# Regulation of Star Formation by a Hot Circumgalactic Medium

Christopher Carr Department of Astronomy, Columbia University, 550 West 120th
Street, New York, NY, 10027, USA  Greg L. Bryan Department of Astronomy,
Columbia University, 550 West 120th Street, New York, NY, 10027, USA  Center
for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Drummond B. Fielding Center for Computational Astrophysics, Flatiron
Institute, New York, NY 10010, USA  Viraj Pandya Hubble Fellow  Columbia
Astrophysics Laboratory, Columbia University, 550 West 120th Street, New York,
NY 10027, USA  Center for Computational Astrophysics, Flatiron Institute, New
York, NY 10010, USA  Rachel S. Somerville Center for Computational
Astrophysics, Flatiron Institute, New York, NY 10010, USA

###### Abstract

Galactic outflows driven by supernovae (SNe) are thought to be a powerful
regulator of a galaxy’s star-forming efficiency. Mass, energy, and metal
outflows ($\eta_{M}$, $\eta_{E}$, and $\eta_{Z}$, here normalized by the star
formation rate, the SNe energy and metal production rates, respectively) shape
galaxy properties by both ejecting gas and metals out of the galaxy and by
heating the circumgalactic medium (CGM), preventing future accretion.
Traditionally, models have assumed that galaxies self-regulate by ejecting a
large fraction of the gas which enters the interstellar medium (ISM), even
though such high mass-loadings are in growing tension with observations. To
better understand how the relative importance of ejective (i.e. high mass-
loading) vs preventative (i.e. high energy-loading) feedback affects the
present-day properties of galaxies, we develop a simple gas-regulator model of
galaxy evolution, where the stellar mass, ISM, and CGM are modeled as distinct
reservoirs which exchange mass, metals, and energy at different rates within a
growing halo. Focusing on the halo mass range from $10^{10}$ to
$10^{12}M_{\odot}$, we demonstrate that, with reasonable parameter choices, we
can reproduce the stellar-to-halo mass relation and the ISM-to-stellar mass
relation with low mass-loaded ($\eta_{M}\sim 0.1-10$) but high energy-loaded
($\eta_{E}\sim 0.1-1$) winds, with self-regulation occurring primarily through
heating and cooling of the CGM. We show that the model predictions are robust
against changes to the mass-loading of outflows but are quite sensitive to our
choice of the energy-loading, preferring $\eta_{E}\sim 1$ for the lowest mass
halos and $\sim 0.1$ for Milky Way-like halos.

Circumgalactic medium (1879), Galactic winds (572), Galaxies (573), Galaxy
evolution (594), Galaxy physics (612), Galactic and Extragalactic astronomy
(563)

##  1 Introduction

Early theories of galaxy formation proposed that galaxies form from the
cooling gas that sinks to the centers of virialized dark matter halos and
self-gravitates to form stars (White & Rees, 1978). However, early models ran
into the longstanding problem of “overcooling”, where the accretion of gas in
halos form galaxies that are too efficient at producing stars, particularly in
low-mass and high-mass halos (Dekel & Silk, 1986; White & Frenk, 1991). This
manifests as a mismatch in the stellar-to-halo mass relation, which compares
the typical stellar masses in galaxies in the real universe to their
respective dark matter halos via abundance matching techniques (Moster et al.,
2010; Behroozi et al., 2019).

Later generations of galaxy formation modeling have attempted to address this
problem of overcooling by introducing feedback from supernova (SNe) explosions
to make star formation less efficient in Milky Way and lower-mass halos
(Somerville & Davé, 2015). Massive stars, upon their death, deposit enormous
quantities of energy, enriched gas, and momentum into the surrounding
interstellar medium (ISM), regulating star formation by providing a source of
effective pressure that counterbalances the local disk weight, supporting the
disk against gravitational collapse (e.g., Thompson et al., 2005; Ostriker &
Kim, 2022). However, feedback not only regulates star formation within the
ISM, by introducing a source of heating and turbulence (Krumholz et al.,
2009), but also through its ability to drive large-scale multi-phase outflows
with mass loss rates comparable to the galaxy’s star formation rate,
quantified by the mass-loading factor $\eta_{M}$ (Heckman et al., 1990;
Strickland & Stevens, 2000; Strickland & Heckman, 2009). The presence of these
galactic winds has been confirmed by observations of outflows in star-forming
galaxies at high and low redshift (for reviews see Veilleux et al., 2005;
Rupke, 2018). Once these outflows leave the ISM, their mass, energy, and
metals are either deposited into the circumgalactic medium (CGM) (Heitsch &
Putman, 2009), recycled and accreted back onto the disk in a ’galactic
fountain’ (Shapiro & Field, 1976; Bregman, 1980; Marasco et al., 2022a), or
even driven out of the halo entirely (Oppenheimer & Davé, 2008; Kereš et al.,
2009; Oppenheimer et al., 2010). SNe feedback, which is the focus of this
work, is important but insufficient to explain the observed star formation
inefficiencies and outflows in halos with masses $\gtrsim 10^{12}M_{\odot}$,
and thus requires additional feedback from active galactic nuclei (AGN)
(Benson et al., 2003; Croton et al., 2006; Somerville et al., 2008; Benson,
2010).

Efforts to properly model these baryon flows on galactic scales typically rely
on three broad categories of techniques. The most direct way involves
numerical hydrodynamic simulations that explicitly track gravity and gas
dynamics for particles or grid cells representing baryons and dark matter.
Despite the ability to make detailed predictions, an immediate downside to
this technique is that it is quite computationally expensive and, except
possibly for the smallest galaxies, does not possess the necessary resolution
to trace individual SNe remnants or properly track the evolution of diffuse,
multiphase galactic winds. These simulations must then rely on simplified
physical subgrid models to follow the physics on scales that are not resolved
– these subgrid models are generally calibrated to reproduce observed galaxy
scaling relations or related observables (e.g. Oppenheimer & Davé, 2006;
Vogelsberger et al., 2014; Crain et al., 2015, and the reviews by Naab &
Ostriker 2017 and Vogelsberger et al. 2019). A less costly alternative is
semi-analytic modeling (SAMs) (e.g. White & Rees, 1978; White & Frenk, 1991;
Kauffmann et al., 1993; Somerville & Primack, 1999; Cole et al., 2000; Baugh,
2006; Somerville et al., 2008, and the reviews by Benson 2010 and Somerville &
Davé 2015). Rather than simulating the hydrodynamics directly, SAMs track
numerous interlocking physical processes in the build-up of galaxies based on
the underlying dark matter merger tree using parameterized recipes. This
method is relatively fast and can efficiently explore different parameter
choices, but it too has its drawbacks, as SAMs can quickly become quite
complex and rely on a host of free parameters that are not well-constrained.

Despite these differences in techniques in galactic modeling, something that
has been common amongst them is their reliance on strong stellar feedback to
reduce the galactic baryon fraction and the overall stellar mass. Large
cosmological simulations that lack the spatial resolution to model wind
generation generally require a significant ejection of mass out of the galaxy
in order to match observed galaxy statistics with a steep dependence on halo
mass or circular velocity in order to match the stellar-to-halo mass halo
relation and the low-mass end slope of the stellar mass function (Somerville &
Davé, 2015). For example, large-scale cosmological simulations such as
Illustris-TNG (Pillepich et al., 2018) or SIMBA (Davé et al., 2019) explicitly
impose high-mass loaded winds (although there are a few exceptions, such as
EAGLE (Schaye et al., 2014) which injects high specific energy feedback
instead). High mass outflow rates also appear in higher resolution ”zoom-in”
cosmological simulations such as FIRE, which aim to produce galactic winds
self-consistently (Muratov et al., 2015; Pandya et al., 2021). Mass loadings
can be even higher in semi-analytic models (10-100 times greater than the FIRE
estimates) in order to compensate for the similarly larger gas accretion rate
into low-mass galaxies (Pandya et al., 2020).

However, this “traditional” approach to achieving low stellar-to-halo mass
fractions, by assuming that galaxies eject a large fraction of the gas which
enters the disc, has come into growing tension from two directions: (i)
observations of galactic winds from low-mass galaxies, and (ii) high-
resolution simulations of self-consistently driven winds.

Observationally derived mass-loading estimates for star-forming galaxies with
stellar masses from $10^{7}<M_{\star}/M_{\odot}<10^{11}$ using UV absorption
line measurements from the Cosmic Origins Spectrograph on board the Hubble
Space Telescope (HST-COS) find $\eta_{M}$ values in the range of $1-10$ with a
negative correlation with stellar mass (Chisholm et al., 2017). Recent deep
H$\alpha$ observations from McQuinn et al. (2019) of the low-surface
brightness features of galactic winds from a sample of 12 low-mass galaxies
derive mass-loadings on the order of $0.2-7$. They find that the majority of
the ejected wind from the dwarfs remain in the halo and, consistent with
Martin (1999), find a much weaker dependence on circular velocity or stellar
mass than what is generally adopted in cosmological simulations. Another study
from Marasco et al. (2022b) using H$\alpha$ kinematics find even smaller
estimates for mass-loading factors for gas that escapes the halo, reporting
values as small as $\log_{\rm 10}\eta_{M}\sim-2$ for systems with stellar mass
from $10^{7}<M_{\star}/M_{\odot}<10^{10}$. The results from Marasco et al.
(2022b) appear in line with similarly low outflow rate estimates from Concas
et al. (2022) at higher redshift ($1.2<z<2.6$) using stacked emission line
data from the KLEVER survey.

On the simulation side, another approach to modeling galactic winds is via
“small-scale” simulations that resolve patches of the ISM down to the $\sim$pc
scale and can resolve the Sedov-Taylor phase of expanding SNe remnants and
their interaction with the ISM, which is crucial to resolving the multiphase
nature of galactic winds (e.g., Girichidis et al., 2016; Li et al., 2017; Kim
et al., 2020). These simulations can track outflows of mass and energy of each
phase of the wind, which reach different heights from the disk as a function
of gas and star formation rate densities. The consensus picture emerging from
these ISM patch simulations is that the hot phase (T $\sim$ $10^{6}$ K)
carries most of the energy, while the cold phase (T $\sim$ 104 K) carries most
of the mass (Fielding et al., 2018; Kim et al., 2020; Li & Bryan, 2020). Since
outflows measured in these simulations are much closer to the disk and
represent a small patch of the ISM, they cannot be directly compared to
observations. Mass loadings in the cool phase measured at the disk scale
height can be as high as $1-100$ times the star formation rate and decreases
with star formation density, but most of this cool gas is moving at low
velocities, meaning that the mass loading of cooler outflows declines
precipitously with increasing height from the galaxy, as cool gas that cannot
escape the galaxy’s potential falls back to the disc. The energy outflows in
the hot phase of the wind, possessing a specific energy an order of magnitude
larger than the cold phase, carry about $\sim 10\%$ of the energy ejected from
SNe, enough energy to break out from the ISM and reach greater heights from
the galaxy, often with Bernoulli velocities greater than the galaxy escape
velocity (Kim et al., 2017, 2020). Lastly, there are high resolution
simulations of dwarf galaxies which also find lower mass loaded winds beyond
the virial radius than what is adopted in cosmological simulations (Hu, 2019).

These hot winds may not eject much mass out of the galaxy, but the energy they
contain may contribute to a preventative aspect of SNe feedback. Instead of
only an ejective mode, energy flux from galactic winds can heat up
circumgalactic material, preventing further radiative cooling in the gas and
reducing the ambient density of the CGM by lifting gas to greater altitudes
through gradual heating (Voit et al., 2017). This heating from a central
feedback source, either from SNe or a central black hole in more massive
galaxies, has been proposed as a means to regulate subsequent star formation
by not only removing baryons from the CGM, preventing them from accreting onto
the galaxy in the first place, but also by keeping gas in the CGM in a quasi-
equilibrium state where precipitation of cold gas clouds via thermal
instabilities out of a hot gaseous halo and accretion onto the disk is limited
by the ratio between the cooling time $t_{\rm cool}$ and the free-fall time
$t_{\rm ff}$ (McCourt et al., 2012; Sharma et al., 2012a, b; Voit et al.,
2015a, b, 2017). These SN-driven winds that escape from halos (or lifted CGM
gas through heating) may also shock-heat gas beyond the CGM and thereby
prevent cosmic accretion (Pandya et al., 2020). The combination of mounting
observations favoring lower mass outflows than what is required in
cosmological simulations and the emerging view from small-scale simulations
emphasizing the multiphase nature of galactic winds may call for a rethinking
of how galaxies regulate their growth, and whether that regulation is
primarily ejective or preventative in nature.

