities for all the CLAMATO$\times$COSTCO
protoclusters, including some that are not typically accessible through
observations such as the matter density field. In addition, this allows us to
estimate the uncertainties related to the derived quantities thanks to the
posterior realisations.

Table 1 lists all the CLAMATO$\times$COSTCO protoclusters we analyse in this
study. Among the entries, ZFIRE was first detect through medium-band imaging
(Spitler et al., 2012) and later confirmed via Keck/MOSFIRE spectroscopy (Yuan
et al., 2014). Hyperion (Diener et al., 2015; Chiang et al., 2015; Casey et
al., 2015; Lee et al., 2016; Wang et al., 2016) is a complex of galaxy
protoclusters identified at redshift $2.423<z<2.507$, where only Hyperion 1,
3, 4 and 5 are enclosed in the volume of CLAMATO. COSTCO-I and COSTCO-III are
newly-discovered galaxy protocluster reported by Ata et al. (2022) based on
constrained-simulation analysis.

Protocluster | RA | Dec | $z_{\mathrm{obs}}$ | Final Mass ($h^{-1}M_{\odot}$)
---|---|---|---|---
ZFIRE | 150.094 | 2.251 | 2.095 | $(1.2\pm 0.3)\times 10^{15}$
Hyperion 1 | 150.093 | 2.404 | 2.468 | $(2.5\pm 0.5)\times 10^{15}$
Hyperion 3 | 149.999 | 2.253 | 2.444
Hyperion 4 | 150.255 | 2.342 | 2.469
Hyperion 5 | 150.229 | 2.338 | 2.507
COSTCO-I | 150.11 | 2.161 | 2.298 | $(4.6\pm 2.2)\times 10^{14}$
COSTCO-III | 150.129 | 2.275 | 2.16 | $(5.3\pm 2.6)\times 10^{14}$

Table 1: CLAMATO$\times$COSTCO galaxy protoclusters analyzed in this work.
Note that Ata et al. (2022) only reported the final mass for Hyperion clusters
as the sum of all member clusters (Hyperion 1-7), including Hyperion 2;
Hyperion 6 and Hyperion 7 are outside the volume of the CLAMATO Lyman-$\alpha$
map.

To derive the quantities of interest, we first measured the matter
overdensity, $\delta_{m}$ around the protoclusters from the N-body outputs. To
do this, we converted the particle data from the simulation output into a grid
of $\delta_{m,\mathrm{raw}}$ and used the mean density of the universe to
normalize the matter density,

$\delta_{m,\mathrm{raw}}=\rho/\bar{\rho}-1$ (2)

Then, we smoothed the $\delta_{m,\mathrm{raw}}$ field with a Gaussian kernel
of 7.5 cMpc/h smoothing radius, and took the value of $\delta_{m}$ at the
positions of protoclusters as the overdensity associated with the
protocluster.

We measured the Ly$\alpha$ forest transmission fluctuation,
$\delta_{F}=F/\bar{F}-1$ (where $F=\exp(-\tau_{\mathrm{Ly}\alpha})$ is the
Ly$\alpha$ transmission, and $\bar{F}$ is the mean Ly$\alpha$ transmission at
$z=2.5$ given by Becker et al. 2013) associated with the protoclusters based
on the Wiener-filtered $\delta_{F}^{\mathrm{WF}}$ from the CLAMATO survey
(Horowitz et al., 2022). To match the measurement of overdensity above, we
simply smoothed the $\delta_{F}^{\mathrm{WF}}$ field with a 7.5 cMpc/h
Gaussian kernel and evaluated $\delta_{F}$ at the position of the
protocluster, which was used to compare with the $\delta_{F}$ matter
overdensity.

###  2.2 Simulations

In order to explore a large range of halo masses, larger simulation boxes are
generally desired. However, this typically requires a trade-off between the
simulation scale and the resolution given finite computational resources. In
our work, we aim to achieve adequate resolution while retaining the ability to
study massive halos ($M_{\mathrm{halo}}>10^{14}h^{-1}M_{\odot}$ at $z=0$) when
picking the runs in each simulation suite. We, therefore, use the large volume
available in the SIMBA suite of cosmological hydrodynamical simulation, i.e.
the $L=100\,h^{-1}\,\mathrm{cMpc}$ box. We also study The Three Hundred, a
series of zoom-in simulations that focus on the evolution of the most massive
galaxy clusters at $z=0$. These simulations have been run with several
different feedback prescriptions. We introduce the simulation suites below
with a focus on their jet feedback implementation and summarize them in Table
2. Due to the limited snapshots available in The Three Hundred, we study the
property of massive protoclusters with the snapshot at $z=2.5$, which
corresponds to the highest redshift in the CLAMATO volume. The choice of
snapshot redshift can be justified for the following reasons. Firstly, the
median redshift of CLAMATO$\times$COSTCO protoclusters is $z=2.444$ (see Table
1), therefore adopting $z=2.5$ is nominally the closest snapshot. Secondly,
most clusters in The Three Hundred sample have larger final masses
($>10^{15}h^{-1}M_{\odot}$) than CLAMATO$\times$COSTCO sample, thus a slightly
higher redshift for the simulated sample facilitates a better match with the
evolutionary stage of the observed protoclusters.

