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

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 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 halo ∼ 1014 M ⊙. We then compare current observational constraints on the M BH–M halo relation with numerical simulations, finding that black hole masses in IllustrisTNG appear to exceed those constraints at M halo < 1013 M ⊙ and that black hole masses in EAGLE fall short of observations at M halo ∼ 1014 M ⊙. 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 BH–M 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.


INTRODUCTION
A galaxy's star formation rate is tied to both the mass of its cosmological halo (M halo ) and the mass of the black hole residing at its center (M BH ).Large galaxy surveys spanning much of cosmic time show that the central galaxies of cosmological halos vigorously form stars until M halo exceeds ∼ 10 12 M ⊙ (e.g., Behroozi et al. 2013Behroozi et al. , 2019)).Star formation then subsides as M halo increases toward ∼ 10 13 M ⊙ .However, suppression of star formation among the central galaxies of present-day cosmological halos correlates more closely with the central velocity dispersion (σ v ) of a galaxy's stars than with M halo (e.g., Wake et al. 2012;Bell et al. 2012;Woo et al. 2015;Teimoorinia et al. 2016;Bluck et al. 2016Bluck et al. , 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 σ v (e.g., Kormendy & Ho 2013), suggest-ing a causal link between galaxy evolution and black hole growth.Observations show that suppression of star formation does indeed correlate with M BH among nearby galaxies of similar stellar mass (Terrazas et al. 2016(Terrazas et al. , 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 BH ∝ σ 5 v ) similar Voit et al.
to the observed one (for a more recent review of similar ideas, see King & Pounds 2015).
A causal connection between M BH and M halo became more plausible when Ferrarese (2002) showed that nearby spiral galaxies follow the same M BH -M 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 halo is defined in terms of a mean matter density, the halo's circular velocity is v c ∝ M 1/3 halo , the specific binding energy of its matter is E B /M halo ∝ v 2 c , and the total binding energy is E B ∝ M halo v 2 c ∝ v 5 c .Black-hole growth limited by baryon lifting should therefore result in M BH ∝ σ 5 v ∝ M 5/3 halo .The Ferrarese (2002) data set supported the baryon lifting hypothesis because it indicated M BH ∝ M γ halo with γ ≈ 1.65-1.82,depending on the methods used to infer M halo from σ v and v 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 halo and indirectly inferred M BH from σ v , finding M BH ∝ M1.55±0.31halo .However, some experts remained deeply skeptical of a direct causal connection between M BH and M halo (e.g., Kormendy & Bender 2011;Kormendy & Ho 2013).
Cosmological simulations then put the proposed relationship between M BH and M 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 BH ∝ M 1.55±0.05halo , 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 powerlaw slope ended up slightly smaller than the 5/3 prediction for identically structured halos because the darkmatter 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(Davies et al. , 2020;;Oppenheimer et al. 2020;Terrazas et al. 2020;Zinger et al. 2020;Appleby et al. 2021).In simulated halos with M halo ≳ 10 12 M ⊙ , black-hole feedback is the prime mover of baryons beyond the virial radius.Furthermore, simulated galaxies centered within halos of mass ∼ 10 12 M ⊙ 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 BHσ v relation among galaxies with active star formation has a power-law slope similar to the M BH -σ v relation among quiescent galaxies but a mass normalization approximately an order of magnitude smaller at fixed σ 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. 1 The proposed connection between baryon lifting and quenching of star formation is theoretically appealing, but then why does quiescence correlate more closely with σ v than with M 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 σ v .The central cooling rate of hot gas in galaxies with large σ v depends primarily on circumgalactic gas pressure.Consequently, as σ v rises above a critical value determined by stellar heating, black hole fueling becomes linked to circumgalactic pressure.Once that link is established, M BH then grows to depend directly on M 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).
This paper presents evidence favoring such a threeway link between black hole growth, baryon lifting, and star-formation quiescence.Section 2 starts things off by examining current observational assessments of the M BH -M 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   Bogdán et al. (2018).In both panels, the relationship kTX = 6 keV × (M200c/10 15 M⊙) 1.7 maps gas temperature onto halo mass.However, masses based on TCGM (left panel) are underestimates in cases where T halo ≫ TCGM. 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 BH -M halo scaling relation.Section 7 summarizes the paper's findings.