In this paper we examine this question using a simplified method that is
distinct from numerical simulations and even more stripped down than
traditional SAMs for modeling the process of galaxy formation. These models
have come to be known as “gas-regulator” or “bathtub” models (e.g., Bouche et
al., 2010; Lilly et al., 2013; Dekel et al., 2013). Explored extensively in
prior works (Davé et al., 2011; Birrer et al., 2014; Peng & Maiolino, 2014;
Dekel & Mandelker, 2014; Mitra et al., 2015; Furlanetto, 2020; Furlanetto &
Mirocha, 2021), these regulator models use simple (but physically-based)
analytic arguments and approximations to model gas flows and the conversion of
gas into stars. The components of the galaxy are treated as mass reservoirs
(“bathtubs”) which exchange mass and metals at different rates. The base
framework comes from a simple set of differential equations which describe the
flows between the various reservoirs. Sometimes, but not always, an
equilibrium condition is also added, which states that the inflow of mass from
the intergalactic medium (IGM) into the ISM balances the sum of gas mass lost
from outflows and converted into stars (e.g., Finlator & Davé, 2008). Usually,
a galaxy is assumed to evolve without mergers, with the mass inflow of dark
matter and baryons set by fits to average mass accretion histories extracted
from dissipationless N-body simulations. An advantage of these models is that
they are simple, making clear the physical underpinnings that connect the
reservoirs, and converge to solutions that are independent of initial
conditions. Although gas regulator models are not meant to replace their more
sophisticated cousins like SAMs or hydrodynamical simulations, these models
have proved to be quite useful at reproducing basic galaxy scaling relations,
and can be implemented and improved upon in simulations or SAMs in the future
(Pandya et al., in prep).

What has not been extensively explored in these regulator models is the role
of the CGM in regulating galaxy formation and the contribution of energy-
loaded preventative feedback. Although Davé et al. (2011), for example,
included a preventative feedback parameter, which quantifies the amount of
baryonic inflow from the IGM prevented from reaching the ISM and thus
remaining in the gaseous halo, the broad properties of cooling and heating in
the CGM as its own reservoir are not explicitly tracked, leaving the CGM
omitted on a phenomenological level. SAM models, which do explicitly include a
halo gas reservoir, often assume that the thermal properties of the CGM are in
equilibrium with those of their dark matter halos (White & Frenk, 1991). There
have been a few efforts to improve the cooling physics of halo gas in SAMs (Lu
et al., 2011; Benson & Bower, 2011; Cousin et al., 2015; Hou et al., 2018) and
generate SAM predictions for the CGM that can be compared to observables
(Faerman et al., 2022), however such SAM models of galaxies and their CGMs
lack an energetically self-consistent treatment of preventative feedback from
galactic winds (Pandya et al., 2020).

While many basic properties of the CGM of the Milky Way and external galaxies
are still not well understood, observations have converged onto a general
picture of the CGM as an extended, multiphase reservoir composed of a hot
phase ($T\sim 10^{6}$ K) directly measurable from X-ray emission, a warm-hot
phase ($T\sim 10^{5-6}$ K) visible in UV absorption lines, a cold neutral and
low-ion phase ($T\sim 10^{4}$ K), and a significant contribution of metals and
dust (for reviews see Putman et al., 2012; Tumlinson et al., 2017). The CGM
contains a substantial budget of baryons, and could play a major role in
regulating galaxy evolution as it supplies fuel to the galaxy for star
formation, but the galaxy in return shapes the CGM, as outflows driven by
feedback processes alter the mass and energy content of the CGM. Thus the
properties of the CGM and galactic winds are deeply intertwined.

The aim of this work is to build on this understanding of the CGM as a
regulator of galaxy evolution, and explore how different compositions of SNe-
driven winds shape properties of galaxies and their CGMs. In particular, we
will track not only the flows of mass and metallicity of the CGM, but also the
flows of energy, allowing for a self-consistent (albeit simplified) regulation
of preventative feedback. We will show that the CGM is the natural seat of
self-regulation in such models, with the key parameter shifting from the mass
outflow rate to the energy outflow rate. Using our new model, we investigate
whether we can match key observational scaling relations with reasonable
assumptions about the energetics of galactic winds.

In the next section, we introduce the regulator model, along with the relevant
equations governing mass evolution, the cooling efficiency of the CGM,
followed by our treatment of preventative feedback from energy-loaded winds
and metallicity evolution. We then show the time evolution of a
$10^{12}M_{\odot}$ halo and test our model over a range of halo masses,
comparing its output to the stellar-to-halo mass relation and the ISM-to-
stellar mass ratio, and its implications for the circumgalactic gas content.
We end our work with a discussion of its implications for self-regulation in
galaxies and feedback, the connection to CGM-precipitation models, and a
comparison to other regulator models. We then summarize our the conclusions.

Figure 1: Schematic of 1D-regulator model. The baryonic components of the
galaxy (the ISM, stellar mass, and the CGM) are distinct reservoirs, which
exchange mass, energy, and metals. The black arrows represent the mass (and
metal) flows, while the red/blue arrows symbolize the positive/negative flows
of energy among the various components.

##  2 Gas-Regulator Model

In this section, we introduce our 1D-regulator model, where the different
components of the galaxy are each treated as reservoirs that exchange baryons
at different rates. We consider a galaxy composed of six reservoirs (see
Figure 1): a dark matter halo ($M_{\rm halo}$), the stellar mass
($M_{\star}$), the interstellar medium ($M_{\rm ISM}$), the circumgalactic
medium ($M_{\rm CGM}$), the energy of the CGM ($E_{\rm CGM}$), and the mass in
metals of the CGM ($M_{\rm Z,CGM}$). Note that, because we are focused on the
CGM, we do not track the metallicity of the ISM or the stars, although this
would be straightforward to do (but would introduce additional free parameters
that do not directly impact the evolution of the CGM).

We write down a simple set of ordinary differential equations (ODEs) for the
evolution of these quantities based on some simple physical reasoning
(including some free parameters), and solve for their evolution using a Runge
Kutta solver (using solve_ivp from the scipy python package). After
introducing the basic equations that define the evolution of each of the mass
reservoirs, we describe our treatment for the cooling rate for CGM gas and
accretion, the lifting of CGM gas out of halo through heating from energetic
winds, and the metallicity enrichment of the CGM from SNe-driven outflows.

###  2.1 Structure of the CGM

We begin by describing our simple model for the structure of the CGM. Our goal
here is to write down a model which retains the basic properties but is
analytic or nearly so. We make two simplifying assumptions – first, we take
the halo to be isothermal, at temperature $T_{\rm CGM}$, which is close to but
not necessarily equal to the virial temperature $T_{\rm vir}$. The temperature
we assume for the CGM is equal to $T_{\rm CGM}=(\mu/k)e_{\rm CGM}$, where k is
the Boltzmann constant and $\mu$ is the mean molecular weight, which we assume
$\mu=0.6$ times the mass of a proton. In fact, rather than the temperature, we
generally use the halo specific energy $e_{\rm CGM}=E_{\rm CGM}/M_{\rm CGM}$,
where $E_{\rm CGM}$ and $M_{\rm CGM}$ are the thermal energy and mass of CGM,
respectively.

In addition, we assume the density profile of the CGM follows a power-law,

$\displaystyle\rho(r)=\rho_{0}\Big{(}\frac{r}{r_{0}}\Big{)}^{-\alpha},$ (1)

where $\alpha=1.4$ so that the entropy profile goes as $K=T/\rho^{2/3}\sim
r^{2/3}$ (Nelson et al., 2016; Fielding et al., 2020; Esmerian et al., 2021).
As we will show, the basic behavior of the model is insensitive to reasonable
variations in $\alpha$. We note that we do not explicitly enforce hydrostatic
equilibrium, nor is it expected due to the presence of heating and cooling
flows (Stern et al., 2019, 2020); however, for $T_{\rm CGM}\approx T_{\rm
vir}$, this profile is close to hydrostatic equilibrium for typical dark
matter halo profiles.

These simple forms allow us to write down expressions for the mass of the CGM,
which we take to range from $r_{0}=0.1r_{\rm vir}$ to $r_{\rm vir}$:

$\displaystyle M_{\rm CGM}$ $\displaystyle=\int_{r_{0}}^{r_{\rm
vir}}\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\alpha}4\pi r^{2}dr$ (2)
$\displaystyle=4\pi\rho_{0}r_{0}^{3}\left(\frac{(r_{\rm
vir}/r_{0})^{3-\alpha}-1}{3-\alpha}\right)$

and similarly for the net radiative cooling rate of the CGM gas:

$\displaystyle\dot{E}_{\rm CGM,cool}$ $\displaystyle=\int_{r_{0}}^{r_{\rm
vir}}\left(\frac{\rho}{\mu}\right)^{2}\Lambda(T_{\rm CGM},Z,z)4\pi r^{2}dr$
(3) $\displaystyle=4\pi\rho_{0}^{2}r_{0}^{3}\left(\frac{\Lambda(T_{\rm
CGM},Z,z)}{\mu^{2}}\right)\left(\frac{(r_{\rm
vir}/r_{0})^{3-2\alpha}-1}{3-2\alpha}\right)$

$\Lambda(T_{\rm CGM},Z,z)$ is the radiative cooling rate, which we take from
Wiersma et al. (2009) and depends on metallicity and redshift (due to a time-
varying metagalactic radiation field), while the mean molecular weight, $\mu$,
equals 0.6 times the mass of a proton. We note that the analytic forms fail
when $\alpha=3$ or $3/2$ (but the solution is well-behaved, with the terms in
parenthesis becoming simply $r_{\rm vir}/r_{0}$).

###  2.2 Mass Evolution

This section describes the evolutionary equations for the various mass
reservoirs (halo, CGM, ISM, stellar) before turning to the CGM energy
reservoir in the next section.

####  2.2.1 Halo Mass

We start with the overall evolution of the dark matter halo $M_{\rm halo}$,
which we assume to grow smoothly and monotonically with time, as given by a
fit to numerical simulations from Dekel et al. (2009):

$\dot{M}_{\rm halo}=0.47M_{\rm halo}\Big{(}\frac{M_{\rm
halo}}{10^{12}M_{\odot}}\Big{)}^{0.15}\Big{(}\frac{1+z}{3}\Big{)}^{2.25}\text{Gyr}^{-1}$
(4)

Since halos of the same mass can have different formation histories, it is
important to note that eq.4 defines the average halo growth rate for halos of
a given halo mass.

####  2.2.2 CGM Mass

We next turn to the evolution of the CGM mass component. Baryons enter the
halo along with dark matter ($\dot{M}_{\rm CGM,in}$), flowing first into the
CGM before cooling and accreting into the ISM ($\dot{M}_{\rm CGM,cool}$). In
addition, star formation can directly eject ISM gas into the CGM
($\dot{M}_{\rm ISM,wind}$) as well as providing energy which can heat the CGM,
causing it to expand and “lift” gas out of the hot CGM and unbind it from the
halo ($\dot{M}_{\rm CGM,out}$). We write this schematically as

$\dot{M}_{\rm CGM}=\dot{M}_{\rm CGM,in}-\dot{M}_{\rm CGM,cool}+\dot{M}_{\rm
ISM,wind}-\dot{M}_{\rm CGM,out},$ (5)

Each of these terms has relatively simple expressions and we discuss each in
turn. To provide the first term, we assume that mass infalls from the IGM
along with the dark matter:

$\dot{M}_{\rm CGM,in}=f_{b}f_{\rm prevent}\dot{M}_{\rm halo}$ (6)

where $f_{b}$ is the cosmic baryon fraction and $f_{\rm prevent}$ is a halo-
level preventive infall factor that we describe below, after discussing the
CGM energy budget. Note that this differs somewhat from preventive feedback in
traditional bathtub models (e.g., Davé et al., 2011) in that $f_{\rm prevent}$
is related to the accretion of gas from the IGM into the CGM, not directly to
the ISM. In particular, our CGM energy model (described next) aims to
explicitly calculate the traditional CGM preventive factor at the halo scale,
in addition to self-consistently accounting for the prevention of accretion to
the ISM by extending the cooling timescale from the CGM.

The second term accounts for radiative energy losses of the CGM that allow
some of the gas to cool to low temperatures ($10^{4}$ K or lower) and accrete
onto the ISM. For this, we write simply,

$\dot{M}_{\rm CGM,cool}=M_{\rm CGM}/t_{\rm cool,eff},$ (7)

where we define the effective cooling time $t_{\rm cool,eff}$ as the sum of
the cooling time and the free fall at the virial radius, the second term
accounting for the time the gas takes to accrete from the CGM into the ISM
(this second term only becomes important at high redshift when the cooling
times are short and the CGM fails to self-regulate):

$t_{\rm cool,eff}=\frac{E_{\rm CGM}}{\dot{E}_{\rm CGM,cool}}+t_{\rm ff}.$ (8)

The third term of Eq. 5 accounts for mass ejected via a galactic wind due to
star formation, which we parameterize with the usual mass loading factor
$\eta_{M}$:

$\dot{M}_{\rm ISM,wind}=\eta_{M}\dot{M}_{\rm SFR}.$ (9)

Mass ejection (generally to a long-lived reservoir) is the primary mode of
self-regulation in classical regulator models, but in our model, mass ejected
into the CGM can cool and accrete back to the ISM, as we will show. Instead,
mass is lifted out of the CGM due to heating (from star formation and other
energy sources). Heating of the CGM can cause overpressurization (i.e. $T_{\rm
CGM}>T_{\rm vir}$) which lifts gas out of the halo into the IGM, where we
assume it is unbound from the halo and doesn’t return.111Gas that is uplifted
out of the halo may cool and return to the halo on some timescale. The
inclusion of this returning gas would require greater suppression of infall
(smaller $f_{\rm prevent}$) than we estimate in this work. We will shortly lay
out the energy equation for the CGM – one term in that equation (eq. 18) is
$\dot{E}_{\rm CGM,out}$, the rate at which energy flows out of the halo due to
this overpressurization. We assume that this energetic outflow brings with it
an equivalent amount of mass such that the outflowing gas has the same
specific energy as the halo gas:

$\dot{M}_{\rm CGM,out}=\dot{E}_{\rm CGM,out}/(E_{\rm CGM}/M_{\rm CGM})$ (10)

####  2.2.3 ISM Mass

The ISM is fed from the CGM and loses mass to star formation and SN-driven
winds. We express the evolution of the gas mass of the galaxy in the form,

$\displaystyle\dot{M}_{\rm ISM}=-\dot{M}_{\rm SFR}(1+\eta_{M}-f_{\rm
rec})+\dot{M}_{\rm CGM,cool},$ (11)

where the first term describes the mass lost from the ISM due to star
formation and mass ejection, and the mass returned to the ISM as stars recycle
a fraction of their mass back to the ISM. We fix this instantaneous stellar
recycling fraction to be $f_{\rm rec}=0.4$ (Kroupa, 2001). The mass-loading
factor $\eta_{M}$ quantifies how efficient star formation is at removing gas
from the ISM. The second (positive) term represents gas accretion into the ISM
from the CGM. The expression for $\dot{M}_{\rm CGM,cool}$ is given in Eq. 7.

