galaxies arrive on the quenched ridgeline, and a second scaling law for $M_{BH}$ versus $\Sigma_1$ emerges having the same slope as the blue line but higher zero point (red line in Figure 4). However, despite the mathematical similarity of the second law to the first, its interpretation is entirely different—galaxies do not evolve along this second line, and it is not an evolutionary track. Rather, it is the boundary where $E_{BH}/E_{bind} \simeq 20$, and $\Sigma_1$ along this line labels the paths that galaxies took through the green valley to get there.

In summary, some scaling laws may represent unique evolutionary tracks but others may be boundaries linking points on different tracks where a third quantity takes on a common value. The scaling law for quenched galaxies in our model is of the second sort. It is not a track like the blue line, and galaxies do not evolve along it; rather, it connects galaxies where $E_{BH}/E_{bind}$ has the common value $\simeq 20$.

With that as background, we now explore briefly whether the blue line might itself be a boundary rather than an evolutionary track. Up to now, we have regarded the blue line as a track that all star-forming galaxies must evolve along while building their BHs. But actually, BHs need not follow this track for their entire lives—they only need to obey the requisite relations as they enter the green valley. The blue line would then express the relationship between $M_{BH}$ and $\Sigma_1$ that needs to obtain only there.

This change of perspective allows us to address a discrepancy that we have thus far glossed over, which is a disagreement between the binding-energy model and state-of-the-art theory. As stressed in Section 3.1, a generally accepted analytical explanation of the BH scaling laws is thus far lacking, which means that the best theory comes from hydro models. However, these models disagree with the binding-energy model in that their BHs in star-forming galaxies lie “dormant” and evolve rapidly only later, approaching the blue line from below (see the dashed green lines in Figure 4). Galaxies no longer evolve along the blue line—instead, it becomes a boundary scaling law linking together different entry points to the green valley along different evolutionary tracks.

This new interpretation has several implications. First, it means that there is no relation like $M_{BH} = \text{const.} \times \Sigma_1^{1.76}$ for star-forming galaxies or, if there is, it is much broader and fuzzier than what we have assumed. The zero point of the blue line was chosen to fit nearby star-forming galaxies with detected BHs. The new interpretation says that many star-forming galaxies with the same $M_*$ and $\Sigma_1$ have even smaller BHs. In other words, the locus of measured BHs in star-forming galaxies, as in Ho & Kim (2014) and Reines & Volonteri (2015), is only the upper envelope of a much larger population with smaller BHs below this locus. This conclusion agrees with estimates that the measured BH relation is biased high for star-forming galaxies, having missed a population of small BHs (Shankar et al. 2016). It would also explain the large scatter in the $M_{BH} \sim \sigma_e^4$ law that is seen among star-forming galaxies (KH13; Ho & Kim 2014) and the anomalously small BHs found in some of them (e.g., M33, Gebhardt et al. 2000).

With the abandonment of the blue line as an evolutionary track, a new theory for the nature of the boundary to the green valley is needed. One possibility is the model of AGN winds by King (2003) and King & Pounds (2015), which says that the structure of quasar outflows fundamentally changes at a certain BH mass. According to this theory, winds change from being

momentum-driven to energy-driven when

$$ \frac{M_{BH}}{10^8 M_{\odot}} \simeq 3.2 \times \left( \frac{f_g}{0.16} \right) \sigma_{200}^4 \quad (26) $$

Here, 0.16 is the cosmic baryon-to-dark-matter ratio and $f_g$ is the total gas mass fraction in the central galaxy potential well (which is assumed to be isothermal with velocity dispersion $\sigma_{200}$). This change makes winds much more efficient at driving gas out of the galaxy and halo. Combining this with Equation (C5) from Appendix C gives

$$ \frac{M_{BH}}{10^8 M_{\odot}} \simeq 0.09 \times \left( \frac{f_g}{0.16} \right) \left( \frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} \right)^{2.0} \quad (27) $$

This can be compared to the equation of the blue line in Figure 4, given by Equation (C6) in Appendix C, which is

$$ \frac{M_{BH}}{10^8 M_{\odot}} = 0.025 \times \left( \frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} \right)^{1.76} \quad (28) $$

If the small difference between the two exponents can be ignored, we see that the King-Pounds expression will lie on or near the blue line provided

$$ \frac{f_g}{0.16} = \frac{0.025}{0.09} \sim 0.3 \quad (29) $$

or

$$ f_g \sim 0.04, \quad (30) $$

which is a reasonable value for the gas mass fraction at the center of a galaxy that is building a BH.