BLACK HOLES AND HALO MASSES
The introduction mentioned some of the observational constraints on the M BH -M 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 ⊙ to 10 14 M ⊙ , because that is where X-ray observations provide both direct evidence for baryon lifting and reliable estimates of M halo .Our review of the literature is therefore neither comprehensive nor complete.

Observations
Figure 1 illustrates several relationships between M BH and M halo .A dotted red line shows the relation (1) corresponding to equation (4) from Ferrarese (2002).A dashed red line shows the relation (2) corresponding to equation (6) from Ferrarese (2002).A dot-dashed purple line shows the relation (3) from Bandara et al. (2009).And a dashed blue line shows the relation derived from observations compiled by Marasco et al. (2021).

Voit et al.
Those four assessments of the M BH -M halo relation generally align with each other and also with the dotdot-dot-dashed magenta line showing the relation (5) that Booth & Schaye (2010) found in cosmological numerical simulations of black hole feedback.However, Ferrarese (2002) inferred M halo from galactic dynamics, not halo properties.Bandara et al. (2009) inferred M BH from σ v , not direct dynamical measurements of M BH .And Marasco et al. (2021) used a heterogeneous set of proxies for M halo , making it difficult to assess the impact of systematic uncertainties on their best fitting M BH normalization.
X-ray analyses have recently provided more direct constraints on the M BH -M 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 BH ).Where possible, that data set provides two distinct X-ray temperatures, one (T CGM ) measured within a few effective radii of the galaxy and another (T halo ) more representative of the halo gas at larger radii (for details, see Gaspari et al. 2019). 2nterestingly, Gaspari et al. (2019) found that M BH correlates more closely with T CGM than with any other observable property, including even σ 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 halo using a relationship based on observations by Sun et al. (2009).The original M halo -T X relation used X-ray data to derive the mass M 500c within a radius encompassing a mean mass density 500 times the universe's critical density.Here, we have recalibrated it by setting M 200c = 1.5M 500c , where M 200c is defined using a density contrast of 200 instead of 500. 3The resulting M BH -M halo relation aligns well with the earlier but less direct constraints (see Figures 1 and 2).
Applying the same M 200c -T X relation to T halo leads to a set of points (grey circles in the right panel of Figure 1) that significantly depart from the M 200c -T CGM relation above ∼ 1.5 keV, where M halo ≳ 10 14 M ⊙ and M BH ≳ 5 × 10 9 M ⊙ .Apparently, the power-law slope of the M BH -M halo relation flattens as halos go from the group scale to the cluster scale.The M BH -T 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 BH -T halo relation (dashed gold line in the right panel of Figure 1) that is less steep than the Gaspari et al. (2019) M BH -T halo relation (dashed black line in the right panel of Figure 1).
The apparent leveling of the M BH -M halo relation above M halo ∼ 10 14 M ⊙ 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 ⊙ ≲ M halo ≲ 10 14 M ⊙ ) 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 halo ≳ 10 14.5 M ⊙ ) 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 BH -M 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 CGM and T 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 CGM and T 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 BH is more highly correlated with T CGM than with T 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 forma- X-ray assessments of the M BH -M halo relation become increasingly difficult as M halo drops below ∼ 10 13 M ⊙ because there are fewer and fewer X-ray photons for making temperature measurements.Also, the link between M 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 BH -M 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 σ v assuming which is equivalent to M 200c for a singular isothermal sphere with an isotropic velocity dispersion identical to the galaxy's observed σ 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 starformation rate ( Ṁ * ) divided by its stellar mass (M * ): • Filled stars, ≥ 10 −10.3 yr −1 • Open stars, 10 −11.0 yr −1 to 10 −10.3 yr −1 • Open triangles, 10 −11.7 yr −1 to 10 −11 yr −1 • Filled triangles, ≤ 10 −11.7 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 BH -T CGM relation.