####  2.2.4 Stellar Mass

Considering next the stellar mass reservoir, the star formation rate,
$\dot{M}_{\rm SFR}$, is dictated by the gas mass of the ISM, $M_{\rm ISM}$,
and the timescale over which that gas mass is converted into stars:

$\displaystyle\dot{M}_{\rm SFR}=\frac{M_{\rm ISM}}{t_{\rm dep}},$ (12)

where $t_{\rm dep}$ is the depletion time. For simplicity, we do not
distinguish between atomic and molecular gas and assume that all ISM gas
participates in star formation. We use an estimate of the depletion time as a
function of stellar mass derived from observations of the cold gas mass and
star formation rates of low surface brightness and star-forming galaxies from
McGaugh et al. (2017). We assume a redshift dependence of $(1+z)^{-3/2}$,
similar to the Hubble timescale adopted in many previous regulator models,
resulting in the depletion time we use for our model:222Since the depletion
times in low mass systems is not well constrained observationally, we assume
that for systems with $M_{\rm star}<5\times 10^{7}M_{\rm\odot}$, below which
the data become sparse, that the depletion time is constant and only involves
with redshift and not with stellar mass.

$t_{\rm
dep}=10^{4.92}\Bigg{(}\frac{M_{\star}}{M_{\odot}}\Bigg{)}^{-0.37}(1+z)^{-3/2}.$
(13)

We can express the full evolution of the stellar mass of the galaxy:

$\displaystyle\dot{M}_{\star}=(1-f_{\rm rec})\frac{M_{\rm ISM}}{t_{\rm dep}},$
(14)

when we consider the star formation rate and the fractional loss of stellar
mass from mass recycled back to the ISM.

###  2.3 CGM Energy Evolution

We turn next to the new feature of our model, an equation for the thermal
energy of the CGM (we neglect here the kinetic energy component, leaving that
for future work in Pandya et al., in prep). As before, we account for all of
the energy gain and loss terms for the evolution of $E_{\rm CGM}$:

$\dot{E}_{\rm CGM}=\dot{E}_{\rm CGM,in}-\dot{E}_{\rm CGM,cool}+\dot{E}_{\rm
ISM,wind}-\dot{E}_{\rm CGM,out},$ (15)

The first term represents the energy of infalling gas (Eq. 6), which we assume
is associated with the specific energy of the halo:

$\dot{E}_{\rm CGM,in}=\left(\frac{kT_{\rm vir}}{\mu}\right)\dot{M}_{\rm
CGM,in}$ (16)

The second term of the energy evolution equation, representing radiative
losses is given in Eq. (3). The third term is positive and comes from the
energy associated with SN-powered galactic winds

$\displaystyle\dot{E}_{\rm CGM,wind}=\eta_{E}\dot{M}_{\rm
SFR}\left(\frac{10^{51}\text{erg}}{100M_{\odot}}\right)$ (17)

where we have parameterized the amount of SN-powered energy that escapes out
of the ISM as $\eta_{E}$. As we will show, this is a key parameter that drives
our CGM-regulated model.

Finally, the fourth term represents the energy that goes into lifting gas out
of the CGM, unbinding it. We assume this only occurs if the CGM is
overpressurized such that its specific energy exceeds the virial value. When
this is true, we compute the outflow rate assuming that gas flows out at the
hot gas sound speed $c_{s}$:

$\dot{E}_{\rm CGM,out}={\rm max}\left(E_{\rm CGM}-kT_{\rm vir}M_{\rm
CGM}/\mu,0\right)\left(\frac{c_{s}}{R_{\rm vir}}\right)$ (18)

This outflow of hot gas also drives a mass flux (see eq. 10). The hot gas
sound speed is the usual $c_{s}=(5kT_{\rm CGM}/3\mu)^{1/2}$.

The inclusion of this mass and energy “lifting” term to the CGM describes the
contribution from preventative feedback. By lifting a fraction of the CGM
beyond the virial radius and (we assume) out of the gravitational clutches of
the dark halo, feedback works to reduce the density of the CGM. This lengthens
the radiative cooling timescale, and thus regulates the rate of accretion out
of the circumgalactic medium onto the galaxy, and any subsequent star
formation. The details of this “lifting” process are clearly complicated and
the simple relation in Eq.(18) is an approximation; however, small
uncertainties in the proportionality can be accommodated by changing the
effective energy-loading factor.

One of the last elements of the model is to define the halo-level preventive
inflow factor $f_{\rm prevent}$ used in Eq. 6. The idea here is that energy
flowing out beyond the virial radius will heat IGM gas and prevent it from
falling in. This effect has previously been found to be operating in the FIRE
simulations (Pandya et al., 2020). We take a deliberately simple form for
this, defining it to be proportional to the ratio of the inflowing to
outflowing energy. Although there is no strong physical justification for this
particular form, it seems reasonable that this ratio of these energies should
be a good measure of the importance of energetic outflows. In particular, we
adopt:333Note that here $\dot{E}_{\rm CGM,in}$ is defined as the energy of
inflowing gas from the IGM at the full cosmic baryon fraction.

$f_{\rm prevent}={\rm min}\left(\alpha_{\rm prevent}\frac{\dot{E}_{\rm
CGM,in}}{\dot{E}_{\rm CGM,out}},1\right)$ (19)

where we have defined a free parameter which we take to be $\alpha_{\rm
prevent}=2$, resulting in $f_{\rm prevent}$ values consistent with Pandya et
al. (2020).

###  2.4 Metallicity of the CGM

The significant share of metals observed in the CGM of galaxies is
incontrovertible evidence for the presence of metals in galactic outflows
(Prochaska et al., 2017). Metals in galactic winds arise, in part by
entrainment of previously enriched ISM gas, and in part to the fact that SNe
enrich the gas as it simultaneously injects energy and momentum. We focus on
the second source and parameterize the fraction of metals produced by SNe that
wind up in outflows using the metal loading factor, defined as
$\eta_{Z}=\frac{\dot{M}_{Z,out,SN}}{\dot{M}_{Z,SN}}$. This definition of the
metallicity-loading factor means that if $\eta_{Z}=1$, then all of the metals
produced in SNe leave the galaxy in outflows. We remark that we are assuming
that this factor is independent of the metallicity of the ISM, which is
clearly an oversimplification; however in simulations which involve a high-
energy loading factor, much of the metals come directly from the SN ejecta,
rather than via entrainment of the ISM, something that will have interesting
implications for the detailed CGM abundances, but is beyond the scope of the
current work.

If we include the metal mass of the CGM ($M_{\rm Z,CGM}$) as its own
reservoir, its evolution takes the form:

$\displaystyle\begin{gathered}\dot{M}_{\rm Z,CGM}=\eta_{Z}y_{\rm
SN}\dot{M}_{\rm SFR}+Z_{\rm IGM}\dot{M}_{\rm CGM,in}\\\ -f_{\rm
Z,CGM}(\dot{M}_{\rm CGM,cool}+\dot{M}_{\rm CGM,out}).\end{gathered}$ (22)

The first term captures the enrichment of metals from SNe-outflows, where the
metal mass enrichment yield produced from one supernova per 100 $M_{\odot}$ is
approximately equivalent to $y_{\rm SN}\sim 0.02$. The second term is the
accretion of metals from the IGM. We assume a constant metallicity for the IGM
inflow of $Z_{\rm IGM}=0.01Z_{\odot}$, where the IGM metallicity is (somewhat
arbitrarily) assumed to be 1$\%$ of the solar metallicity $Z_{\odot}=0.0134$
(Asplund et al., 2009). This is different from the usual assumption that the
IGM metallicity is ”pristine” and has only been enriched by Pop III SNe. Here
we imagine that some of the cosmic accretion has been further pre-enriched
either by the galaxy’s own outflows or from galactic neighbors.

Lastly, the third term captures the metals lost to accretion into the ISM and
those lifted out of the halo, with $f_{\rm Z,CGM}=M_{\rm Z,CGM}/M_{\rm CGM}$
representing the fraction of the CGM mass in metals.

##  3 Results

Now that we have described our basic model, we explore it in detail for a
single halo, before turning to the scaling relations that it predicts as the
halo mass is varied.

Figure 2: Time evolution of the mass of each reservoir in the regulator model
(upper), the total energy of the CGM (solid) compared to its virial binding
energy (dashed) (middle), and the metallicity of gas in the CGM with respect
to the solar metallicity (lower) for a $10^{12}M_{\odot}$ mass halo from
$z\simeq 6$ to $z\simeq 0$. The mass, energy, and metallicity loadings are
equal to $\eta_{M}=0.1$, $\eta_{E}=0.1$, and $\eta_{Z}=0.2$ respectively.

###  3.1 Evolution for a $10^{12}$ M⊙ Halo

To get a sense of whether the regulator model can reproduce galaxy properties
by the present day, we run our model on a Milky Way-like galaxy. The top panel
of Figure 2 shows the time-evolution of the mass in each of our four
reservoirs, for a $10^{12}M_{\odot}$ halo from a redshift of $z\simeq 6$ to
the present-day. The middle panel charts the evolution of the total CGM energy
compared to its virial binding energy ($kT_{\rm vir}M_{\rm CGM}$). The bottom
panel displays the CGM metallicity as a function of time with respect to solar
metallicity. This run uses parameter values of $\eta_{M}=0.1$, $\eta_{E}=0.1$,
and $\eta_{Z}=0.2$ respectively, in line with typical values for the loading
factors in MW-like galaxies seen in FIRE-2 (Pandya et al., 2021).

There is rapid growth in the CGM and ISM masses early on, as pristine gas from
the IGM accretes onto the CGM before rapidly cooling and accreting onto the
central galaxy. This prompts a steep initial rise in the gas mass of the
galaxy, and this newly available fuel triggers star formation on a timescale
of order $t_{\rm dep}$. The birth of new stars increases the stellar mass, but
their subsequent death through supernovae begins to drive energy and mass
outflows back into the CGM. Due to the relatively small mass-loading factor
adopted here, the major contribution to the CGM is due to the energy-loading,
which heats the CGM and lifts mass out of the halo, into the IGM. This loss of
mass from the CGM not only serves as a counterbalance to the mass gains from
early IGM accretion by limiting the mass of the CGM across a wide range of
redshifts, but also regulates the density of the CGM and therefore its overall
cooling efficiency. This regulation of cooling limits the flow of gas into the
ISM, resulting in generally slow growth of the ISM and stellar masses.

The model produces stellar and ISM masses on the order of $\sim
10^{10}M_{\odot}$, estimates that are slightly lower but broadly consistent
with the total stellar mass of the Milky Way of $\sim 5\times
10^{10}M_{\rm\odot}$ (Bland-Hawthorn & Gerhard, 2016). The mass estimate of
the CGM is also on order of $\sim 10^{10}M_{\odot}$, lying within the
observational constraints of the CGM mass of the Milky Way for reasonable
choices of $\eta_{M}$ and $\eta_{E}$ (e.g., Salem et al., 2015; Tumlinson et
al., 2017). However, we note that the total CGM mass is poorly constrained
observationally.

The total energy of the CGM grows rapidly at early times from the infall of
gas from the IGM. Radiative losses cause the gas to cool and accrete quickly
into the ISM, where the early burst of star formation drives energy outflows
back into the CGM. The energy-loading of these outflows heats the CGM, raising
its specific energy and limiting its cooling efficiency. Comparing the time
evolution of the total energy of the CGM against its virial energy, we find
that the total energy of the CGM exceeds its virial value, resulting in a CGM
that is overpressurized. This overpressurization expands the CGM, lowering its
density and lifting mass and energy beyond the virial radius. This brings the
energy of the CGM closer to its virial value over time. This process of
heating the CGM and lifting gas out of the halo through overpressurization
regulates the energy content of the CGM against radiative loses and energy
gains from cosmic accretion and star formation, thus limiting the growth of
the CGM energy to the present-day.