####  2.2.1 SIMBA

The SIMBA simulation (Davé et al., 2019) is a cosmological simulation suite
based on a forked version of the Gizmo (Hopkins, 2015) Gravity+hydrodynamics
solver. The meshless-finite-mass (MFM) scheme in Gizmo gets rid of artificial
viscosity in its hydrodynamic implementation, which handles shocks in a more
precise and realistic way. The SIMBA simulation includes a variety of input
physics. The radiative cooling and photo-ionisation heating are implemented
using Grackle-3.1 (Smith et al., 2017) with an on-the-fly, and self-consistent
self-shielding treatment (Rahmati et al., 2013). SIMBA adopts the ionizing
background in Haardt & Madau (2012), which is further modified to account for
self-shielding. The star formation formalism is similar to its precursor,
MUFASA (Davé et al., 2016). The star formation rate (SFR) is calculated with
an $H_{2}$-based Schmidt (1959) law, where the SFR is proportional to the
$H_{2}$ density and the inverse of dynamic time. SIMBA incorporates the
stellar feedback from Type Ia & Type II Supernovae (SNe) and Asymptotic Giant
Branch (AGB) stars. The SIMBA applied the decoupled two-phase winds for
stellar feedback, of which the effect is aligned with high-resolution FIRE
simulation. (Feedback In Realistic Environments, Hopkins et al. 2014)

The major improvement in SIMBA sub-grid models relative to MUFASA is the black
hole physics. For the black hole accretion, SIMBA employs a combination of
Bondi-Hoyle-Lyttleton accretion Bondi (1952) (from hot gas) and Torque-limited
accretion Hopkins & Quataert (2011) (from cold gas). The AGN feedback is
modelled in a two-phase approach, featuring criteria on the black hole mass
$M_{\mathrm{BH}}$ and Eddington ratio $f_{\mathrm{Edd}}$. The “radiative" mode
is activated for the black holes in high accretion state
($f_{\mathrm{Edd}}>0.2$) or with low mass
($M_{\mathrm{BH}}<10^{7.5}M_{\odot}$), in which the black holes exert kinetic
kick on surrounding gas to create “AGN winds" with a speed of $\sim
10^{3}\,\mathrm{km/s}$. For more massive black holes with lower accretion
rate, the feedback enters “jet" mode, where the gas at black hole vicinity is
heated to $\gtrsim 10^{7}\,\mathrm{K}$ and launched with a speed of $\sim
7000\,\mathrm{km/s}$. The crucial feature of SIMBA’s AGN feedback arises from
the directionality of momentum ejection. For both wind and jet feedback, the
ejected gas launches from the direction perpendicular to the inner disc around
the SMBH with an opening angle of zero. Given the high velocity of the jet
feedback, it leads to a highly collimated bipolar outflow, which has an impact
on the scale of observed radio lobes. The authors argue the highly collimated
implementation of feedback is later alleviated by other processes on longer
time scales, such as the precession of the inner disk and the interaction of
outflow and ambient gas. For more detail, we refer the readers to the
introductory paper of SIMBA (Davé et al., 2019).

SIMBA has simulation boxes with settings different in different box sizes and
resolutions publicly available. In this study, We adopt the flagship run
(hereafter SIMBA-100) with the box size of 100 cMpc/h and $2\times 1024^{3}$
particles to increase the volume and hence halo mass coverage compared with
the 50 $h^{-1}\,\mathrm{cMpc}$ versions.

####  2.2.2 The THREE HUNDRED

The Three Hundred project (Cui et al., 2018, 2022) is a set of re-simulations
of 324 large galaxy clusters. These clusters were first selected from the 1
cGpc/h MDPL2 dark-matter-only simulation (Klypin et al., 2016) based on their
halo mass at $z=0$ ($M_{h,z=0}>6.42\times 10^{14}M_{\odot}$). A spherical
volume with a radius $r=15$ cMpc/h was then selected as the re-simulation
region. The initial conditions were generated by the GINNUNGAGAP code with the
re-simulation region split into gas and dark matter particles, and the outer
regions are resampled using low-resolution particles to save the computation
cost. The Three Hundred resimulation set provides a mass-complete sample of
the extremely massive halos with flexibility in baryon model prescriptions.
Our study includes the runs with GadgetX, GadgetMUSIC and GIZMO-SIMBA models
for all 324 clusters.

The detail of three prescriptions in The Three Hundred is listed as follows.

  * •

GadgetMUSIC (hereafter 300-GadgetMUSIC): This prescription is based on
Sembolini et al. (2012), running with GADGET-3 Tree-PM code, an updated
version of GADGET-2 (Springel, 2005) The classic entropy-conserving SPH
formulation with a 40 neighbour M3 interpolation kernel is used in the
hydrodynamic solver of 300-GadgetMUSIC. It adopts a metal-independent approach
for radiative cooling and a homogeneous UV background (UVB) by Haardt & Madau
(2001) for photo-ionisation heating. On star formation, it follows the hybrid
multi-phase model in Springel & Hernquist (2003) with the Salpeter (1955)
initial mass function (IMF). The kinetic stellar feedback and thermal stellar
feedback in 300-GadgetMUSIC are realized based on Springel & Hernquist (2003)
and the 2-phase model in Yepes et al. (1997), respectively. There is neither
black hole accretion nor black hole feedback implemented in 300-GadgetMUSIC.

