# AGN feedback in isolated galaxies with a SMUGGLE multiphase ISM

Aneesh Sivasankaran1, Laura Blecha1, Paul Torrey2, Luke Zoltan Kelley3, Aklant
Bhowmick1,Mark Vogelsberger4, Lars Hernquist5, Federico Marinacci6,7 and Laura
V. Sales8
1Department of Physics, University of Florida, Gainesville, Florida 32601, USA
2Department of Astronomy, University of Virginia, 530 McCormick Road,
Charlottesville, VA 22903, USA
3Department of Astronomy, University of California at Berkeley, 501 Campbell
Hall, Berkeley, CA 94720, USA
4Department of Physics, Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
5Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
6Department of Physics & Astronomy “Augusto Righi”, University of Bologna, via
Gobetti 93/2, 40129 Bologna, Italy
7INAF, Astrophysics and Space Science Observatory Bologna, Via P. Gobetti
93/3, I-40129 Bologna, Italy
8University of California, Riverside, 900 University Ave., Riverside, CA
92521, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Feedback from active galactic nuclei (AGN) can strongly impact the host
galaxies by driving high-velocity winds that impart substantial energy and
momentum to the interstellar medium (ISM). In this work, we study the impact
of these winds in isolated galaxies using high-resolution hydrodynamics
simulations. Our simulations use the explicit ISM and stellar evolution model
called Stars and MUltiphase Gas in GaLaxiEs (SMUGGLE). Additionally, using a
super-Lagrangian refinement scheme, we resolve AGN feedback coupling to the
ISM at $\sim$10-100 pc scales. We find that AGN feedback efficiently regulates
the growth of SMBHs. However, its effect on star formation and outflows
depends strongly on the relative strengths of AGN vs local stellar feedback
and the geometrical structure of the gas disk. When the energy injected by AGN
is subdominant to that of stellar feedback, there are no significant changes
in the star formation rates or mass outflow rates of the host galaxy.
Conversely, when the energy budget is dominated by the AGN, we see a
significant decline in the star formation rates accompanied by an increase in
outflows. Galaxies with thin gas disks like the Milky Way allow feedback to
escape easily into the polar directions without doing much work on the ISM. In
contrast, galaxies with thick and diffuse gas disks confine the initial
expansion of the feedback bubble within the disk, resulting in more work done
on the ISM. Phase space analysis indicates that outflows primarily comprise
hot and diffuse gas, with a lack of cold and dense gas.

###### keywords:

methods: numerical – black hole physics – quasars: supermassive black holes –
galaxies: ISM – galaxies: interactions

††pubyear: 2024††pagerange: AGN feedback in isolated galaxies with a SMUGGLE
multiphase ISM–AGN feedback in isolated galaxies with a SMUGGLE multiphase ISM

##  1 Introduction

It is widely accepted that supermassive black holes (SMBHs) exist at the
centers of the majority of massive galaxies. These BHs are thought to start as
$\sim 10^{2}-10^{5}$ ${\rm M_{\odot}}$ seeds and grow in size by many orders
of magnitude primarily through gas accretion (Soltan, 1982). During accretion,
a fraction of the accreted energy is converted into radiation resulting in
luminosities as high as $\sim 10^{48}$ergs s-1 emanating from milliparsec size
regions. These objects are called Active Galactic Nuclei (AGNs) and the
coupling of this energy to the surrounding interstellar medium (ISM) of the
host galaxy is called AGN feedback. AGN feedback is postulated to play a major
role in the evolution of the host galaxy by regulating star formation by
expelling cold gas or heating up gas. AGN feedback is also thought to self-
regulate the growth of SMBHs by suppressing gas inflows. These postulates are
motivated by observational data which show strong correlations between the
mass of the SMBHs and the properties of their host galaxies such as bulge
mass, stellar velocity dispersion and luminosity (Kormendy & Richstone, 1995;
Magorrian et al., 1998; Ferrarese & Merritt, 2000; Gültekin et al., 2009;
McConnell & Ma, 2013; Kormendy & Ho, 2013; Reines & Volonteri, 2015; Bennert
et al., 2015; Savorgnan et al., 2016). Apart from these correlations, there
are also observations of strong outflows (with outflow rates $\sim 1000\,$
${\rm M_{\odot}}$yr-1 and velocities $>1000\,$km s-1) emanating from galactic
nuclei (Rupke & Veilleux, 2011; Sturm et al., 2011; Cicone et al., 2014; Fiore
et al., 2017; Fluetsch et al., 2019), supporting the existence of AGN feedback
and its importance in the evolution of the host galaxy. Nevertheless, the role
of AGN feedback in galaxy evolution is still a debated topic due to the
uncertainties in our understanding of how the feedback actually couples to the
interstellar medium (ISM). This coupling can happen through various mechanisms
such as jets, winds and radiation pressure. Presently, the contributions of
these distinct modes and their corresponding coupling efficiencies remain
poorly constrained.

Theoretical studies of galaxy formation and evolution using numerical
simulations have been very successful at reproducing many observational data
(e.g., Vogelsberger et al., 2014b, a; Genel et al., 2014; Schaye et al., 2015;
McAlpine et al., 2017; Pillepich et al., 2018; Nelson et al., 2018; Davé et
al., 2019). AGN feedback is an essential component in these simulations, and
it has been found to be necessary for regulating star formation in massive
galaxies, particularly at low redshifts (Teyssier et al., 2011; Dubois et al.,
2013; Sijacki et al., 2015; Weinberger et al., 2018). However, some studies
suggest that the observed correlations between BH and galaxy properties can be
explained by a common gas supply fueling the BHs and star formation (Anglés-
Alcázar et al., 2013; Anglés-Alcázar et al., 2015, 2017). Hence, more work is
needed in this area to fully understand the role of AGNs in galaxy evolution.

The biggest challenge in modelling AGN feedback in these simulations is the
large range of scales involved: from the accretion disks around the SMBHs at
sub-parsec scales to the galactic outflows which happen at kpc to Mpc scales.
In cosmological simulations, because of resolution limitations, AGN feedback
is typically injected as thermal or kinetic energy directly into the ISM at
$\sim$kpc scales without resolving the actual mechanism that heats up the gas.
These crude modeling of such an important important physical mechanisms
generates uncertainties that limit the predictive power of the simulations.

In this work, we use high-resolution idealized simulations of AGN feedback in
a range of galactic environments, with the aim of resolving the scale at which
the feedback first couples to the surrounding ISM. The AGN feedback mode we
consider in this work is fast nuclear winds with velocities of the order of
$\sim 0.1c$. This type of winds can be driven by radiation from the accretion
disks around SMBHs and can result in broad absorption lines in the UV or X-ray
(Weymann et al., 1981; Turnshek, 1988; Chartas et al., 2003). Such high energy
winds can have a significant impact on the galaxy; for example by shock-
heating the surrounding ISM and sweeping up matter into large-scale outflows.
The amount of work done by the wind on the ISM depend on several factors, such
as the geometry and porosity of the ISM (Wagner et al., 2012, 2013; Bieri et
al., 2017; Torrey et al., 2020). Feedback is expected to do more work on a
dense and uniform ISM compared to a porous one, which can allow the hot gas to
escape through low density channels (Faucher-Giguère & Quataert, 2012; Torrey
et al., 2020).

Therefore, an explicit ISM model that can accurately resolve the multiphase
structure, geometry and density variations of the ISM is crucial for accurate
studies of AGN feedback. In this work we employ the recently developed Stars
and MUltiphase Gas in GaLaxiEs (SMUGGLE, Marinacci et al. 2019) model for the
moving mesh hydrodynamics code AREPO (Springel, 2010). SMUGGLE is an explicit
and comprehensive stellar feedback model which has been shown to produce an
ISM with well resolved multiphase structure, observationally consistent star
formation rates, galactic outflows (Marinacci et al., 2019), H$\alpha$
emission line profiles (Tacchella et al., 2022; Smith et al., 2022) and star
cluster properties (Li et al., 2020, 2021), and constant density cores in
dwarf galaxies (Jahn et al., 2021). In Sivasankaran et al. (2022) (Paper I
hereafter) we analyzed BH accretion (without including AGN feedback) in
isolated and merging galaxies with the SMUGGLE ISM. We showed that the gas
turbulence in the ISM leads to highly variable accretion rates, which is
qualitatively different from the way BHs accrete in simulations with widely
used effective equation-of-state models of the ISM.