The growth of $M_{\star}$ produces metals, and after a steep initial climb
from the first SNe-driven outflows of metals, the metallicity of the CGM
slowly decreases with time. The on-going star formation continues to increase
the supply of metals being delivered to the CGM. Metal-rich outflows work to
enhance future cooling from the CGM, while efficient energy-loading work to
reduce it. The metallicity of the CGM with respect to the solar metallicity
approaches $Z/Z_{\odot}\simeq 0.07$ by $z=0$, lower than existing estimates of
the metallicity of the Milky Way’s CGM and similar hosts (Prochaska et al.,
2017). This suggests that either larger metallicity outflows are needed to
match observations or a consequence of the fact that we don’t consider the
metallicity evolution of the ISM and its outflows.

Our regulator model can produce galaxies by the present-day with properties
that are in qualitative agreement with observations of the Milky Way-like
galaxies. This was achieved through mass outflows out of the ISM and energy
regulation of the CGM acting as a mode of preventative feedback by
constraining the total baryonic content of the galaxy.

###  3.2 Galaxy Scaling Relations

Next we compare the $z=0$ galaxy properties predicted from our model to galaxy
scaling relations in the range $M_{\rm halo}=10^{10}-10^{12}M_{\odot}$. SNe
are believed to drive the primary mode of feedback and star formation self-
regulation for galaxies in halos below $\sim 10^{12}M_{\odot}$, so the
regulator model should be compared in this range. Since the effects of AGN-
driven feedback are not considered in this work, disparities at the high mass
end of the considered range may be present between the data and the model.

Figure 3: Log of $M_{\star}/M_{\rm h}$ from the regulator model as a function
of halo mass, compared to the median observed stellar mass with uncertainties
in gray and their derived halo mass from abundance matching from Behroozi et
al. (2019). Estimates of $M_{\star}/M_{\rm h}$ are presented as a function of
increasing the power-law dependence of wind mass-loading $\eta_{M}$ (left),
energy-loading $\eta_{E}$ (middle), and metallicity-loading $\eta_{Z}$
(right). The annotations of each plot display which parameters were kept
constant in the iterations of the model on display.

####  3.2.1 $M_{\star}/M_{\rm halo}$-$M_{\rm halo}$ Relation

We display the predicted $M_{\star}/M_{\rm halo}-M_{\rm halo}$ relation over
this range for a set of different outflow parameters in Figure 3, plotted
alongside the estimated observed stellar masses with uncertainties and their
derived halo mass from abundance matching from Behroozi et al. (2019). The
relation rises as a power-law until peaking at a mass of $10^{12}M_{\odot}$,
where the stellar mass fraction is largest, and then steadily falls for more
massive halos as the quenching effects of AGN take hold. We choose an initial
set of outflow parameters, setting $\eta_{M}=1$, $\eta_{E}=0.1$, and
$\eta_{Z}=0.5$, which are typical In our exploration, we modulate the strength
of one outflow parameter at a time, while keeping the values of the other two
set to their fiducial values.

As the left panel demonstrates, the predicted stellar mass for low-mass halos
is remarkably robust against changes in $\eta_{M}$, as the mass-loading factor
is increased by a factor of 200; the only significant impact is to reduce the
stellar mass of the highest mass halos when using very highly mass-loaded
winds. This suggests that the system self-regulates to similar galactic
properties despite large changes in $\eta_{M}$. Self-regulation in this
instance means that enhancements in the mass-loading of outflows increase the
mass and density of the CGM, which increases the CGM’s ability to cool and
accrete gas back to the galaxy. When $\eta_{E}$ is held constant, highly mass-
loaded outflows also lower the specific energy of the CGM. This lowers the
overpressurization of the CGM and thus the flux of mass and energy leaving the
halo, resulting in a larger $f_{\rm prevent}$ factor at the virial radius.
Both of these processes$-$weakening the suppression of mass infall from the
IGM and enhancing radiative losses of CGM gas$-$restrict the ability of mass-
loaded outflows to meaningfully reduce the present-day stellar mass of
galaxies on their own. The largest $\eta_{M}$ values explored in Figure 3 do
appear to be more effective at reducing the overall stellar mass in more
massive halos, which may seem counter-intuitive. The $M_{\star}/M_{\rm halo}$
ratio flattens at halo masses beyond a threshold mass, which becomes lower as
$\eta_{M}$ increases. This occurs when $\eta_{M}$ is large enough at a given
halo mass (and given $\eta_{E}$) such that the outflow specific energy (and so
$e_{\rm CGM}$) drops below $kT_{\rm vir}/\mu$, which shuts off all energy and
mass outflows at the virial radius (Eq. 18) and with that, any preventative
feedback at the halo scale ($f_{\rm prevent}=1$). The impact of larger values
of $\eta_{M}$ on the stellar mass beyond that threshold are still countered by
enhancing CGM density and cooling, but are no longer affected by any cosmic
outflows.

The center panel of Figure 3 displays the influence of changing $\eta_{E}$.
Increasing the energy-loading factor of the outflows has the most significant
impact of all outflow parameters, reducing $M_{\star}$ for larger $\eta_{E}$.
Increasing $\eta_{E}$ has the effect of enhancing the mass and energy lifted
out of the CGM, as well as heating a larger share of IGM gas beyond $R_{\rm
vir}$ and preventing its accretion. This reduces the CGM mass and density, and
thus its ability to cool. Lowering the accretion rate reduces the overall star
formation rate, resulting in much lower estimates of the present-day stellar
mass for all halos. Since the mass lifted out of the halo is assumed to never
return to the galaxy in our model, increasing $\eta_{E}$ has the effect of
permanently removing baryons from the galactic system. It is also worthy of
note that the model is able to produce stellar masses in the lowest mass
galaxies approaching those of observations with $\eta_{E}\sim 1$. This is
despite having mass-loadings below unity, which is at least an order of
magnitude smaller than what is usually required to match galaxy scaling
relations at the low-mass end. Interestingly, plotting the $M_{\star}/M_{\rm
halo}-M_{\rm halo}$ relation from the regulator model for different constant
values of $\eta_{E}$ against the observed relation reveals that in order to
reproduce the halo mass dependence of the observed relation, $\eta_{E}$ must
depend on halo mass, favoring $\eta_{E}$ close to unity for the lowest mass
halos and $\sim 0.1$ for Milky Way-like halos.

As for changing $\eta_{Z}$ (rightmost plot), we find only a very small effect
on the stellar mass of the galaxy in the halo mass range explored. Increasing
the metallicity of the winds increases the overall cooling efficiency of the
CGM. However, because of the tight energy regulation, this enhanced cooling
does not result in a significantly larger stellar mass (the mean density of
the CGM is decreased by feedback to counterbalance the enhanced metallicity,
resulting in nearly the same total cooling and star formation). The
cancellation is such that this parameter appears to only make a meaningful
difference in the accretion rate in halos with high star formation (and more
massive halos).

The outflow parameter with the most significant influence on the present-day
stellar mass is $\eta_{E}$, likely pointing to its prominent role in
regulating the density of the CGM and limiting future gas cooling through
preventative feedback. This form of preventative feedback becomes increasingly
important in lower mass halos. We will parameterize and discuss this
dependence between $\eta_{E}$ and $M_{\rm halo}$ in Section 3.3.

Figure 4: Log of $M_{\rm ISM}/M_{\star}$ from the regulator model as a
function of stellar mass, compared to the best-fit single power-law fit for
the total cold gas mass as a function of $M_{\star}$ over the range
$7.3\lesssim\log{(M_{\star}/M_{\rm h})}\lesssim 11.2$ for late-type galaxies
from Calette et al. (2018). Estimates of $M_{\rm ISM}/M_{\star}$ are presented
as a function of increasing the power-law dependence of wind mass-loading
$\eta_{M}$ (left), energy-loading $\eta_{E}$ (middle), and metallicity-loading
$\eta_{Z}$ (right). The annotations of each plot display which parameters were
kept constant in the iterations of the model on display.

####  3.2.2 $M_{\rm ISM}/M_{\star}$-$M_{\star}$ Relation

Now we turn to the relation between the ISM gas and the stellar mass, which
encodes star formation efficiency and can be constraining on models of galaxy
evolution. Figure 4 compares the best-fit single power-law fit for the total
cold gas mass as a function of $M_{\star}$ over the range
$7.3\lesssim\log{(M_{\star}/M_{\rm halo})}\lesssim 11.2$ for late-type
galaxies from Calette et al. (2018) to the gas-to-stellar mass relation at
$z=0$ for the regulator model.

Figure 4 shows the $M_{\rm ISM}/M_{\star}$-$M_{\star}$ relation as a function
of outflow parameters. Changes in the energy (middle panel) and metallicity
(right panel) composition of the outflows makes negligible difference to the
proportion of gas mass to stellar mass over the parameter and halo range
studied. The mass-loading factor (left panel) makes the most discernible
difference in the $M_{\rm ISM}/M_{\star}$-$M_{\star}$ relation, where more
efficient mass loadings somewhat increase the ratio of gas mass to stellar
mass for all galaxies and leads to a flatter dependence with stellar mass. The
larger mass loading lowers the specific energy of the CGM and inhibits the
suppression of cosmic inflow (larger $f_{\rm prevent}$ means more gas falls
in), especially in more massive halos. This results in a more massive CGM and
inevitably a more massive ISM reservoir due to the prompt radiative loses of
the halo gas.

Our current treatment of the star formation produces a shallower gradient with
increasing stellar mass than the observed power-law fit in Calette et al.
(2018). This means that we predict an excess of gas with increasing halo mass.
Although this could stem from a sampling bias in the data where we derive our
depletion time, or from uncertainties in HI or molecular gas content, the size
of the mismatch seems significant. However, as we discuss in the appendix, the
slope and amplitude of this relation in our model depends greatly on our
choice of the depletion time. This dependence makes sense because the
depletion timescale governs the rate at which gas in the ISM is converted to
stars. Additional feedback from AGN may also help to ameliorate these
conflicts at the higher mass end, providing another means of reducing the ISM
mass and quenching star formation. As demonstrated in the appendix, we are
able to vary the depletion time to match this relation and are still able to
produce galaxies with stellar masses in qualitative agreement with
observational constraints.

Figure 5: Estimates of $\log(M_{\rm CGM}/f_{b}M_{\rm h})$ and the CGM
metallicity with respect to the solar metallicity from the regulator model as
a function of halo mass. Estimates of $M_{\rm CGM}/M_{\rm halo}$ and
$Z/Z_{\odot}$ are presented as a function of increasing the power-law
dependence of wind mass-loading $\eta_{M}$ (left column), energy-loading
$\eta_{E}$ (middle column), and metallicity-loading $\eta_{Z}$ (right column).
The annotations of each plot display which parameters were kept constant in
the iterations of the model on display on a column.

####  3.2.3 Implications for the CGM gas content

The properties of the CGM and the properties of galactic outflows are
intertwined, such that changes to the composition of galactic winds ought to
greatly influence the properties of the CGM. Figure 5 plots the mass (upper
panel) and metallicity of the CGM (lower panel) as a function of halo mass for
different wind mass-loadings. Broadly we find a that the CGM mass is mostly
dependent on the mass-loading while less sensitive to the adopted energy-
loading factor (in contrast to what was seen for the stellar-mass halo-mass
relation), and that the metallicity drops as $\eta_{M}$ or $\eta_{E}$
increases, while it (unsurprisingly) increases with increasing $\eta_{Z}$. In
the following paragraphs, we examine these trends more carefully.

First, we see with the leftmost panels of Figure 5, that increasing $\eta_{M}$
has the fairly predictable effect of increasing the overall CGM mass (although
not linearly with $\eta_{M}$). This is true for all halo masses in our range.
Increasing the mass loading has the effect of reducing the overall metallicity
of the CGM in our model. Since $\eta_{Z}$ is kept constant and the star
formation rate is largely insensitive to wind mass-loading, large mass-loading
dilutes the average metallicity content of the wind and thus its contribution
to the CGM metallicity. However, since we don’t consider the metallicity
evolution of the ISM and in turn, the metallicity of the mass-loaded outflow,
the full consequences for the CGM metallicity in real galaxies for increasing
$\eta_{M}$ alone are not completely captured by our model.

Exploring the variation of the energy-loading in the middle column, we see
that more efficient energy-loading has the opposite effect, reducing the
overall mass of the CGM as energetic flows from SNe heat the CGM and lift gas
out of the halo. The reduction of star formation in the high $\eta_{E}$
scenario also stymies overall metal production in the galaxy, leaving less to
be ejected into the CGM in the first place. Similar to the case of raising
$\eta_{M}$, the lowest mass halos are quite metal-poor, producing
metallicities by the present-day that are not much higher than what we assumed
for the background metallicity of the IGM of $Z_{\rm IGM}/Z_{\odot}=0.01$.
This is consistent with the interpretation that dwarfs easily lose most of the
metals that their stars produced in outflows to the IGM (Muratov et al., 2017;
Pandya et al., 2021).