  * •

GadgetX (hereafter 300-GadgetX): This prescription has a few differences
compared with 300-GadgetMUSIC. It uses an updated SPH scheme (Beck et al.,
2015) with improvement in artificial thermal diffusion, time-dependent
artificial viscosity, high-order Wendland C4 interpolating kernel and wake-up
scheme. On the gas heating and cooling, 300-GadgetX adopts Haardt & Madau
(1996) UVB and a metal-dependent cooling calculation Wiersma et al. (2009). It
models star formation using Tornatore et al. (2007) and Chabrier (2003) IMF.
300-GadgetX also implemented kinetic stellar feedback following Springel &
Hernquist (2003). Contrary to 300-GadgetMUSIC, 300-GadgetX suite has a black
hole physics implementation by Steinborn et al. (2015), which includes a
multi-phase gas accretion and thermal AGN feedback.

  * •

GIZMO-SIMBA (hereafter 300-SIMBA): This prescription adopted a slightly
modified model based on the SIMBA cosmological runs. Because the resolution of
The Three Hundred is downgraded compared to SIMBA-100, Cui et al. (2022) re-
calibrated the feedback model based on a series of criteria related to stellar
mass and halo mass in their Section 3.2. The modification is mainly motivated
by the deficit of star formation in satellite galaxies. The star formation
efficiency was then tuned higher to alleviate the deficit, which led to an
overproduction of stars in central galaxies. In turn, the jet feedback was
tuned more powerful (with a max speed of $\sim 15000\,\mathrm{km/s}$) to
regulate stellar mass in the central galaxies. As a result, the jets in
300-SIMBA are the strongest among the simulation suites in this study, which
results in a relatively flat gas density profile at z=0 (Li et al., 2023).

Apart from the snapshot at $z=2.5$, we also use the output of halo finder AHF
to determine the position of the most massive halos in each re-simulation.

Simulation | Box size (cMpc/h) | Cosmology ($h,\sigma_{8},\Omega_{m},\Omega_{b}$) | Mass resolution ($h^{-1}M_{\odot}$) | Feedback model
---|---|---|---|---
SIMBA-100 | 100 | ($0.68,0.82,0.3,0.048$) | $6.51\times 10^{7}$ | SIMBA full physics
300-SIMBA | zoom-in | ($0.678,0.823,0.307,0.048$) | $1.27\times 10^{9}$ | SIMBA full physics
300-GadgetX | zoom-in | ($0.678,0.823,0.307,0.048$) | $1.27\times 10^{9}$ | Stellar & AGN thermal feedback
300-GadgetMUSIC | zoom-in | ($0.678,0.823,0.307,0.048$) | $1.27\times 10^{9}$ | Stellar feedback

Table 2: Hydrodynamical simulations used in this work. We specify the alias,
box size, cosmology, resolution and feedback model in each run.

###  2.3 Simulation Observables

As all the simulations used in this study utilized GADGET as their backbone,
we use a light-weighted Python package pygad to handle the simulation output.
The package integrates the modules for reading simulation data with
automatically inferred units, producing maps of quantities with the given SPH
kernel, and generating the Lyman-$\alpha$ optical depth.

For the simulations, we first identified the positions of massive halos
($M_{h}>10^{14}M_{\odot}$ at present day) in SIMBA-100 and the most massive
halo in each of the The Three Hundred zoom-in simulation at $z=2.5$, then
calculated the same Lyman-$\alpha$ forest and mass proxies as for
CLAMATO$\times$COSTCO protoclusters: $\delta_{m}$ and $\delta_{F}$. This
involved computing the gridded matter density and Lyman-$\alpha$ transmission
for each zoom-in simulation snapshot using pygad. We then applied the same
smoothing kernels as described in 2.1, and finally evaluated the value at the
position of corresponding halos. Some of the central halos in The Three
Hundred were at the edge of the simulation zoom-in regions and were not
properly simulated at full resolution, so we removed these halos from our
subsequent analysis.

##  3 Results

Figure 1 illustrates the distribution of gas particles on the temperature-
density phase diagram. As a "baseline" run for this study, SIMBA-100
reproduces the power-law temperature-density relation in the diffuse cold gas
regime, and shows a population of WHIM gas on $T>10^{5}$ K. All The Three
Hundred phase diagrams exhibit a more notable WHIM component since The Three
Hundred runs are zoom-in simulations around extremely massive regions rather
than a regular cosmological volume used in SIMBA-100. Among these phase
diagrams, the WHIM fraction of 300-SIMBA run exceeds 300-GadgetX run, and
300-GadgetX outmatches 300-GadgetMUSIC run, which reflects the latter has
relatively weak feedback.

Figure 1: The mass-weighted temperature-density relation for SIMBA-100 (upper
left) and three feedback prescriptions (rest panels) on The Three Hundred
cluster #0001 at $z=2.5$. In each panel, the vertical grey dash line
represents $\log_{10}\rho_{b}/\bar{\rho}_{b}=2$ while the horizontal line is
$T=10^{5}\mathrm{K}$. The two dash lines divide the $\rho-T$ phase space into
diffused cold gas (lower left), warm-hot intergalactic medium (WHIM, upper
left), hot halo gas (upper right) and condensed gas (lower right). Among the
The Three Hundred runs, the fraction of the WHIM component in each panel
reflects the strength of feedback.