There are a few other explicit ISM models in literature such as the Feedback
In Realistic Environments (FIRE; Hopkins et al. 2018) framework in the
meshless magnetohydrodynamics code GIZMO (Hopkins, 2015) and the Agertz et al.
(2013) ISM in the adaptive mesh refinement code RAMSES (Teyssier, 2002).
Recently, the FIRE model has been used to perform ultra high resolution
($\lesssim 10$pc) but short timescale (10-50 Myr) studies of SMBH fueling
(Anglés-Alcázar et al., 2021) and AGN feedback (Cochrane et al., 2023;
Mercedes-Feliz et al., 2023) in cosmological zoom-in simulations.

In this paper we implement a novel prescription for AGN feedback in the
SMUGGLE framework in the simulations presented in Paper I. Our study aims to
investigate the impact of feedback on the growth of SMBHs, generation of gas
outflows, and star formation in isolated galaxies over $\sim$Gyr timescales .
Additionally, we aim to explore how the properties of the host galaxy, such as
the geometry of the disc and its gas fraction, affect the extent of feedback’s
impact on the galaxy.

In addition to having an explicitly modelled ISM, very high resolution is also
needed to accurately resolve the coupling of AGN feedback to the ISM, which is
expected to happen at sub-parsec to tens of parsec scales. However, running
galactic or cosmological scale simulations at such high resolutions is
generally not practical. To address this issue, we implemented a super-
Lagrangian refinement scheme in Paper I, which increases the gas mass
resolution in a small region around the BHs. By utilizing this method in
lower-resolution simulations, we were able to reproduce the central gas
dynamics and accretion rates of very high, uniform resolution simulations.
This refinement scheme not only improves the estimation of accretion rates,
but also enables us to resolve the initial coupling of AGN feedback to the ISM
at scales as close to the radius of influence of the BH as possible by
capturing finer gas density variations around the BHs. Similar refinement
techniques have been applied in other studies to investigate BH accretion and
feedback (Curtis & Sijacki, 2015; Anglés-Alcázar et al., 2021).

This paper is structured as follows: In Section 2, we provide a brief overview
of the simulation code AREPO, the SMUGGLE model, the BH accretion and feedback
subgrid models, and the super-Lagrangian refinement scheme. Section 3 presents
the findings from our isolated galaxy simulations. Finally, in Section 4, we
discuss our results and draw our conclusions.

##  2 Methods

###  2.1 AREPO and SMUGGLE overview

The simulations in the paper are carried out using the moving-mesh
magnetohydrodynamics code AREPO (Springel, 2010; Pakmor et al., 2016). AREPO
solves magnetohydrodynamics equations on an unstructured Voronoi mesh using a
finite volume method. The mesh points are allowed to move along with the fluid
flow allowing an automatic adjustment of spatial resolution. AREPO models dark
matter and stars as collisionless particles obeying the Vlasov equation
coupled to Poisson’s equations (gravity). The ISM is treated as an ideal gas
following Euler’s equations. The ISM physics, such as star formation, stellar
feedback, gas heating and cooling, are modelled explicitly in our simulations
using the SMUGGLE model (Marinacci et al., 2019).

In SMUGGLE, gas heating and cooling includes mechanisms such as cosmic rays,
photoelectric heating, and low temperature atomic and molecular cooling, which
can create a multiphase ISM with temperatures from $\sim 10$K to $\sim
10^{8}$K. Star formation and stellar feedback are implemented locally in
SMUGGLE. Star particles are allowed to form only from cold, dense and
gravitationally bound gas cells. A density threshold of 100 cm-3 is adopted
for star formation in our simulations. Stellar feedback (including supernova
energy and momentum, radiative feedback from stars, and AGN and OB wind
momentum) is injected into the nearest gas cells of the star particles in a
kernel-weighted fashion. All stellar feedback inputs are calculated by
sampling individual stars from the star particles using the Chabrier (2003)
IMF. We refer the reader to Marinacci et al. (2019) for more details of the
included physical processes and their implementation.

Figure 1: BH mass (left) and Eddington ratio (right) as functions of time for
the MW, Sbc and SMC isolated galaxy runs with wind velocities of $0.1c$ and
$0.03c$. BH mass growth is effectively regulated by the fast AGN winds in all
galaxies. $\rm{v_{w}=0.1c}$ results in a factor of $2-3$ BH mass growth over
2.5 Gyr in all galaxies. Lowering the wind velocity leads to higher accretion
rates and higher BH masses. Eddington ratios have large fluctuations due to
transient cavities created primarily by bursty stellar feedback. Most of the
BH growth happens during the peak Eddington ratios. When the wind velocity is
lowered, the BHs spend more time in the high accretion state.

###  2.2 Black Hole Subgrid Models

####  2.2.1 BH Accretion

In our simulations that include AGN feedback, BH accretion rates are
calculated based on the Eddington-limited Bondi-Hoyle prescription. In this
approach the gas around the BH is assumed to be spherically symmetric and the
accretion rate is estimated from the density and temperature of the gas as
given below:

$\dot{M}_{\rm
BH}=\mathrm{min}(\dot{M}_{\mathrm{Bondi}},\dot{M}_{\mathrm{Edd}}),$ (1)

where $\dot{M}_{\mathrm{Bondi}}$ is the Bondi-Hoyle accretion rate,

$\dot{M}_{\mathrm{Bondi}}=\frac{4\pi G^{2}M^{2}_{\rm BH}\rho}{c^{3}_{s}},$ (2)

and $\dot{M}_{\mathrm{Edd}}$ is the Eddington limit,

$\dot{M}_{\mathrm{Edd}}=\frac{4\pi GM_{\rm
BH}m_{p}}{\epsilon_{r}\sigma_{T}c},$ (3)

which represents the accretion rate at which the radiation pressure becomes
equal to the gravitational pull on the gas. In Equations 2-3, $\rho$ and
$c_{s}$ are the kernel-weighted (see Paper I for details) gas density and
sound speed of the gas near the BH, $G$ is Newton’s constant, $c$ is the speed
of light, $\epsilon_{r}$ is the radiative efficiency, which is set to 0.1,
$M_{\rm BH}$ is the mass of the BH, $m_{p}$ is the mass of the proton and
$\sigma_{T}$ is the Thompson cross section. We will often use the Eddington
ratio, defined as $\chi_{\rm Edd}\equiv\dot{M}_{\rm BH}/\dot{M}_{\rm Edd}$, to
parameterize the accretion rates.

In addition to our fiducial simulations in which $\dot{M}_{\rm BH}$ is
calculated from the BH and gas properties according to Equations 1-3, we also
run a set of simulations with the accretion rate fixed at all times to 20% of
the initial Eddington accretion rate. While artificial, this subset of
simulations helps to disentangle the impacts of AGN feedback on the galaxy
from the impacts of AGN feedback on the BH fueling rate. These results are
described in § 3.2.

####  2.2.2 AGN feedback

In this work we model AGN feedback via fast nuclear winds, which are assumed
to be driven by physics happening on scales unresolved in our simulations such
as the accretion disc and broad line region. In our model we assume that a
fraction of the AGN luminosity drives fast kinetic winds with velocities of
the order of $\sim 0.1c$ radially away from the BH. Our model involves two
free parameters which prescribe the strength of the wind: (i) the wind
velocity ($v_{\rm w}$) and (ii) the wind mass loading factor $(\eta_{\rm w})$,
which is the ratio of the wind mass outflow rate $(\dot{M}_{\rm w})$ and the
BH growth rate $(\dot{M}_{\rm BH})$. At each time-step the wind mass is
calculated as $M_{w}=\eta_{w}\dot{M}_{\rm BH}\Delta t$, and the gas cells
around the BHs are kicked stochastically. Specifically each gas cell in the
BH’s kernel is kicked with a velocity of $v_{w}$ in a random direction with a
probability $p=M_{w}/M_{\rm kernel}$ where $M_{\rm kernel}$ is the total mass
of all gas cells in the kernel. This ensures energy and momentum conservation
in a time-averaged sense. We inject feedback only when the Eddington ratio is
above 0.05.

Note that it is possible to put a mass and(or) distance dependent weight in
the kick probability, but we find that this can lead to a slight (factor of
few) violation of energy conservation. Thus, we keep the probability uniform
inside the kernel. It is also possible to inject the feedback energy in a
continuous manner at every time-step, but this method might not be able to
capture the effects of the shocks created by the time-varying, high-velocity
wind with the surrounding gas. Note than $v_{w}\sim 0.1c$ is roughly
consistent with the quasar-mode feedback coupling efficiency assumed in
literature. A coupling efficiency of $5\%$ results in a velocity of $0.15c$.

We also simulate each of the three isolated galaxies with no AGN feedback, to
better quantify the impact of AGN feedback on the galactic environment. In
these simulations, the BHs are not allowed to accrete at all and act merely as
passive particles. This avoids unphysically high accretion rates in the
absence of AGN feedback, an issue described in Paper I and in Section 3 below.