Increasing $\eta_{Z}$ has a weak impact on the CGM mass but a strong influence
on the metallicity: building up the metallicity of the CGM in Milky Way halos
to $\sim 35\%$ of the solar metallicity, in rough agreement with metallicity
estimates from COS-Halos survey (Prochaska et al., 2017), but higher
$\eta_{Z}$ still leaves the CGM metallicity around $10\%-15\%$ of the solar
metallicity in the lowest mass halos of our range. When considering the mass,
metal-enriched winds do have the effect of increasing the cooling efficiency
in the CGM, leading to more mass lost via cooling and accretion in the galaxy.
This increase in cooling efficiency is small in the low-mass halos, but
becomes slightly more important for halos above $\log(M_{\rm
halo}/M_{\odot})\gtrsim 10.5$.

When we assume $\eta_{E}\lesssim 0.1$ and/or $\eta_{M}>1$, our CGM mass
estimates for Milky-Way mass galaxies approach those of the models presented
in Faerman et al. (2020, 2022), where they find MW CGM mass estimates in the
range of $\sim 3-10\times 10^{10}M_{\odot}$. These models have been shown to
agree with a broad variety of observational constraints on the Milky Way CGM.
However, our mass estimates for dwarf CGMs with those fixed parameters
disagree with the values found in the FIRE-2 simulations (Pandya et al.,
2020). CGM masses at the low-mass end are sensitive to not only the mass
loading, which in FIRE is at least a factor of 2 higher than our maximum
assumed $\eta_{M}=20$ but also the shape of the cooling curve. The lowest CGM
masses are found in dwarfs where the CGM temperature approaches the peak of
the cooling curve.

Since we assume an isothermal CGM with temperature derived from the specific
energy, we cannot model the variations in gas phases captured by observations,
but our work does have important implications for the CGM’s multiphase
structure. Gas that cools and precipitates out of a hot halo onto the galaxy
will be transitioning to cooler, more dense gas phases, forming structures
that could resemble the high-velocity cool cloud complexes observed in the
Milky Way’s CGM (Maller & Bullock, 2004; Fraternali et al., 2015; Li &
Tonnesen, 2020). On the hotter end, energetic outflows could heat gas from
cool to warm phases (preventing its accretion) as well as heat warmer gas to
super-virial temperatures before it ultimately becomes unbound from the halo
(Das et al., 2021).

###  3.3 The Halo-Mass Dependence of Wind Energy-loading

Figure 6: Energy loading $\eta_{E}$ as a function of halo mass from recursive
bisection (points) plotted with its best-fit power-law fit (red) (top). The
middle shows the preventative inflow factor $f_{\rm prevent}$ as a function of
halo mass for the optimal choice of $\eta_{E}$. Log of $M_{\star}/M_{h}$ from
the regulator model computed with $\eta_{E}$ from recursive bisection as a
function of halo mass, compared to the median observed stellar mass with
uncertainties in gray and their derived halo mass from abundance matching from
Behroozi et al. (2019) (bottom). The annotations on the bottom plot display
which parameters were kept constant.

Comparisons of the regulator model with the $M_{\star}/M_{\rm halo}-M_{\rm
halo}$ relation in Figure 3 indicate that $\eta_{E}$ must depend strongly on
halo mass. To ascertain the form of this dependence, we perform a recursive
bisection using optimize.bisect from the scipy python package to find the
optimal value of $\eta_{E}$ that would yield the closest agreement between the
model and the $M_{\star}/M_{\rm halo}-M_{\rm halo}$ relation at z=0 from
Behroozi et al. (2019).

Previous work on stellar feedback in low-mass galaxies from simulations and
observations find that mass loadings likely depend on halo mass (see recent
review Collins & Read, 2022). For our recursive bisection on $\eta_{E}$, we
assume that the mass-loading factor of outflows has a simple power-law
dependence with halo mass with the form

$\displaystyle\eta_{M}=\Big{(}\frac{M_{\rm
halo}(z=0)}{10^{12}M_{\odot}}\Big{)}^{-\beta}$ (23)

where $\beta$ is a power-law index that we set to $\beta=0.5$. This value of
$\beta$ translates to larger mass-loadings for lower mass halos, where
$\eta_{M}=10$ and $\eta_{M}=1$ for halos of mass $\sim 10^{10}M_{\odot}$ and
$\sim 10^{12}M_{\odot}$ respectively. We assume this scaling for $\beta$ such
that our values for our mass-loading factor are comparable to those derived
from observations of galactic winds in nearby dwarf and Milky Way-like
galaxies (Chisholm et al., 2017; McQuinn et al., 2019). We also assume
$\eta_{Z}=0.5$, the same as in section 3. Metal-loading may also scale with
halo mass in real galaxies, but as we demonstrated in section 3.2.1, our
results for the stellar-to-halo mass relation are not strongly dependent on
our choices for $\eta_{M}$ and $\eta_{Z}$, and should not significantly alter
our fitted value for $\eta_{E}$.

We plot the optimal values of $\eta_{E}$ from our recursive bisection to match
the median stellar mass at a given halo mass at $z=0$ in the top panel of
Figure 6, along with a best fit power law relation of $\eta_{E}$ from our
best-fit model as a function of halo mass using optimize.curve_fit. This power
law relation takes the form:

$\displaystyle\eta_{E}=A\Big{(}\frac{M_{\rm
halo}(z=0)}{10^{12}M_{\odot}}\Big{)}^{-\lambda},$ (24)

and find good agreement with fitted parameter values of $A=0.051\pm 0.007$ and
$\lambda=0.65\pm 0.04$. We find that our model prefers $\eta_{E}\sim 1$ for
the lowest mass halos and $\sim 0.1$ for MW-mass halos.

Our best-fit model also yields an estimate for the total CGM preventative
inflow factor $f_{\rm prevent}$ as a function of halo mass. Displayed in the
middle panel of Figure 6, we find that mass and energy outflows from the halo
suppresses cosmic inflow to well below the cosmic baryon fraction, reaching
$f_{\rm prevent}\sim 0.3$ in the lowest mass halos. Preventative inflow
factors of $f_{\rm prevent}\sim 0.3$ are also found in FIRE for similarly low-
mass halos (Pandya et al., 2020), and rise to $f_{\rm prevent}\sim 1$ for
Milky Way mass halos, slightly higher than the results of our model where
$f_{\rm prevent}\sim 0.8$ in those halos. In the bottom panel, we show the
resulting model stellar mass-halo mass.

The relation we find, in which $\eta_{E}$ decreases in more massive halos
appears consistent with the results of FIRE-2, which also find larger energy-
loadings in low-mass dwarfs relative to MW-like halos at both high and low
redshifts (Pandya et al., 2021). This result may conflict with results from
“tall-box” simulations like TIGRESS which show a flat relationship between
$\eta_{E}$ and star formation density and lower values of $\eta_{E}$ overall
($\eta_{E}\sim 0.1$) Li & Bryan (2020). These ISM patch simulations were done
under Milky Way-like conditions with solar metallicities (Kim et al., 2020).
Simulations that can probe the interaction between SNe and the ISM under low-
metallicity conditions will be needed to better constrain $\eta_{E}$ in low-
mass galaxies.

##  4 Discussion

Figure 7: Time evolution of the logarithmic values of the flows in energy
(topmost row) measured in ergs Gyr-1 and mass flows (middle-top row) in
$M_{\odot}$ Gyr-1 for each sink and source term for the CGM reservoir in halos
of mass $M_{\rm halo}=10^{10}M_{\odot}$ (left), $10^{11}M_{\odot}$ (center),
and $10^{12}M_{\odot}$ (right) respectively. Positive source terms are colored
red while negative sink terms are in blue. Solid lines distinguish mass and
energy flows at the galaxy-CGM interface, while dashed lines represent flows
across the boundary of the CGM and IGM. The middle-bottom row tracks the time
evolution of the mass of each reservoir in the regulator model, and the bottom
row tracks the evolution of the preventative inflow parameter, $f_{\rm
prevent}$, for each halo mass to the present-day.

###  4.1 How the Model Self-Regulates

The insensitivity we see to varying the mass loading factor of galactic
outflows suggest that the model is self-regulated to a significant extent, by
which we mean that changes to the composition of outflows prompts a change in
the cooling efficiency of the CGM, which in turn shapes the properties of the
galaxy through regulating future gas accretion. Increasing $\eta_{M}$ serves
to enhance future accretion by increasing the density of circumgalactic gas,
shortening the cooling timescale and enabling the rapid precipitation of gas
back into the ISM in short order. The timescale of this re-accretion of gas in
real galaxies is not well-understood, but likely could persist for multiple
cycles, providing a significant source of gas accretion to galaxies at low
redshift (Oppenheimer et al., 2010; Ford et al., 2014; Anglés-Alcázar et al.,
2017). Our model assumes the CGM to be well-mixed, supporting the picture of
cool gas precipitating out of a hot halo (Fraternali et al., 2015), where
mass-loaded material from galactic winds can trigger condensation in excess of
the amount of gas delivered to the CGM in outflows.

This form of self-regulation is not the case for $\eta_{E}$; instead
increasing $\eta_{E}$ enhances the heating of the CGM for a given amount of
star formation, allowing more mass to be lifted out of the CGM, which prevents
subsequent accretion by increasing $t_{\rm cool}$ of the CGM. What makes the
model so sensitive to the energy-loading of the winds is its ability to remove
baryons from the halo entirely, while adjusting $\eta_{M}$ or $\eta_{Z}$ only
serves to shift baryons to a different component of the galaxy for a short
while before they return back to the galaxy. We note that, in real galaxies,
material unbound from the halo by energy-driven winds may eventually cool and
return back to the halo on some (potentially long) timescale. Building a
better understanding of both the timescales of recycled gas from outflows and
the return of gas that has been expelled beyond the viral radius are required
to better link observed galaxy scaling relations and the properties of SNe-
driven winds.

To understand how this self-regulation works in detail, we examine Fig. 7,
which shows the energy and mass flows in three halos spanning the mass range
of interest (with $10^{10}M_{\odot}$, $10^{11}M_{\odot}$, and
$10^{12}M_{\odot}$ in the left, center and right columns, respectively). In
each case, there is an initial period of adjustment between rapid cooling and
heating by early star formation while the system comes into a quasi-
equilibrium (on a timescale dictated by the depletion time) and then a
generally slow evolution of both the energy and mass flows over time, with the
evolution timescale set mostly by changes in the accretion rate (i.e. on a
Hubble time). For the two lower mass halos, equilibrium is largely dictated by
a balance between the energetic input from supernovae and the energy required
to lift mass out of the CGM. Indeed, from the close balance of the cosmic
inflow and CGM outflow terms in the $\dot{M}_{\rm CGM}$ curves (second row of
panels), we see that the star formation rate is being set by the requirement
that the CGM mass be held roughly constant (so that its cooling rate provides
just enough mass to power the star formation). This simple balance of two
terms is more complicated in the highest mass ($10^{12}M_{\odot}$) halo as the
energy provided by the inflowing gas starts to become important (this is
driven by the fact that the specific energy of the inflowing gas scales as
$M^{2/3}$); however, the self-regulated quasi-equilibrium behaviour still
holds.

The result of this self-regulation can be seen in the bottom two rows. The
mass builds up relatively slowly and energetic preventative feedback is
working on two levels: regulating the CGM mass but also preventing the inflow
of gas from the IGM. We note that the details of this evolution depend
somewhat on the redshift evolution of the depletion time, which is not well
constrained at high redshift (particularly for low mass halos). Our default
selection is a relatively rapid evolution of $t_{\rm dep}$ such that the star
formation is high at high redshift and the ISM mass is kept low; a more
gradual evolution results in a slower build-up of the stellar component and a
high ISM mass at high redshift. However, the overall self-regulatory behaviour
is robust to changes in the depletion time, with the exception that longer
depletion times at early times cause a longer period of oscillatory self-
adjustment and, in extreme cases, can cause a large fraction of the overall
gas accretion into the ISM to occur at early times, particularly for low-mass
halos.

Figure 8: Log of $M_{\star}/M_{\rm h}$ from the regulator model as a function
of halo mass, compared to the median observed stellar mass with uncertainties
in gray and their derived halo mass from abundance matching from (Behroozi et
al., 2019). The left plot shows the estimates of log of $M_{\star}/M_{\rm h}$
in the ”No Feedback” case, where $\eta_{M}=\eta_{E}=\eta_{Z}=0$. The middle
and right plots show the same relation, but in the altered scenarios where
winds of increasing mass-loading eject all of their mass to the CGM or mass is
ejected directly to the IGM respectively, while $\eta_{E}=0$. The annotation
of each plot displays which parameters were kept constant in the iterations of
the model on display.

###  4.2 Importance of Preventative Feedback Compared to Ejective Feedback

The leftmost panel of Figure 8 displays what happens to the $M_{\star}/M_{\rm
halo}$-$M_{\rm halo}$ relation when we remove all feedback$-$setting
$\eta_{M}$, $\eta_{E}$, and $\eta_{Z}$ to equal 0. We recover the classical
result from early galaxy formation models of overcooling the baryons into the
center of the dark matter halo and the over-abundant production of stars
(White & Rees, 1978; Dekel & Silk, 1986; White & Frenk, 1991).