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 EA-GLE points (orange circles) overlap with the observational points up to M 200c ∼ 10 13.5 M ⊙ but predict smaller black hole masses in more massive halos.A power-law fit to the EAGLE points with M 200c > 10 12.3 M ⊙ gives 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 2 illustrates the two pieces of this piecewise powerlaw fit.The high-mass flattening of the EAGLE relation qualitatively agrees with observations but sets in nearer to M halo ∼ 10 12.3 M ⊙ than to ∼ 10 14 M ⊙ .Interestingly, fitting all of the EAGLE points having M 200c > 10 11.5 M ⊙ with a single power law yields a relation with essentially the same slope found by Booth & Schaye (2010) but a slightly greater M BH normalization.
The IllustrisTNG points (small purple triangles) representing M BH -M 200c are less well aligned with the observational constraints.A thick purple line shows the power-law fit  (2016,2017) and Habouzit et al. (2021), in the context of the M BH -M * relation.

Accretion versus Mergers
Black holes in IllustrisTNG halos above M halo ∼ 10 12 M ⊙ accumulate mass primarily through mergers with other black holes (Weinberger et al. 2018).Mergerdominated growth therefore results in a sub-linear M BH -M halo relation (Truong et al. 2021).However, Figure 1 shows that the observed M BH -T CGM relation indicates that the M BH -M halo relation is super-linear up to M halo ∼ 10 14 M ⊙ .Those observations therefore imply either (1) that M BH grows in proportion to M 1.6  halo as halo mass evolves up to ∼ 10 14 M ⊙ , or (2) that black holes in halos that will eventually merge to form a ∼ 10 14 M ⊙ 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 blackhole 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 BH -M halo relation at high masses are less clear and may indicate a combination of merger-driven and accretion-driven growth beyond M halo ∼ 10 12.3 M ⊙ .

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

MBH and σv
The left panel of Figure 3 illustrates the M BH -M halo relations obtained using σ 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 BHσ v relation has been transmuted into M BH -M halo using equation ( 8).It follows the power-law M BH -M halo relation predicted by the baryon lifting hypothesis up to σ v ∼ 240 km s −1 , at which M halo ∼ 10 13.1 M ⊙ and M BH ∼ 10 9 M ⊙ .Beyond there, the M BH -σ v relation becomes much steeper than M BH ∝ σ 5 v .However, the observed M BH -T X relations show no such break at the same location.Comparing with Figure 2 demonstrates that the steeper trend arises because σ v is no longer a good proxy for M halo .This upturn in the M BH -σ 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)  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 * ∼ 10 10.7 M ⊙ also have M BH > 10 8 M ⊙ and seem to be consistent with the IllustrisTNG M BH -M halo relation.However, both Figure 2 and the left panel of Figure 3 show no data points in that region, because both σ v and kT X for those galaxies indicate greater halo masses.The median temperature among those nine galaxies is kT X ≈ 0.3 keV, and the median velocity dispersion is σ v ≈ 238 km s −1 , implying a median halo mass (∼ 10 13 M ⊙ ) that places those same galaxies closer to the EAGLE M BH -M halo relation.Systematic uncertainties among various sets of M halo proxies might therefore explain why the apparent dispersion of M BH at M halo ∼ 10 12−12.5 M ⊙ 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 BH found by Terrazas et al. (2016Terrazas et al. ( , 2017) ) in their sample provides a clue as to why the scatter in M BH at fixed M * is so large.At any given halo mass, stellar masses within the starforming 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 BH near M * ∼ 10 10.7 M ⊙ , where the quiescent and star-forming populations strongly overlap.

Feedback and TCGM
Another possibility to assess is that the Gaspari et al. (2019) galaxies with kT CGM ∼ 0.3 keV are indicating halo masses that are approximately an order of magnitude too large.If M halo is indeed overestimated by that much, then correcting for the overestimate would place those galaxies on the IllustrisTNG relation, with For example, such an overestimate might happen if kinetic feedback produces temperature fluctuations sev-eral times greater than what T 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 σ v and kT CGM in that sample, along with three lines representing the hydrostatic relation for α = 1, 1.5, and 2, given an isotropic velocity dispersion.Those values of α are representative for this sample and account for the spread in kT CGM at fixed σ 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 α = 1 line at low σ v , but there is just one outlier there.It is NGC 7331, which has σ v = 115 km s −1 near its center but v c ≈ 250 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 σ v ≈ 180 km s −1 , making the CGM temperature of NGC 7331 consistent with hydrostatic equilibrium at α ≈ 1.