####  2.2.3 Super-Lagrangian refinement

We recently implemented a super-Lagrangian refinement scheme in AREPO to
accurately resolve accretion rates and gas dynamics near the BH in low
resolution simulations (Sivasankaran et al., 2022). We use the same refinement
scheme in all the simulations in this work. In this method, the target gas
mass in the simulation is varied as a function of distance from the BH as
follows:

$\displaystyle m(r)=\begin{cases}\displaystyle\frac{m_{0}}{F}&r\leq r_{\rm
min}\\\ \displaystyle\frac{m_{0}}{F}\left(1+(F-1)\frac{r-r_{\rm min}}{r_{\rm
max}-r_{\rm min}}\right)&r_{\rm min}<r\leq r_{\rm max}\\\ \displaystyle
m_{0}&r_{\rm max}<r\end{cases}$ (4)

where r is the distance of the gas cells from the nearest BH, $m_{0}$ is the
uniform target mass, and $F$ is the refinement factor. The two radii $r_{\rm
min}$ and $r_{\rm max}$ are set to 2.86 kpc (= 2 kpc/h) and 14.3 kpc (=10
kpc/h), and the refinement factor $F$ is set to 10 in all our simulations. For
more details the reader is referred to Sivasankaran et al. (2022). Gas cells
in the refinement region of our fiducial simulations have a median size of
$\sim 30\,$pc and a minimum softening length of 45 pc. The kernel radius of
the BHs with the weighted number of neighbors set to 64 is $\sim 100\,$pc.

Name | ${\rm M_{200}}$ | ${\rm M_{BH}}$ | ${\rm M_{bulge}}$ | ${\rm M_{disk}}$ | Disk gas
---|---|---|---|---|---
| $({\rm M_{\odot}})$ | $({\rm M_{\odot}})$ | $({\rm M_{\odot}})$ | $({\rm M_{\odot}})$ | fraction
MW | $1.43\times 10^{12}$ | $1.14\times 10^{6}$ | $1.50\times 10^{10}$ | $4.73\times 10^{10}$ | 0.16
Sbc | $2.14\times 10^{11}$ | $1.14\times 10^{6}$ | $1.43\times 10^{9}$ | $5.71\times 10^{9}$ | 0.59
SMC | $2.86\times 10^{10}$ | $5.00\times 10^{4}$ | $1.43\times 10^{7}$ | $1.86\times 10^{8}$ | 0.86
Table 1: Initial parameters for the isolated galaxy simulations. Columns 2 - 6
show the total mass of the galaxy, initial BH mass, bulge mass, disk mass and
disk gas fraction of the three isolated galaxy initial conditions. These
initial conditions are the same as the ones used in Sivasankaran et al.
(2022). Figure 2: Instantaneous mass outflow rates (left) and cumulative mass
outflow rates (right) normalized by the initial gas mass within the stellar
half mass radius for MW (top), Sbc (middle) and SMC (bottom) galaxies. Each
panel shows the time evolution for runs with wind velocities set to 0.1$c$,
$0.03c$ and without BH feedback. Note that y-axis scales are different on the
three panels on the left. With $\mathrm{v_{w}=0.1c}$, outflows are strongest
in SMC. MW has slightly weaker outflows than SMC whereas Sbc shows no
difference with feedback. None of the galaxies show significant impact in the
case of $\mathrm{v_{w}=0.03c}$

###  2.3 Initial conditions

We employ the same three different initial conditions (ICs) used in
Sivasankaran et al. (2022) in our isolated galaxy simulations: A Milky-Way
like galaxy (MW), a Small Magellanic Cloud like dwarf galaxy (SMC) and a
luminous infrared type galaxy (Sbc). All three initial conditions contain a
dark matter halo, stellar and gaseous disks, a stellar bulge and a central BH.
The parameters of the ICs are given in Table 1. The stellar bulge and dark
matter halo follow the Hernquist (1990) profile and the gaseous and stellar
disks follow an exponential radial profile. The vertical structure of the
gaseous disk is set by hydrostatic equilibrium and the stellar disk has a
${\rm sech}^{2}(z)$ profile. In all our simulations we set the (unrefined)
target gas mass to $2.5\times 10^{4}$ ${\rm M_{\odot}}$, dark matter particle
mass to $2.64\times 10^{5}$ ${\rm M_{\odot}}$, stellar bulge particle mass to
$2.59\times 10^{4}$ ${\rm M_{\odot}}$, and stellar disk particle mass to
$2.78\times 10^{4}$ ${\rm M_{\odot}}$. We also use the super-Lagrangian
refinement with a refinement factor of 10 in all the runs at all times. This
resolution is the same as the resx0.1F10 resolution level in Sivasankaran et
al. (2022). We also allow an initial relaxation period of 0.36 Gyr for the
star formation and BH accretion to reach an approximate steady state. BH
accretion and feedback are turned on only after this period. The gas disks
have scale heights of 0.31, 1.34 and 1.07 kpc and scale radii of 21.0, 8.6 and
6.9 kpc for MW, Sbc and SMC respectively. We define these scale lengths as the
distances at which the gas density decreases by a factor of 10 after the disks
have reached steady state. Note that although Sbc and SMC are much smaller
compared to the overall size of MW, their gas disks are much thicker than that
of the MW. We refer to this set of three simulations as our fiducial
simulation suite, the results of which are discussed below in § 3.1. We
separately simulate the same three galaxies with a fixed accretion rate and
present these results in § 3.2. We also run the fiducial simulations with a
refinement factor of 30 and also with a 10 times higher uniform resolution to
test resolution convergence.

##  3 Results

Our study involves the simulations of three galaxies with and without AGN
feedback. In the runs without feedback, BHs are not allowed to accrete gas
from the ISM. The feedback runs consist of two sets of simulations, each with
different wind velocities ($v_{w}=0.1c$ and $0.03c$). Additionally, we also
perform the $v_{w}=0.1c$ feedback runs with fixed accretion rates (20% of
initial Eddington rate). In subsequent sections, we examine the effects of the
wind feedback on each of the three galaxies.

###  3.1 Fiducial simulation suite

####  3.1.1 Black hole mass growth and Eddington ratios

In the left panels of Figure 1, we compare the BH mass growth in the MW, Sbc,
and SMC isolated galaxy runs with wind velocities of $0.1c$ and $0.03c$. We
can see that feedback with a wind velocity of $v_{w}=0.1c$ effectively
regulates BH growth, resulting in moderate BH mass growth in all three
galaxies over a period of $2.5\,$Gyr. The BHs in the MW, Sbc, and SMC grow by
factors of 2.85, 2.13, and 2.12, respectively. With a lower AGN wind velocity
of 0.03c, BH growth is more efficient, especially in MW and Sbc. The BHs grow
by factors of 11.9 and 8.57 in these galaxies, respectively, over the same
period, while the BH mass in SMC grows by a factor of 3.48. While this may
provide a hint that BH growth in SMC is more efficiently regulated by feedback
than in the other two galaxies, it also reflects the $M_{\rm BH}^{2}$ scaling
of the Bondi accretion rate, given the lower initial BH mass in the SMC
galaxy. In the following sections we explore the balance between accretion,
feedback, and galactic environment in more detail.

In the right panels of Figure 1, we plot the Eddington ratio of the BHs as a
function of time. In all three galaxies, the Eddington ratio fluctuates by
many orders of magnitude over few-Myr timescales. In Paper I we showed that
bursty stellar feedback can produce transient cavities in the ISM that lead to
these large fluctuations in background gas density and corresponding BH
fueling rates. Figure 1 shows that this bursty stellar feedback strongly
influences BH fueling rates even in the presence of AGN feedback. Nonetheless,
with a AGN wind velocity of $0.1c$, the average Eddington ratios are a factor
of 2 - 3 lower than in the $v_{w}=0.03c$ case. The average Eddington ratios of
the BHs in MW, Sbc and SMC are 0.04, 0.07 and 0.03 for $v_{w}=0.1c$ and 0.12,
0.15 and 0.06 for $v_{w}=0.03c$. The BHs accrete above 10% of the Eddington
limit for $\sim 8\%,10\%$ and $7\%$ of the simulated time with $v_{w}=0.1c$ in
MW, Sbc and SMC, respectively. With the lower wind velocity this fraction
increases to $\sim 22\%,18\%$ and $8\%$.