In Figure 2 we display the projected matter density, gas temperature, and
Lyman-$\alpha$ transmission maps of The Three Hundred cluster #0001. The
matter density distribution is almost identical among the different feedback
prescriptions, which supports the idea that even strong feedback does not
significantly alter the overall matter distribution (Sunseri et al., 2023).
The gas temperature maps, however, show distinct differences. The 300-SIMBA
AGN jet-feedback runs show the highest overall IGM temperature in The Three
Hundred, which is centered on hot ($>10^{7}$ K) bubbles at the nodes of the
cosmic web and elevated gas temperature ($\sim 10^{5}$ K) that extends over
several Megaparsecs into regions with milder overdensity. As a result, the
Lyman-$\alpha$ forest absorption is almost completely suppressed in this
300-SIMBA protocluster (top-right panel of 2), and the protocluster appears as
a roughly mean-density region when viewed through the Lyman-$\alpha$ forest. A
search for protoclusters using only Lyman-$\alpha$ forest data (for example,
Stark et al. 2015; Lee et al. 2016; Cai et al. 2016) would probably miss
protoclusters like this.

Hot bubbles are also seen in the 300-GadgetX thermal AGN run, but they do not
extend into underdensities and the gas temperature in those regions remain
cool ($\sim 10^{4}$ K). As a result, these regions exhibit some Ly$\alpha$
forest absorption such that the 300-GadgetX case is not as transparent as in
300-SIMBA.

In the 300-GadgetMUSIC runs, high temperature gas is seen only in the direct
vicinity of filaments within $\lesssim 1$Mpc from galaxies, and the
Lyman-$\alpha$ transmission value is therefore the lowest among the different
feedback prescriptions of The Three Hundred. The Lyman-$\alpha$ absorption of
this protocluster is clearly evident and would be easily detected in a search
of the Ly-$\alpha$ forest.

Figure 2: $20\times 20\times 2$ cMpc/h slice of The Three Hundred # 0001. The
left column shows matter density maps, the middle panels are the gas
temperature maps, and the right panel are the maps of Lyman-$\alpha$
transmission fluctuation $\delta_{F}$. The $\delta_{F}$ map is smoothed with a
Gaussian kernel with $R=1$ cMpc/h. From top to bottom, the maps are from
GIZMO-SIMBA, GadgetX and GadgetMUSIC prescriptions, respectively. The centres
of halos are marked with red plus signs; the red circles illustrate the $7.5$
cMpc/h scale used in Section 2.1 and Section 2.3. The jet feedback of the
300-SIMBA run significantly heats up the surrounding IGM, leading to a high
overall Lyman-$\alpha$ transmission even with the massive galaxy protocluster
overdensity.

Figure 3 shows the smoothed Lyman-$\alpha$ forest transmission, $\delta_{F}$,
plotted against the correspondingly smoothed matter overdensity, $\delta_{m}$,
measured for the ensemble of simulated protoclusters at $z=2.5$. We find a
correlation between the two quantities within the individual simulation
suites: $\delta_{F}$ generally decreases when a halo has higher $\delta_{m}$.
The 300-SIMBA suite has the highest overall Lyman-$\alpha$ transmission among
The Three Hundred runs, while the $\delta_{F}$ value of 300-GadgetX
protoclusters shows an offset of $\Delta\delta_{F}\sim-0.1$, i.e. more
Lyman-$\alpha$ opacity. The 300-GadgetMUSIC suite exhibits the least
transparent Lyman-$\alpha$ forest, which is probably due to the lack of AGN
feedback in its feedback prescription. For SIMBA-100, we show the matter-
transmission relationship over the entire box as a contour in Figure 3, i.e.
its fluctuating Gunn-Peterson relationship (Kooistra et al., 2022b). The
corresponding values centred on individual protoclusters within SIMBA-100 are
also indicated separately as purple points. The FGPA of SIMBA-100 appears to
flatten out in $\delta_{F}$ with increasing $\delta_{m}$, but interestingly
the points in 300-GadgetMUSIC seem to continue following the FGPA relation
traced out by the lower-density SIMBA-100 points. This is in qualitative
agreement with Viel et al. (2013b), who argued that stellar feedback does not
have a significant effect on the power spectrum (i.e. fluctuations) of the
Lyman-$\alpha$ forest.

The most massive halos in the SIMBA-100 box are generally not as massive as
those in The Three Hundred, but they do not appear to follow a continuous
relationship with those in 300-SIMBA. This is suspicious as one would expect
the halos in SIMBA-100 to have similar properties as halos in 300-SIMBA due to
the similar feedback prescriptions; we believe this to be due to the coarser
resolution and the somewhat strengthened feedback 300-SIMBA, which we will
discuss further in Section 4. The data point for COSTCO-I is in broad
agreement with 300-SIMBA simulations: its overdensity is comparable with the
less massive halos in The Three Hundred, while its transmission falls in the
range of $\delta_{F}$ of 300-SIMBA.

We have also overplotted in Figure 3 the measured $\delta_{F}$ from the
CLAMATO Wiener-filtered maps and the $\delta_{m}$ derived from the COSTCO
constrained simulations, for the observed protoclusters listed in Table 1. The
error bars were estimated from the ensemble of posterior realizations in the
COSTCO simulation suite.