The King-Pounds theory therefore predicts a wind-enhancement boundary in the vicinity of the blue line.<fnref n="49" /> Hydro models approaching this boundary from below would experience a large increase in wind efficiency at this point such that feedback energy from the BH is suddenly transferred more efficiently to the halo gas. With higher feedback, quenching becomes perceptible, and galaxies enter the green valley along the steep green tracks in Figure 4.

We therefore have an analytic theory that predicts a boundary in the right place at the entrance to the green valley, together with a plausible way of feeding that boundary (from below) as in the hydro simulations. However, a major ingredient is still missing: the theory predicts a boundary in $M_{BH}-\Sigma_1$ only, whereas matching quenching boundaries are also seen in $M_{BH}-M_*$ and $\Sigma_1-M_*$ (see Figures 2(b) and (c)).<fnref n="50" /> In other words, to match all the data, a galaxy needs to cross over the King-Pounds boundary in $M_{BH}-\Sigma_1$ with the right $M_*$.

In summary, the existence of a quenching boundary involving three separate variables ($M_{BH}$, $\Sigma_1$, and $M_*$) is a major constraint whose significance has not been appreciated until now. The original binding-energy model met this constraint by making the $M_{BH}-\Sigma_1$ relation a tight evolutionary track obeyed by all star-forming galaxies, which removed one degree of freedom. If the $M_{BH}-\Sigma_1$ relation is instead a boundary that can be crossed in different places by galaxies with different $M_*$, a separate process is needed to align $\Sigma_1$ and

<footnote n="49">⁴⁹ This point was also made by King (2003) and King & Pounds (2015).</footnote>

<footnote n="50">⁵⁰ The boundary in $R_e-M_*$ in Figure 2(a) is not fundamental, being a mapping of the $\Sigma_1-M_*$ boundary into $R_e-M_*$ via the p, q relation.</footnote>
$M_{BH}$ with the proper value of $M_*$ at the crossing point. Meeting this second constraint is a new test for hydro models.

## 7.9. Comparison to the Model of Lilly & Carollo (2016)

We close this Discussion by returning to the model of Lilly et al. (2016; L16), which was mentioned in the Introduction. This model is distinguished by the assumption that the quenching probability is a function of $M_*$ only, not of, say, $M_*$ and $R_e$. Despite this, the model still manages to reproduce the sloping contours of equal quenched fraction in $R_e-M_*$ by exploiting two factors: (1) galaxies that quench earlier have smaller $R_e$ and thus lower the quenched ridgeline zero point at fixed $M_*(\text{progenitor bias})$, and (2) $R_e$ values are light weighted and shrink upon fading in this model owing to lower-$M/L$ stars in their outskirts (also mentioned by F13). These two effects together increase the fraction of quenched galaxies at smaller $R_e$ at fixed $M_*$.

Despite this success, the model fails to explain other data:

(1) The difference in $R_e$ between quenched versus star-forming galaxies in the model is smaller at high redshift than at low redshift (see L16, Figure 3), whereas the reverse is true in real data: quenched galaxies, particularly lower-mass ones, are much smaller than star-forming galaxies at $z \sim 2$ than at later times (see Figure 1). This is naturally accommodated in the binding-energy model because $R_e$ is free to shrink more at early times due to compaction (Section 7.4), whereas the L16 model does not have this extra degree of freedom.

(2) The L16 model predicts a $10 \times$ radial increase in SSFR from 1 to 10 kpc within star-forming galaxies of mass $10^{10.75} M_{\odot}$ at all redshifts, which is generated by the assumption of strong inside-out star formation. A gradient of this magnitude is needed in order to produce the required amount of $R_e$ shrinkage upon fading (see above). Current data on distant galaxies do not support gradients of this magnitude within 10 kpc—the SSFR profiles of most galaxies are flat out to this radius (Wang et al. 2017; Tacchella et al. 2018; Woo & Ellison 2019).

