# Black Hole Growth, Baryon Lifting, Star Formation, and IllustrisTNG

G. Mark Voit Michigan State University, Department of Physics and Astronomy,
East Lansing, MI 48824, USA  Benjamin D. Oppenheimer University of Colorado,
Center for Astrophysics and Space Astronomy, 389 UCB, Boulder, CO 80309, USA
Eric F. Bell University of Michigan, Department of Astronomy, Ann Arbor, MI
48109, USA  Bryan Terrazas Columbia University, Department of Astronomy, New
York, NY 10027, USA  Megan Donahue Michigan State University, Department of
Physics and Astronomy, East Lansing, MI 48824, USA

(Received 14 May 2023; Revised 1 Oct 2023; Accepted 3 Oct 2023)

###### Abstract

Quenching of star formation in the central galaxies of cosmological halos is
thought to result from energy released as gas accretes onto a supermassive
black hole. The same energy source also appears to lower the central density
and raise the cooling time of baryonic atmospheres in massive halos, thereby
limiting both star formation and black hole growth, by lifting the baryons in
those halos to greater altitudes. One predicted signature of that feedback
mechanism is a nearly linear relationship between the central black hole’s
mass ($M_{\rm BH}$) and the original binding energy of the halo’s baryons. We
present the increasingly strong observational evidence supporting a such a
relationship, showing that it extends up to halos of mass $M_{\rm halo}\sim
10^{14}M_{\odot}$. We then compare current observational constraints on the
$M_{\rm BH}$–$M_{\rm halo}$ relation with numerical simulations, finding that
black hole masses in IllustrisTNG appear to exceed those constraints at
$M_{\rm halo}<10^{13}M_{\odot}$ and that black hole masses in EAGLE fall short
of observations at $M_{\rm halo}\sim 10^{14}M_{\odot}$. A closer look at
IllustrisTNG shows that quenching of star formation and suppression of black
hole growth do indeed coincide with black hole energy input that lifts the
halo’s baryons. However, IllustrisTNG does not reproduce the observed $M_{\rm
BH}$–$M_{\rm halo}$ relation because its black holes gain mass primarily
through accretion that does not contribute to baryon lifting. We suggest
adjustments to some of the parameters in the IllustrisTNG feedback algorithm
that may allow the resulting black hole masses to reflect the inherent links
between black hole growth, baryon lifting, and star formation among the
massive galaxies in those simulations.

††journal: ApJ

##  1 Introduction

A galaxy’s star formation rate is tied to both the mass of its cosmological
halo ($M_{\rm halo}$) and the mass of the black hole residing at its center
($M_{\rm BH}$). Large galaxy surveys spanning much of cosmic time show that
the central galaxies of cosmological halos vigorously form stars until $M_{\rm
halo}$ exceeds $\sim 10^{12}\,M_{\odot}$ (e.g., Behroozi et al., 2013, 2019).
Star formation then subsides as $M_{\rm halo}$ increases toward $\sim
10^{13}\,M_{\odot}$. However, suppression of star formation among the central
galaxies of present-day cosmological halos correlates more closely with the
central velocity dispersion ($\sigma_{v}$) of a galaxy’s stars than with
$M_{\rm halo}$ (e.g., Wake et al., 2012; Bell et al., 2012; Woo et al., 2015;
Teimoorinia et al., 2016; Bluck et al., 2016, 2020), implying that star-
formation quiescence depends more directly on galactic structure than on halo
mass.

The mass of a galaxy’s central black hole also closely correlates with
$\sigma_{v}$ (e.g., Kormendy & Ho, 2013), suggesting a causal link between
galaxy evolution and black hole growth. Observations show that suppression of
star formation does indeed correlate with $M_{\rm BH}$ among nearby galaxies
of similar stellar mass (Terrazas et al., 2016, 2017). Both galactic structure
and black hole growth therefore seem to conspire in the shutdown of star
formation known as quenching.

Eruptions of feedback energy as a galaxy’s central black hole grows are
thought to be crucial for limiting black hole growth and perhaps also galactic
star formation. An early analysis by Silk & Rees (1998) proposed that the
energy released as a galaxy’s central black hole grows would limit the black
hole’s growth once it surpassed the energy required to lift baryons out of the
galaxy’s bulge, or perhaps even out of the galaxy’s entire potential well (see
also Haehnelt et al., 1998). The predicted result: a scaling relation ($M_{\rm
BH}\propto\sigma_{v}^{5}$) similar to the observed one (for a more recent
review of similar ideas, see King & Pounds, 2015).

A causal connection between $M_{\rm BH}$ and $M_{\rm halo}$ became more
plausible when Ferrarese (2002) showed that nearby spiral galaxies follow the
same $M_{\rm BH}$–$M_{\rm halo}$ scaling relation as their more massive
elliptical counterparts, based on assuming that a galaxy’s central velocity
dispersion and rotation speed are proportional to each other. When $M_{\rm
halo}$ is defined in terms of a mean matter density, the halo’s circular
velocity is $v_{\rm c}\propto M_{\rm halo}^{1/3}$, the specific binding energy
of its matter is $E_{\rm B}/M_{\rm halo}\propto v_{\rm c}^{2}$, and the total
binding energy is $E_{\rm B}\propto M_{\rm halo}v_{\rm c}^{2}\propto v_{\rm
c}^{5}$. Black-hole growth limited by baryon lifting should therefore result
in $M_{\rm BH}\propto\sigma_{v}^{5}\propto M_{\rm halo}^{5/3}$. The Ferrarese
(2002) data set supported the baryon lifting hypothesis because it indicated
$M_{\rm BH}\propto M_{\rm halo}^{\gamma}$ with $\gamma\approx 1.65$–1.82,
depending on the methods used to infer $M_{\rm halo}$ from $\sigma_{v}$ and
$v_{\rm c}$. A few years later, Bandara et al. (2009) strengthened the
evidence for such a power-law relation, through a survey that used lensing
observations to obtain $M_{\rm halo}$ and indirectly inferred $M_{\rm BH}$
from $\sigma_{v}$, finding $M_{\rm BH}\propto M_{\rm halo}^{1.55\pm 0.31}$.
However, some experts remained deeply skeptical of a direct causal connection
between $M_{\rm BH}$ and $M_{\rm halo}$ (e.g., Kormendy & Bender, 2011;
Kormendy & Ho, 2013).

Cosmological simulations then put the proposed relationship between $M_{\rm
BH}$ and $M_{\rm halo}$ on a firmer theoretical footing. Booth & Schaye (2010)
demonstrated that energy released by the black-hole feedback algorithm in
their simulations led to the scaling relation $M_{\rm BH}\propto M_{\rm
halo}^{1.55\pm 0.05}$, with a normalization coefficient proportional to the
assumed ratio of accreted mass to energy output. This relationship arose
because the simulated black holes grew through accretion until they released
an energy comparable to the gravitational binding energy of all the halo’s
baryons. The accumulating energy then lifted the halo’s baryons, thereby
lowering the density, pressure, and cooling time of baryons in the black
hole’s vicinity, reducing its long-term accretion rate and limiting its
growth. The resulting power-law slope ended up slightly smaller than the 5/3
prediction for identically structured halos because the dark-matter density
profiles of lower-mass halos tend to be more centrally concentrated than those
of higher-mass halos, leading to a shallower dependence of specific binding
energy on halo mass.

More recent simulations incorporating many more astrophysical details have
demonstrated that lifting of a halo’s baryons via black-hole feedback may also
be critical for suppressing star formation (Davies et al., 2019, 2020;
Oppenheimer et al., 2020; Terrazas et al., 2020; Zinger et al., 2020; Appleby
et al., 2021). In simulated halos with $M_{\rm halo}\gtrsim 10^{12}M_{\odot}$,
black-hole feedback is the prime mover of baryons beyond the virial radius.
Furthermore, simulated galaxies centered within halos of mass $\sim
10^{12}M_{\odot}$ tend to have star formation rates that correlate with the
proportion of the halo’s baryons remaining within the virial radius.

A similar story has emerged from analyses of correlations between star-
formation quenching, the structural properties of galaxies, and the masses of
their central black holes. According to Chen et al. (2020), the $M_{\rm
BH}$–$\sigma_{v}$ relation among galaxies with active star formation has a
power-law slope similar to the $M_{\rm BH}$–$\sigma_{v}$ relation among
quiescent galaxies but a mass normalization approximately an order of
magnitude smaller at fixed $\sigma_{v}$. The transition from active to
quenched star formation therefore appears to be associated with rapid black
hole mass growth. It is also consistent with an amount of black hole growth
that is proportional to the halo’s baryonic binding energy, suggesting that
quenching results from lifting of the halo’s baryons via black hole
feedback.111 In the interpretation presented by Chen et al. (2020), the amount
of injected feedback energy needed to quench star formation is 4 times the
halo’s baryonic binding energy, but the numerical factor is degenerate with
the conversion efficiency of accreted mass-energy to feedback energy, which
they assume to be $0.01$.

The proposed connection between baryon lifting and quenching of star formation
is theoretically appealing, but then why does quiescence correlate more
closely with $\sigma_{v}$ than with $M_{\rm halo}$? Voit et al. (2020) have
argued that baryon lifting via black hole feedback is an inevitable
consequence of structural evolution that raises a galaxy’s central stellar
mass density, as reflected by $\sigma_{v}$. The central cooling rate of hot
gas in galaxies with large $\sigma_{v}$ depends primarily on circumgalactic
gas pressure. Consequently, as $\sigma_{v}$ rises above a critical value
determined by stellar heating, black hole fueling becomes linked to
circumgalactic pressure. Once that link is established, $M_{\rm BH}$ then
grows to depend directly on $M_{\rm halo}$ as cumulative black hole energy
injection rises to scale with the halo’s baryonic binding energy (for an
extensive review, see Donahue & Voit, 2022).