A Closer Look at TNG
We therefore conclude that the IllustrisTNG M BH -M 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 BH -M 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 BH -M halo relation is anomalous and discuss what might be done to improve it.

LIFTING AND QUENCHING
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. 2019Davies et al. , 2020;;Oppenheimer et al. 2020;Terrazas et al. 2020;Zinger et al. 2020;Piotrowska et al. 2022).Figure 5 illustrates one For an atmosphere in HSE and isotropic !v: of the key findings: central galaxies with quenched star formation in the TNG100 simulation have less halo gas than galaxies with active star formation. 4The figure plots the halo gas mass fraction (f gas ≡ M gas,500 /M 500c ) as a function of M 500c .Colors indicate the median sSFR at each combination of f gas and M 500c .Red squares representing suppressed star formation are prevalent among halos of mass M 500c ≳ 10 12.5 M ⊙ across the redshift range 0 ≤ z ≤ 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 gas in EAGLE is generally a factor of ∼ 3 smaller at M halo ≲ 10 12 M ⊙ than in IllustrisTNG (Davies et al. 2020).Baryon lifting in lowmass halos must therefore proceed somewhat differently in the two simulation environments.
Figure 6 shows that star-formation rates in Illus-trisTNG are also closely related to the central black hole's cumulative kinetic energy input (E 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 which is an estimate of the initial binding energy of the halo's baryons, for the cosmic mean baryon fraction f 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 ≤ z ≤ 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 kin exceeds the gravitational binding energy of gaseous baryons currently within the galaxy (E bind,gal ) and declines much more rapidly once E kin ≳ 10 E bind,gal .Most of the quenched galaxies end up with E kin ≫ 100 E bind,gal .Also, Figure 6 in Terrazas et al. (2020) shows that E 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 bind,halo ).However, neither E bind,gal nor E bind,halo scales linearly with E 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 kin to an atmospheric binding energy scale (E B ) that remains steady while feedback rapidly acts to lift Interestingly, the distribution of E kin at fixed M 500c among IllustrisTNG halos with quenched central galaxies becomes narrower as halo mass increases.Meanwhile, the median value of E kin at fixed M 500c converges toward E B , becoming nearly equal to it as halo mass 5 Bounded by a radius ∝ M 1/3 halo ρ −1/3 cr approaches ∼ 10 14 M ⊙ .In halos that are even more massive, E B exceeds E kin .This outcome is qualitatively consistent with the observed rise in f gas as halo masses go from ∼ 10 13.5 M ⊙ to ∼ 10 14.5 M ⊙ (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 entropy6 of the circumgalactic atmosphere rises above ∼ 10 keV cm 2 and its cooling time rises above ∼ 1 Gyr.
In halos of mass ≲ 10 13.5 M ⊙ , 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 b times the total matter density.Making the transition happen therefore requires an energy input roughly equivalent to E 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.