The $\delta_{m}$ values of the observed CLAMATO$\times$COSTCO protoclusters
are generally $\delta_{m}>1$ as expected for overdensities, although those for
Hyperion 3 and Hyperion 5 are only mildly overdense. This is not in
contradiction with their identification as protoclusters, because
protoclusters are often only mildly overdense over scales of multiple
Megaparsecs (Chiang et al., 2013). Hyperion 3 and Hyperion 5 are components of
the Hyperion proto-supercluster (Cucciati et al., 2018), which was shown by
Ata et al. (2022) to be the progenitor of an eventual $L\sim 100$Mpc long
filamentary supercluster with a total mass of $M_{h,z=0}\approx 2.5\times
10^{15}\,h^{-1}\,M_{\odot}$. However, when compared with the individual
protoclusters identified within SIMBA-100, Hyperion 3 and Hyperion 5 appear to
be quite consistent with the FGPA-like $\delta_{m}$-$\delta_{F}$ relationship
traced by these simulated protoclusters.

On the other hand, one sees clear deviations from FGPA in the other
CLAMATO$\times$COSTCO structures. The majority of the protoclusters (COSTCO-
III, ZFIRE, Hyperions 1 and 4) overlaps with the $\delta_{m}$-$\delta_{F}$
points from the 300-GadgetX cluster simulations, which incorporates the effect
of thermal AGN feedback. It indicates a clear departure from the global FGPA-
like power-law relationship for these protoclusters, in that the amount of
observed Lyman-$\alpha$ absorption is not commensurate with the matter
overdensity traced by the COSMOS galaxies.

COSTCO-I exhibits an even more significant departure from the FGPA
relationship, with a slightly positive $\delta_{F}$ value more usually
indicative of underdensities. Its $\delta_{m}$-$\delta_{F}$ relationship does
not appear consistent with those of the 300-GadgetX protoclusters, and lies
among those of the 300-SIMBA simulations where AGN jet feedback has been
implemented.

Figure 3: The relation between $\delta_{m}$ and $\delta_{F}$ in simulations as
well as data. The scatter plots with blue, orange and green dots are measured
from 300-SIMBA, 300-GadgetX, and 300-GadgetMUSIC, respectively. The shadow
region and contours show the global relation of the two quantities in
SIMBA-100 simulation box, where the purple points are the values of massive
halos in SIMBA-100. The red points with error bars are the measurements from
the CLAMATO$\times$COSTCO protocluster. The left histogram shows the
distribution of $\delta_{F}$measured from massive protoclusters in our
simulation samples, with the same color code as in the right panel. Most of
the CLAMATO$\times$COSTCO protoclusters appear to be deviating from the
standard transmission-density relationship, while COSTCO-I shows a good match
with the 300-SIMBA protoclusters that are implemented with jet feedback.

##  4 Discussion

###  4.1 Gravitational Collapse as the Origin of Protocluster Heating?

In this work, we made a comparison of the large-scale Lyman-$\alpha$ forest
transmission of several galaxy protoclusters observed within the COSMOS field
(Horowitz et al., 2022; Ata et al., 2022) with several hydrodynamical
simulation suites that have varied galaxy and AGN feedback prescriptions. In
particular, we aimed to investigate the anomalously low amount of absorption
seen in the COSTCO-I protocluster at $z$=2.30 (Dong et al., 2023).

Interestingly, the different feedback prescriptions lead to clearly distinct
Lyman-$\alpha$ transmission values $\delta_{F}$ in the simulated
protoclusters, especially if the controlling variable of underlying matter
overdensity $\delta_{m}$ is taken into account. This indicates that measuring
the FGPA-like relationship between Lyman-$\alpha$ flux and matter overdensity
within galaxy protoclusters can be a sensitive probe of galaxy feedback. In a
previous paper, Kooistra et al. (2022b) showed that hydrodynamical simulations
with different feedback distributions exhibit different effective slopes in
their global $\delta_{F}-\delta_{m}$ relationships, but this is a subtle
effect that would probably require next-generation telescopes with $\sim 30$m
diameter apertures to obtain sufficient signal to beat down the observational
uncertainties. The results of Figure 3, on the other hand, suggests that
focusing on galaxy protoclusters could provide clearer constraints on the
feedback mechanisms.

Only two of the observed protoclusters within the CLAMATO$\times$COSTCO volume
appear to follow the global power-law FGPA relationship, while most of the
protoclusters exhibit clear deviations from FGPA. The majority of our
protoclusters appear to follow the $\delta_{F}-\delta_{m}$ relationship traced
by the 300-GadgetX simulations, which incorporate the effect of AGN thermal
X-ray feedback. The COSTCO-I protocluster, on the other hand, exhibits an even
more extreme deviation from the FGPA, which places it within the relationship
traced by the AGN jet-feedback model of the 300-SIMBA runs. Indeed, COSTCO-I
is at a $\sim 5\sigma$ discrepancy with the simulated 300-GadgetMUSIC
protoclusters, which model only stellar/supernova feedback as well as the
generic hydrodynamical effect of gravitational shock-heating.