Figure 1: Observed relationships between black hole mass ($M_{\rm BH}$), halo
mass ($M_{\rm halo}$), and atmospheric temperature ($T$). Dotted and dashed
red lines show two $M_{\rm BH}$–$M_{\rm halo}$ relations from Ferrarese
(2002). Dot-dashed purple lines show the $M_{\rm BH}$–$M_{\rm halo}$ relation
from Bandara et al. (2009). Dot-dot-dot-dashed magenta lines show the
simulated $M_{\rm BH}$–$M_{\rm halo}$ relation from Booth & Schaye (2010).
Dashed blue lines show the $M_{\rm BH}$–$M_{\rm halo}$ relation from Marasco
et al. (2021). Solid grey lines separating shaded from unshaded regions
indicate a linear $M_{\rm BH}$–$M_{\rm halo}$ correlation. Black points in the
left panel show the $M_{\rm BH}$–$T_{\rm CGM}$ measurements from Gaspari et
al. (2019). The best fitting power law relation ($M_{\rm BH}\propto M_{\rm
halo}^{1.6}$) shown by the solid black line is clearly super-linear. Grey
points in the right panel show $M_{\rm BH}$–$T_{\rm halo}$ measurements from
Gaspari et al. (2019), and a dashed black line shows the best fitting power
law relation ($M_{\rm BH}\propto M_{\rm halo}^{1.3}$). An additional dashed
gold line in the right panel shows the $M_{\rm BH}$–$T_{\rm halo}$ relation
from the sample of Bogdán et al. (2018). In both panels, the relationship
$kT_{\rm X}=6\,{\rm keV}\times(M_{\rm 200c}/10^{15}\,M_{\odot})^{1.7}$ maps
gas temperature onto halo mass. However, masses based on $T_{\rm CGM}$ (left
panel) are underestimates in cases where $T_{\rm halo}\gg T_{\rm CGM}$.

This paper presents evidence favoring such a three-way link between black hole
growth, baryon lifting, and star-formation quiescence. Section 2 starts things
off by examining current observational assessments of the $M_{\rm BH}$–$M_{\rm
halo}$ relation, comparing them with the results of numerical simulations, and
finding general support for the three-way link, except in IllustrisTNG, which
requires a deeper examination. Section 3 establishes that quenching of star
formation in IllustrisTNG does, in fact, coincide with kinetic feedback input
sufficient to lift a halo’s baryons. Section 4 analyzes the contrasting roles
that the thermal (“quasar”) and kinetic (“radio”) feedback modes of
IllustrisTNG play in baryon lifting. Section 5 briefly discusses how the
feedback efficiency parameters employed in numerical simulations determine the
“price” of feedback, as reflected by black hole mass growth. Section 6
speculates about how “price” changes might bring IllustrisTNG black hole
masses into better agreement with both observations and the predicted $M_{\rm
BH}$–$M_{\rm halo}$ scaling relation. Section 7 summarizes the paper’s
findings.

##  2 Black Holes and Halo Masses

The introduction mentioned some of the observational constraints on the
$M_{\rm BH}$–$M_{\rm halo}$ relation. Now we will take a closer look at those
observations and compare them with what emerges from the IllustrisTNG and
EAGLE cosmological simulations. We will focus most closely on the halo mass
range from $10^{12.5}M_{\odot}$ to $10^{14}M_{\odot}$, because that is where
X-ray observations provide both direct evidence for baryon lifting and
reliable estimates of $M_{\rm halo}$. Our review of the literature is
therefore neither comprehensive nor complete.

###  2.1 Observations

Figure 1 illustrates several relationships between $M_{\rm BH}$ and $M_{\rm
halo}$. A dotted red line shows the relation

$M_{\rm BH}=10^{8.25}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.82}$ (1)

corresponding to equation (4) from Ferrarese (2002). A dashed red line shows
the relation

$M_{\rm BH}=10^{8.50}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.65}$ (2)

corresponding to equation (6) from Ferrarese (2002). A dot-dashed purple line
shows the relation

$M_{\rm BH}=10^{8.18}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.55}$ (3)

from Bandara et al. (2009). And a dashed blue line shows the relation

$M_{\rm BH}=10^{8.68}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.62}$ (4)

derived from observations compiled by Marasco et al. (2021).

Those four assessments of the $M_{\rm BH}$–$M_{\rm halo}$ relation generally
align with each other and also with the dot-dot-dot-dashed magenta line
showing the relation

$M_{\rm BH}=10^{8.01}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.55}$ (5)

that Booth & Schaye (2010) found in cosmological numerical simulations of
black hole feedback. However, Ferrarese (2002) inferred $M_{\rm halo}$ from
galactic dynamics, not halo properties. Bandara et al. (2009) inferred $M_{\rm
BH}$ from $\sigma_{v}$, not direct dynamical measurements of $M_{\rm BH}$. And
Marasco et al. (2021) used a heterogeneous set of proxies for $M_{\rm halo}$,
making it difficult to assess the impact of systematic uncertainties on their
best fitting $M_{\rm BH}$ normalization.

X-ray analyses have recently provided more direct constraints on the $M_{\rm
BH}$–$M_{\rm halo}$ relation (Bogdán et al., 2018; Lakhchaura et al., 2019;
Gaspari et al., 2019). Among those analyses, the Gaspari et al. (2019) sample
relies on the largest data set (85 galaxies with dynamical measurements of
$M_{\rm BH}$). Where possible, that data set provides two distinct X-ray
temperatures, one ($T_{\rm CGM}$) measured within a few effective radii of the
galaxy and another ($T_{\rm halo}$) more representative of the halo gas at
larger radii (for details, see Gaspari et al., 2019).222In our notation,
$T_{\rm CGM}$ corresponds to their $T_{\rm x,g}$ and $T_{\rm halo}$
corresponds to their $T_{\rm x,c}$.

Interestingly, Gaspari et al. (2019) found that $M_{\rm BH}$ correlates more
closely with $T_{\rm CGM}$ than with any other observable property, including
even $\sigma_{v}$, among a large set of observable galactic and X-ray
characteristics. Black circles in the left panel of Figure 1 show that
correlation, with X-ray temperature mapped onto $M_{\rm halo}$ using a
relationship

$M_{\rm 200c}=10^{15}M_{\odot}\,\left(\frac{kT_{\rm X}}{6\,{\rm
keV}}\right)^{1.7}$ (6)

based on observations by Sun et al. (2009). The original $M_{\rm
halo}$–$T_{\rm X}$ relation used X-ray data to derive the mass $M_{\rm 500c}$
within a radius encompassing a mean mass density 500 times the universe’s
critical density. Here, we have recalibrated it by setting $M_{\rm
200c}=1.5M_{\rm 500c}$, where $M_{\rm 200c}$ is defined using a density
contrast of 200 instead of 500.333This conversion factor assumes a Navarro-
Frenk-White mass profile with a concentration parameter $c_{200}\approx 4$
(e.g., Merten et al., 2015). The resulting $M_{\rm BH}$–$M_{\rm halo}$
relation

$M_{\rm BH}=10^{8.3}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}M_{\odot}}\right)^{1.6}$ (7)

aligns well with the earlier but less direct constraints (see Figures 1 and
2).

Figure 2: Comparisons between numerical simulations and the $M_{\rm
BH}$–$M_{\rm halo}$ relations inferred from observations. All lines and
symbols in common with Figure 1 represent identical quantities. Black and grey
points representing the same black hole from Gaspari et al. (2019) are
connected by horizontal dotted lines. Blue stars and inverted red triangles
represent galaxies from Terrazas et al. (2017), with shapes and shading
encoding sSFR as described in §2, and rely on the $M_{\rm halo}(\sigma_{v})$
relation in equation (8). Purple triangles show $M_{\rm BH}$ and $M_{\rm
halo}$ from the TNG100 simulation, with a thick purple line illustrating the
power-law fit for $M_{\rm halo}>10^{12}M_{\odot}$ from Truong et al. (2021).
Small orange circles show $M_{\rm BH}$ and $M_{\rm halo}$ from the EAGLE
simulation, with a thick orange line illustrating a piecewise power-law fit to
EAGLE.

Applying the same $M_{\rm 200c}$–$T_{\rm X}$ relation to $T_{\rm halo}$ leads
to a set of points (grey circles in the right panel of Figure 1) that
significantly depart from the $M_{\rm 200c}$–$T_{\rm CGM}$ relation above
$\sim 1.5\,{\rm keV}$, where $M_{\rm halo}\gtrsim 10^{14}M_{\odot}$ and
$M_{\rm BH}\gtrsim 5\times 10^{9}M_{\odot}$. Apparently, the power-law slope
of the $M_{\rm BH}$–$M_{\rm halo}$ relation flattens as halos go from the
group scale to the cluster scale. The $M_{\rm BH}$–$T_{\rm halo}$ relation
from Bogdán et al. (2018) supports this conclusion. Their sample spans a
narrower mass range than the Gaspari et al. (2019) sample, is dominated by
high-mass halos, and obtains an $M_{\rm BH}$–$T_{\rm halo}$ relation (dashed
gold line in the right panel of Figure 1) that is less steep than the Gaspari
et al. (2019) $M_{\rm BH}$–$T_{\rm halo}$ relation (dashed black line in the
right panel of Figure 1).

The apparent leveling of the $M_{\rm BH}$–$M_{\rm halo}$ relation above
$M_{\rm halo}\sim 10^{14}M_{\odot}$ may result from a qualitative change in
how supermassive black holes interact with their environments, for two
reasons: (1) it coincides with the mass scale at which halos appear to retain
nearly all of their baryons, and (2) it coincides with the mass scale at which
the circular velocity of a central galaxy no longer reflects the circular
velocity of its dark-matter halo. Observational inventories of baryons in
galaxy groups ($10^{13}M_{\odot}\lesssim M_{\rm halo}\lesssim
10^{14}M_{\odot}$) show that they contain only about half the cosmic baryon
fraction (e.g., Sun et al., 2009; Lovisari et al., 2015; Eckert et al., 2021),
while similar inventories of galaxy clusters ($M_{\rm halo}\gtrsim
10^{14.5}M_{\odot}$) find essentially all of the expected baryons (e.g., Pratt
et al., 2009). Among galaxy clusters, radiative losses plausibly balance the
black hole’s energy input (e.g., McNamara & Nulsen, 2007, 2012), but the
baryon lifting observed in lower mass halos implies that black hole power,
when integrated over time, greatly exceeds cumulative radiative losses
(Donahue & Voit, 2022). Leveling of the $M_{\rm BH}$–$M_{\rm halo}$ relation
therefore appears to happen where black hole power is no longer capable of
lifting a halo’s baryons and instead dissipates through radiative losses.
Furthermore, the pronounced differences between $T_{\rm CGM}$ and $T_{\rm
halo}$ observed among galaxy clusters reflect a disruption of the usual link
between the circular velocity of a cosmological halo and the circular velocity
of its central galaxy. In galaxy groups, $T_{\rm CGM}$ and $T_{\rm halo}$ are
typically more similar because the circular velocity of a group’s potential
well is closer to the circular velocity of its central galaxy.