MODES OF BLACK HOLE GROWTH
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 kin that exceeds the halo's baryonic binding energy scale.The transition to a quiescent state generally happens near M halo ∼ 10 12 M ⊙ , at which E B ∼ 10 59 erg.The corresponding amount of black hole mass growth is in which ϵ kin is the conversion efficiency of accreted restmass energy into kinetic feedback energy and c is the speed of light.In IllustrisTNG, this relationship results in given ϵ 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 BH -M 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 ( ṀBH ) onto a halo's central black hole compares with the limiting Eddington rate where G is the gravitational constant, m p is the proton mass, σ T is the Thomson electron scattering cross section, and ϵ 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 ṀBH ṀEdd > min 0.002 Otherwise, the feedback mode is kinetic.The massdependent factor in equation ( 16) favors kinetic feedback as M BH rises above ∼ 10 8 M ⊙ .However, the thermal mode is still active when ṀBH > 0.1 ṀEdd , even if the black hole is very massive.
Figure 7 shows the joint dependence of feedback mode on both M 500c and E 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 kin surpasses E 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 ⊙ 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 kin is rising toward E B .Those kinetic feedback episodes become increasingly likely as M 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 ṀBH in IllustrisTNG is taken to be the local Bondi accretion rate (Bondi 1952), which depends strongly on the specific entropy of accreting gas.7 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 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 BH rises above ∼ 10 8.2 M ⊙ .Terrazas et al. (2020)   halo show the characteristic scale of baryonic binding energy (EB) 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⊙ ≲ M halo ≲ 10 14 M⊙.
a similar conclusion and also showed that f gas dramatically declines at the same black hole mass threshold.
Superficially, quenching of star formation in Illus-trisTNG may seem to depend most strongly on the threshold value of M BH marking the onset of kinetic feedback, but cumulative kinetic energy input (E 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 BH and E kin .As in Figure 7, the transition to kinetic mode depends most directly on how E kin compares with E B .If a threshold in M 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 Figure 8 also shows that the transitional values of M BH are larger in higher-redshift galaxies.Given how ṀBH / ṀEdd 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 ⊙ at z ≈ 3.At lower redshifts, the transitional value of M BH clearly correlates with E kin and does not exceed M BH ≈ 10 8.3 M ⊙ at z = 0 for any value of E kin .This decline with time in the maximum M BH at which thermal mode feedback occurs implies that the typical specific entropy (K ∝ kT n −2/3 ) of gas near black holes of mass M BH ∼ 10 8.3−8.5 M ⊙ is lower at z ∼ 3 than at z ∼ 0, corresponding to greater pressure at a given gas temperature, presumably because of greater gas accretion rates onto those high-redshift galaxies.
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 BH ∼ 10 7.5 M ⊙ have managed to generate E kin ≳ 10 54 erg.As their cumulative kinetic output then grows to reach ∼ 10 59 erg, those black holes accrete another ∼ 10 8 M ⊙ .Only ∼ 3 × 10 5 M ⊙ of that mass increase comes from accretion associated with kinetic feedback (see equation 13), corresponding to ≲ 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 gas depends jointly on M BH and E kin .In general, the red and dark orange points indicating large reductions in halo gas density lie above the purple dashed lines marking E kin = E 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 kin and M BH in the z = 0 panel of Figure 9 (to be discussed in §5), there is no systematic dependence of f gas on either E kin or M BH below the purple dashed lines.The lack of dependence on M 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 BH .In contrast, reductions in halo gas content clearly depend on how E kin compares to E B , with the greatest reductions in f gas corresponding to E kin ≫ E B (see also Terrazas et al. 2020).At M BH ≈ 10 8.3 M ⊙ in the z = 0 panel of Figure 9, the median E kin exceeds E B by an order of magnitude, and that is where the reductions in f gas are greatest.The excess of E kin over E B then declines with increasing M BH , until the two quantities are almost equal at M BH ≈ 10 9.5 M ⊙ , where E B ≈ 10 61.5 erg.In that part of Figure 9 the squares indicating f gas are typically yellow.Therefore, baryon lifting is minimal for E kin ≲ E B at both low and high halo masses.
Convergence of f 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 BH ≈ 10 9.5 M ⊙ corresponds to M 500c ≈ 10 14 M ⊙ at z = 0 in the IllustrisTNG universe.In the real universe, halos with similar masses currently have X-ray luminosities ∼ 10 44 erg s −1 , meaning that they can radiate ∼ 10 61.5 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 halo ∼ 10 14−15 M ⊙ ) can selfregulate 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 Il-lustrisTNG, 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 ⊙ , but notably Illus-trisTNG 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 halo ∼ 10 12.5 -10 14 M ⊙ , 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.