In Paper I we simulated the same three galaxies without AGN feedback. However,
the Bondi accretion rate in those simulations had to be scaled down by a
factor of $10^{5}$, owing to the extremely high nuclear densities in the
absence of AGN feedback. With this factor of $10^{5}$ scaling, reasonable
average Eddington ratios of 0.15 for MW, 0.13 for Sbc, and 0.11 for SMC were
achieved in these no-feedback runs. In contrast, when using the AGN feedback
model presented here, we find that the average central gas densities near the
BH are about 3 orders of magnitude lower, and sound speeds are increased by
about an order of magnitude. Consequently, AGN feedback reduces the average
Bondi accretion rates by $\sim 5$ orders of magnitude, leading to BH growth
rates similar to the scaled Bondi rates seen in Paper I. Therefore, in the
presence of AGN feedback, we are able to obtain reasonable accretion rates
with the Bondi prescription without any artificial scaling. (As noted in
Section 2, in the no-AGN-feedback runs presented in this paper, the BHs are
not allowed to accrete gas at all.)

Figure 3: Total star formation rate (left), specific star formation rate
(middle) and cumulative stellar mass formed normalized by initial stellar mass
(right) in MW, Sbc and SMC runs with wind velocities set to 0.1$c$, $0.03c$
and without BH feedback. Note that the $y$-axis scales are different for each
simulation in the left and middle panels. SMC and Sbc have more bursty star
formation than MW. AGN feedback reduces the burstiness of star formation
somewhat, which lowers the final stellar mass in the SMC and the MW galaxies,
but has minimal impact on star formation in Sbc.

####  3.1.2 Mass outflow rates

Next we look at the mass outflow rates in these simulations. We calculate the
outflow rates using a shell method: we sum the radial momentum ($m_{i}v_{{\rm
rad},i}$) of all the gas cells inside a thin shell centered around the BH and
then divide it by the width of the shell ($\Delta r$), giving

$\dot{M}_{\rm out}=\frac{\sum_{i}m_{i}v_{{\rm rad},i}}{\Delta r}.$ (5)

It is also possible to calculate the outflow rate by integrating the radial
flux across the surface of the sphere as $\dot{M}_{\rm out}=\int\rho v_{\rm
rad}dS$. We find that both methods give similar results in our simulations,
and the shell method is computationally cheaper. We set the radius of the
shell $(r_{\rm shell})$ to be 1.5 times the initial stellar half-mass radius
($r_{\rm half,\star}$) of the galaxies, which is $r_{\rm shell}=10.3\,$kpc for
MW, $3.73\,$kpc for Sbc and $2.61\,$kpc for SMC. We set the width of the shell
to be $\Delta r=0.71\,$kpc (= 0.5 kpc/h) for all the galaxies.

In Figure 2 we compare the mass outflow rates of each galaxy, for wind
velocities of 0.1c and 0.03c, as well as runs without AGN feedback. We show
both instantaneous and cumulative outflow rates, each normalized by the
initial gas mass within $1.5\,r_{\rm half,\star}$, which is $2.8\times 10^{9}$
${\rm M_{\odot}}$ in MW, $2.1\times 10^{9}$ ${\rm M_{\odot}}$ in Sbc and
$2.4\times 10^{8}$ ${\rm M_{\odot}}$ in SMC. In all three galaxies, the
instantaneous mass flux oscillates between inflow and outflow over $\sim
100\,$ Myr timescales, even in the absence of AGN feedback. However, the
magnitude of these fluctuations is significantly smaller in the MW than in the
other two galaxies. The normalized peak outflow rates reach values of $\sim 1$
Gyr-1 in MW, 10-20 Gyr-1 in Sbc, and $\sim 40$ Gyr-1 in SMC. (The
corresponding non-normalized peak outflow rates are $\rm{\sim
3\;M_{\odot}yr^{-1}}$ in MW, $\rm{\sim 20-40\;M_{\odot}yr^{-1}}$ in Sbc and
$\rm{\sim 10\;M_{\odot}yr^{-1}}$ in SMC.)

The cumulative mass fluxes (integrated over the entire simulation length of
2.86 Gyr) on the right side of Figure 2 more clearly highlight qualitative
differences between the three types of galaxies. The MW galaxy experiences a
net inflow of gas through $r_{\rm shell}$, regardless of whether AGN feedback
is included. AGN feedback reduces the average inflow rate (instantaneous flux
averaged over the entire run) from $0.1$ Gyr-1 to $0.07$ Gyr-1 or $0.02$ Gyr-1
(for $v_{w}=0.03$c & $0.1$c, respectively). MW shows a clear trend of
decreasing inflow with increasing feedback strength. The Sbc galaxy, in
contrast, has a net outflow through $r_{\rm shell}$ in all cases, including
the no-AGN-feedback run. In the two feedback runs the net flux becomes
negative for very brief time intervals. There is no clear long-term impact of
AGN feedback in the Sbc galaxy; the cumulative outflow is very similar in the
runs with and without AGN feedback. This is because of the extreme bursty star
formation in Sbc, as we will see below.

The SMC galaxy presents still another qualitatively different scenario in
which AGN feedback has a significant impact on the cumulative mass flux. In
all three runs the cumulative flux has large oscillations. Without AGN
feedback, the cumulative flux is mostly negative until $\sim 2$ Gyr after
which it has large oscillations around zero. Feedback with $v_{\rm w}=0.03$c
has minimal impact on the cumulative flux, which is negative throughout the
run except a few brief intervals. (Note that these small differences are
within the random fluctuations in the flux expected in bursty galaxies like
Sbc and SMC.) In the feedback run with $v_{\rm w}=0.1$c we do see some
significant differences. The cumulative flux is positive throughout most of
the simulation except during brief periods of time where it decreases slightly
below zero. By the end of the run it reaches values as large as $\sim 0.9$.
The feedback also prevents the large inflows seen at later times in the no-
feedback and $v_{\rm w}=0.03$c runs.

Figure 4: Cumulative SNII energy injected into the central 2 kpc region (solid
lines) and cumulative BH feedback energy (dashed lines) in the MW, Sbc and SMC
runs with $v_{w}=0.1c$. AGN feedback is subdominant to stellar feedback in
Sbc, resulting in minimal impact of the AGN on this host galaxy. In the MW
simulation, the AGN dominates the feedback energy budget, and in SMC AGN and
stellar feedback energies are comparable.  Figure 5: Edge-on projections of
gas density (column 1), temperature (column 2), pressure (column 3) and speed
(column 4) during a typical AGN outflow event in the MW (left 4$\times$4
block) and SMC (right 4$\times$4 block) galaxies with $v_{w}=0.1c$. Snapshot
time increases from top to bottom (different for the two galaxies). A slice
thickness of 1.4kpc is used for averaging the quantities. MW has a very thin
disk which allows feedback to escape it with minimal work done resulting in
low coupling efficiency. The thicker disk of SMC allows the feedback to do
more work before leaving the disk resulting in more efficient coupling.

####  3.1.3 Star formation rates

In Figure 3 we compare the total star formation rates (SFRs), specific star
formation rates (sSFRs), and the cumulative stellar mass (normalized by the
initial stellar mass) formed in the three galaxies for $v_{w}=0.1c,\;0.03c$
and in the runs without AGN feedback. The (s)SFRs highlight key differences
between these three galaxies. The Sbc galaxy has a $\sim 10$ times higher sSFR
than the MW, despite the fact that they have similar average SFRs (0.89 ${\rm
M_{\odot}}$ yr-1, 1.53 ${\rm M_{\odot}}$ yr-1, respectively, in the absence of
AGN feedback). Star formation is also much burstier in the Sbc galaxy than in
the MW, as indicated by the larger stochastic variations. The SMC has
similarly bursty star formation and a high sSFR, despite the much lower
average SFR in this low-mass galaxy (0.059 ${\rm M_{\odot}}$ yr-1). The high
sSFRs in the Sbc and SMC galaxies reflect their higher initial gas fractions.

The introduction of AGN feedback results in a slight decline in SFRs in the MW
and SMC feedback runs after $\sim 0.5$ Gyr. There is also a noticeable decline
in the burstiness of star formation. However, in Sbc there is no significant
change in SFR when AGN feedback is introduced. These effects of AGN feedback
are clearer in the cumulative stellar mass plots on the right. In the SMC
simulation, AGN feedback with $v_{w}=0.03c$ and $0.1c$ results in a decrease
of 11% and 37% in stellar mass formed, respectively. In the MW simulation,
star formation is decreased by 12% for $v_{w}=0.03c$ and 26% for $v_{w}=0.1c$.
The impact in Sbc is even smaller. The cumulative stellar mass is roughly the
same in all three runs until $t\sim 2$ Gyr. After this, the increased BH mass
results in stronger feedback which decreases the cumulative stellar mass by 8%
and 10% in the $v_{w}=0.03c$ and $v_{w}=0.1c$ runs, respectively. These trends
align with the results of previous sections, and we conclude that the overall
impact of AGN feedback on star formation is small to moderate in all three
galaxies, but is especially weak in the Sbc galaxy.