Removing gas from the galaxy through galactic outflows with high mass-loaded
winds has been a key part of most previous solutions to this problem (e.g.,
Kereš et al., 2009). To demonstrate that we can recover this result, that
ejective feedback ($\eta_{M}>0$) alone is capable of addressing the problems
of overcooling, we set $\eta_{E}=0$, removing the primary source of
preventative feedback within the regulator model. We explore two instances
where $\eta_{E}=0$: one where all mass ejected from the central galaxy is
deposited into the CGM and remains within the circulation of baryon flows
(middle), and the extreme (but traditional) scenario where SNe-driven outflows
launch all their gas out of the ISM such that it escapes to the IGM (right)

When ejected gas is deposited into the CGM with our simple power-law
dependence of $\eta_{M}$ with halo mass, mass-loadings in low-mass galaxies
only begin to touch the Behroozi et al. (2019) constraints when $\beta$
approaches $1.3$. This occurs because more efficient mass-loading is able to
slightly reduce the stellar mass, but in the absence of a means to remove
baryons from the halo, most gas recently ejected from the galaxy accretes back
to the ISM in short-order and provides fuel for star formation. In the
scenario where gas is driven directly out of the galaxy and does not return,
we can significantly reduce the stellar mass, but large values for $\beta$ in
eq.23 are required to drive down the stellar mass such that it agrees with
observed galaxies, requiring mass-loadings as high as $\eta_{M}\sim 100$ in
the lowest mass halo explored by our model, similar to values in previous
cosmological simulations and SAM models (e.g., Muratov et al., 2015; Anglés-
Alcázar et al., 2017; Pillepich et al., 2018; Nelson et al., 2019; Pandya et
al., 2020, 2021).

The difference between the two scenarios in the middle and left panels of
Figure 8, SNe-driven winds ejecting gas into the CGM or ejecting all of the
gas into the IGM, demonstrate the prominent role of the CGM in regulating the
baryonic content of galaxies through cooling and accretion. High specific
energy feedback provides a means of preventing further accretion by heating
the gas and providing additional thermal support against radiative losses,
suppressing cosmic infall from the IGM, and lifting heated gas out of the
halo.

###  4.3 Connection to precipitation: the $t_{\rm cool}/t_{\rm ff}$-$M_{\rm
halo}$ relation

There has been considerable evidence from observations and simulations of the
intracluster medium (ICM) that precipitation-regulated feedback from a central
heating mechanism of the cluster suspends the ICM gas in a state of critical
balance with the ratio of the cooling time to the freefall time $\gtrsim 10$
in massive galaxies (for a recent review, see Donahue & Voit, 2022). Gas in
the ICM becomes increasingly susceptible to condensation as the ratio falls
below this precipitation limit. When this occurs, the condensing gas will fuel
accretion onto the cluster central black hole and the resulting energetic
feedback from the black hole (in the form of a jet) will heat the gas and
lengthen the cooling timescale of the gas, bringing the circumgalactic gas
back to equilibrium such that the ratio exceeds 10. In clusters, this process
is thought to be the primary mechanism which keeps the galaxy quenched and the
ICM regulated.

Figure 9: The ratio of the cooling time to the freefall time, $t_{\rm
cool}/t_{\rm ff}$, evaluated at $R=0.1R_{\rm vir}$ as a function of halo mass
with the optimal choice of $\eta_{E}$ from recursive bisection. The
annotations on the bottom plot display which parameters were kept constant.
The orange dashed line marks critical balance, the predicted $t_{\rm
cool}/t_{\rm ff}\gtrsim 10$ of gas in precipitation models of the CGM.

This same mechanism has been suggested to be operating in galaxies, with the
ICM replaced by the CGM, and SN playing the role of the central black hole
(Voit et al., 2015a). Fielding et al. (2017) showed this mechanism could be at
play in the CGM when $\eta_{E}/\eta_{M}$ is sufficiently high. The model we
described in this paper is similar in many respects but is not directly
connected to precipitation (although that is a natural manifestation of our
CGM cooling). In this section, we explore the connection to see if the CGM
properties we find can be matched on to this critical balance picture.

The cooling time extends to longer timescales as the density drops farther out
in the halo, while the freefall time of the gas also increases with distance
(generally more slowly), giving the ratio a slowly rising radial profile. For
Figure 9, we choose to evaluate the $t_{\rm cool}/t_{\rm ff}$ ratio as a
function of halo mass at a common radius of 0.1$R_{\rm vir}$ for our optimal
model for the energy-loading found in section 3.3. Generally, if we evaluated
it at small radii, the ratio would be lower (although this depends on the
density and temperature profiles we adopt).

We find that $t_{\rm cool}/t_{\rm ff}$ is somewhat larger than the cannonical
precipitaiton prediction in low mass galaxies, approaching values of $\sim
30-40$, but this ratio drops with increasing halo mass, reaching ratios close
to $t_{\rm cool}/t_{\rm ff}\simeq 10$ for Milky Way mass halos. The long
cooling times in the low-mass halos seen in our model arise because of the
high energy-loading (and low mass-loading) in the galactic winds from
supernovae, which raises the CGM temperature well above the virial temperature
and makes the CGM strongly over-pressurized, as well as lowering the CGM
density due to high mass and energy outflows out of the halo. The amount of
energy and mass lifted out of the CGM is sensitive to what we assume for the
specific energy of these outflows, which is not well understood. For
simplicity we assume that the specific energy of these halo outflows is equal
to the average specific energy of the entire CGM, but this need not be the
case in real galaxies, where it is possible that the gas lifted out of the
halo could have a much higher specific energy. Such outflows would lower the
average temperature in the CGM and potentially bring the remaining gas closer
to the precipitation picture than what our model predicts for low mass halos.

Our ratios for $t_{\rm cool}/t_{\rm ff}$ can also be compared with certain
cooling flow models of the CGM, where ineffective heating allows cool gas to
cascade inward on a cooling timescale (Stern et al., 2019). Such models for a
Milky Way halo also find ratios between $t_{\rm cool}$ and $t_{\rm ff}$ that
approximately equal $\sim 7-10$ when gas accretion and the star formation are
approximately equal. Such cooling flow models also predict shorter cooling
times in lower mass halos, which is in contrast to our model, where energy
outflows from supernovae in these galaxies is very effective at heating the
CGM and constraining gas accretion.

###  4.4 Comparison to Other Regulator Models

Similar analytic regulator models in the past have also considered treatments
of preventative and ejective feedback. Although they do not explicitly solve
for the CGM reservoir, Davé et al. (2011) included a preventative feedback
parameter to quantify the amount of gas that enters the halo but is prevented
from reaching the ISM. This preventative feedback includes not only
contributions from winds, but also photoionization, quenching, and virial
shocks. Although their inclusion of preventative feedback is largely
illustrative, they do converge onto a picture of preventative feedback being a
stronger contributor that limits gas accretion in lower mass galaxies ($\sim
10^{10}M_{\odot}$) and declining in importance for more massive halos).

Key differences between our regulator model and past constructions lay in
determining how feedback processes set the stellar-to-halo mass relation and
its low-mass end slope. Lilly et al. (2013) find with their gas-regulator
model that $\sim 40\%$ of baryons that flow into the halos of galaxies across
a stellar mass range from $10^{9}$ to $10^{11}M_{\odot}$ are processed within
galaxies. Their fit to SDSS data from Mannucci et al. (2010) yield $\eta_{M}$
values of $0.2-0.3$ for $M_{\star}=10^{10}M_{\odot}$ systems, which are
comparable to values assumed in this work for similar systems, but this
agreement would not be true for lower-mass systems due to the steep inverse
dependence between mass-loading factor and stellar mass required in their fit
to match the data. The equilibrium model used in Mitra et al. (2015),
employing a similar preventative feedback scheme to Davé et al. (2011) and
fitting $\eta_{M}$ to galaxy scaling relations, find $\eta_{M}\propto
M_{\star}^{-0.5}$, not much larger than our $\beta=0.5$ scaling, however
progressively larger mass outflows are required to match observations at
higher redshift. More recent work from Mitchell & Schaye (2022) build a
regulator-like model based on reproducing the flow terms from the EAGLE
cosmological simulation, finding ejective feedback out of galaxies and halos
to be the primary process shaping the stellar-to-halo mass relation, although
with preventative feedback also playing a significant role. High mass
loadings, particularly in low-mass galaxies, are a persistent feature of most
galactic regulator models, which makes our model, accounting for the energy
content of the CGM and self-consistent treatment of preventative feedback, a
significant departure from past regulator models.

Other preventative feedback models have placed the location of galaxy
regulation in the IGM, where pre-heating in the IGM prevents a portion of
baryons from collapsing into halos due to their enhanced thermal pressure (Lu
et al., 2015a). These models rely predominantly on preventative feedback with
low mass-loading at late times, and have also found some success in matching
certain galaxy scaling relations, such as the size evolution of galactic discs
(Lu et al., 2015b) and the mass-metallicity relation of low-mass galaxies (Lu
et al., 2017). The role each of these preventative mechanisms named above (or
others) play in regulating star formation and their interactions with one
another remains largely unexplored territory in galactic modeling.

In our model of the energy regulation of the CGM, we left out non-thermal
contributions to the energy, but the CGM is also home to a significant kinetic
energy reservoir. This kinetic energy may manifest as turbulence or bulk flows
which may provide their own form of pressure support against gravity in the
CGM that is not entirely thermal (Fielding et al., 2017). Incorporating this
kinetic energy component will be the subject of a future paper (Pandya et al.,
in prep.). In addition, cosmic rays can potentially operate as a preventative
mechanism in the CGM, but we leave this for later work.

##  5 Summary & Conclusion

Wind outflows driven by supernovae are a powerful mechanism for regulating the
baryonic contents of galaxies. We adopt a simple gas regulator model to
understand how different mass and energy wind loading factors affect the
global properties of galaxies and their scaling relations. We include the CGM
as a reservoir; adopting a simple radial density profile, and track the flows
of mass, metallicity, and energy, along with the gas mass and stellar mass of
the central galaxy. We run the regulator model over a halo mass range of
$\log{M_{\rm halo}/M_{\odot}}=10$ to 12 and compare the galaxy properties at
the present-day to a set of galaxy scaling relations. The major conclusions of
our work are described below:

  * •

A gas-regulator model tracking the flows of mass, metallicity, and energy in
and out of the CGM reservoir is able to qualitatively reproduce the broad
features of galaxy scaling relations such as the $M_{\star}/M_{\rm
halo}-M_{\rm halo}$ relation and the $M_{\rm ISM}/M_{\star}-M_{\star}$
relation over the range $\log{M_{\rm halo}/M_{\odot}}=10$ to 12, where SNe are
believed to be the dominant source of baryonic feedback.

  * •

The estimated stellar mass near the present-day redshift is robust against
choices for $\eta_{M}$ and $\eta_{Z}$, but is sensitive to the value of
$\eta_{E}$, suggesting that preventative feedback from energetic outflows
could play a significant role in limiting subsequent gas accretion into the
ISM. High energy-loaded winds heat the CGM and lift heated gas out of the
halo, reducing the density of the CGM and suppressing cosmic infall from the
IGM. This limits the cooling efficiency of the CGM and accretion to the ISM.
The $M_{\star}/M_{\rm halo}-M_{\rm halo}$ relation favors increasing values of
$\eta_{E}$ for decreasing $M_{\rm halo}$, where $\eta_{E}\sim 1$ for
$10^{10}M_{\odot}$ mass halos and $\sim 0.1$ for Milky Way-like halos.

  * •

The CGM plays a significant role in regulating mass outflows from the galaxy
when the CGM is overpressurized. Highly mass-loaded outflows from the galaxy
increase the mass and density of the CGM and lower the CGM’s specific energy,
inhibiting its ability to suppress cosmic inflow from the IGM. These processes
enhance future accretion from the CGM through shortening the cooling timescale
and enables the rapid accretion of gas back into the ISM.

  * •

Our model of a hot CGM with a radial density profile going as $\propto
r^{-1.4}$ produces mass estimates of the CGM that are broadly consistent with
observational constraints from the Milky Way. The mass and metallicity of the
CGM is strongly linked to the composition of galactic outflows.

  * •

Larger values of $\eta_{M}$ increase the mass left over in the CGM reservoir
by the present-day, whereas greater $\eta_{E}$ and $\eta_{Z}$ diminish the CGM
mass through lifting heated gas out of the halo and enhancing accretion from
radiative cooling respectively.

  * •

SNe-driven winds regulate the cooling efficiency of the CGM, and the relation
between the cooling time, $t_{\rm cool}$, and the free-fall time, $t_{\rm ff}$
of circumgalactic gas. When evaluating $t_{\rm cool}/t_{\rm ff}$ at a common
radius of 0.1 $R_{\rm vir}$, the value of the $t_{\rm cool}/t_{\rm ff}$ is
largest in the CGM of low-mass halos, reaching as high as $30-40$ due to very
effective heating from supernovae which raises the CGM temperature well above
the virial temperature. The ratio $t_{\rm cool}/t_{\rm ff}$ decreases with
increasing halo mass and approaches the predicted ratio from precipitation and
cooling flow models of $\approx 10$ in Milky Way-like halos.

  * •

Feedback through strong mass ejection from the ISM alone is unable to
reproduce the $M_{\star}/M_{\rm halo}-M_{\rm halo}$ relation in the scenario
if material ejected from the galaxy is deposited to the CGM, unless mass-
loadings in low-mass galaxies are considerable larger than those considered in
this work. It is sufficient however in the (more extreme) scenario in which
the ejected gas is delivered directly to the IGM.