Donahue & Voit (2022) have hypothesized that $M_{\rm BH}$ is more highly
correlated with $T_{\rm CGM}$ than with $T_{\rm halo}$ because it more closely
represents the halo’s baryonic binding energy at the time black hole feedback
lifted those baryons and quenched the central galaxy’s star formation. For
example, the most massive black hole in the Gaspari et al. (2019) sample
resides in NGC 4889, the central galaxy of the Coma Cluster. Its atmospheric
temperature ($T_{\rm CGM}\approx 2.4\,{\rm keV}$) is considerably lower than
the cluster’s atmospheric temperature ($T_{\rm halo}\approx 7.2\,{\rm keV}$),
which indicates $M_{\rm halo}\sim 10^{15}\,M_{\odot}$. Presumably, the baryon-
lifting event that quenched star formation in NGC 4889 predated the Coma
Cluster’s growth to such a large mass, explaining why its black hole’s mass
falls below the power-law $M_{\rm BH}$–$M_{\rm halo}$ relation followed by
lower-mass halos in the right panel of Figure 1.

X-ray assessments of the $M_{\rm BH}$–$M_{\rm halo}$ relation become
increasingly difficult as $M_{\rm halo}$ drops below $\sim 10^{13}M_{\odot}$
because there are fewer and fewer X-ray photons for making temperature
measurements. Also, the link between $M_{\rm halo}$ and X-ray temperatures
measurements may become weaker because of transient temperature fluctuations
produced by feedback events (e.g., Truong et al., 2021). However, the $M_{\rm
BH}$–$M_{\rm halo}$ relation can be extended toward lower masses using other
mass proxies.

As an example, the blue stars and inverted red triangles in Figure 2 show an
extension based on $\sigma_{v}$ assuming

$M_{\rm halo}=10^{12.9}M_{\odot}\left(\frac{\sigma_{v}}{200\,{\rm
km\,s^{-1}}}\right)^{3}$ (8)

which is equivalent to $M_{\rm 200c}$ for a singular isothermal sphere with an
isotropic velocity dispersion identical to the galaxy’s observed $\sigma_{v}$.
The stars and triangles represent galaxies from the Terrazas et al. (2017)
sample that do not appear in the Gaspari et al. (2019) sample. Their shapes
and shading represent specific star-formation rates (sSFR) equal to each
galaxy’s star-formation rate ($\dot{M}_{*}$) divided by its stellar mass
($M_{*}$):

  * •

Filled stars, $\geq 10^{-10.3}\,{\rm yr}^{-1}$

  * •

Open stars, $10^{-11.0}{\rm yr}^{-1}$ to $10^{-10.3}\,{\rm yr}^{-1}$

  * •

Open triangles, $10^{-11.7}{\rm yr}^{-1}$ to $10^{-11}\,{\rm yr}^{-1}$

  * •

Filled triangles, $\leq 10^{-11.7}\,{\rm yr}^{-1}$ .

We will return to the significance of sSFR in §2.4.2. For now, we will simply
note that those points align with the $M_{\rm BH}$–$T_{\rm CGM}$ relation.

###  2.2 Simulations

Figure 2 shows how the EAGLE (Schaye et al., 2015) and TNG100 (Pillepich et
al., 2018; Nelson et al., 2018a) simulations compare with the observations.
The EAGLE points (orange circles) overlap with the observational points up to
$M_{\rm 200c}\sim 10^{13.5}M_{\odot}$ but predict smaller black hole masses in
more massive halos. A power-law fit to the EAGLE points with $M_{\rm
200c}>10^{12.3}M_{\odot}$ gives

$M_{\rm BH}=10^{8.1}M_{\odot}\left(\frac{M_{\rm
200c}}{10^{13}M_{\odot}}\right)^{1.0}\;\;,$ (9)

but the EAGLE power-law slope is steeper at lower halo masses (Rosas-Guevara
et al., 2016). For example, the best fitting power law for
$10^{11.5}M_{\odot}<M_{\rm 200c}<10^{12.3}M_{\odot}$ is $M_{\rm BH}\propto
M_{\rm halo}^{2.0}$. A thick orange line in Figure 2 illustrates the two
pieces of this piecewise power-law fit. The high-mass flattening of the EAGLE
relation qualitatively agrees with observations but sets in nearer to $M_{\rm
halo}\sim 10^{12.3}M_{\odot}$ than to $\sim 10^{14}M_{\odot}$. Interestingly,
fitting all of the EAGLE points having $M_{\rm 200c}>10^{11.5}M_{\odot}$ with
a single power law yields a relation with essentially the same slope found by
Booth & Schaye (2010) but a slightly greater $M_{\rm BH}$ normalization.

The IllustrisTNG points (small purple triangles) representing $M_{\rm
BH}$–$M_{\rm 200c}$ are less well aligned with the observational constraints.
A thick purple line shows the power-law fit

$M_{\rm BH}=10^{8.8}M_{\odot}\left(\frac{M_{\rm
200c}}{10^{13}M_{\odot}}\right)^{0.76}$ (10)

from Truong et al. (2021) for halos with $M_{\rm 200c}>10^{12}M_{\odot}$. It
agrees with the $M_{\rm BH}$ observations among the most massive halos
($M_{\rm halo}\gtrsim 10^{14}M_{\odot}$) but exceeds them at $M_{\rm
halo}\lesssim 10^{13.5}M_{\odot}$, ending up near $M_{\rm BH}\sim
10^{8}M_{\odot}$ at $M_{\rm halo}\sim 10^{12}M_{\odot}$. The anomalously large
IllustrisTNG black hole masses at $M_{\rm halo}\sim 10^{12}M_{\odot}$ have
previously been noted by Li et al. (2020), in the context of the $M_{\rm
BH}$–$\sigma_{v}$ relation, and by both Terrazas et al. (2016, 2017) and
Habouzit et al. (2021), in the context of the $M_{\rm BH}$–$M_{*}$ relation.

###  2.3 Accretion versus Mergers

Black holes in IllustrisTNG halos above $M_{\rm halo}\sim 10^{12}M_{\odot}$
accumulate mass primarily through mergers with other black holes (Weinberger
et al., 2018). Merger-dominated growth therefore results in a sub-linear
$M_{\rm BH}$–$M_{\rm halo}$ relation (Truong et al., 2021). However, Figure 1
shows that the observed $M_{\rm BH}$–$T_{\rm CGM}$ relation indicates that the
$M_{\rm BH}$–$M_{\rm halo}$ relation is super-linear up to $M_{\rm halo}\sim
10^{14}M_{\odot}$. Those observations therefore imply either (1) that $M_{\rm
BH}$ grows in proportion to $M_{\rm halo}^{1.6}$ as halo mass evolves up to
$\sim 10^{14}M_{\odot}$, or (2) that black holes in halos that will eventually
merge to form a $\sim 10^{14}M_{\odot}$ halo grow through accretion to greater
masses than black holes forming in halos destined to reach lower halo masses.
This latter possibility would imply that black-hole accretion early in time is
influenced by environmental effects extending beyond the borders of its own
cosmological halo. The implications of EAGLE’s nearly linear $M_{\rm
BH}$–$M_{\rm halo}$ relation at high masses are less clear and may indicate a
combination of merger-driven and accretion-driven growth beyond $M_{\rm
halo}\sim 10^{12.3}M_{\odot}$.

###  2.4 Halo Mass Proxies

Figure 3: Relationships between $M_{\rm BH}$ and $M_{\rm halo}$ inferred from
mass proxies other than X-ray temperature. Left panel: The $M_{\rm
BH}$–$M_{\rm halo}$ relation inferred from $M_{\rm halo}(\sigma_{v})$ using
equation (8). Right panel: The $M_{\rm BH}$–$M_{\rm halo}$ relation inferred
from $M_{*}$ using abundance matching of $M_{*}$ with $M_{\rm halo}$ at
$z\approx 0$ via the Universe Machine (Behroozi et al., 2019). All symbols
represent the same quantities as in Figure 2, except that the entire Terrazas
et al. (2017) sample is shown, not just the subset without X-ray measurements
from Gaspari et al. (2019).

According to Figure 2, black holes with $M_{\rm BH}\approx 10^{8}\,M_{\odot}$
tend be found in halos close to $10^{13}\,M_{\odot}$ in mass, but in
IllustrisTNG they reside in halos an order of magnitude less massive. Is it
possible that the $M_{\rm halo}$ proxies shown in Figure 2 overestimate the
mean halo masses of black holes with $M_{\rm BH}\approx 10^{8}M_{\odot}$ by
nearly an order of magnitude? That is the size of the adjustment needed to
align the observations with the IllustrisTNG $M_{\rm BH}$–$M_{\rm halo}$
relation. To explore that possibility, we can consider what happens when halo
masses are inferred from mass proxies other than X-ray temperature.

####  2.4.1 $M_{\rm BH}$ and $\sigma_{v}$

The left panel of Figure 3 illustrates the $M_{\rm BH}$–$M_{\rm halo}$
relations obtained using $\sigma_{v}$ as a mass proxy. Points based on X-ray
data have been removed, but all other symbols remain as they were in Figure 2.
The $M_{\rm BH}$–$\sigma_{v}$ relation has been transmuted into $M_{\rm
BH}$–$M_{\rm halo}$ using equation (8). It follows the power-law $M_{\rm
BH}$–$M_{\rm halo}$ relation predicted by the baryon lifting hypothesis up to
$\sigma_{v}\sim 240\,{\rm km\,s^{-1}}$, at which $M_{\rm halo}\sim
10^{13.1}M_{\odot}$ and $M_{\rm BH}\sim 10^{9}M_{\odot}$. Beyond there, the
$M_{\rm BH}$–$\sigma_{v}$ relation becomes much steeper than $M_{\rm
BH}\propto\sigma_{v}^{5}$. However, the observed $M_{\rm BH}$–$T_{\rm X}$
relations show no such break at the same location. Comparing with Figure 2
demonstrates that the steeper trend arises because $\sigma_{v}$ is no longer a
good proxy for $M_{\rm halo}$. This upturn in the $M_{\rm BH}$–$\sigma_{v}$
relation is well known (e.g., McConnell & Ma, 2013; Bogdán et al., 2018; Sahu
et al., 2019), and indicates that some physical process (such as the “black
hole feedback valve” outlined in Voit et al., 2020) prevents the velocity
dispersion of a halo’s central galaxy from rising in proportion with the
halo’s maximum circular velocity once it reaches $\sigma_{v}\sim 240\,{\rm
km\,s^{-1}}$.