Figure 6: Host galaxy properties in simulations with the BH accretion rate
artificially held to a constant value and a wind velocity of $0.1c$, compared
to the fiducial $v_{\rm w}=0.1c$ runs and the no-AGN-feedback runs. Left:
Cumulative mass outflow rate normalized by the initial gas mass within
1.5$\times$ the stellar half mass radius for MW, Sbc and SMC runs. Middle: Gas
mass within 1.5$\times$ the stellar gas mass radius normalized by the initial
value for the same runs. Right: Cumulative stellar mass formed as a fraction
of total initial stellar mass for the same runs. The higher accretion rates in
the fixed-$\dot{M}$ runs lead to slightly higher outflows and lower star
formation in MW. In contrast, SMC and Sbc shows much larger differences with
most of the gas expelled by the end of the runs.

####  3.1.4 Energetics and coupling strengths

To better understand the physics behind these trends, in Figure 4 we compare
the energy injected into the central region of the galaxies by stellar
feedback and by the AGN with $v_{w}=0.1$c. The dashed lines represent the
cumulative energy injected by BH feedback, and the solid lines represent the
cumulative SNII energy injected into the central 2 kpc region of the galaxies.
Only in the MW does the AGN feedback energy dominate; by the end of the
simulation, the AGN has injected several times more energy than stellar
feedback. In contrast, stellar feedback energy consistently dominates over AGN
feedback in both Sbc and SMC. This is especially true in the Sbc galaxy, which
has strong, bursty star formation that is negligibly modulated by AGN winds.
In SMC, the AGN feedback energy is still subdominant but is only slightly
lower than the stellar feedback energy. This results in a more noticeable
impact of AGN feedback in the SMC than in the Sbc galaxy.

Despite the fact that the AGN is energetically dominant in the MW and is
subdominant in the SMC, we saw in the previous subsections that star formation
is more strongly suppressed by AGN feedback in the SMC, at least for the
$v_{\rm w}=0.1c$ case. This is primarily due to the different coupling
efficiencies of the feedback in the two systems. To illustrate this, Figure 5
shows the edge-on view of the gas temperature, pressure, density and velocity
in the central 4 kpc region of the MW and SMC galaxies during a typical AGN
outflow event in each galaxy. By comparing the initial gas density plots of MW
and SMC, we see that MW is a thin and concentrated disk compared to SMC, which
is thicker and more diffuse. This allows the feedback winds in MW to escape
more easily in the vertical direction without doing much $PdV$ work on the gas
disk. During the initial stage, the hot pressurized gas expands mostly in the
vertical direction, along the path of least resistance. After leaving the disk
plane it expands in the horizontal directions. There is only a little impact
on the dense gas in the disk plane.

However, in SMC, the hot pressurized bubble of gas created by the AGN feedback
expands in all three directions inside the disk plane to a larger volume
before escaping in the vertical direction. It also has much more impact on the
dense gas in the disk. In the third snapshot of the gas density distribution,
we can see that the feedback ejects some of the dense gas above and below the
BH, completely disrupting the morphology of gas in the central region around
the BH. Thus, the diffuse vertical structure of the gaseous disk of SMC allows
a more efficient coupling of the AGN feedback to the galaxy, whereas the
thinner disk of MW results in a weaker coupling. We also note that the
potential well of SMC is much shallower than that of the more massive MW and
Sbc galaxies. This factor also contributes to the larger outflows detected in
SMC.

###  3.2 Simulations with fixed accretion rate

The outflow rates and quenching effects of AGN feedback on host galaxies are
influenced by the surrounding ISM conditions, which in turn affect both the BH
accretion rates and the AGN feedback coupling strength. Decoupling these two
factors is very challenging due to their highly nonlinear interdependence. To
isolate and compare the effects of feedback coupling to the ISM, we simulated
the same three galaxies with BH accretion rates set to 20% of the initial
Eddington accretion rate throughout the simulation and the wind speed set to
0.1c. We choose a slightly higher accretion rate than the average Eddington
ratio of the fiducial runs in order to increase the visibility of feedback
effects, while still avoiding artificial effects such as persistent cavities
around the BH due to extreme feedback injection.

####  3.2.1 Outflow and star formation rates

In the leftmost column of Figure 6, we show the cumulative mass outflow rates
normalized with respect to the initial gas mass within $1.5\times r_{\rm
half,\star}$ of each galaxy for the fixed-$\dot{M}$ runs, compared to the
corresponding fiducial runs with and without AGN feedback. We also show the
normalized gas masses within the same radii in the middle column of Figure 6.
With the increased accretion rates, the feedback energy budget is dominated by
the AGNs in all three galaxies. In MW, the increased accretion rate only
causes a marginal increase in the impact of feedback. The average normalized
outflow increases to 0.03 Gyr-1 compared to -0.01 Gyr-1 in the fiducial run
and -0.1 Gyr-1 in the no-feedback run. (Recall that these rates are normalized
by the initial gas mass inside $1.5\times r_{\rm half,\star}$).

AGN feedback has an even more dramatic effect in the fixed-$\dot{M}$ Sbc and
SMC runs, relative to their fiducial counterparts. In Sbc, the fiducial
simulations had modest cumulative mass outflows with and without AGN feedback.
In the fixed-$\dot{M}$ Sbc simulation, however, we see large outflows with the
cumulative mass outflow rate more than twice that of the no-AGN-feedback run.
By the end of the simulation almost all of the gas in the central region has
been evacuated by the feedback. (Note that the normalized, cumulative mass
outflow in the fixed-$\dot{M}$ Sbc and SMC runs exceeds unity at the end of
those simulations, which results from sampling errors due to the lack of
enough gas cells.) In the case of SMC, the fiducial feedback run suppressed
the inflows and produced a net outflow. With the higher (constant) accretion
rate, we see much stronger net outflows, which are also larger than that of
the Sbc fixed-$\dot{M}$ run. Similar to the case of Sbc, essentially all the
gas in the central region of the SMC is evacuated by the feedback. This
underscores our finding in § 3.1 that increasing the AGN feedback strength
will result in larger outflows in thicker disks due to the high coupling
efficiency, especially in low-mass galaxies, while in thinner disks, the
increase in outflows will be more modest.

In the rightmost column of Figure 6 we show the cumulative stellar mass formed
normalized by the initial stellar mass as a function of time in the same
galaxies. In fixed-$\dot{M}$ case, the suppression in star formation due to
AGN feedback is stronger in relation to their fiducial counterparts, with the
largest effect seen for the SMC. Relative to the no-feedback runs, star
formation is lowered by 38% in MW, 49% in Sbc and 70% in SMC for these
fixed-$\dot{M}$ runs. In comparison, the fiducial AGN feedback suppressed star
formation by 26% in MW, 10% in Sbc, and 37% in SMC, respectively, relative to
the no-feedback case. Thus the star formation rates are much more sensitive to
the accretion rates in Sbc and SMC compared to MW, which is similar to the
trend seen in the outflow rates. These two trends again imply that the
feedback coupling is much stronger in Sbc and SMC compared to MW.

Overall, these findings reinforce a major conclusion from the fiducial
simulations: namely, that the impact of AGN feedback depends strongly on the
morphology and nuclear environment of the host galaxy. In both the fiducial
and fixed-$\dot{M}$ runs, the SMC galaxy has the strongest coupling of AGN
feedback to the surrounding ISM. This trend seen in the fiducial simulations
persists when the accretion rate is fixed to a constant value demonstrates
that the differences in host environment, rather than self-regulation of BH
fueling, are driving the differences in AGN feedback coupling.

Figure 7: Temperature - Density phase space diagram of the outflowing gas
averaged over the entire simulation and color coded by mass (column 1), mass-
weighted outward radial velocity (column 2) and mass-weighted distances in the
vertical direction scaled by the disk height (column 3) in MW, Sbc and SMC
runs without feedback and with $v_{w}=0.1c$ at t=2 Gyr. Only gas cells with
outward radial velocity $>$ 3 times the virial velocity and located within the
virial radius are shown. All three feedback runs have more outflowing gas than
the corresponding no-feedback runs. Runs with AGN feedback have bimodal
distributions separated by 1-2 orders of magnitude in temperature. MW produces
the highest outflow velocities. Feedback heats up the gas in all three
galaxies with largest shift in SMC and smallest shift in MW. SMC and Sbc also
have more gas within the disk compared to MW.