(3) The L16 model does not include BHs and therefore does not explain the $M_{BH}-M_*$ scaling relation of quenched galaxies. Quenching is assumed to occur with uniform probability per unit stellar mass growth, but no mechanism is offered to achieve this, and the role of BHs in quenching is unaddressed.

(4) Most important, the L16 model has not yet attempted to match the $\Sigma_1-M_*$ diagram. $\Sigma_1$ is likely to remain rather constant through and after quenching in the L16 picture. But $\Sigma_1$ is set by the *p*, *q* mapping for star-forming galaxies, and hence by $R_e$ and $M_*$. If quenching is equally probable for all galaxies at a given mass regardless of $R_e$, the $\Sigma_1$ distributions of green valley and star-forming galaxies at a given $M_*$ should be the same. Instead, the $\Sigma_1$ distributions of real green-valley galaxies are shifted to much higher values and are much narrower than those of star-forming galaxies at all redshifts (F13; vD15; B17; Woo et al. 2017; Lee et al. 2018; Luo et al. 2020). The key point is that green-valley galaxies are observed close to the beginning of quenching, so that the evolutionary time difference between them and their star-forming progenitor galaxies should be small. Even so, progenitor bias might induce a small offset in average $\Sigma_1$, but it cannot explain the large difference in the widths of the loci of green-valley galaxies versus star-forming galaxies. Moreover, as a mass density, $\Sigma_1$ is not affected by fading. Thus, $\Sigma_1$ sidesteps both effects that the L16 model uses to convert quenching contours that are

intrinsically vertical at fixed mass into sloping contours in $R_e$ versus $M_*$. The sloping behavior of $\Sigma_1$ versus $M_*$ signals that a second physical variable is involved in quenching beyond $M_*$, which we have suggested is $R_e/\langle R_e \rangle$.

Parenthetically, we note that all quenching models that rely largely on stellar (or halo) mass meet the same objection, of which the obvious examples are the pure “halo quenching” models (e.g., Ostriker & Rees 1977; Blumenthal et al. 1984; Cattaneo et al. 2006; Dekel & Birnboim 2006; Zu & Mandelbaum 2015, 2018). Being one-dimensional theories in which the only variable is mass, they inherently fail to reproduce the observed diversity of properties of galaxies at a given stellar mass.

# 8. Summary

A schematic model is presented for the growth of BHs in central star-forming galaxies and their role in causing central galaxies to quench. The simplicity of the model is useful for appreciating how various factors in galaxy evolution—such as structural evolution, BH growth rules, and quenching criteria— interact and how results depend on these inputs. The impetus for the new model is the belief that the observed scaling laws of galaxies contain important clues to quenching that can be elucidated from a proper model with the right degrees of freedom. The paper starts with a list of 18 separate observations for quenched and star-forming galaxies that a successful model should explain. The final binding-energy model presented here explains 11 of these items fairly well and sheds useful light on the remaining 7.

The main feature that enables this success is a new degree of freedom in which larger star-forming galaxies at fixed mass have smaller BHs owing to their lower central densities, $\Sigma_1$. A second key assumption is that galaxies enter the green valley when the total effective energy emitted by their central BHs equals some fixed multiple, $f_B$, of the binding energy of the gas in their dark halos. Together, these assumptions imply that large-$R_e$ star-forming galaxies must evolve to higher stellar masses before AGN feedback terminates the gaseous infall from their halos, and they accordingly have higher values of $M_{BH}$, $\Sigma_1$, and $R_e$ once quenched. In other words, the simple addition of an $R_e$ dependence to BH mass explains the otherwise mysterious sloped quenching boundaries in $\Sigma_1-M_*$, $R_e-M_*$, and $M_{BH}-M_*$ whereby more massive quenched galaxies have larger central densities, larger effective radii, and larger BHs. For star-forming galaxies, these three spaces are different projections of the same 4D parent space, and mappings are presented to transform among them: $\Sigma_1 \sim R_e^{-p}M_*^q$ is the transformation between $\Sigma_1$ and $R_e$, where $p=1$ and $q=1.1$, while $M_{BH} \sim \Sigma_1^v$ is the transformation between $\Sigma_1$ and $M_{BH}$, where $v=1.76$.