In Dong et al. (2023), we discussed the possibility that the transparent
Lyman-$\alpha$ forest of COSTCO-I might simply be due to shock-heating of the
cosmic web driven by nonlinear gravitational collapse within the overdense
protocluster region. Given that stellar feedback appears to have only a minor
effect on the FGPA-like photo-ionisation equilibrium of the Lyman-$\alpha$
forest in the 300-GadgetMUSIC protoclusters (Viel et al. 2013a), the
300-GadgetMUSIC protoclusters set an upper limit to the impact of
gravitational shock heating in protoclusters at this redshift. While there is
a possibility that The Three Hundred has not fully converged for the
Lyman-$\alpha$ forest (see Appendix A, also the discussion on simulation
resolution in Bourne et al. 2015), the clear deviations from FGPA in the
majority of the observed protoclusters suggest that effects beyond simple
gravitational collapse are likely at play. Upcoming datasets from The Three
Hundred with higher resolutions (Cui et al., in prep.) would help to
distinguish the effect of the unresolved baryon process in the case. The fact
also appears to be true for the COSTCO-III, Hyperion 1, Hyperion 4, and ZFIRE
protoclusters: although their $\delta_{F}$ are not as high as COSTCO-I, they
appear to deviate clearly from the adiabatic $\delta_{m}$-$\delta_{F}$
relationship. Even for the most massive simulated protocluster progenitors
($M_{h,z=0}>10^{15}M_{\odot}$) in the weak feedback model of 300-GadgetMUSIC,
the Megaparsec-scale overdensities have not yet collapsed sufficiently at
Cosmic Noon to drive enough gravitational shock heating to suppress the
Lyman-$\alpha$ forest signal to the level seen in the majority of the
CLAMATO$\times$COSTCO protoclusters. This should also true for lower-mass,
COSTCO-I-like halos $M_{h,z=0}=(4.6\pm 2.2)\times 10^{14}\,h^{-1}\,M_{\odot}$,
as they are even further from virialization.

In the literature, the importance of gravitational heating has mostly been
discussed in the context of star formation quenching in galaxies. The
pioneering work by Dekel & Birnboim (2006) suggests that the occurrence of
gravitational shocks leads to the bimodality of galaxies, as the gravitational
shocks shut down the continuous gas supply via cold streams. However, such a
scenario is largely restricted by the redshift evolution of halo mass and
shock heating mass (c.f., Figure 7 of Dekel & Birnboim 2006), limiting the
effect of gravitational shocks to $z<2$ galaxies. More recent works, for
example, Liu & Cen (2017); Zhu & Feng (2017), also set the scope of such
shock-heating-induced quenching to $z<2$. In addition to the redshift
restriction of gravitational-induced shocks, the shock radii of these $z\sim
2.5$ halos also suggest that the shock scenario is not preferable. As a rough
estimation, the most massive halo in the $z=2.5$ snapshot of SIMBA-100 has a
total mass of $8.53\times 10^{13}M_{\odot}/h$, which translates into
$r_{200c}=946\,\mathrm{ckpc/h}$. The central galaxy group in COSTCO-I
($M_{h}\approx 6\times 10^{13}\,h^{-1}\,M_{\odot}$; Dong et al. 2023) has a
similar halo mass, thus the extent of the central core halo spans a
significantly smaller region than the observed scale of enhanced
Lyman-$\alpha$ transparency in COSTCO-I, which is a few comoving Megaparsecs.
These scales would need to encompass the filamentary structures within the
protocluster region, which are at lower densities than halos and therefore
expected to collapse and shock-heat later at $z\sim 1$ (for example, Martizzi
et al. 2019).

###  4.2 Can AGN jet feedback explain COSTCO-I?

Among the feedback prescriptions implemented in The Three Hundred, we find
that the 300-SIMBA AGN jet feedback model yields the best match with the
observed Lyman-$\alpha$ transmission of COSTCO-I. We also find a discrepancy
between 300-SIMBA zoom-in runs and the SIMBA-100 cosmological volume, which
ostensibly have similar feedback prescriptions.

The massive cluster environment and the modified parameters from the original
SIMBA model to 300-SIMBA contribute to the difference in transmission, but the
simulation resolution might also be an issue. In Appendix A we compare the
SIMBA-100 run with two additional tests with the 300-SIMBA prescription. We
find that the stronger jet feedback gives rise to the WHIM gas with $T\sim
10^{7}$ K by $z=2.5$, whereas the downgraded resolution changes the power-law
$\rho-T$ relation in the diffuse cold gas regime. As stated in Section 4.1,
the fact suggests that the 300-SIMBA runs have yet to reach resolution
convergence, but this will not undermine the robustness of comparison between
the different models implemented for The Three Hundred, nor does it affect our
discovery that the majority of our observed protoclusters are in disagreement
with FGPA. Understanding the impact of resolution better necessitates zoom-in
simulation suites with improved resolution (for example, TNG-Cluster; Nelson
et al. 2023).

We conclude that extreme transmission in the 300-SIMBA sample stems from the
collective effect of a massive cluster environment, stronger feedback and low
resolution, thus it is premature to claim that this particular AGN jet
feedback model is necessarily an accurate depiction of the Lyman-$\alpha$
forest observed in COSTCO-I. Nevertheless, 300-SIMBA results are still useful
for qualitative investigations into mechanisms that might facilitate large-
scale heating of the IGM. In particular, it confirms the highly-collimated jet
feedback can boost the large-scale heating of surrounding gas, and make
massive protoclusters transparent in the Ly$\alpha$ forest. This is broadly
consistent with results from previous works on the effect of the different
feedback prescriptions in the SIMBA suite of simulations on the distribution
and thermal state of gas in the IGM and CGM. (Sorini et al., 2022; Khrykin et
al., 2023).

Within the next few years, the IGM tomography component of the Subaru Prime
Focus Spectrograph (PFS, Takada et al. 2014; Greene et al. 2022) will discover
$\gtrsim 100$ galaxy protoclusters at $2.2<z<2.7$ to enable a statistical
sample to compare with the hydrodynamical feedback models. By then, we aim to
have resolved suites of simulations to compare with the observations.