####  3.2.2 Phase space structure

The strong impact of AGN feedback in the fixed-$\dot{M}$ simulations allows us
to identify key differences in the gas phase-space structure in galaxies with
and without AGN feedback. In Figure 7, we compare the phase space properties
of the high-velocity outflowing gas in the three galaxies with fixed $\dot{M}$
and without AGN feedback. The phase space plots are averaged over the entire
simulation ($t>0.36\,$Gyr) and the bins in $T-\rho$ space are color-coded in
each panel to indicate the average mass, the mass-weighted average radial
velocity, and the average vertical distance in units of disk scale height. To
separate outflowing gas from the bulk motion in the galaxy, we consider only
high-velocity outflowing gas, defined as gas cells with radial outward
velocity greater than 3 times $v_{200}$ of the system. The corresponding
velocity cutoffs are 452 km/s for MW, 245 km/s for Sbc and 123 km/s for SMC.
We also plot only gas cells that are within the virial radius of the galaxies
to avoid boundary effects at the edges of the simulation box.

From the plots in the first column we can see that most of the rapidly
outflowing gas falls within the temperature range of $10^{4}$K to $10^{8}$K
and has densities between $10^{-8}\mathrm{cm}^{-3}$ and
$10^{-3}\mathrm{cm}^{-3}$, with a higher-density tail. The only exception is
the SMC run without feedback, which has very little outflowing gas at all. In
each set of simulations, those with and without feedback, Sbc has the most
outflowing material and SMC has the least.

As expected, the runs with AGN feedback have more outflowing gas at higher
temperatures, as well as more massive outflows overall. The total outflowing
gas mass is greater by factors of 4.8, 2.6 and 14 in the MW, Sbc and SMC runs
with feedback relative to their counterparts without feedback. (Note that
these numbers will vary with the adopted velocity cutoffs.) The mass-weighted
median temperature of the outflowing gas also increased by factors of 3.9, 2.7
and 28 in MW, Sbc and SMC relative to the no-feedback runs. These numbers are
consistent with the trends seen in the previous section, namely that the
strongest AGN feedback coupling occurs in the SMC galaxy.

The most noticeable difference between the distributions in the feedback vs
no-feedback runs is observed in the low density ($<10^{-3}\mathrm{cm}^{-3}$)
region of the phase space, which is significantly more occupied in the
feedback runs. These gas cells constitute $>95\%$ of the outflowing gas in all
three galaxies. From the velocity and $z$ color coding we can see that most of
these gas cells are located far ($\sim\mathrm{R_{vir}}$) from the galactic
plane. These cells are also flowing out with velocities significantly higher
than that of the cells in the same phase space region of the corresponding no-
feedback runs. The feedback runs also feature some outflowing gas with
temperatures exceeding $10^{8}$ K, which also possess the highest velocities
within the distributions. This portion of the phase space is nearly vacant in
the no-feedback runs. These signatures indicate that these gas cells were
shock heated by AGN feedback.

From the plots in the second column we can see that the highest radial
velocities are obtained for outflowing gas in the MW feedback run. Both MW and
Sbc feedback runs have significant amounts of gas with velocities of the order
of several thousands of km/s, but MW has roughly $\sim 8$ times more gas with
velocities $>4000$ km/s. Note that both galaxies have very similar initial gas
content. Because of the thin disk of the MW, feedback does work on a much
smaller amount of the ISM gas compared to that of Sbc, allowing the outflowing
gas to retain high velocity. In SMC, there is only a negligible amount of gas
with these velocities.

By comparing the vertical distribution of the gas between the three feedback
runs in the third column, we find that around 0.06%, 0.8% and 1.4% of the
outflowing gas in MW, Sbc and SMC, respectively, is located within $\sim 4$
times the disk scale height $(|z|<2h)$. This again suggests that most of the
feedback energy in MW escapes into the vertical direction, whereas in Sbc and
in SMC, relatively more work is being done within the disk plane (with the
largest impact in SMC).

By comparing the three feedback runs we see that MW has no outflowing material
in the atomic or molecular phase (expected with T$<10^{4}$K and
$\rho>10^{-1}\mathrm{cm^{-3}}$) whereas Sbc and SMC feature some material in
this phase. This is somewhat expected since a higher disk thickness makes it
easier for the feedback to entrain cold and dense gas.

Figure 8: Results of resolution convergence tests are shown for the fiducial
MW, Sbc, and SMC galaxies with the $v_{w}=0.1c$ AGN feedback model. The left
column shows BH mass growth in the same manner as Figure 1, the middle column
shows the cumulative mass outflow rate normalized to the initial gas mass
within $1.5\times$ the stellar half-mass radius, in the same manner as Figure
2, and the right column shows the cumulative stellar mass formed, normalized
to the total initial stellar mass, in the same manner as Figure 3. In each
column, panels show results for the MW (top), Sbc (middle) and SMC (bottom)
galaxies, each simulated with the fiducial resolution (resx0.1F10, dashed
lines), with a $3\times$ higher resolution near the BH (resx0.1F30, dot-dashed
lines), and with a resolution throughout the whole simulation volume that
matches the fiducial resolution near the BH (resx1, solid lines). The final BH
mass varies by $\sim 50\%$ for different resolution simulations of each
galaxy. Outflows and star formation rates are well converged in the MW runs,
but the SMC and Sbc galaxies show somewhat poorer convergence due to the
stochasticity induced by extreme bursty star fomation.

###  3.3 Resolution Convergence

In Figure 8 we analyze the resolution convergence of three quantities: BH mass
growth, cumulative (normalized) mass outflow rates and cumulative (normalized)
star formation rates in the three galaxies. We compare the fiducial resolution
runs (resx0.1F10) with ${v_{\rm w}=0.1c}$ to runs with a uniform resolution
(resx1) that is the same as the central resolution of the fiducial run, as
well as runs with same background resolution but with a 3 times larger
refinement factor (resx0.1F30). Because these additional simulations
alternately vary the background and the central resolution, relative to the
fiducial setup, these comparisons do not constitute a study of the rate of
convergence. Nonetheless, they serve to demonstrate that our conclusions are
robust to variations in simulation resolution.

We first note that, while variations in these quantities with resolution do
exist, the variations do not exhibit discernible systematic trends. Some
variation is expected between different realizations of similar initial
conditions, especially given the highly nonlinear and stochastic nature of the
explicitly resolved, multiphase ISM as well as the $M_{\rm BH}^{2}$ scaling of
the Bondi accretion rate. Accordingly, we see that MW shows the best
resolution convergence among the three galaxies. This can be attributed to the
relatively weak stellar feedback in MW, leading to reduced stochasticity and
randomness within the ISM. The BH mass growth, which again is notoriously
sensitive to small changes in initial conditions within a Bondi accretion
scenario, exhibits $\sim 50\%$ variation by the end of the MW simulation. In
contrast, the cumulative mass outflow rate and stellar mass formed have almost
no variations between the different resolutions.

The BH mass growth curves in Sbc and SMC have fluctuations with similar
magnitudes to those in MW. However, because Sbc and SMC have much more bursty
star formation than MW as shown in Figure 3, these galaxies are more
susceptible to stochastic fluctuations in the gas dynamics. As a result, the
integrated outflow and star formation rates show significantly more
fluctuation between the different resolution simulations, with roughly $\sim
100\%$ and $30\%$ spread respectively in both galaxies. The normalized,
cumulative flux in SMC switches from a slight net outflow of 0.14 Gyr-1 in the
fiducial run to slight net inflows of 0.12 Gyr-1 in the resx0.1F30 run and
0.06 Gyr-1 in the resx1 run. In the case of Sbc, all three resolutions show
net outflows. However, the net flux changes from 0.17 Gyr-1 in the fiducial
run to 0.23 Gyr-1 in the resx0.1F30 run and 0.05 Gyr-1 in the resx1 run. We
again emphasize that no systematic trend is seen in these variations with
resolution. Thus, although our quantitative results are susceptible to these
fluctuations with different resolutions, the qualitative results from the
comparisons between the feedback vs. no-feedback runs and the different ICs
remain minimally affected.

##  4 Discussions and Conclusions

In this paper we modeled AGN feedback in the form of fast nuclear winds in
AREPO hydrodynamics simulations of idealized isolated galaxies. Our
simulations utilized the explicit ISM and stellar feedback model SMUGGLE,
which generates a multiphase ISM and implements localized stochastic star
formation and stellar feedback. We simulated three types of galaxies: a MW-
type, gas-poor galaxy with a thin disk, a gas-rich, SMC type galaxy with a
thick gas disk, and a LIRG-like galaxy (Sbc) that is also gas rich and has a
thick gas disk. Using a super-Lagrangian refinement scheme, we were able to
resolve gas dynamics at $\sim 10-100$ pc scales in the vicinity of the BHs.