As long as galaxies are star-forming, they are assumed to evolve along approximately log-linear parallel tracks in this 4D space. Each track is labeled with two numbers: the starting redshift, $z_s$, when the galaxy passes $M_* = 10^9 M_\odot$, and the factor by which its radius $R_e$ differs from the mean $\langle R_e \rangle$ at that mass and time. These two numbers make central star-forming galaxies a two-parameter family, which can also be labeled by $M_*(\text{or equivalently, } M_{vir})$ and $R_e/\langle R_e \rangle$ today. The physical origin of the scatter in $R_e$ is not known, but halo concentration and/or halo formation time is a promising candidate. If further tests support this, then $M_{vir}$ and $c$ (or formation time) would be
the two principal halo parameters that predict the evolutionary paths of central galaxies.

All star-forming galaxies in a given slice of $M_*$ are assumed to have the same halo mass, $M_{vir}$, at all times, and the mean effective $\langle R_e \rangle$ in that slice is always $0.02 \times R_{vir}$. The star-forming branch of the SMHM relation is represented by a power law, $M_* = \text{const.} \times M_{vir}^s$, with zero scatter, where $s = 1.75$ and the zero point is constant with time. Both slope and zero point are taken from abundance matching (RP17). Another key assumption is that all galaxies in a single mass slice make stars at the same rate regardless of $R_e/\langle R_e \rangle$, in agreement with observational data. These assumptions, together with the physics of evolving halos, yield mean evolutionary track slopes and zero points for star-forming galaxies in $R_e-M_*$ and $\Sigma_1-M_*$ that generally agree with observations.

Galaxies are assumed to enter the green valley when they pass over a boundary in $\Sigma_1-M_*$ (and matching boundaries in $R_e-M_*$ and $M_{BH}-M_*$). The empirically observed boundary, shown as the red lines in Figure 1, is based on the well-measured quenched ridgeline in $\Sigma_1-M_*$, which is offset downwards by 0.2 dex in $\Sigma_1$ to represent entry into the green valley. In the evolving empirical power-law model in Section 4, this boundary is defined as the point where the total energy deposited in the halo by the BH, $\eta M_{BH}c^2$, equals some empirical energy quota needed to impact the halo and stop gas infall. To fit the observed quenching boundary slopes in $\Sigma_1-M_*$ and $M_{BH}-M_*$ forces the energy quota to have the form $E_{\text{quench}} = \text{const.} \times h(z)^{0.66v} M_{\text{vir}}^t$, where $h(z)$ is the normalized Hubble constant $H(z)/H(0)$ and $t = 2.03$. This prescription fits the red lines precisely, and the log-linear slope of the boundary in $\Sigma_1-M_*$ is $t/vs = 0.66$, in $R_e-M_*$ it is $q-pt/vs = 0.44$, and in $M_{BH}-M_*$ it is $t/s = 1.16$.

The above empirical expression for $E_{\text{quench}}$ is then compared to $4 \times$ the total binding-energy content of the hot halo gas, $E_{\text{bind}}$, and it is shown that their behaviors versus mass and time are similar (red lines versus blue lines in Figure 1). Hence, nearly the same quenching boundary is predicted using this more physically based expression for halo gas-binding energy.

Four variations on this basic picture are considered in Section 6, including two that express the BH-halo balance in terms of an energy-rate balance instead of total energy equality. Such rate prescriptions have been employed in SAM models in the past but fail badly (as implemented here) to match the observed width of the green valley, the sloping power-law nature of the quenching boundaries, or the steady flow of galaxies from the blue cloud to the red sequence since $z \sim 3$.

The formula for BH mass growth, $M_{BH} = \text{const.} \times \Sigma_1^{1.76}$, with constant zero point, applies only to galaxies on the star-forming main sequence (i.e., down to -0.45 dex below the ridgeline). Evidence is given that the BH growth relation steepens as galaxies enter the green valley, indicating an acceleration in BH growth relative to $\Sigma_1$ at that point (Figure 4). Increased BH accretion in the green valley is also seen in X-ray luminosities and in hydro models. Total BH mass growth while galaxies are in the green valley is at least $10 \times$. The star-forming slope, $M_{BH} = \text{const.} \times \Sigma_1^{1.76}$, is reacquired when galaxies come to rest on the quenched ridgeline, but now with a different zero point that is $\sim 10$ times larger than the star-forming zero point. This scenario for BH growth is summarized in Figure 4.