##  5 Summary and Conclusions

In this paper, we examined the effect of various galaxy and AGN feedback
mechanisms on the large-scale ($\sim$Mpc) Lyman-$\alpha$ forest transmission
of galaxy protoclusters at Cosmic Noon ($z=2.5$). This was motivated by the
recent observational discovery of a $z=2.30$ galaxy protocluster, dubbed
COSTCO-I (Dong et al., 2023), which exhibits excess Lyman-$\alpha$
transmission (i.e. too little absorption) given the significant matter
overdensity that it represents.

In addition to the $100\,h^{-1}\,\mathrm{cMpc}$ SIMBA-100 cosmological
hydrodynamical simulation that features multiple stellar- and AGN-feedback
mechanisms, we also analyzed a sample of cluster zoom-in simulations from The
Three Hundred suite, in which each cluster was also simulated with three
different feedback prescriptions (stellar feedback, AGN thermal feedback, and
AGN jet feedback). We focused on the redshift snapshot at $z=2.50$ when most
of the clusters were still in the protocluster stage. The simulated
protocluster sample from The Three Hundred suite covers a range of cluster
masses up to $3\times 10^{15}M_{\odot}/h$ at $z=0$, providing a good
statistical sample to study COSTCO-I.

The observable we focused on was the smoothed Lyman-$\alpha$ transmission,
$\delta_{F}$, as a function of the similarly-smoothed underlying matter
overdensity $\delta_{m}$. While these quantities can be straightforwardly
calculated in the simulations, on the observational side we can do direct
comparisons thanks to the combination of the 3D Lyman-$\alpha$ forest
tomographic maps from the CLAMATO survey, as well as the COSTCO constrained
simulations covering the same volume, that were based on the observed
spectroscopic galaxy distribution in the COSMOS field.

We summarize our findings as follows:

  * •

When averaged over apertures of several Megaparsecs around individual galaxy
protoclusters, we find clear differences in the Ly$\alpha$ transmission
depending on the feedback model. There is a weak dependence of Ly$\alpha$
transmission on underlying matter density, but different feedback models
studied here all occupy distinct populations in the matter density-
transmission ($\delta_{m}-\delta_{F}$) plane. This establishes that the
Lyman-$\alpha$ forest around protoclusters is a potentially powerful probe of
feedback mechanisms at Cosmic Noon ($z\sim 2-3$).

  * •

From the $\delta_{m}-\delta_{F}$ relation for several of the observed
CLAMATO$\times$COSTCO protoclusters, we find clear deviations from the global
power-law relationship that is expected to hold for the photo-ionised IGM at
these epochs, i.e. there is a deficit of Lyman-$\alpha$ absorption in these
structures. These protoclusters are in disagreement with the simulated
protoclusters that incorporate stellar feedback only (300-GadgetMUSIC model).
Since this model also sets an upper limit on the effect of shock heating from
gravitational collapse, we argue that neither gravitational shock-heating nor
stellar feedback is a likely cause of the transparent Ly$\alpha$ forest seen
in these protoclusters. Some form of AGN feedback appears to be necessary to
explain this discrepancy.

  * •

The COSTCO-I protocluster, in particular, exhibits an even stronger deviation
($\sim 5\sigma$) from the FGPA relationship and 300-GadgetMUSIC. At face
value, COSTCO-I is best matched to the protocluster population simulated with
the AGN jet feedback model (300-SIMBA). However, lower-mass protoclusters in
the SIMBA-100 cosmological hydrodynamical box, which nominally has a similar
feedback model, do not form a continuous relation with the 300-SIMBA
population in the $\delta_{m}-\delta_{F}$ plane. This is probably due to the
large-scale extreme environment within the 300-SIMBA regions even in early
times, but there is a possibility that The Three Hundred is not fully
converged concerning the Lyman-$\alpha$ forest. Future analyses will need to
be careful to ensure simulation convergence.

Strong AGN feedback in galaxy protoclusters has potential implications toward
the use of the Lyman-$\alpha$ forest as a cosmological probe. Cosmological
analyses of the Lyman-$\alpha$ forest power spectrum typically make the
implicit assumption that the IGM at Cosmic Noon follows the FGPA, which
stellar feedback largely does not perturb (Viel et al., 2013b). Galaxy
protoclusters represent regions of the Universe with the strongest density
fluctuations at the $z\gtrsim 2$ epoch, therefore widespread AGN feedback in
these structures could potentially smooth out the corresponding fluctuations
that manifest into the Lyman-$\alpha$ forest. While Chabanier et al. (2020)
analyzed the effect of AGN jet feedback on the 1D Lyman-$\alpha$ forest power
spectrum in the context of the HorizonAGN simulations, they did not check the
specific case of galaxy protoclusters. It is therefore unclear whether their
AGN model would reproduce the protocluster effects seen in this paper. A
recent paper by Rogers & Poulin (2023) found an anomalous small-scale
suppression in the Lyman-$\alpha$ forest power spectrum in tension with
Planck, but our findings indicate that the effect of AGN feedback needs to be
taken into account in such analyses.

In the near future, larger samples of Cosmic Noon galaxy protoclusters, such
as those to be observed by the Subaru PFS Galaxy Evolution survey, in
combination with converged hydrodynamical simulations, will provide strong
constraints on the AGN feedback mechanisms at play in galaxy protoclusters.