####  2.4.2 $M_{\rm BH}$ and $M_{*}$

On the right side of Figure 3, $M_{\rm halo}$ is inferred from $M_{*}$ via the
abundance-matching fit of Behroozi et al. (2019) at $z\approx 0$. The scatter
in $M_{\rm BH}$ at fixed $M_{*}$ is impressively large, indicating that
$M_{*}$ is a poor halo mass proxy for this purpose. Stellar bulge mass might
be a better proxy for halo mass, given its tighter correlation with $M_{\rm
BH}$ (e.g., Häring & Rix, 2004; Gültekin et al., 2009; Kormendy & Ho, 2013;
McConnell & Ma, 2013; Savorgnan et al., 2016), but certain features of the
$M_{\rm BH}$–$M_{*}$ relation suggest that $M_{\rm BH}$ may anticorrelate with
$M_{*}$ at fixed halo mass.

For example, consider just the red triangles (both filled and unfilled)
representing quenched galaxies, which tend to be bulge-dominated. Nine such
triangles near $M_{*}\sim 10^{10.7}M_{\odot}$ also have $M_{\rm
BH}>10^{8}M_{\odot}$ and seem to be consistent with the IllustrisTNG $M_{\rm
BH}$–$M_{\rm halo}$ relation. However, both Figure 2 and the left panel of
Figure 3 show no data points in that region, because both $\sigma_{v}$ and
$kT_{\rm X}$ for those galaxies indicate greater halo masses. The median
temperature among those nine galaxies is $kT_{\rm X}\approx 0.3\,{\rm keV}$,
and the median velocity dispersion is $\sigma_{v}\approx 238\,{\rm
km\,s^{-1}}$, implying a median halo mass ($\sim 10^{13}M_{\odot}$) that
places those same galaxies closer to the EAGLE $M_{\rm BH}$–$M_{\rm halo}$
relation. Systematic uncertainties among various sets of $M_{\rm halo}$
proxies might therefore explain why the apparent dispersion of $M_{\rm BH}$ at
$M_{\rm halo}\sim 10^{12-12.5}M_{\odot}$ is so large in data sets that combine
several different halo-mass proxies (see e.g., Figure 1 of Marasco et al.,
2021).

The anti-correlation between sSFR and $M_{\rm BH}$ found by Terrazas et al.
(2016, 2017) in their sample provides a clue as to why the scatter in $M_{\rm
BH}$ at fixed $M_{*}$ is so large. At any given halo mass, stellar masses
within the star-forming subset of galaxies are still increasing, while the
stellar masses of the quiescent subset are not. It is therefore likely that
some of the quiescent galaxies have stellar masses that are unusually small
for their halo mass. Additionally, black hole masses in the quiescent
population might be unusually large for their stellar mass, precisely because
they have already experienced episodes of rapid black-hole growth that have
lifted the halo’s baryons and quenched star formation, resulting in a large
dispersion in $M_{\rm BH}$ near $M_{*}\sim 10^{10.7}M_{\odot}$, where the
quiescent and star-forming populations strongly overlap.

####  2.4.3 Feedback and $T_{\rm CGM}$

Another possibility to assess is that the Gaspari et al. (2019) galaxies with
$kT_{\rm CGM}\sim 0.3\,{\rm keV}$ are indicating halo masses that are
approximately an order of magnitude too large. If $M_{\rm halo}$ is indeed
overestimated by that much, then correcting for the overestimate would place
those galaxies on the IllustrisTNG relation, with $M_{\rm BH}\sim
10^{8}M_{\odot}$ corresponding to $M_{\rm halo}\sim 10^{12}M_{\odot}$. For
example, such an overestimate might happen if kinetic feedback produces
temperature fluctuations several times greater than what $T_{\rm CGM}$ would
be in hydrostatic equilibrium.

Truong et al. (2021) have performed mock X-ray observations of IllustrisTNG
galaxies showing that the TNG feedback mechanism does produce biases in
apparent temperature large enough to account for the apparent offset in halo
mass. However, CGM temperatures in the Gaspari et al. (2019) sample show no
evidence for such large departures from hydrostatic equilibrium. Figure 4
presents the relationship between $\sigma_{v}$ and $kT_{\rm CGM}$ in that
sample, along with three lines representing the hydrostatic relation

$kT=\frac{2\mu
m_{p}\sigma_{v}^{2}}{\alpha}\;\;\;\;\;,\;\;\;\;\;\alpha\equiv\left|\frac{d\ln
P}{d\ln r}\right|\;\;,$ (11)

for $\alpha=1$, 1.5, and 2, given an isotropic velocity dispersion. Those
values of $\alpha$ are representative for this sample and account for the
spread in $kT_{\rm CGM}$ at fixed $\sigma_{v}$. If there were a feedback-
induced departure from the hydrostatic relations below 0.5 keV, one would
expect to see an excess of galaxies above the $\alpha=1$ line at low
$\sigma_{v}$, but there is just one outlier there. It is NGC 7331, which has
$\sigma_{v}=115\,{\rm km\,s^{-1}}$ near its center but $v_{c}\approx 250\,{\rm
km\,s^{-1}}$ at 30 kpc (Bottema, 1999), indicating a greater halo mass than
its central stellar velocity dispersion implies. That circular velocity is
equivalent to $\sigma_{v}\approx 180\,{\rm km\,s^{-1}}$, making the CGM
temperature of NGC 7331 consistent with hydrostatic equilibrium at
$\alpha\approx 1$.

####  2.4.4 A Closer Look at TNG

We therefore conclude that the IllustrisTNG $M_{\rm BH}$–$M_{\rm halo}$
relation is in strong tension with the available observational constraints.
Those simulations consequently seem to be inconsistent with the proposed
three-way link between black hole growth, baryon lifting, and quenching of
star formation, but they are not. The rest of the paper looks more closely at
IllustrisTNG and shows that both black hole growth and quenching of star
formation are indeed linked to baryon lifting, despite the anomalous $M_{\rm
BH}$–$M_{\rm halo}$ relation. Sections 3 and 4 outline how baryon lifting in
IllustrisTNG is linked to star formation and black hole growth. Sections 5 and
6 explain why the IllustrisTNG $M_{\rm BH}$–$M_{\rm halo}$ relation is
anomalous and discuss what might be done to improve it.

Figure 4: Relationship between $\sigma_{v}$ and $kT_{\rm CGM}$ in the Gaspari
et al. (2019) galaxy sample. Colored lines indicate hydrostatic temperatures
corresponding to $\sigma_{v}$ for $\alpha\equiv|d\ln P/d\ln r|$ equal to 1,
1.5, and 2, as labeled. These values of $\alpha$ are representative of the
range observed among massive elliptical galaxies. There is no obvious
departure from those relations at low $\sigma_{v}$ that would indicate a
temperature enhancement produced by black-hole feedback. The only outlier is
NGC 7331, which has an observed rotation speed $v_{\rm c}\approx 250\,{\rm
km\,s^{-1}}$ (Bottema, 1999), indicating that $\sigma_{v}$ does not reflect
its halo mass. A green arrow shows where NGC 7331 ends up if $v_{\rm
c}/\sqrt{2}$ is used instead of $\sigma_{v}$.

##  3 Lifting and Quenching

Figure 5: Dependence of specific star formation rate (sSFR) on halo gas
fraction and halo mass ($M_{\rm 500c}$) across the redshift range $0\leq z\leq
3$ in the TNG100 simulation. Solid black lines show the median halo gas
fraction at each halo mass, and dotted lines show the 10th and 90th
percentiles. Colored squares show the typical sSFR associated with each
combination of halo mass and halo gas fraction. Red squares representing low
sSFR are systematically associated with lower halo gas fraction, indicating
that star-formation quenching in IllustrisTNG galaxies is linked to feedback
that lifts a halo’s baryons to greater altitudes. (The halo gas fractions
plotted here and in Figure 9 include gas at all temperatures.)

Previous work has already established that baryon lifting coincides with star-
formation quenching in both the IllustrisTNG and EAGLE cosmological
simulations (Bower et al., 2017; Davies et al., 2019, 2020; Oppenheimer et
al., 2020; Terrazas et al., 2020; Zinger et al., 2020; Piotrowska et al.,
2022). Figure 5 illustrates one of the key findings: central galaxies with
quenched star formation in the TNG100 simulation have less halo gas than
galaxies with active star formation.444 Figures 5 through 9 come from the
IllustrisTNG plotting tool at https://www.tng-project.org/data/groupcat/
thanks to Nelson et al. (2019). The figure plots the halo gas mass fraction
($f_{\rm gas}\equiv M_{\rm gas,500}/M_{\rm 500c}$) as a function of $M_{\rm
500c}$. Colors indicate the median sSFR at each combination of $f_{\rm gas}$
and $M_{\rm 500c}$. Red squares representing suppressed star formation are
prevalent among halos of mass $M_{\rm 500c}\gtrsim 10^{12.5}\,M_{\odot}$
across the redshift range $0\leq z\leq 3$. Among lower-mass halos, blue
squares representing active star formation correspond to larger halo gas
fractions than the red squares representing suppressed star formation.
Galaxies in the EAGLE simulation follow the same qualitative trend (Davies et
al., 2019), but $f_{\rm gas}$ in EAGLE is generally a factor of $\sim 3$
smaller at $M_{\rm halo}\lesssim 10^{12}\,M_{\odot}$ than in IllustrisTNG
(Davies et al., 2020). Baryon lifting in low-mass halos must therefore proceed
somewhat differently in the two simulation environments.