Rapid BH growth in the green valley implies that AGN activity should be high there. AGNs are indeed frequent in the green valley, but integrated X-ray luminosity is not as high as expected. Perhaps AGNs are obscured, or accretion may be radiatively inefficient. This conflict with the X-ray data is the biggest unanswered challenge for the binding-energy model.

A surprise from the model is that the derived galaxy evolutionary tracks in $\Sigma_1-M_*$ and $R_e-M_*$ are nearly parallel to the quenching boundary, and most galaxies would not cross it if it were stationary. In practice, it is the decline in halo gas-binding energy (at fixed $M_{vir}$) that brings the quenching boundary down and across the tracks. The angle between the tracks and boundary remains acute, however, and the small value of this angle is responsible for the $1^+$ dex width of the green valley.

An important requirement at the heart of any successful model is that the quenching boundary and the star-forming galaxy cloud must maintain a constant relationship to one another over time in $\Sigma_1-M_*$, i.e., their zero points must evolve together in lockstep (B17). This constancy is observed and is responsible for the unchanged appearance of the $\Sigma_1-M_*$ diagram over time (except for the downward evolution in both zero points) and is also why the green-valley quenching channel has nearly the same mass at all redshifts. An exact match between the boundary and the star-forming cloud is imposed by fiat in the empirically derived quenching boundary, $E_{\text{quench}}$, in Equation (20). The more physically motivated quenching boundary based on halo gas-binding energy, $E_{\text{bind}}$, in Equation (23), also moves down but not quite fast enough, and the average green-valley mass would vary slightly on that account from -0.2 dex at $z = 3$ to +0.2 dex at $z = 0.5$.

For most of the paper, we assume that star-forming galaxies build their BHs by evolving along the relation $M_{BH} = \text{const.} \times \Sigma_1^{1.76}$ (blue line in Figure 4). If this is true, then BH masses in star-forming galaxies can be read off directly from the $\Sigma_1-M_*$ diagram using $M_{BH} = \text{const.} \times \Sigma_1^{1.76}$, or from the $R_e-M_*$ diagram using $M_{BH} = \text{const.} \times R_e^{-1.76} M_*^{1.94}$.

However, inspired by hydro models, Section 7.8 explores an alternate picture in which BHs in star-forming galaxies spend most of their lives below the blue line and grow rapidly only later. The wind theory of King (2003) and King & Pounds (2015) predicts that AGN winds should suddenly become more efficient as galaxies cross a relation very near the blue line, setting the stage for entry into the green valley. The blue line would then no longer be an evolutionary track but rather the boundary where the King-Pounds relation is satisfied. This picture is still incomplete, however, because galaxies must cross the boundary with the right combination of $\Sigma_1$, $M_{BH}$, and $M_*$, and the tight connection to $M_*$ is still theoretically unexplained.

The major takeaway messages from the binding-energy model are:

1. Central galaxies perceptibly quench when the total effective energy emitted by their BHs equals some constant multiple, $f_B$, of the binding energy of gas in their dark halos. Galaxies enter the green valley when $f_B$ is near 4 (Section 5) and are fully quenched when $f_B$ is near 20 (Section 7.5).

2. Effective radius is an important second parameter in shaping the life histories of galaxies and the masses of their BHs. Promising candidates for the origin of radius scatter (at fixed stellar mass) are halo concentration and/or halo formation time (Section 7.2).
(3) In the simple version of the theory, star-forming galaxies build their BHs along the relation $M_{BH} = \text{const.} \times \Sigma_{1}^{1.76}$ (blue line in Figure 4). The positive dependence on central stellar density means that BHs are smaller in galaxies with larger $R_e$ at fixed mass (Section 3.1). Points (1) and (2) then explain the sloping boundaries between quenched and star-forming galaxies in key structural diagrams (Section 4.3). In this interpretation, the blue line is an evolutionary track.