GLB acknowledges support from the NSF (AST-2108470, XSEDE grant MCA06N030),
NASA TCAN award 80NSSC21K1053, and the Simons Foundation (grant 822237).
Support for this work was provided by NASA through the NASA Hubble Fellowship
grant HST-HF2-51489 awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555. RSS is supported by the Simons
Foundation.

## References

  * Anglés-Alcázar et al. (2017) Anglés-Alcázar, D., Faucher-Giguère, C.-A., Kereš, D., et al. 2017, MNRAS, 470, 4698, doi: 10.1093/mnras/stx1517
  * Asplund et al. (2009) Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481, doi: 10.1146/annurev.astro.46.060407.145222
  * Baugh (2006) Baugh, C. M. 2006, Reports on Progress in Physics, 69, 3101, doi: 10.1088/0034-4885/69/12/R02
  * Behroozi et al. (2019) Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182
  * Benson (2010) Benson, A. J. 2010, Physics Reports, 495, 33, doi: 10.1016/j.physrep.2010.06.001
  * Benson & Bower (2011) Benson, A. J., & Bower, R. 2011, MNRAS, 410, 2653, doi: 10.1111/j.1365-2966.2010.17641.x
  * Benson et al. (2003) Benson, A. J., Bower, R. G., Frenk, C. S., et al. 2003, ApJ, 599, 38, doi: 10.1086/379160
  * Birrer et al. (2014) Birrer, S., Lilly, S., Amara, A., Paranjape, A., & Refregier, A. 2014, The Astrophysical Journal, 793, 12, doi: 10.1088/0004-637X/793/1/12
  * Bland-Hawthorn & Gerhard (2016) Bland-Hawthorn, J., & Gerhard, O. 2016, ARA&A, 54, 529, doi: 10.1146/annurev-astro-081915-023441
  * Bouche et al. (2010) Bouche, N., Dekel, A., Genzel, R., et al. 2010, The Astrophysical Journal, 718, 1001, doi: 10.1088/0004-637X/718/2/1001
  * Bregman (1980) Bregman, J. N. 1980, ApJ, 236, 577, doi: 10.1086/157776
  * Calette et al. (2018) Calette, A. R., Avila-Reese, V., Rodríguez-Puebla, A., Hernández-Toledo, H., & Papastergis, E. 2018, arXiv:1803.07692 [astro-ph].  http://arxiv.org/abs/1803.07692
  * Chisholm et al. (2017) Chisholm, J., Tremonti, C. A., Leitherer, C., & Chen, Y. 2017, MNRAS, 469, 4831, doi: 10.1093/mnras/stx1164
  * Cole et al. (2000) Cole, S., Lacey, C. G., Baugh, C. M., & Frenk, C. S. 2000, MNRAS, 319, 168, doi: 10.1046/j.1365-8711.2000.03879.x
  * Collins & Read (2022) Collins, M. L. M., & Read, J. I. 2022, Nature Astronomy, 6, 647, doi: 10.1038/s41550-022-01657-4
  * Concas et al. (2022) Concas, A., Maiolino, R., Curti, M., et al. 2022, MNRAS, 513, 2535, doi: 10.1093/mnras/stac1026
  * Cousin et al. (2015) Cousin, M., Lagache, G., Bethermin, M., & Guiderdoni, B. 2015, A&A, 575, A33, doi: 10.1051/0004-6361/201424462
  * Crain et al. (2015) Crain, R. A., Schaye, J., Bower, R. G., et al. 2015, MNRAS, 450, 1937, doi: 10.1093/mnras/stv725
  * Croton et al. (2006) Croton, D. J., Springel, V., White, S. D. M., et al. 2006, MNRAS, 365, 11, doi: 10.1111/j.1365-2966.2005.09675.x
  * Das et al. (2021) Das, S., Mathur, S., Gupta, A., & Krongold, Y. 2021, The Astrophysical Journal, 918, 83, doi: 10.3847/1538-4357/ac0e8e
  * Davé et al. (2019) Davé, R., Anglés-Alcázar, D., Narayanan, D., et al. 2019, MNRAS, 486, 2827, doi: 10.1093/mnras/stz937
  * Davé et al. (2011) Davé, R., Finlator, K., & Oppenheimer, B. D. 2011, Monthly Notices of the Royal Astronomical Society, no, doi: 10.1111/j.1365-2966.2011.20148.x
  * Dekel & Mandelker (2014) Dekel, A., & Mandelker, N. 2014, Monthly Notices of the Royal Astronomical Society, 444, 2071, doi: 10.1093/mnras/stu1427
  * Dekel & Silk (1986) Dekel, A., & Silk, J. 1986, The Astrophysical Journal, 303, 39, doi: 10.1086/164050
  * Dekel et al. (2013) Dekel, A., Zolotov, A., Tweed, D., et al. 2013, Monthly Notices of the Royal Astronomical Society, 435, 999, doi: 10.1093/mnras/stt1338
  * Dekel et al. (2009) Dekel, A., Birnboim, Y., Engel, G., et al. 2009, Nature, 457, 451, doi: 10.1038/nature07648
  * Donahue & Voit (2022) Donahue, M., & Voit, G. M. 2022, Phys. Rep., 973, 1, doi: 10.1016/j.physrep.2022.04.005
  * Esmerian et al. (2021) Esmerian, C. J., Kravtsov, A. V., Hafen, Z., et al. 2021, MNRAS, 505, 1841, doi: 10.1093/mnras/stab1281
  * Faerman et al. (2022) Faerman, Y., Pandya, V., Somerville, R. S., & Sternberg, A. 2022, ApJ, 928, 37, doi: 10.3847/1538-4357/ac4ca6
  * Faerman et al. (2020) Faerman, Y., Sternberg, A., & McKee, C. F. 2020, ApJ, 893, 82, doi: 10.3847/1538-4357/ab7ffc
  * Fielding et al. (2018) Fielding, D., Quataert, E., & Martizzi, D. 2018, MNRAS, 481, 3325, doi: 10.1093/mnras/sty2466
  * Fielding et al. (2017) Fielding, D., Quataert, E., McCourt, M., & Thompson, T. A. 2017, MNRAS, 466, 3810, doi: 10.1093/mnras/stw3326
  * Fielding et al. (2020) Fielding, D. B., Tonnesen, S., DeFelippis, D., et al. 2020, ApJ, 903, 32, doi: 10.3847/1538-4357/abbc6d
  * Finlator & Davé (2008) Finlator, K., & Davé, R. 2008, MNRAS, 385, 2181, doi: 10.1111/j.1365-2966.2008.12991.x
  * Ford et al. (2014) Ford, A. B., Davé, R., Oppenheimer, B. D., et al. 2014, MNRAS, 444, 1260, doi: 10.1093/mnras/stu1418
  * Fraternali et al. (2015) Fraternali, F., Marasco, A., Armillotta, L., & Marinacci, F. 2015, MNRAS, 447, L70, doi: 10.1093/mnrasl/slu182
  * Furlanetto (2020) Furlanetto, S. R. 2020, Monthly Notices of the Royal Astronomical Society, 500, 3394, doi: 10.1093/mnras/staa3451
  * Furlanetto & Mirocha (2021) Furlanetto, S. R., & Mirocha, J. 2021, arXiv:2109.04488 [astro-ph].  http://arxiv.org/abs/2109.04488
  * Girichidis et al. (2016) Girichidis, P., Walch, S., Naab, T., et al. 2016, MNRAS, 456, 3432, doi: 10.1093/mnras/stv2742
  * Heckman et al. (1990) Heckman, T. M., Armus, L., & Miley, G. K. 1990, The Astrophysical Journal Supplement Series, 74, 833, doi: 10.1086/191522
  * Heitsch & Putman (2009) Heitsch, F., & Putman, M. E. 2009, ApJ, 698, 1485, doi: 10.1088/0004-637X/698/2/1485
  * Hou et al. (2018) Hou, J., Lacey, C. G., & Frenk, C. S. 2018, MNRAS, 475, 543, doi: 10.1093/mnras/stx3218
  * Hu (2019) Hu, C.-Y. 2019, MNRAS, 483, 3363, doi: 10.1093/mnras/sty3252
  * Kauffmann et al. (1993) Kauffmann, G., White, S. D. M., & Guiderdoni, B. 1993, MNRAS, 264, 201, doi: 10.1093/mnras/264.1.201
  * Kereš et al. (2009) Kereš, D., Katz, N., Davé, R., Fardal, M., & Weinberg, D. H. 2009, Monthly Notices of the Royal Astronomical Society, 396, 2332, doi: 10.1111/j.1365-2966.2009.14924.x
  * Kim et al. (2017) Kim, C.-G., Ostriker, E. C., & Raileanu, R. 2017, ApJ, 834, 25, doi: 10.3847/1538-4357/834/1/25
  * Kim et al. (2020) Kim, C.-G., Ostriker, E. C., Somerville, R. S., et al. 2020, ApJ, 900, 61, doi: 10.3847/1538-4357/aba962
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Krumholz et al. (2009) Krumholz, M. R., McKee, C. F., & Tumlinson, J. 2009, ApJ, 699, 850, doi: 10.1088/0004-637X/699/1/850
  * Li & Bryan (2020) Li, M., & Bryan, G. L. 2020, The Astrophysical Journal, 890, L30, doi: 10.3847/2041-8213/ab7304
  * Li et al. (2017) Li, M., Bryan, G. L., & Ostriker, J. P. 2017, ApJ, 841, 101, doi: 10.3847/1538-4357/aa7263
  * Li & Tonnesen (2020) Li, M., & Tonnesen, S. 2020, ApJ, 898, 148, doi: 10.3847/1538-4357/ab9f9f
  * Lilly et al. (2013) Lilly, S. J., Carollo, C. M., Pipino, A., Renzini, A., & Peng, Y. 2013, The Astrophysical Journal, 772, 119, doi: 10.1088/0004-637X/772/2/119
  * Lu et al. (2017) Lu, Y., Benson, A., Wetzel, A., et al. 2017, ApJ, 846, 66, doi: 10.3847/1538-4357/aa845e
  * Lu et al. (2011) Lu, Y., Kereš, D., Katz, N., et al. 2011, MNRAS, 416, 660, doi: 10.1111/j.1365-2966.2011.19072.x
  * Lu et al. (2015a) Lu, Y., Mo, H. J., & Lu, Z. 2015a, arXiv e-prints, arXiv:1504.02109.  https://arxiv.org/abs/1504.02109
  * Lu et al. (2015b) Lu, Y., Mo, H. J., & Wechsler, R. H. 2015b, MNRAS, 446, 1907, doi: 10.1093/mnras/stu2215
  * Maller & Bullock (2004) Maller, A. H., & Bullock, J. S. 2004, MNRAS, 355, 694, doi: 10.1111/j.1365-2966.2004.08349.x
  * Mannucci et al. (2010) Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., & Gnerucci, A. 2010, MNRAS, 408, 2115, doi: 10.1111/j.1365-2966.2010.17291.x
  * Marasco et al. (2022a) Marasco, A., Fraternali, F., Lehner, N., & Howk, J. C. 2022a, MNRAS, 515, 4176, doi: 10.1093/mnras/stac1172
  * Marasco et al. (2022b) Marasco, A., Belfiore, F., Cresci, G., et al. 2022b, arXiv e-prints, arXiv:2209.02726.  https://arxiv.org/abs/2209.02726
  * Martin (1999) Martin, C. L. 1999, ApJ, 513, 156, doi: 10.1086/306863
  * McCourt et al. (2012) McCourt, M., Sharma, P., Quataert, E., & Parrish, I. J. 2012, MNRAS, 419, 3319, doi: 10.1111/j.1365-2966.2011.19972.x
  * McGaugh et al. (2017) McGaugh, S. S., Schombert, J. M., & Lelli, F. 2017, ApJ, 851, 22, doi: 10.3847/1538-4357/aa9790
  * McQuinn et al. (2019) McQuinn, K. B. W., van Zee, L., & Skillman, E. D. 2019, ApJ, 886, 74, doi: 10.3847/1538-4357/ab4c37
  * Mitchell & Schaye (2022) Mitchell, P. D., & Schaye, J. 2022, MNRAS, 511, 2948, doi: 10.1093/mnras/stab3339
  * Mitra et al. (2015) Mitra, S., Davé, R., & Finlator, K. 2015, MNRAS, 452, 1184, doi: 10.1093/mnras/stv1387
  * Moster et al. (2010) Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, ApJ, 710, 903, doi: 10.1088/0004-637X/710/2/903
  * Muratov et al. (2015) Muratov, A. L., Kereš, D., Faucher-Giguère, C.-A., et al. 2015, MNRAS, 454, 2691, doi: 10.1093/mnras/stv2126
  * Muratov et al. (2017) —. 2017, MNRAS, 468, 4170, doi: 10.1093/mnras/stx667
  * Naab & Ostriker (2017) Naab, T., & Ostriker, J. P. 2017, Annual Review of Astronomy and Astrophysics, 55, 59, doi: 10.1146/annurev-astro-081913-040019
  * Nelson et al. (2016) Nelson, D., Genel, S., Pillepich, A., et al. 2016, MNRAS, 460, 2881, doi: 10.1093/mnras/stw1191
  * Nelson et al. (2019) Nelson, D., Pillepich, A., Springel, V., et al. 2019, MNRAS, 490, 3234, doi: 10.1093/mnras/stz2306
  * Oppenheimer & Davé (2006) Oppenheimer, B. D., & Davé, R. 2006, MNRAS, 373, 1265, doi: 10.1111/j.1365-2966.2006.10989.x
  * Oppenheimer et al. (2010) Oppenheimer, B. D., Davé, R., Kereš, D., et al. 2010, MNRAS, 406, 2325, doi: 10.1111/j.1365-2966.2010.16872.x
  * Oppenheimer & Davé (2008) Oppenheimer, B. D., & Davé, R. 2008, Monthly Notices of the Royal Astronomical Society, 387, 577, doi: 10.1111/j.1365-2966.2008.13280.x
  * Ostriker & Kim (2022) Ostriker, E. C., & Kim, C.-G. 2022, arXiv e-prints, arXiv:2206.00681.  https://arxiv.org/abs/2206.00681
  * Pandya et al. (2020) Pandya, V., Somerville, R. S., Anglés-Alcázar, D., et al. 2020, ApJ, 905, 4, doi: 10.3847/1538-4357/abc3c1
  * Pandya et al. (2021) Pandya, V., Fielding, D. B., Anglés-Alcázar, D., et al. 2021, arXiv e-prints, arXiv:2103.06891.  https://arxiv.org/abs/2103.06891
  * Peng & Maiolino (2014) Peng, Y.-j., & Maiolino, R. 2014, Monthly Notices of the Royal Astronomical Society, 443, 3643, doi: 10.1093/mnras/stu1288
  * Pillepich et al. (2018) Pillepich, A., Springel, V., Nelson, D., et al. 2018, MNRAS, 473, 4077, doi: 10.1093/mnras/stx2656
  * Prochaska et al. (2017) Prochaska, J. X., Werk, J. K., Worseck, G., et al. 2017, The Astrophysical Journal, 837, 169, doi: 10.3847/1538-4357/aa6007
  * Putman et al. (2012) Putman, M., Peek, J., & Joung, M. 2012, Annual Review of Astronomy and Astrophysics, 50, 491, doi: 10.1146/annurev-astro-081811-125612
  * Rupke (2018) Rupke, D. S. N. 2018, Galaxies, 6, 138, doi: 10.3390/galaxies6040138
  * Salem et al. (2015) Salem, M., Besla, G., Bryan, G., et al. 2015, The Astrophysical Journal, 815, 77, doi: 10.1088/0004-637X/815/1/77
  * Schaye et al. (2014) Schaye, J., Crain, R. A., Bower, R. G., et al. 2014, Monthly Notices of the Royal Astronomical Society, 446, 521, doi: 10.1093/mnras/stu2058
  * Shapiro & Field (1976) Shapiro, P. R., & Field, G. B. 1976, ApJ, 205, 762, doi: 10.1086/154332
  * Sharma et al. (2012a) Sharma, P., McCourt, M., Parrish, I. J., & Quataert, E. 2012a, MNRAS, 427, 1219, doi: 10.1111/j.1365-2966.2012.22050.x
  * Sharma et al. (2012b) Sharma, P., McCourt, M., Quataert, E., & Parrish, I. J. 2012b, MNRAS, 420, 3174, doi: 10.1111/j.1365-2966.2011.20246.x
  * Somerville & Davé (2015) Somerville, R. S., & Davé, R. 2015, Annual Review of Astronomy and Astrophysics, 53, 51, doi: 10.1146/annurev-astro-082812-140951
  * Somerville et al. (2008) Somerville, R. S., Hopkins, P. F., Cox, T. J., Robertson, B. E., & Hernquist, L. 2008, MNRAS, 391, 481, doi: 10.1111/j.1365-2966.2008.13805.x
  * Somerville & Primack (1999) Somerville, R. S., & Primack, J. R. 1999, MNRAS, 310, 1087, doi: 10.1046/j.1365-8711.1999.03032.x
  * Stern et al. (2019) Stern, J., Fielding, D., Faucher-Giguère, C.-A., & Quataert, E. 2019, MNRAS, 488, 2549, doi: 10.1093/mnras/stz1859
  * Stern et al. (2020) —. 2020, MNRAS, 492, 6042, doi: 10.1093/mnras/staa198
  * Strickland & Heckman (2009) Strickland, D. K., & Heckman, T. M. 2009, The Astrophysical Journal, 697, 2030, doi: 10.1088/0004-637X/697/2/2030
  * Strickland & Stevens (2000) Strickland, D. K., & Stevens, I. R. 2000, Monthly Notices of the Royal Astronomical Society, 314, 511, doi: 10.1046/j.1365-8711.2000.03391.x
  * Thompson et al. (2005) Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ, 630, 167, doi: 10.1086/431923
  * Tumlinson et al. (2017) Tumlinson, J., Peeples, M. S., & Werk, J. K. 2017, Annual Review of Astronomy and Astrophysics, 55, 389, doi: 10.1146/annurev-astro-091916-055240
  * Veilleux et al. (2005) Veilleux, S., Cecil, G., & Bland-Hawthorn, J. 2005, Annual Review of Astronomy and Astrophysics, 43, 769, doi: 10.1146/annurev.astro.43.072103.150610
  * Vogelsberger et al. (2019) Vogelsberger, M., Marinacci, F., Torrey, P., & Puchwein, E. 2019, arXiv:1909.07976 [astro-ph].  http://arxiv.org/abs/1909.07976
  * Vogelsberger et al. (2014) Vogelsberger, M., Genel, S., Springel, V., et al. 2014, Nature, 509, 177, doi: 10.1038/nature13316
  * Voit et al. (2015a) Voit, G. M., Bryan, G. L., O’Shea, B. W., & Donahue, M. 2015a, ApJ, 808, L30, doi: 10.1088/2041-8205/808/1/L30
  * Voit et al. (2015b) Voit, G. M., Donahue, M., Bryan, G. L., & McDonald, M. 2015b, Nature, 519, 203, doi: 10.1038/nature14167
  * Voit et al. (2017) Voit, G. M., Meece, G., Li, Y., et al. 2017, ApJ, 845, 80, doi: 10.3847/1538-4357/aa7d04
  * White & Frenk (1991) White, S. D. M., & Frenk, C. S. 1991, The Astrophysical Journal, 379, 52, doi: 10.1086/170483
  * White & Rees (1978) White, S. D. M., & Rees, M. J. 1978, Monthly Notices of the Royal Astronomical Society, 183, 341, doi: 10.1093/mnras/183.3.341
  * Wiersma et al. (2009) Wiersma, R. P. C., Schaye, J., & Smith, B. D. 2009, MNRAS, 393, 99, doi: 10.1111/j.1365-2966.2008.14191.x