Figure 6: Dependence of specific star formation rate (sSFR) on cumulative
kinetic black hole feedback ($E_{\rm kin}=\int E_{\rm injected,low}$) and halo
mass ($M_{\rm 500c}$) across the redshift range $0\leq z\leq 3$ in the TNG100
simulation. Solid black lines show the median amount of cumulative kinetic
energy injection at each halo mass, and dotted lines show the 10th and 90th
percentiles. Colored squares show the typical sSFR associated with each
combination of halo mass and injected energy. Purple dashed lines show the
characteristic scale of baryonic binding energy ($E_{\rm B}$) at each halo
mass, and purple dotted lines in the lower right panels show the $M_{\rm
500c}$–$E_{\rm B}$ relation at $z=0$. At each halo mass and across all
redshifts, the dark red squares indicating quenched star formation
($\rm{sSFR}\lesssim 10^{-2}\,{\rm Gyr}^{-1}$) are almost entirely above the
dashed lines, and the lower edge of the quenched galaxy population tracks
those lines. This correspondence implies that IllustrisTNG galaxies become
quenched when kinetic feedback injects energy sufficient to lift the halo’s
baryons, thereby lowering the gas pressure and increasing the cooling time of
hot gas near the central black hole.

Figure 6 shows that star-formation rates in IllustrisTNG are also closely
related to the central black hole’s cumulative kinetic energy input ($E_{\rm
kin}$), which includes the kinetic energy released by smaller black holes that
have merged with the central one. A purple dashed line in each panel
represents the quantity

$E_{\rm B}\equiv\frac{3}{5}\frac{GM_{\rm 200c}^{2}}{r_{\rm 200}}f_{\rm
b}\;\;,$ (12)

which is an estimate of the initial binding energy of the halo’s baryons, for
the cosmic mean baryon fraction $f_{\rm b}$ (Nelson et al., 2018b). It
corresponds to a uniform sphere and should not be considered exact. But
notably, the transition to highly suppressed star formation (dark red squares)
lies close to that line across the redshift range $0\leq z\leq 3$, indicating
that star formation becomes quenched when the kinetic energy input associated
with black hole accretion exceeds the amount of energy required to lift the
circumgalactic gas.

Terrazas et al. (2020) presented similar results. Their Figure 4 shows that
the sSFR of an IllustrisTNG galaxy starts to decline when $E_{\rm kin}$
exceeds the gravitational binding energy of gaseous baryons currently within
the galaxy ($E_{\rm bind,gal}$) and declines much more rapidly once $E_{\rm
kin}\gtrsim 10\,E_{\rm bind,gal}$. Most of the quenched galaxies end up with
$E_{\rm kin}\gg 100\,E_{\rm bind,gal}$. Also, Figure 6 in Terrazas et al.
(2020) shows that $E_{\rm kin}$ among the quenched galaxies is typically an
order of magnitude greater than the gravitational binding energy of the
gaseous baryons remaining within the halo ($E_{\rm bind,halo}$). However,
neither $E_{\rm bind,gal}$ nor $E_{\rm bind,halo}$ scales linearly with
$E_{\rm B}$ because their values decline precipitously as black hole feedback
starts to lift baryons out of both the galaxy and the halo that contains it.

Figure 6 of this paper is therefore complementary to the figures in Terrazas
et al. (2020), because it compares $E_{\rm kin}$ to an atmospheric binding
energy scale ($E_{\rm B}$) that remains steady while feedback rapidly acts to
lift the halo’s baryons. The value of $E_{\rm B}$ at fixed $M_{\rm 500c}$
declines slowly with time because the specific binding energy of a
cosmological halo555Bounded by a radius $\propto M_{\rm halo}^{1/3}\rho_{\rm
cr}^{-1/3}$ is $\propto M_{\rm halo}^{2/3}\rho_{\rm cr}^{1/3}$ and the
cosmological critical density $\rho_{\rm cr}$ declines as the universe ages.
Dotted purple lines in Figure 6 show how much greater $E_{\rm B}(M_{\rm
500c})$ is at $z\geq 2$ than at $z=0$. The lower envelope of the dark red
squares representing quenching is correspondingly at greater $E_{\rm B}$.

Interestingly, the distribution of $E_{\rm kin}$ at fixed $M_{\rm 500c}$ among
IllustrisTNG halos with quenched central galaxies becomes narrower as halo
mass increases. Meanwhile, the median value of $E_{\rm kin}$ at fixed $M_{\rm
500c}$ converges toward $E_{\rm B}$, becoming nearly equal to it as halo mass
approaches $\sim 10^{14}\,M_{\odot}$. In halos that are even more massive,
$E_{\rm B}$ exceeds $E_{\rm kin}$. This outcome is qualitatively consistent
with the observed rise in $f_{\rm gas}$ as halo masses go from $\sim
10^{13.5}M_{\odot}$ to $\sim 10^{14.5}M_{\odot}$ (e.g., Pratt et al., 2009;
Sun et al., 2009; Lovisari et al., 2015; Eckert et al., 2021).

Zinger et al. (2020) have shown how black hole feedback in IllustrisTNG alters
the central entropy and cooling time in massive halos as star formation shuts
down. Early feedback is overwhelmingly thermal and relatively ineffective at
quenching star formation. The transition to quiescence does not happen until
kinetic feedback becomes significant. During that transition to kinetic
feedback, the entropy666 As represented by $K=kTn^{-2/3}$, where $kT$ is the
gas temperature in energy units and $n$ is the number density of gas particles
of the circumgalactic atmosphere rises above $\sim 10\,{\rm keV\,cm^{2}}$ and
its cooling time rises above $\sim 1$ Gyr.

In halos of mass $\lesssim 10^{13.5}\,M_{\odot}$, baryon lifting is a
necessary consequence of the transition to kinetic feedback, because those
increased entropy levels and cooling times correspond to gas densities smaller
than $f_{\rm b}$ times the total matter density. Making the transition happen
therefore requires an energy input roughly equivalent to $E_{\rm B}$.
Initially, suppression of star formation in IllustrisTNG may result from
“ejective” feedback that expels cool gas clouds from the galaxy, but long-term
quiescence requires “preventative” feedback that limits the galaxy’s supply of
cold gas by increasing the entropy and cooling time of the circumgalactic
medium (CGM), which entails lifting of the entire atmosphere.

##  4 Modes of Black Hole Growth

Figure 7: Dependence of black hole feedback mode on cumulative kinetic black
hole feedback ($E_{\rm kin}=\int E_{\rm injected,low}$) and halo mass ($M_{\rm
500c}$) across the redshift range $0\leq z\leq 3$ in the TNG100 simulation.
Solid black lines show the median amount of cumulative kinetic energy
injection at each halo mass, and dotted lines show the 10th and 90th
percentiles. Colored squares show the typical feedback mode associated with
each combination of halo mass and injected energy: Green squares represent the
thermal feedback mode associated with higher accretion rates, and yellow
squares represent the kinetic feedback mode associated with lower accretion
rates. Purple dashed lines approximately proportional to $M_{\rm halo}^{5/3}$
show the characteristic scale of baryonic binding energy ($E_{\rm B}$) at each
halo mass. At each halo mass and across all redshifts, the yellow squares
representing the kinetic feedback mode are almost entirely above that line,
and the lower edge of the population in which kinetic feedback dominates
tracks that line. The upper edge of that population tracks long-dashed green
lines that are proportional to halo mass. Those bounds imply that the kinetic
feedback mode in IllustrisTNG tunes itself to supply a total energy that is
tied to the halo’s baryonic binding energy in the mass range
$10^{12}\,M_{\odot}\lesssim M_{\rm halo}\lesssim 10^{14}\,M_{\odot}$.

Now we turn to the connection between black hole growth and suppression of
star formation. The previous section showed that quenching in IllustrisTNG
coincides with a cumulative kinetic energy input $E_{\rm kin}$ that exceeds
the halo’s baryonic binding energy scale. The transition to a quiescent state
generally happens near $M_{\rm halo}\sim 10^{12}\,M_{\odot}$, at which $E_{\rm
B}\sim 10^{59}\,{\rm erg}$. The corresponding amount of black hole mass growth
is

$\Delta M_{\rm BH}\>\gtrsim\>\frac{E_{\rm B}}{\epsilon_{\rm kin}c^{2}}$ (13)

in which $\epsilon_{\rm kin}$ is the conversion efficiency of accreted rest-
mass energy into kinetic feedback energy and $c$ is the speed of light. In
IllustrisTNG, this relationship results in

$\Delta M_{\rm BH}\>\gtrsim\>3\times 10^{5}\,M_{\odot}\left(\frac{E_{\rm
B}}{10^{59}\,{\rm erg}}\right)\;\;,$ (14)

given $\epsilon_{\rm kin}=0.2$ (Weinberger et al., 2017). The total mass
accumulated during kinetic-mode accretion is always subdominant compared to
the mass accumulated during prior thermal-mode accretion because the
efficiency factor assigned to kinetic-mode accretion is so large (Weinberger
et al., 2018). Interpreting the $M_{\rm BH}$–$M_{\rm halo}$ relations emerging
from IllustrisTNG therefore requires close attention to what governs the
transition between feedback modes.

IllustrisTNG feedback modes depend on how the instantaneous accretion rate
($\dot{M}_{\rm BH}$) onto a halo’s central black hole compares with the
limiting Eddington rate

$\dot{M}_{\rm Edd}=\frac{4\pi GM_{\rm BH}m_{p}}{\epsilon_{\rm rad}\sigma_{\rm
T}c}\;\;,$ (15)

where $G$ is the gravitational constant, $m_{p}$ is the proton mass,
$\sigma_{\rm T}$ is the Thomson electron scattering cross section, and
$\epsilon_{\rm rad}$ is the conversion efficiency of accreted rest mass to
radiative energy. In the fiducial IllustrisTNG model (Weinberger et al.,
2017), black hole feedback is in thermal mode when

$\frac{\dot{M}_{\rm BH}}{\dot{M}_{\rm
Edd}}\>>\>\min\left[0.002\left(\frac{M_{\rm
BH}}{10^{8}\,M_{\odot}}\right)^{2},0.1\right]\;\;.$ (16)

Otherwise, the feedback mode is kinetic. The mass-dependent factor in equation
(16) favors kinetic feedback as $M_{\rm BH}$ rises above $\sim
10^{8}\,M_{\odot}$. However, the thermal mode is still active when
$\dot{M}_{\rm BH}>0.1\dot{M}_{\rm Edd}$, even if the black hole is very
massive.

Figure 7 shows the joint dependence of feedback mode on both $M_{\rm 500c}$
and $E_{\rm kin}$ in the TNG100 simulation. The axes are identical to Figure
6. Comparing the two figures shows that kinetic feedback prevails among
galaxies with quenched star formation. Just as in Figure 6, galaxies undergo a
transition as $E_{\rm kin}$ surpasses $E_{\rm B}$, switching to predominantly
kinetic feedback.