## Acknowledgements

Kavli IPMU is supported by World Premier International Research Center
Initiative (WPI), MEXT, Japan. This work was performed in part at the Center
for Data-Driven Discovery, Kavli IPMU (WPI). This research was supported by
Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a
World-leading Innovative Graduate Study (WINGS) Program, the University of
Tokyo. WC is supported by the STFC AGP Grant ST/V000594/1, the Atracción de
Talento Contract no. 2020-T1/TIC-19882 granted by the Comunidad de Madrid in
Spain and the science research grants from the China Manned Space Project. He
also thanks the Ministerio de Ciencia e Innovación (Spain) for financial
support under Project grant PID2021-122603NB-C21, ERC: HORIZON-TMA-MSCA-SE for
supporting the LACEGAL-III project with grant number 101086388, and the
science research grants from the China Manned Space Project with No. CMS-
CSST-2021-A01, and CMS-CSST-2021-A03..

This work has been made possible by the ‘The Three Hundred’
collaboration.111https://www.the300-project.org The 300 simulations were
performed at the MareNostrum Supercomputer of the BSC-CNS through The Red
Española de Supercomputación grants (AECT-2022-3-0027, AECT-2023-1-0013), and
at the DIaL – DiRAC machines at the University of Leicester through the RAC15
grant: Seedcorn/ACTP317. 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 results shown in this work use data from The Three Hundred galaxy clusters
sample. These data are available on request following the guidelines of The
Three Hundred collaboration, at https://www.the300-project.org. The data
specifically shown in this paper will be shared upon request to the authors.
The SIMBA-100 cosmological simulation is publicly available at
http://simba.roe.ac.uk/.

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##  Appendix A Consistency test between SIMBA-100 and 300-SIMBA

In this appendix, we show our observations on consistency between SIMBA-100
and 300-SIMBA. To achieve this, we run two test simulation boxes with randomly
generated initial conditions (i.e., not in the environment of massive
clusters) but keep the feedback prescription identical as in The Three
Hundred. One simulation box has the identical resolution as SIMBA-100 and a
box size of $25$ cMpc/h, while another box has a degraded resolution as in The
Three Hundred, and a box size of $33$ cMpc/h.

Fig. 4 illustrates the mass-weighted temperature-density relation of gas in
the simulation (TDR, e.g., Cen & Ostriker (2006), Martizzi et al. (2019)) in
SIMBA-100 and two test runs at $z=2.5$. For the connivance of discussion, we
adopted the IGM phase classification of Lukić et al. (2014). The IGM gas is
divided into four phases according to their temperature and density: diffuse
cold gas (T $<10^{5}K,\rho_{b}/\bar{\rho}_{b}<100$), warm-hot intergalactic
medium (WHIM, T $>10^{5}K,\rho_{b}/\bar{\rho}_{b}<100$), hot halo gas (T
$>10^{5}K,\rho_{b}/\bar{\rho}_{b}>100$) and condensed gas (T
$<10^{5}K,\rho_{b}/\bar{\rho}_{b}>100$). We find, in the diffuse regime, all
the three simulation boxes reproduced a power-law $\rho$-T relation
$T=T_{0}(1+\delta)^{\gamma-1}$, but the low-resolution run has a larger power-
law index $\gamma$, which may be relevant to the high overall Ly$\alpha$
transmission shown in Section 3. The pattern of gas distribution in the WHIM
regime is different from simulation to simulation: most of WHIM in SIMBA-100
has $T\leq 10^{6}$ K, though there is a small fraction of WHIM reaching a
temperature of $10^{7}-10^{8}$ K, which can be attributed to the high
resolution and large box size of SIMBA-100. In the 300-SIMBA high-resolution
test run, we find a branch of WHIM gas spanning from
$(\log_{10}\rho_{b}/\bar{\rho}_{b},\log_{10}T)=(1,6)$ to
$(\log_{10}\rho_{b}/\bar{\rho}_{b},\log_{10}T)=(0,7)$, implying the effect of
stronger jet feedback in 300-SIMBA. The branch, however, only contributes a
negligible portion of the total gas mass. A similar trend is found in the
300-SIMBA low-resolution test run, but the limitation on resolution makes the
pattern less notable. We conclude that the modification of feedback strength
mainly leads to a population of WHIM gas with $T>10^{7}$ K at redshift
$z=2.5$, and a lower resolution is likely to have an impact on the diffuse
cold gas.

Figure 4: From left to right: the mass-weighted $z=2.5$ temperature-density
relation for SIMBA-100 and two (high resolution and low resolution) 300-SIMBA
runs with initial conditions that are roughly at mean cosmic density. In each
panel, the vertical grey dash line represents
$\log_{10}\rho_{b}/\bar{\rho}_{b}=2$ while the horizontal line is
$T=10^{5}\mathrm{K}$. The two dash lines divide the $\rho-T$ phase space into
diffused cold gas (lower left), warm-hot intergalactic medium (WHIM, upper
left), hot halo gas (upper right) and condensed gas (lower right). In each
panel, we add a reference dashed line to emphasize the linear $\rho-T$
relation of the diffuse cold gas regime and list the best-fit parameters,
$\log_{10}T_{0}$ (in Kelvin) and $\gamma$ (defined in Section 1). All three
panels show a similar pattern in the WHIM regime, while the low-resolution run
has a distinct $\rho-T$ relation in the diffuse cold gas regime.