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 B ) into the CGM.The injected energy comes at a "price" of at least in black hole mass growth that depends on the conversion efficiency ϵ 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 BH -T CGM relation obtained by Gaspari et al. (2019)  for star-formation quenching via baryon lifting.In the present-day universe, that price is when written in terms of halo mass, for halos in the mass range 10 12.5 M ⊙ ≲ M halo ≲ 10 14 M ⊙ .
The simulations of Booth & Schaye (2010) produced a similar relationship using a feedback efficiency factor ϵ fb = 0.015 and obtained black hole masses a factor of ∼ 2 smaller at fixed M halo .The motivation for that choice of ϵ fb was to reproduce both the M BH -M * and M BH -σ v relations observed at z ≈ 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 B .Its black hole mass must therefore exceed ∼ 10 8 M ⊙ , so that the condition in equation ( 16) allows the kinetic mode to prevail.That is the fixed cost, and it establishes a ratio M BH /M halo ∼ 10 −4 at the time of quenching in halos of mass ∼ 10 12 M ⊙ .In comparison, the marginal cost of the kinetic feedback that actually quenches star formation is miniscule, amounting to 3 × 10 5 M ⊙ for every 10 59 erg of energy injection (see equation 14).The total (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 andTruong 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 ∼ 10 62 erg during the history of a ∼ 10 14 M ⊙ halo, at a black hole mass cost of ∼ 3×10 8 M ⊙ .Meanwhile, the central black hole's mass approaches ∼ 10 10 M ⊙ by consuming smaller black holes that grew to contain a fraction ∼ 10 −4 of their halo's mass during the time of quenching.
Another consequence of hierarchical merging in Illus-trisTNG is a mass-dependent upper limit on the value of E kin .Dashed green lines in Figure 7 show that E kin remains ≲ 10 60 erg (M halo /10 12 M ⊙ ) as halo masses increase toward ∼ 10 14 M ⊙ .In IllustrisTNG, black hole mergers preserve the sum of cumulative kinetic energy injection, and so E kin reflects the entire history of kinetic energy injection associated with a particular halo.The upper edge of the relation between E kin and halo mass therefore reflects the kinetic energy requirements for quenching at M halo ∼ 10 12−12.5 M ⊙ .
Before we consider how the IllustrisTNG feedback parameters might be adjusted to bring the simulated M BH -M 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 BH and E 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 BH and E kin that is not present in TNG50.Also, the diagonal climb of the green region to the transition at E kin ≈ E B is steepest in TNG50.Apparently, the incidence of kinetic feedback episodes while M BH < 10 8 M ⊙ 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 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.

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(Weinberger et al. , 2018)).Adjustments of some of those parameters could potentially improve agreement with observational constraints on the M BH -M 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 BH -M halo relation in IllustrisTNG are ϵ kin and the 10 8 M ⊙ pivot mass in equation ( 16).Reduction of ϵ kin would appear necessary for better agreement with observational constraints, as its fiducial value (ϵ 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 BH -T CGM correlation presented by Gaspari et al. (2019).The EAGLE simulations, which adopt the Booth & Schaye (2010) feedback efficiency, indicate that ϵ kin ≈ 0.015 might yield an M BH -M halo relation in better alignment with observations.Equation (18) implies a lower limit of ϵ kin ≳ 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 ⊙ -10 11 M ⊙ range of stellar mass have a median black hole mass M BH = 10 7.15 M ⊙ .The standard deviation around that median is 0.19 dex, and none of those galaxies has M BH > 10 8 M ⊙ (see Figure 3).
Reducing the pivot mass in equation ( 16) would lower the maximum black-hole masses in star-forming Illus-trisTNG galaxies.Truong et al. (2021) have explored the consequences of a reduction to M piv = 10 6.4 M ⊙ .That change results in a nearly linear M BH -M halo relationship close to M BH ≈ 10 −5 M halo for 10 11.5 M ⊙ ≲ M halo ≲ 10 13.5 M ⊙ .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 ϵ kin ≈ 0.015 and M piv ≈ 10 7 M ⊙ in IllustrisTNG could potentially result in black hole masses that agree with M BH -M 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 BH ≈ 10 7 M ⊙ in the IllustrisTNG simulations corresponds to M halo ∼ 10 11.3 M ⊙ and M * ∼ 10 9.3 M ⊙ , 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 ṀBH / ṀEdd .In the current incarnation of IllustrisTNG, feedback during periods when ṀBH ≳ 0.1 ṀEdd 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).