The twin transitions in both feedback mode and star formation behavior depend
on two factors. First, a black hole’s mass needs to approach
$10^{8}\,M_{\odot}$ for the TNG implementation of kinetic feedback to come
into play. In the green regions of Figure 7, where the thermal mode dominates,
episodes of kinetic feedback must still sometimes occur, because $E_{\rm kin}$
is rising toward $E_{\rm B}$. Those kinetic feedback episodes become
increasingly likely as $M_{\rm BH}$ grows, because of the relationship in
equation (16). Eventually the kinetic mode dominates, resulting in both star-
formation quenching and baryon lifting. Second, thermal mode feedback becomes
strongly disfavored as a transitioning galaxy loses its cold, dense clouds.
The reason is that $\dot{M}_{\rm BH}$ in IllustrisTNG is taken to be the local
Bondi accretion rate (Bondi, 1952), which depends strongly on the specific
entropy of accreting gas.777 The Bondi accretion rate is $\propto M_{\rm
BH}^{2}K^{-3/2}$. Whenever the black hole is surrounded by the hot, high-
entropy ambient gas characteristic of a quenched galaxy, accretion is slower,
making the kinetic feedback more likely.

Previous analyses of quenching and feedback mode in IllustrisTNG have focused
more closely on the role of $M_{\rm BH}$ than on multiphase gas and its role
in black hole fueling. For example, Weinberger et al. (2018) showed that the
median sSFR of IllustrisTNG galaxies dramatically drops as a direct result of
kinetic feedback as $M_{\rm BH}$ rises above $\sim 10^{8.2}\,M_{\odot}$.
Terrazas et al. (2020) came to a similar conclusion and also showed that
$f_{\rm gas}$ dramatically declines at the same black hole mass threshold.

Superficially, quenching of star formation in IllustrisTNG may seem to depend
most strongly on the threshold value of $M_{\rm BH}$ marking the onset of
kinetic feedback, but cumulative kinetic energy input ($E_{\rm kin}$) turns
out to be even more critical (Terrazas et al., 2020). Figure 8 shows the joint
dependence of black hole feedback mode on both $M_{\rm BH}$ and $E_{\rm kin}$.
As in Figure 7, the transition to kinetic mode depends most directly on how
$E_{\rm kin}$ compares with $E_{\rm B}$. If a threshold in $M_{\rm BH}$ were
more critical, then the boundary between the green and yellow regions would be
vertical in Figure 8. Instead, the boundary is diagonal and closely coincides
with the line marking $E_{\rm kin}=E_{\rm B}$.

Figure 8: Dependence of black hole feedback mode on cumulative kinetic black
hole feedback ($E_{\rm kin}=\int E_{\rm injected,low}$) and black hole mass
($M_{\rm BH}$) across the redshift range $0\leq z\leq 3$ in the TNG100
simulation. Solid black lines show the median amount of cumulative kinetic
energy injection at each halo mass, and dotted lines show the 10th and 90th
percentiles. Colored squares show the typical feedback mode associated with
each combination of halo mass and injected energy: Green squares represent the
thermal feedback mode associated with higher accretion rates, and yellow
squares represent the kinetic feedback mode associated with lower accretion
rates. Purple dashed lines show the characteristic scale of baryonic binding
energy ($E_{\rm B}$) at each black hole mass. The transition to dominant
kinetic feedback occurs as the cumulative kinetic energy input surpasses the
halo’s baryonic binding energy and happens at lower black hole masses within
less massive halos. Consequently, cumulative kinetic feedback input prior to
the transition is the primary cause of that transition.

Figure 8 also shows that the transitional values of $M_{\rm BH}$ are larger in
higher-redshift galaxies. Given how $\dot{M}_{\rm BH}/\dot{M}_{\rm Edd}$
determines the feedback mode, this redshift dependence indicates that the
Bondi accretion rates onto the most massive black holes in IllustrisTNG are
generally greater early in time than later in time, causing the thermal mode
to dominate among black holes with masses approaching $10^{8.5}\,M_{\odot}$ at
$z\approx 3$. At lower redshifts, the transitional value of $M_{\rm BH}$
clearly correlates with $E_{\rm kin}$ and does not exceed $M_{\rm BH}\approx
10^{8.3}\,M_{\odot}$ at $z=0$ for any value of $E_{\rm kin}$. This decline
with time in the maximum $M_{\rm BH}$ at which thermal mode feedback occurs
implies that the typical specific entropy ($K\propto kTn^{-2/3}$) of gas near
black holes of mass $M_{\rm BH}\sim 10^{8.3-8.5}\,M_{\odot}$ is lower at
$z\sim 3$ than at $z\sim 0$, corresponding to greater pressure at a given gas
temperature, presumably because of greater gas accretion rates onto those
high-redshift galaxies.

Figure 9: Dependence of halo gas fraction on cumulative kinetic black hole
feedback ($E_{\rm kin}=\int E_{\rm injected,low}$) and black hole mass
($M_{\rm BH}$) across the redshift range $0\leq z\leq 3$ in the TNG100
simulation. Solid black lines show the median amount of cumulative kinetic
energy injection at each halo mass, and dotted lines show the 10th and 90th
percentiles. Colored squares show the halo gas fraction associated with each
combination of halo mass and injected energy: Redder squares represent halo
gas fractions substantially lower than the cosmic mean baryon fraction,
indicating that feedback has lifted the halo’s baryons. Purple dashed lines
show the characteristic scale of baryonic binding energy ($E_{\rm B}$) at each
black hole mass. The transition to low halo gas fractions occurs as the
cumulative kinetic energy input surpasses the halo’s baryonic binding energy
and coincides with the transitions to both star-formation quenching and a
shutdown in thermal mode feedback.

The diagonal trend of each green region in Figure 8, sharply rising from lower
left to upper right, indicates that episodes of kinetic feedback still
sometimes occur among black holes below the transitional mass, but they must
be rare. In the panels of Figure 8, some of the black holes near $M_{\rm
BH}\sim 10^{7.5}\,M_{\odot}$ have managed to generate $E_{\rm kin}\gtrsim
10^{54}\,{\rm erg}$. As their cumulative kinetic output then grows to reach
$\sim 10^{59}\,{\rm erg}$, those black holes accrete another $\sim
10^{8}\,M_{\odot}$. Only $\sim 3\times 10^{5}\,M_{\odot}$ of that mass
increase comes from accretion associated with kinetic feedback (see equation
13), corresponding to $\lesssim 0.3$% of the total.

Figure 9 confirms that star-formation quenching does indeed coincide with
baryon lifting brought about by kinetic feedback. It illustrates how $f_{\rm
gas}$ depends jointly on $M_{\rm BH}$ and $E_{\rm kin}$. In general, the red
and dark orange points indicating large reductions in halo gas density lie
above the purple dashed lines marking $E_{\rm kin}=E_{\rm B}$. Those same
lines also mark the transition to a quenched state in Figure 6. Except for the
spurious tail of dark red squares at low $E_{\rm kin}$ and $M_{\rm BH}$ in the
$z=0$ panel of Figure 9 (to be discussed in §5), there is no systematic
dependence of $f_{\rm gas}$ on either $E_{\rm kin}$ or $M_{\rm BH}$ below the
purple dashed lines. The lack of dependence on $M_{\rm BH}$ implies that
thermal mode feedback does not contribute to baryon lifting, because
cumulative thermal energy injection by the thermal mode is proportional to
$M_{\rm BH}$.

In contrast, reductions in halo gas content clearly depend on how $E_{\rm
kin}$ compares to $E_{\rm B}$, with the greatest reductions in $f_{\rm gas}$
corresponding to $E_{\rm kin}\gg E_{\rm B}$ (see also Terrazas et al., 2020).
At $M_{\rm BH}\approx 10^{8.3}\,M_{\odot}$ in the $z=0$ panel of Figure 9, the
median $E_{\rm kin}$ exceeds $E_{\rm B}$ by an order of magnitude, and that is
where the reductions in $f_{\rm gas}$ are greatest. The excess of $E_{\rm
kin}$ over $E_{\rm B}$ then declines with increasing $M_{\rm BH}$, until the
two quantities are almost equal at $M_{\rm BH}\approx 10^{9.5}\,M_{\odot}$,
where $E_{\rm B}\approx 10^{61.5}\,{\rm erg}$. In that part of Figure 9 the
squares indicating $f_{\rm gas}$ are typically yellow. Therefore, baryon
lifting is minimal for $E_{\rm kin}\lesssim E_{\rm B}$ at both low and high
halo masses.

Convergence of $f_{\rm gas}$ back toward the cosmic mean at high masses is
consistent with the general trend observed among real galaxy groups and
clusters and can be understood in terms of radiative cooling. Figures 6 and 7
show that $M_{\rm BH}\approx 10^{9.5}\,M_{\odot}$ corresponds to $M_{\rm
500c}\approx 10^{14}\,M_{\odot}$ at $z=0$ in the IllustrisTNG universe. In the
real universe, halos with similar masses currently have X-ray luminosities
$\sim 10^{44}\,{\rm erg\,s^{-1}}$, meaning that they can radiate $\sim
10^{61.5}\,{\rm erg}$ over the course of cosmic time, thereby converting a
comparable amount of injected feedback energy into photons rather than into
atmospheric gravitational potential energy. Consequently, black hole feedback
in galaxy clusters ($M_{\rm halo}\sim 10^{14-15}\,M_{\odot}$) can self-
regulate by balancing radiative losses, without much baryon lifting.

However, different simulations make strikingly different predictions for the
radial distributions of baryons in and around massive halos (Oppenheimer et
al., 2021; Sorini et al., 2022). Recently, Ayromlou et al. (2023) compared the
baryon distributions emerging from IllustrisTNG, EAGLE, and also the SIMBA
simulation (Davé et al., 2019), finding that black hole feedback lifts a
halo’s baryons least effectively in EAGLE and most effectively in SIMBA. The
radial profiles of those baryon distributions are largely unconstrained by
existing observations of halos below $10^{13.5}\,M_{\odot}$, but notably
IllustrisTNG and EAGLE appear to overlap X-ray observations of massive groups
more closely than SIMBA (Oppenheimer et al., 2021).