Our simulations show that the AGN wind feedback is very efficient at
regulating the BH mass growth in all three galaxies. In the absence of AGN
feedback, the cold and dense gas clumps in the central region of the galaxies
result in peak Bondi accretion rates that are many orders of magnitude above
the Eddington limit (Sivasankaran et al., 2022). Our wind feedback mechanism
results in lower gas densities and higher temperatures around the BHs,
producing reasonable accretion rates with the Bondi prescription –average
Eddington ratios in our fiducial simulations range from 0.03 to 0.15, with
lower Eddington ratios corresponding to higher AGN wind velocities. Over a
period of 2.5 Gyr, the BHs grow by factors of 2-3 with a wind velocity of
0.1c. When the wind velocity is lowered to $0.03$c, the BH mass in MW and Sbc
increases by factors of $\gtrsim 10$ over the same period. In SMC, which has a
smaller initial BH mass, BH growth is similarly enhanced with weaker AGN
feedback but to a lesser degree, owing to the $M_{\rm BH}^{2}$ scaling of the
Bondi accretion rate.

Despite the clear long-timescale impacts of AGN feedback on BH fueling, we
find that the BH accretion rate still fluctuates by orders of magnitude on
short ($<100$ Myr) timescales, as in (Sivasankaran et al., 2022). The
characteristic variability timescales appear to be independent of AGN feedback
strength and are instead dominated by rapid variations in the central gas
density, owing to stellar-feedback-driven turbulence in the ISM.

Turning to examine the impact of AGN feedback on the host galaxy, we find
significant qualitative differences between the three types of simulated
galaxies. The SMC experiences the strongest AGN-driven outflows and
suppression of star formation, with a 37% lower mass in newly formed stars in
the $v_{\rm w}=0.1c$ case compared to the no-AGN-feedback run. In the MW
galaxy, this high-velocity AGN feedback led to a 26% decline in new star
formation. Unlike the SMC, however, the MW galaxy does not experience large
fluctuations between massive inflows and outflows, and the cumulative mass
flux through the nuclear region is a net inflow throughout the simulations,
even in the presence of strong AGN feedback. Large variations in central mass
flux are also seen in the Sbc run, but the impact of AGN feedback is more
modest: only 10% fewer new stars form with $v_{\rm w}=0.1c$ AGN feedback
compared to the no-AGN-feedback case.

We find that these trends are controlled by two main factors: the relative
strength of AGN vs. stellar feedback in the galactic nucleus, and the host
galaxy morphology (specifically, the thickness of the gas disk and the depth
of the central potential). In the MW, which has relatively quiescent star
formation, the AGN energy input easily dominates over the stellar feedback
energy in the galactic nucleus, leading to a significant suppression in star
formation. However, the MW has a very thin disk, which allows the much of the
AGN feedback to escape easily through the path of least resistance in the
vertical direction with minimal work being done on the gas disk. Thus, the
cumulative mass flux remains a net inflow, even in the presence of
energetically dominant AGN feedback. This result is in agreement with the
findings of Torrey et al. 2020 using the Feedback in Realistic Environments
(FIRE) framework. They also found that AGN feedback couples inefficiently to
thin disc galaxies, with a significant fraction of the energy venting in the
polar directions.

Contrary to MW, the thick and diffuse gas disks of Sbc and SMC allow for a
much more efficient coupling of AGN feedback to the ISM. Feedback does more
work inside the disk during the initial expansion phase before escaping from
the disk plane. At the same time, the Sbc and SMC also have much burstier star
formation than the MW, and the resulting stellar feedback also couples more
efficiently to their thicker gas disks than in the MW. In the Sbc galaxy, AGN
feedback energy is subdominant to stellar feedback in the nuclear region;
thus, we see minimal impact of AGN feedback on star formation and gas outflow
rates, despite its efficient coupling. In the SMC simulations, meanwhile, the
total energy injected by the AGN versus stellar feedback in the central region
is comparable. The SMC is also a low-mass galaxy with a shallower potential,
from which outflowing gas can more easily escape. These factors combined with
the efficient feedback coupling in the SMC galaxy yield the strongest AGN-
driven outflows and star formation quenching among the three galaxy types
considered.

To further isolate the effects of the host galaxy environment on the impact of
AGN feedback, we simulated the three galaxies with fixed BH accretion rate and
with the AGN wind velocity set to $0.1c$. This exercise effectively eliminates
the non-linear dependence of AGN feedback coupling on the BH accretion rates
from our analysis. By choosing a fixed $\dot{M}$ of 20% of the initial
Eddington rate, which is somewhat higher than the average Eddington ratio in
our fiducial runs, we also amplify the impact of AGN feedback to make trends
more readily apparent. In MW the effects are modest, changing the cumulative
central mass flux from a slight inflow to a slight outflow and decreasing new
star formation by 38% relative to the no-AGN-feedback (versus 26% with
fiducial AGN feedback). In Sbc and SMC, however, the higher, fixed accretion
rate and efficient coupling leads to strong outflows and quenching effects.
Here the AGN feedback in the Sbc finally becomes energetically dominant, and
the impact on new stellar mass formed is five times larger than in the
fiducial run. Little gas remains in the central region of the galaxy by the
end of the run. Regardless, the SMC again experiences the strongest impact
from AGN feedback in these fixed-$\dot{M}$ runs, with 70% less star formation
than in the no-AGN-feedback run, and almost no gas remaining in the central
region of the galaxy at the end of the run. The changes in the outflow and
star formation rates with the fixed-$\dot{M}$ are consistent with our findings
from the fiducial simulation, further validating our conclusions.

Our phase space analysis of the fast outflowing gas in the fixed-$\dot{M}$
runs also revealed similar trends. The fraction of outflowing material located
within the gas disk was much higher in SMC and Sbc compared to MW, indicating
more work done on the ISM by AGN feedback in these galaxies. Additionally, we
find that the outflows generated by the AGNs are mainly made up of hot and low
density gas in all three galaxies. In Sbc and SMC galaxies we found small
amounts of gas with temperatures and densities corresponding to atomic and
molecular phase. Outflows in MW did not have any gas in the atomic and
molecular phases. The phase space distributions of the feedback runs also
exhibited signatures of shock heated gas with very high temperatures
$(>10^{8}\mathrm{K})$ and velocities ($>1000\mathrm{\,km\,s^{-1}}$) which were
absent in the outflows generated by stellar feedback alone.

In summary, we have studied the impact of AGN feedback in a set of three
isolated galaxies with varying AGN wind velocities. While the AGN are highly
variable within the stochastic, multiphase ISM, they have moderate average
Eddington ratios. We avoid simulating the regime of massive elliptical
galaxies with extreme quenching, where AGN feedback is expected to have strong
global effects on the host (e.g., Springel et al., 2005b; Dubois et al.,
2013). Instead, we deliberately focus on a small suite of isolated galaxies to
probe in detail the interplay of BH fueling, AGN feedback, star formation, and
stellar feedback, in the absence of extreme quasar phases or secular effects
such as galaxy interactions. These simulations reveal efficient self-
regulation of BH growth and a complex picture of BH/galaxy co-evolution. Our
investigation reveals that AGN luminosity and wind velocity, gas disk
morphology, host potential, and stellar feedback strength are the key factors
determining the AGN-galaxy coupling. In certain regimes, specifically in
galaxies with thick gas discs and luminous AGNs, the feedback can
significantly reduce star formation and alter the gas content and morphology
of the host.

In forthcoming work, we extend our analysis to include merging galaxies.
Galaxy mergers are thought to be important drivers of BH/galaxy co-evolution.
They can trigger bursts of star formation and AGN activity by rapidly
funneling gas into galactic nuclei (e.g., Sanders et al., 1988; Hernquist,
1989; Barnes & Hernquist, 1991, 1996; Mihos & Hernquist, 1996). The resulting
feedback can quench star formation, while the merging galaxies simultaneously
undergo morphological transformations (Springel et al., 2005a; Hernquist,
1992, 1993). However, the extent of the connection between galaxy mergers and
AGN fueling remains a subject of debate, underscoring the need for improved
constraints from simulations (e.g., Satyapal et al., 2014; Weston et al.,
2016; Goulding et al., 2018; Grogin et al., 2005; Pierce et al., 2007;
Villforth et al., 2019). The ability to explicitly resolve localized,
stochastic star formation and a multiphase ISM with models like SMUGGLE has
opened a new discovery space for studies of BH/galaxy co-evolution. Our
findings in this work highlight the rich and complex physics that such studies
will continue to reveal.