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 BH and the binding energy of the halo's baryons (E B ), in alignment with models of selfregulated black hole growth (e.g., Silk & Rees 1998;Haehnelt et al. 1998).Among identically structured halos, the expected relationship would be M BH ∝ M 5/3 halo , but Booth & Schaye (2010) found a slightly shallower relationship (M BH ∝ M 1.55 halo ) among cosmological halos with a more realistic dependence of halo concentration on halo mass.
X-ray analyses of the M BH -T X relation (e.g., Bogdán et al. 2018;Lakhchaura et al. 2019;Gaspari et al. 2019) have recently provided additional insights, because T X supplies the most reliable estimates of M halo for nearby galaxies with dynamical M BH measurements.Notably, Gaspari et al. (2019) found that M BH correlates more closely with circumgalactic gas temperature (T CGM ) than with any other observable galactic or halo property.Figure 1 shows that the M BH -M halo relation found by converting T CGM to M halo using an observational M halo -T X relation gives M BH ∝ M 1.6 halo , in excellent alignment with earlier constraints.It also extends that power-law M BH -M halo relationship up to M halo ∼ 10 14 M ⊙ , 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 BH -M 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 halo ∼ 10 11.5−13.5 M ⊙ but underpredict M BH in more massive halos.IllustrisTNG, on the other hand, produces black hole masses at M halo ∼ 10 14 M ⊙ in apparent agreement with observations but overpredicts M BH at M halo ∼ 10 12 M ⊙ .That happens because the M BH -M halo relation that emerges from Il-lustrisTNG (M BH ∝ M 0.76 halo ) 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 ≤ z ≤ 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 Voit et al. in IllustrisTNG is linked to baryon lifting in a manner consistent with the observed M BH -T 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 ∼ 10 8 M ⊙ 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 ∼ 10 12 M ⊙ (Figure 7), where the observed ratio of M * to M halo peaks.The onset of kinetic feedback therefore lifts the galaxy's atmosphere when the ratio of black hole mass to halo mass reaches ∼ 10 −4 (Figures 7, 8, and 9).
As baryon lifting happens, the efficiency parameter for kinetic feedback (ϵ kin ) determines the associated amount of black hole mass growth.Its fiducial value in Illus-trisTNG is ϵ kin = 0.2, meaning that the black hole mass "price" required to lift the galaxy's atmosphere and quench star formation is only ∼ 3×10 5 M ⊙ in a 10 12 M ⊙ 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 BH /M halo ratio that is in place at the time the kinetic mode comes to dominate.
Reduction of the ϵ kin parameter in IllustrisTNG would increase the black hole mass price paid for baryon lifting in proportion to ϵ −1 kin .The observed M BH -T CGM relation implies a lower limit of ϵ kin ≳ 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 ϵ fb = 0.015 maximized agreement with the M BH -M 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 * ∼ 10 10.5−11 M ⊙ is observed to be M BH ∼ 10 7 M ⊙ , whereas M BH ∼ 10 8 M ⊙ is typical for similar galaxies in IllustrisTNG (Terrazas et al. 2020).Given the anticorrelation observed between M BH and sSFR at fixed stellar mass (Terrazas et al. 2016(Terrazas et al. , 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 starformation quenching when M halo ∼ 10 12 M ⊙ and M * ∼ 10 10.5 M ⊙ .In IllustrisTNG, the corresponding central black hole mass is ∼ 10 8 M ⊙ near z = 0 and ∼ 10 8.5 M ⊙ 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 BH -M halo relation but would also substantially reduce the values of M * and M halo at which quenching occurs.A different solution is therefore needed, one that limits black hole masses in star-forming galaxies to ∼ 10 7 M ⊙ prior to quenching of star formation and allows them to rise to ≳ 10 8 M ⊙ as black hole feedback lifts the surrounding galactic atmosphere and shuts down star formation.