Differences among the simulations are to be expected, given how crude their
black-hole feedback implementations remain. In EAGLE, that feedback is purely
thermal, at a temperature chosen to minimize radiative losses (Schaye et al.,
2015; Crain et al., 2015). The kinetic feedback mode of IllustrisTNG injects
energy through a series of randomly oriented impulses (Weinberger et al.,
2017). SIMBA’s kinetic black hole feedback is bipolar (Davé et al., 2019).
None of those simulations reproduces the distinctive jet-lobe radio
morphologies observed among massive halos with active feedback (Donahue &
Voit, 2022). Refinements of their feedback algorithms will benefit from paying
close attention to observations of jets, X-ray cavities, and radio lobes,
which reflect the jet power and zone of influence more directly than they
reflect cumulative feedback energy. Nevertheless, cumulative black hole
feedback energy in all of them suffices to lift the atmospheres of halos with
$M_{\rm halo}\sim 10^{12.5}$–$10^{14}\,M_{\odot}$, meaning that atmospheric
lifting is linked to black hole growth in all such simulations, as long as
feedback energy couples to the circumgalactic medium without significant
radiative losses.

##  5 The Price of Feedback

Whether or not the masses of real black holes reflect the energy input
required for quenching of star formation depends on the price of feedback.
Assuming that baryon lifting is necessary for long-term quenching implies that
a central black hole must inject an amount of energy at least as great as the
halo’s baryonic binding energy ($E_{\rm B}$) into the CGM. The injected energy
comes at a “price” of at least

$\Delta M_{\rm BH}\,\sim\,\frac{E_{\rm B}}{\epsilon_{\rm fb}c^{2}}$ (17)

in black hole mass growth that depends on the conversion efficiency
$\epsilon_{\rm fb}$ of accreted rest-mass energy into feedback energy. If
coupling of feedback energy to the CGM is highly inefficient, as happens
during episodes of thermal mode feedback in IllustrisTNG, then the price can
be much greater.

Donahue & Voit (2022) show that the $M_{\rm BH}$–$T_{\rm CGM}$ relation
obtained by Gaspari et al. (2019) is consistent with $M_{\rm BH}\approx
200E_{\rm B}/c^{2}$. If this observed relationship does indeed reflect a
connection between black hole mass and baryonic binding energy, then it
implies a price of

$\Delta M_{\rm BH}\,\approx\,10^{7}\,M_{\odot}\left(\frac{E_{\rm
B}}{10^{59}\,{\rm erg}}\right)$ (18)

for star-formation quenching via baryon lifting. In the present-day universe,
that price is

$\Delta M_{\rm BH}\,\approx\,10^{8.3}\,M_{\odot}\left(\frac{M_{\rm
halo}}{10^{13}\,M_{\odot}}\right)^{1.6}$ (19)

when written in terms of halo mass, for halos in the mass range
$10^{12.5}M_{\odot}\lesssim M_{\rm halo}\lesssim 10^{14}\,M_{\odot}$.

The simulations of Booth & Schaye (2010) produced a similar relationship using
a feedback efficiency factor $\epsilon_{\rm fb}=0.015$ and obtained black hole
masses a factor of $\sim 2$ smaller at fixed $M_{\rm halo}$. The motivation
for that choice of $\epsilon_{\rm fb}$ was to reproduce both the $M_{\rm
BH}$–$M_{*}$ and $M_{\rm BH}$–$\sigma_{v}$ relations observed at $z\approx 0$
(Booth & Schaye, 2009). Feedback from black holes in their simulations (and
the EAGLE simulations that ensued) is purely thermal but episodic and is
released in pulses great enough to limit radiative losses of the injected
feedback energy. It therefore couples far more efficiently with the CGM than
the thermal mode feedback in IllustrisTNG.

The price of star-formation quenching in IllustrisTNG is considerably greater
than the one in equation (18) because it contains both a fixed cost and a
marginal cost. Quenching doesn’t happen in a halo’s central galaxy until the
kinetic feedback mode introduces a cumulative energy comparable to $E_{\rm
B}$. Its black hole mass must therefore exceed $\sim 10^{8}\,M_{\odot}$, so
that the condition in equation (16) allows the kinetic mode to prevail. That
is the fixed cost, and it establishes a ratio $M_{\rm BH}/M_{\rm halo}\sim
10^{-4}$ at the time of quenching in halos of mass $\sim 10^{12}\,M_{\odot}$.
In comparison, the marginal cost of the kinetic feedback that actually
quenches star formation is miniscule, amounting to $3\times 10^{5}\,M_{\odot}$
for every $10^{59}\,{\rm erg}$ of energy injection (see equation 14). The
total black-hole mass price for quenching in IllustrisTNG

$\Delta M_{\rm BH}\>\sim\>10^{8}M_{\odot}+10^{5.5}M_{\odot}\left(\frac{M_{\rm
halo}}{10^{12}M_{\odot}}\right)^{5/3}$ (20)

therefore depends almost entirely on the pivot mass in equation (16) and can
be lowered by reducing that pivot mass (see Terrazas et al. 2020 and Truong et
al. 2021).

Once the cost to activate the kinetic mode has been paid, hierarchical merging
ensures that the majority of a black hole’s mass in IllustrisTNG still comes
from thermal-mode accretion (Weinberger et al., 2018). Figure 7 shows that
kinetic feedback injects $\sim 10^{62}\,{\rm erg}$ during the history of a
$\sim 10^{14}\,M_{\odot}$ halo, at a black hole mass cost of $\sim 3\times
10^{8}\,M_{\odot}$. Meanwhile, the central black hole’s mass approaches $\sim
10^{10}M_{\odot}$ by consuming smaller black holes that grew to contain a
fraction $\sim 10^{-4}$ of their halo’s mass during the time of quenching.

Another consequence of hierarchical merging in IllustrisTNG is a mass-
dependent upper limit on the value of $E_{\rm kin}$. Dashed green lines in
Figure 7 show that $E_{\rm kin}$ remains $\lesssim 10^{60}\,{\rm erg}\>(M_{\rm
halo}/10^{12}\,M_{\odot})$ as halo masses increase toward $\sim
10^{14}\,M_{\odot}$. In IllustrisTNG, black hole mergers preserve the sum of
cumulative kinetic energy injection, and so $E_{\rm kin}$ reflects the entire
history of kinetic energy injection associated with a particular halo. The
upper edge of the relation between $E_{\rm kin}$ and halo mass therefore
reflects the kinetic energy requirements for quenching at $M_{\rm halo}\sim
10^{12-12.5}\,M_{\odot}$.

Figure 10: Dependence of black hole feedback mode on spatial resolution. Each
panel is a version of the upper left ($z=0$) panel of Figure 8, with TNG300 at
left, TNG100 in the center, and TNG50 at right. Spatial resolution improves
from left to right, and the total number of halos declines. The tail of points
toward low $M_{\rm BH}$ and low $E_{\rm kin}$ in the left two panels is absent
from the right panel, indicating that the tail results from insufficient
spatial resolution.

Before we consider how the IllustrisTNG feedback parameters might be adjusted
to bring the simulated $M_{\rm BH}$–$M_{\rm halo}$ relation into better
agreement with observations, it is worth noting that the history of kinetic
feedback immediately preceding star-formation quenching depends somewhat on
numerical resolution. Figure 10 shows the dependence of feedback mode on both
$M_{\rm BH}$ and $E_{\rm kin}$ in the TNG300, TNG100, and TNG50 simulations,
proceeding toward finer spatial resolution from left to right. In the two
lower resolution simulations, there is a tail of points at low $M_{\rm BH}$
and $E_{\rm kin}$ that is not present in TNG50. Also, the diagonal climb of
the green region to the transition at $E_{\rm kin}\approx E_{\rm B}$ is
steepest in TNG50. Apparently, the incidence of kinetic feedback episodes
while $M_{\rm BH}<10^{8}\,M_{\odot}$ is smallest in TNG50, implying that
improvements in spatial resolution raise the probability that there will be
some low-entropy clouds, capable of fueling large Bondi accretion rates, close
to the black hole. The points with low $f_{\rm gas}$ in the $z=0$ tail of
Figure 9 are therefore likely to be spurious results of insufficient spatial
resolution near the central black hole.

##  6 What Price is Right?

The IllustrisTNG feedback model depends on several parameters that were tuned
to optimize agreement with observations of the stellar populations of galaxies
(Weinberger et al., 2017, 2018). Adjustments of some of those parameters could
potentially improve agreement with observational constraints on the $M_{\rm
BH}$–$M_{\rm halo}$ relation. However, care must be taken not to degrade many
other aspects of IllustrisTNG that agree with observations of galaxy
evolution.

The analyses in §4 and §5 imply that the two most critical feedback parameters
determining the $M_{\rm BH}$–$M_{\rm halo}$ relation in IllustrisTNG are
$\epsilon_{\rm kin}$ and the $10^{8}\,M_{\odot}$ pivot mass in equation (16).
Reduction of $\epsilon_{\rm kin}$ would appear necessary for better agreement
with observational constraints, as its fiducial value ($\epsilon_{\rm
kin}=0.2$) results in a black hole mass price for baryon lifting at least an
order of magnitude smaller than the one that Donahue & Voit (2022) infer from
the $M_{\rm BH}$–$T_{\rm CGM}$ correlation presented by Gaspari et al. (2019).
The EAGLE simulations, which adopt the Booth & Schaye (2010) feedback
efficiency, indicate that $\epsilon_{\rm kin}\approx 0.015$ might yield an
$M_{\rm BH}$–$M_{\rm halo}$ relation in better alignment with observations.
Equation (18) implies a lower limit of $\epsilon_{\rm kin}\gtrsim 0.005$,
because further reduction would result in a black hole mass price for baryon
lifting that exceeds observations.

More importantly, the pivot mass for switching to kinetic feedback in
IllustrisTNG results in a black hole mass price prior to quenching that
appears to be an order of magnitude larger than observations indicate. For
example, the 24 star-forming galaxies in the Terrazas et al. (2017) sample
belonging to the $10^{10.5}\,M_{\odot}$–$10^{11}\,M_{\odot}$ range of stellar
mass have a median black hole mass $M_{\rm BH}=10^{7.15}\,M_{\odot}$. The
standard deviation around that median is 0.19 dex, and none of those galaxies
has $M_{\rm BH}>10^{8}\,M_{\odot}$ (see Figure 3).