##  Appendix A Depletion Time Sensitivity

Here we test the sensitivity of our regulator model to assumed parameters
outside our characterization of wind outflows. Firstly, the depletion time
sets the timescale for the conversion of gas into stars. This conversion is
observed to be more efficient in Milky Way-mass galaxies (i.e. higher stellar
masses). This is expressed in eq. 13 used for $t_{\rm dep}$ referenced in
section 2.2.4. We can modulate the normalization of $t_{\rm dep}$, and its
power-law dependence. In prior sections, we use the depletion time estimate
derived from McGaugh et al. (2017), where the logarithmic depletion time is
normalized by a prefactor equal to 4.92 and and a dependence on stellar mass
with a power law index of 0.37. We define the normalization of depletion time
as a parameter called $C$ and its power law index as $\gamma$. For this
section, we keep one parameter set to its fiducial value, while varying the
other. Since the influence of $t_{\rm dep}$ should primarily cover the
conversion of gas into stars in the galaxy, we place a special emphasis on the
$M_{\star}/M_{\rm halo}$-$M_{\rm halo}$ relation, and the $M_{\rm
ISM}/M_{\star}$-$M_{\star}$ relation. For these tests and those in Appendix B,
we use the power law formula dependent on halo mass introduced in eq 23 with a
power law index of $\beta=0.5$, the power law fit of $\eta_{E}$ from eq. 24
and its best-fit values, and set $\eta_{Z}=0.2$.

Figure 10: Log of $M_{\star}/M_{\rm h}$ from the regulator model as a function
of halo mass, compared to the median observed stellar mass with uncertainties
in gray and their derived halo mass from abundance matching from Behroozi et
al. (2019). Estimates of $M_{\star}/M_{\rm halo}$ are presented as a function
of the normalization of the depletion time $C$ (left), and its power-law
dependence, $\gamma$ (right). Figure 11: Log of $M_{\rm ISM}/M_{\star}$ from
the regulator model as a function of stellar mass, compared to the best-fit
single power-law fit for the total cold gas mass as a function of $M_{\star}$
over the range $7.3\lesssim\log{(M_{\star}/M_{\rm h})}\lesssim 11.2$ for late-
type galaxies from Calette et al. (2018). Shaded lines correspond to
alterations in the normalization of the depletion time $C$ (left) and its
power-law dependence with stellar mass, $\gamma$ (right) and their respective
influence on the $M_{\rm ISM}-M_{\star}$ relation.

We show in the left-hand plot of Figure 10 that modulating the normalization
of $t_{\rm dep}$ over an order of magnitude from $C=4.25-5.5$ does have a
noticeable effect on the $M_{\star}/M_{\rm halo}$-$M_{\rm halo}$ relation,
such that larger (smaller) values result in a reduced (greater) stellar
masses, especially in more massive halos. This is partly to be expected,
considering that a larger normalization would lead to larger depletion
timescales and a reduced star formation rate. The less efficient conversion of
gas into stars leads a greater proportion of gas to stars in all galaxies as a
function of halo mass, manifesting as a vertical shift in the $M_{\rm
ISM}/M_{\star}$-$M_{\star}$ relation of the model (left-hand plot of Figure
11).

The right-hand plot of Figure 10 shows how modifying the power-law index of
$t_{\rm dep}$ has very little effect on the $M_{\star}/M_{\rm halo}$-$M_{\rm
halo}$ relation. The most apparent consequence of modifying $\gamma$ in seen
in Figure 11, where it is shown that the power law index of the depletion time
alters the slope of the $M_{\rm ISM}/M_{\star}$-$M_{\star}$ relation produced
by the model. Increasing the index reduces the star formation efficiency in
low-mass halos and increases its efficiency in more massive halos and vice-
versa for its reduction. Larger values of $\gamma$ are in better agreement
with the slope of the PL fit from Calette et al. (2018), with $\gamma\geq 0.5$
producing the closest match with their observational constraints.

The robustness of the stellar mass to the power law dependence of the
depletion time and even its normalization at the lower mass end in Figure 10
may point to the important role that $t_{\rm dep}$ plays in setting the
timescale for self-regulation to arise at high redshift. Longer depletion
times, especially in low mass halos, delay the ignition of star formation and
the high specific energy outflows which prevent accretion into the CGM and the
ISM. This means that although star formation in the disk becomes less
efficient, the increased baryon fraction in the halo leaves the resulting
stellar mass largely unchanged. Shorter depletion times in low-mass halos
(lower power-law index) means that they are more efficient in their star
production, but suppress cosmic inflow through self-regulation much earlier,
leaving less gas in the galaxy to be processed into stars.

##  Appendix B Density Profile Sensitivity

Figure 12: Log of $M_{\star}/M_{\rm h}$ from the regulator model as a function
of halo mass, compared to the median observed stellar mass with uncertainties
in gray and their derived halo mass from abundance matching from (Behroozi et
al., 2019). Shaded lines represent different variations of the power-law index
for the CGM radial density profile, $\alpha$, and its effect on the
$M_{\star}/M_{\rm halo}$-$M_{\rm halo}$ relation.

Another important quantity in our model is our assumed density profile of the
CGM. In eq.(1), we assume a power-law dependence with distance from the center
with an index of $\alpha=1.4$. Since the density profile of the CGM is not
well-constrained, it is important to observe how our models responds to
different assumptions of the gas density profile. Varying $\alpha$ in our
parameterization of the density profile changes how $\rho(r)$ depends on
radius, where larger values of $\alpha$ produce a steeper density profile
where a greater concentration of gas is placed in the inner CGM relative to
its outskirts.

Figure 13: Estimates of $\log(M_{\rm CGM}/M_{\rm h})$ as a function of halo
mass. Estimates of $M_{\rm CGM}/M_{\rm halo}$ are presented as a function of
the power-law index for the CGM radial density profile, $\alpha$.

After plotting the $M_{\star}/M_{\rm halo}$-$M_{\rm halo}$ relation for
varying forms of the CGM density profile in Figure 12, we find that larger
values of $\alpha$ increase the stellar mass at $z=0$ for all halos. With a
greater concentration of mass in the inner CGM, steeper density profiles
elevate the overall cooling efficiency of gas near the galaxy and result in
greater accretion and more fuel for star formation. This rise in star
formation should also work to increase the total outflow of energy into the
CGM, but this countering response to enhanced accretion is not enough to fully
oppose the significant radiative losses in the inner gas halo. Steeper density
profiles have a predictable effect on the CGM. In Figure 13, larger values of
$\alpha$ result in less mass in the CGM at late times due to the greater
accretion from the CGM to the ISM.