Figure 1 .
Figure 1.Observed relationships between black hole mass (MBH), halo mass (M halo ), and atmospheric temperature (T ).Dotted and dashed red lines show two MBH-M halo relations from Ferrarese (2002).Dot-dashed purple lines show the MBH-M halo relation from Bandara et al. (2009).Dot-dot-dot-dashed magenta lines show the simulated MBH-M halo relation from Booth & Schaye (2010).Dashed blue lines show the MBH-M halo relation from Marasco et al. (2021).Solid grey lines separating shaded from unshaded regions indicate a linear MBH-M halo correlation.Black points in the left panel show the MBH-TCGM measurements from Gaspari et al. (2019).The best fitting power law relation (MBH ∝ M 1.6 halo ) shown by the solid black line is clearly super-linear.Grey points in the right panel show MBH-T halo measurements from Gaspari et al. (2019), and a dashed black line shows the best fitting power law relation (MBH ∝ M 1.3 halo ).An additional dashed gold line in the right panel shows the MBH-T halo relation from the sample ofBogdán et al. (2018).In both panels, the relationship kTX = 6 keV × (M200c/10 15 M⊙) 1.7 maps gas temperature onto halo mass.However, masses based on TCGM (left panel) are underestimates in cases where T halo ≫ TCGM.
10) fromTruong et al. (2021) for halos with M 200c > 10 12 M ⊙ .It agrees with the M BH observations among the most massive halos (M halo ≳ 10 14 M ⊙ ) but exceeds them at M halo ≲ 10 13.5 M ⊙ , ending up near M BH ∼ 10 8 M ⊙ at M halo ∼ 10 12 M ⊙ .The anomalously large Il-lustrisTNG black hole masses at M halo ∼ 10 12 M ⊙ have previously been noted byLi et al. (2020), in the context of the M BH -σ v relation, and by both Terrazas et al.

Figure 6 .
Figure 6.Dependence of specific star formation rate (sSFR) on cumulative kinetic black hole feedback (E kin = E injected,low ) and halo mass (M500c) across the redshift range 0 ≤ z ≤ 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 (EB) at each halo mass, and purple dotted lines in the lower right panels show the M500c-EB relation at z = 0.At each halo mass and across all redshifts, the dark red squares indicating quenched star formation (sSFR ≲ 10 −2 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 7 .
Figure 7. Dependence of black hole feedback mode on cumulative kinetic black hole feedback (E kin = E injected,low ) and halo mass (M500c) across the redshift range 0 ≤ z ≤ 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 5/3

Figure 8 .
Figure 8. Dependence of black hole feedback mode on cumulative kinetic black hole feedback (E kin = E injected,low ) and black hole mass (MBH) across the redshift range 0 ≤ z ≤ 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 (EB) 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 9 .
Figure 9. Dependence of halo gas fraction on cumulative kinetic black hole feedback (E kin = E injected,low ) and black hole mass (MBH) across the redshift range 0 ≤ z ≤ 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 (EB) 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.

Figure 10 .
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 MBH and low E kin in the left two panels is absent from the right panel, indicating that the tail results from insufficient spatial resolution.
prevents the velocity dispersion of a halo's central galaxy from rising in propor- Häring & Rix 2004;Gültekin et al. 2009;Kormendy & Ho 2013;McConnell & Ma 2013;Savorgnan et al. 2016 temperature.Left panel: The MBH-M halo relation inferred from M halo (σv) using equation (8).Right panel: The MBH-M halo relation inferred from M * using abundance matching of M * with M halo at z ≈ 0 via the Universe Machine(Behroozi et al. 2019).All symbols represent the same quantities as in Figure2, except that the entire Terrazas et al. (2017) sample is shown, not just the subset without X-ray measurements fromGaspari et al. (2019).tionwith the halo's maximum circular velocity once it reaches σ v ∼ 240 km s −1 .2.4.2.MBH and M *On the right side of Figure3, M halo is inferred from M * via the abundance-matching fit ofBehroozi et al. (2019)at z ≈ 0. The scatter in M 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 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 BH -M * relation suggest that M BH may anticorrelate with M * at fixed halo mass.
(Bottema 1999)tionship between σv and kTCGM in the Gaspari et al. (2019) galaxy sample.Colored lines indicate hydrostatic temperatures corresponding to σv for α ≡ |d ln P/d ln r| equal to 1, 1.5, and 2, as labeled.These values of α are representative of the range observed among massive elliptical galaxies.There is no obvious departure from those relations at low σv that would indicate a temperature enhancement produced by black-hole feedback.The only outlier is NGC 7331, which has an observed rotation speed vc ≈ 250 km s −1(Bottema 1999), indicating that σv does not reflect its halo mass.A green arrow shows where NGC 7331 ends up if vc/ √ 2 is used instead of σv.
is consistent with M BH ≈ 200E 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 ∆M BH ≈ 10 7 M ⊙