Reducing the pivot mass in equation (16) would lower the maximum black-hole
masses in star-forming IllustrisTNG galaxies. Truong et al. (2021) have
explored the consequences of a reduction to $M_{\rm piv}=10^{6.4}\,M_{\odot}$.
That change results in a nearly linear $M_{\rm BH}$–$M_{\rm halo}$
relationship close to $M_{\rm BH}\approx 10^{-5}\,M_{\rm halo}$ for
$10^{11.5}\,M_{\odot}\lesssim M_{\rm halo}\lesssim 10^{13.5}\,M_{\odot}$. It
therefore improves agreement with the Terrazas et al. (2017) sample at the
low-mass end but produces black hole masses that fall short of observations at
the high-mass end.

Simultaneously implementing $\epsilon_{\rm kin}\approx 0.015$ and $M_{\rm
piv}\approx 10^{7}\,M_{\odot}$ in IllustrisTNG could potentially result in
black hole masses that agree with $M_{\rm BH}$–$M_{\rm halo}$ observations at
both the low- and high-mass ends. However, a large potential downside could be
a reduction in both the halo mass and stellar mass at which star-formation
quenching sets in, once black hole feedback becomes primarily kinetic. For
example, $M_{\rm BH}\approx 10^{7}\,M_{\odot}$ in the IllustrisTNG simulations
corresponds to $M_{\rm halo}\sim 10^{11.3}\,M_{\odot}$ and $M_{*}\sim
10^{9.3}\,M_{\odot}$, both of which are significantly smaller than the
observed quenching scales for central galaxies.

Another conceivable modification to the black hole feedback algorithm would be
a transition from thermal to kinetic feedback that is not a step function of
$\dot{M}_{\rm BH}/\dot{M}_{\rm Edd}$. In the current incarnation of
IllustrisTNG, feedback during periods when $\dot{M}_{\rm BH}\gtrsim
0.1\,\dot{M}_{\rm Edd}$ is entirely thermal, even though many quasars are
known to have powerful winds and jets. Adding a kinetic feedback channel to
the “quasar” mode could qualitatively change how that feedback mode interacts
with the surrounding atmosphere, even if the proportion of feedback energy in
kinetic form is relatively small (see, e.g., Meece et al., 2017).

##  7 Summary

Observations gathered over the last couple of decades have long suggested that
the masses of supermassive black holes are linked to the masses of the
cosmological halos in which they reside (e.g., Ferrarese, 2002; Bandara et
al., 2009; Marasco et al., 2021). Those observations have repeatedly indicated
a nearly linear relationship between $M_{\rm BH}$ and the binding energy of
the halo’s baryons ($E_{\rm B}$), in alignment with models of self-regulated
black hole growth (e.g., Silk & Rees, 1998; Haehnelt et al., 1998). Among
identically structured halos, the expected relationship would be $M_{\rm
BH}\propto M_{\rm halo}^{5/3}$, but Booth & Schaye (2010) found a slightly
shallower relationship ($M_{\rm BH}\propto M_{\rm halo}^{1.55}$) among
cosmological halos with a more realistic dependence of halo concentration on
halo mass.

X-ray analyses of the $M_{\rm BH}$–$T_{\rm X}$ relation (e.g., Bogdán et al.,
2018; Lakhchaura et al., 2019; Gaspari et al., 2019) have recently provided
additional insights, because $T_{\rm X}$ supplies the most reliable estimates
of $M_{\rm halo}$ for nearby galaxies with dynamical $M_{\rm BH}$
measurements. Notably, Gaspari et al. (2019) found that $M_{\rm BH}$
correlates more closely with circumgalactic gas temperature ($T_{\rm CGM}$)
than with any other observable galactic or halo property. Figure 1 shows that
the $M_{\rm BH}$–$M_{\rm halo}$ relation found by converting $T_{\rm CGM}$ to
$M_{\rm halo}$ using an observational $M_{\rm halo}$–$T_{\rm X}$ relation
gives $M_{\rm BH}\propto M_{\rm halo}^{1.6}$, in excellent alignment with
earlier constraints. It also extends that power-law $M_{\rm BH}$–$M_{\rm
halo}$ relationship up to $M_{\rm halo}\sim 10^{14}M_{\odot}$, implying that
the masses of black holes in galaxy groups reflect the energy input needed to
lift their baryons (see also Donahue & Voit, 2022).

However, the $M_{\rm BH}$–$M_{\rm halo}$ relations emerging from cosmological
numerical simulations are not as well aligned with the observational
constraints (Figure 2). Central black hole masses in EAGLE are close to the
observational constraints for $M_{\rm halo}\sim 10^{11.5-13.5}M_{\odot}$ but
underpredict $M_{\rm BH}$ in more massive halos. IllustrisTNG, on the other
hand, produces black hole masses at $M_{\rm halo}\sim 10^{14}M_{\odot}$ in
apparent agreement with observations but overpredicts $M_{\rm BH}$ at $M_{\rm
halo}\sim 10^{12}M_{\odot}$. That happens because the $M_{\rm BH}$–$M_{\rm
halo}$ relation that emerges from IllustrisTNG ($M_{\rm BH}\propto M_{\rm
halo}^{0.76}$) is much flatter than the one predicted by baryon lifting
models.

This paper therefore looked more deeply into the relationship between black
hole mass and baryon lifting in IllustrisTNG, focusing on the TNG100
simulation, to determine the reason for the discrepancy. Previous work has
already shown that quenching of star formation in IllustrisTNG is closely
related to baryon lifting (Davies et al., 2020; Terrazas et al., 2020), as
reflected in a reduction of the halo gas fraction (Figure 5). Throughout the
redshift range $0\leq z\leq 3$, the transition from active star formation to
quiescence occurs when the cumulative kinetic energy input from black hole
feedback becomes comparable to the halo’s baryonic binding energy (Figure 6).
Those findings imply that black hole mass growth during the period of star-
formation quenching in IllustrisTNG is linked to baryon lifting in a manner
consistent with the observed $M_{\rm BH}$–$T_{\rm CGM}$ relation.

However, early black hole mass growth associated with thermal mode feedback in
IllustrisTNG vastly exceeds later mass growth coinciding with baryon lifting.
During a halo’s early period of thermal mode feedback, the mass of its central
black hole grows to exceed $\sim 10^{8}\,M_{\odot}$ before much lifting
occurs. That mass threshold is built into the switch that determines whether
black hole feedback is in thermal mode or kinetic mode (see equation 16). The
switch starts to favor kinetic feedback over thermal feedback as a halo’s mass
approaches $\sim 10^{12}\,M_{\odot}$ (Figure 7), where the observed ratio of
$M_{*}$ to $M_{\rm halo}$ peaks. The onset of kinetic feedback therefore lifts
the galaxy’s atmosphere when the ratio of black hole mass to halo mass reaches
$\sim 10^{-4}$ (Figures 7, 8, and 9).

As baryon lifting happens, the efficiency parameter for kinetic feedback
($\epsilon_{\rm kin}$) determines the associated amount of black hole mass
growth. Its fiducial value in IllustrisTNG is $\epsilon_{\rm kin}=0.2$,
meaning that the black hole mass “price” required to lift the galaxy’s
atmosphere and quench star formation is only $\sim 3\times 10^{5}\,M_{\odot}$
in a $10^{12}\,M_{\odot}$ halo (see equation 13). That amount of mass growth
is negligible compared to the black hole mass growth needed to switch on
kinetic feedback. Subsequent black hole mass growth via mergers in
IllustrisTNG therefore largely preserves the $M_{\rm BH}/M_{\rm halo}$ ratio
that is in place at the time the kinetic mode comes to dominate.

Reduction of the $\epsilon_{\rm kin}$ parameter in IllustrisTNG would increase
the black hole mass price paid for baryon lifting in proportion to
$\epsilon_{\rm kin}^{-1}$. The observed $M_{\rm BH}$–$T_{\rm CGM}$ relation
implies a lower limit of $\epsilon_{\rm kin}\gtrsim 0.005$ (Donahue & Voit,
2022). However, the parameter value employed in simulations may need to be
greater because of inefficiencies in coupling between kinetic feedback and
baryon lifting. For example, Booth & Schaye (2010) showed that setting the
equivalent feedback parameter in their simulations to $\epsilon_{\rm
fb}=0.015$ maximized agreement with the $M_{\rm BH}$–$M_{\rm halo}$ relations
inferred from the observations available at that time.

Initiating baryon lifting at a lower black hole mass in IllustrisTNG also
seems necessary to improve agreement with observations. For example, the
typical black hole mass in a star-forming galaxy with $M_{*}\sim
10^{10.5-11}\,M_{\odot}$ is observed to be $M_{\rm BH}\sim 10^{7}\,M_{\odot}$,
whereas $M_{\rm BH}\sim 10^{8}\,M_{\odot}$ is typical for similar galaxies in
IllustrisTNG (Terrazas et al., 2020). Given the anticorrelation observed
between $M_{\rm BH}$ and sSFR at fixed stellar mass (Terrazas et al., 2016,
2017), it would appear that the majority of black hole mass growth in star-
forming galaxies happens during the quenching process, not prior to it (e.g.,
Chen et al., 2020).

Exactly how to adjust the black hole feedback algorithm in IllustrisTNG
remains an open question. The current algorithm initiates baryon lifting and
star-formation quenching when $M_{\rm halo}\sim 10^{12}\,M_{\odot}$ and
$M_{\rm*}\sim 10^{10.5}\,M_{\odot}$. In IllustrisTNG, the corresponding
central black hole mass is $\sim 10^{8}\,M_{\odot}$ near $z=0$ and $\sim
10^{8.5}\,M_{\odot}$ near $z=3$. Simply reducing the pivot mass in equation
(16) by an order of magnitude might bring about better agreement with
observational constraints on the $M_{\rm BH}$–$M_{\rm halo}$ relation but
would also substantially reduce the values of $M_{*}$ and $M_{\rm halo}$ at
which quenching occurs. A different solution is therefore needed, one that
limits black hole masses in star-forming galaxies to $\sim 10^{7}\,M_{\odot}$
prior to quenching of star formation and allows them to rise to $\gtrsim
10^{8}\,M_{\odot}$ as black hole feedback lifts the surrounding galactic
atmosphere and shuts down star formation.

This paper is dedicated to the memory of Richard Bower, whose pioneering work
on black-hole feedback and galactic atmospheres inspired many of the ideas we
have presented. GMV is supported in part by grant AAG-2106575 from the NSF.
BDO’s contribution was supported by Chandra Grant TM2-23004X. The authors are
grateful to Dylan Nelson, Annalisa Pillepich, and Nhut Truong, whose comments
significantly improved the paper.

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