## Acknowledgements

LB acknowledges support from NASA Astrophysics Theory Program awards
80NSSC20K0502 and 80NSSC22K0808. The simulations in this study were performed
using the supercomputing cluster HiPerGator at University of Florida.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Agertz et al. (2013) Agertz O., Kravtsov A. V., Leitner S. N., Gnedin N. Y., 2013, ApJ, 770, 25
  * Anglés-Alcázar et al. (2013) Anglés-Alcázar D., Özel F., Davé R., 2013, ApJ, 770, 5
  * 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., Davé R., Faucher-Giguère C.-A., Özel F., Hopkins P. F., 2017, MNRAS, 464, 2840
  * Anglés-Alcázar et al. (2021) Anglés-Alcázar D., et al., 2021, ApJ, 917, 53
  * Barnes & Hernquist (1991) Barnes J. E., Hernquist L. E., 1991, ApJ, 370, L65
  * Barnes & Hernquist (1996) Barnes J. E., Hernquist L., 1996, ApJ, 471, 115
  * Bennert et al. (2015) Bennert V. N., et al., 2015, ApJ, 809, 20
  * Bieri et al. (2017) Bieri R., Dubois Y., Rosdahl J., Wagner A., Silk J., Mamon G. A., 2017, MNRAS, 464, 1854
  * Chabrier (2003) Chabrier G., 2003, Publications of the Astronomical Society of the Pacific, 115, 763
  * Chartas et al. (2003) Chartas G., Brandt W., Gallagher S., 2003, ApJ, 595, 85
  * Cicone et al. (2014) Cicone C., et al., 2014, A & A, 562, A21
  * Cochrane et al. (2023) Cochrane R., et al., 2023, arXiv preprint arXiv:2303.12858
  * Curtis & Sijacki (2015) Curtis M., Sijacki D., 2015, MNRAS, 454, 3445–3463
  * Davé et al. (2019) Davé R., Anglés-Alcázar D., Narayanan D., Li Q., Rafieferantsoa M. H., Appleby S., 2019, MNRAS, 486, 2827
  * Dubois et al. (2013) Dubois Y., Gavazzi R., Peirani S., Silk J., 2013, MNRAS, 433, 3297
  * Faucher-Giguère & Quataert (2012) Faucher-Giguère C.-A., Quataert E., 2012, MNRAS, 425, 605
  * Ferrarese & Merritt (2000) Ferrarese L., Merritt D., 2000, ApJ Letters, 539, L9
  * Fiore et al. (2017) Fiore F., et al., 2017, A & A, 601, A143
  * Fluetsch et al. (2019) Fluetsch A., et al., 2019, MNRAS, 483, 4586
  * Genel et al. (2014) Genel S., et al., 2014, MNRAS, 445, 175
  * Goulding et al. (2018) Goulding A. D., et al., 2018, PASJ, 70, S37
  * Grogin et al. (2005) Grogin N., et al., 2005, ApJ Letters, 627, L97
  * Gültekin et al. (2009) Gültekin K., Cackett E. M., Miller J. M., Di Matteo T., Markoff S., Richstone D. O., 2009, ApJ, 706, 404
  * Hernquist (1989) Hernquist L., 1989, Nature, 340, 687
  * Hernquist (1990) Hernquist L., 1990, ApJ, 356, 359
  * Hernquist (1992) Hernquist L., 1992, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 400, no. 2, p. 460-475., 400, 460
  * Hernquist (1993) Hernquist L., 1993, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 409, no. 2, p. 548-562., 409, 548
  * Hopkins (2015) Hopkins P. F., 2015, MNRAS, 450, 53
  * Hopkins et al. (2018) Hopkins P. F., et al., 2018, MNRAS, 480, 800
  * Jahn et al. (2021) Jahn E. D., et al., 2021, arXiv preprint arXiv:2110.00142
  * Kormendy & Ho (2013) Kormendy J., Ho L. C., 2013, Annual Review of Astronomy and Astrophysics, 51, 511
  * Kormendy & Richstone (1995) Kormendy J., Richstone D., 1995, Annual Review of Astronomy and Astrophysics, 33, 581
  * Li et al. (2020) Li H., Vogelsberger M., Marinacci F., Sales L. V., Torrey P., 2020, MNRAS, 499, 5862
  * Li et al. (2021) Li H., Vogelsberger M., Bryan G. L., Marinacci F., Sales L. V., Torrey P., 2021, arXiv preprint arXiv:2109.10356
  * Magorrian et al. (1998) Magorrian J., et al., 1998, The Astronomical Journal, 115, 2285
  * Marinacci et al. (2019) Marinacci F., Sales L. V., Vogelsberger M., Torrey P., Springel V., 2019, MNRAS, 489, 4233
  * McAlpine et al. (2017) McAlpine S., Bower R. G., Harrison C. M., Crain R. A., Schaller M., Schaye J., Theuns T., 2017, MNRAS, 468, 3395
  * McConnell & Ma (2013) McConnell N. J., Ma C.-P., 2013, ApJ, 764, 184
  * Mercedes-Feliz et al. (2023) Mercedes-Feliz J., et al., 2023, arXiv preprint arXiv:2301.01784
  * Mihos & Hernquist (1996) Mihos J. C., Hernquist L., 1996, ApJ, 464, 641
  * Nelson et al. (2018) Nelson D., et al., 2018, MNRAS, 475, 624
  * Pakmor et al. (2016) Pakmor R., Springel V., Bauer A., Mocz P., Munoz D. J., Ohlmann S. T., Schaal K., Zhu C., 2016, MNRAS, 455, 1134
  * Pierce et al. (2007) Pierce C., et al., 2007, ApJ Letters, 660, L19
  * Pillepich et al. (2018) Pillepich A., et al., 2018, MNRAS, 475, 648
  * Reines & Volonteri (2015) Reines A. E., Volonteri M., 2015, ApJ, 813, 82
  * Rupke & Veilleux (2011) Rupke D. S., Veilleux S., 2011, ApJ Letters, 729, L27
  * Sanders et al. (1988) Sanders D. B., Soifer B. T., Elias J. H., Neugebauer G., Scoville N. Z., 1988, ApJ, 325, 74
  * Satyapal et al. (2014) Satyapal S., Ellison S. L., McAlpine W., Hickox R. C., Patton D. R., Mendel J. T., 2014, MNRAS, 441, 1297
  * Savorgnan et al. (2016) Savorgnan G. A., Graham A. W., Marconi A., Sani E., 2016, ApJ, 817, 21
  * Schaye et al. (2015) Schaye J., et al., 2015, MNRAS, 446, 521
  * Sijacki et al. (2015) Sijacki D., Vogelsberger M., Genel S., Springel V., Torrey P., Snyder G. F., Nelson D., Hernquist L., 2015, MNRAS, 452, 575
  * Sivasankaran et al. (2022) Sivasankaran A., et al., 2022, MNRAS, 517, 4752
  * Smith et al. (2022) Smith A., et al., 2022, MNRAS, 517, 1
  * Soltan (1982) Soltan A., 1982, MNRAS, 200, 115–122
  * Springel (2010) Springel V., 2010, MNRAS, 401, 791
  * Springel et al. (2005a) Springel V., Di Matteo T., Hernquist L., 2005a, MNRAS, 361, 776
  * Springel et al. (2005b) Springel V., Di Matteo T., Hernquist L., 2005b, The Astrophysical Journal, 620, L79
  * Sturm et al. (2011) Sturm E., et al., 2011, ApJ Letters, 733, L16
  * Tacchella et al. (2022) Tacchella S., et al., 2022, MNRAS, 513, 2904
  * Teyssier (2002) Teyssier R., 2002, A & A, 385, 337
  * Teyssier et al. (2011) Teyssier R., Moore B., Martizzi D., Dubois Y., Mayer L., 2011, MNRAS, 414, 195
  * Torrey et al. (2020) Torrey P., et al., 2020, MNRAS, 497, 5292
  * Turnshek (1988) Turnshek D. A., 1988, QSO absorption lines: probing the Universe, p. 17
  * Villforth et al. (2019) Villforth C., Herbst H., Hamann F., Hamilton T., Bertemes C., Efthymiadou A., Hewlett T., 2019, MNRAS, 483, 2441
  * Vogelsberger et al. (2014a) Vogelsberger M., et al., 2014a, MNRAS, 444, 1518
  * Vogelsberger et al. (2014b) Vogelsberger M., et al., 2014b, Nature, 509, 177
  * Wagner et al. (2012) Wagner A., Bicknell G., Umemura M., 2012, arXiv preprint arXiv:1205.0542
  * Wagner et al. (2013) Wagner A., Umemura M., Bicknell G., 2013, ApJ Letters, 763, L18
  * Weinberger et al. (2018) Weinberger R., et al., 2018, MNRAS, 479, 4056
  * Weston et al. (2016) Weston M. E., McIntosh D. H., Brodwin M., Mann J., Cooper A., McConnell A., Nielsen J. L., 2016, MNRAS, 464, 3882
  * Weymann et al. (1981) Weymann R. J., Carswell R. F., Smith M. G., 1981, Annual Review of Astronomy and Astrophysics, 19, 41