(4) An alternate version of the theory says that BHs in star-forming galaxies spend most of their time below the relation $M_{BH} = \text{const.} \times \Sigma_{1}^{1.76}$ and grow only later when supernova winds fail, as in hydro models. The King-Pounds wind theory then predicts a jump in wind feedback efficiency when BHs reach the blue line, which would propel host galaxies into the green valley. In this interpretation, the blue line is a boundary not an evolutionary track, and many small BHs should exist below the blue line in star-forming galaxies that have not yet been detected (Section 7.8).

(5) The quenching boundary moves with time in structural diagrams because the gas-binding energies of halos at fixed $M_{\text{vir}}$ are lower at late times (Section 7.1). The evolutionary tracks of star-forming galaxies are intrinsically parallel to the boundary, and it is mainly the sideways motion of the boundary that causes them to quench (Sections 4.3 and 5).

(6) The downward motion of the quenching boundary in $\Sigma_{1}-M_{*}$ is matched by a similar motion of the star-forming cloud. This holds the quenching and star-forming populations in the same relationship to one another, which keeps the green-valley mass nearly constant. In physical terms, BH masses are getting smaller with time owing to the growth in galaxy radii (at fixed $M_{*}$) while dark halos are getting easier to quench with time owing to the decline in halo gas-binding energy (at fixed $M_{\text{vir}}$; Section 7.1). This approximate equality is why quenching has looked so similar since $z \sim 3$ (Section 7.1).

(7) BH scaling laws and their relationships to dark halos are well defined but are different for star-forming galaxies, the green valley, and quenched galaxies (Section 7.7). At least 90% of BH mass is gained in the green valley (Sections 3.1 and 7.5).

(8) The quenching process is highly uniform from halo to halo. The rms raw scatter in the factor $f_B$ is at most 0.28 dex based on the observed scatter in the $\Sigma_1-M_*$ quenched ridgeline and will probably be considerably less than this once progenitor bias is fully allowed for (Section 5).

(9) Despite the successes of the binding-energy model, the physical basis of the model remains incomplete. The foundation of the model is the way that BHs grow in star-forming galaxies, and the default version assumes that a tight evolutionary track, $M_{BH} = \text{const.} \times \Sigma_{1}^{1.76}$, is followed by all star-forming galaxies. This assumption has empirical support but disagrees with recent hydro models, which predict more scatter and generally smaller BHs in star-forming galaxies. An alternative theory predicts that AGN wind efficiency should increase strongly with $\Sigma_1$ in star-forming galaxies, but this creates a boundary in $M_{BH}-\Sigma_1$, not an evolutionary track. It explains the boundary in $M_{BH}-\Sigma_1$ but not the associated boundaries in $\Sigma_1-M_*$ or $M_{BH}-M_*$ because $M_*$ is not linked to these other two quantities. The existence of three separate, matched quenching boundaries—in $\Sigma_1-M_{BH}$, $\Sigma_1-M_*$, and $M_{BH}-M_*$—is currently the biggest challenge to theoretical quenching models.

The major impetus for the model in this paper was the structural and star-forming data on galaxies from the CANDELS program on the Hubble Space Telescope. We thank the many workers who helped to create this unique database. The main CANDELS observations were supported under program number HST-GO-12060, provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. C.Z. acknowledges support from NFSC grants 11403016, 11433003, 11673018, and Innovation Program 2019-01-07-00-02-E00032 of SMEC. C.Z. thanks the support by Science and Technology Commission of Shanghai Municipality to the Key Lab for Astrophysics, International Joint Lab, and Project No. 18590780100. Members of the CANDELS team at UCSC acknowledge support from NASA HST grant GO-12060.10-A and from NSF grants AST-0808133 and AST-1615730. We thank the anonymous referee for raising important questions that stimulated several significant and beneficial improvements.

# Appendix A

## The *p*, *q* Mapping in $\Sigma_1-R_e-M_*$ for Star-forming Galaxies

Figures A1(a) and (b) show observed versus model values of $\Sigma_1$ and $R_e$ predicted by our adopted power-law *p*, *q* mapping $\Sigma_1 \sim R_e^{-p} M_*^q$. The sample shown is described in F18; basic parameters are $H \le 24.5$ mag, $M_* \ge 10^{9.0} M_\odot$, and axis ratio $b/a \ge 0.5$. $R_e$ (SMA) comes from GALFIT fits by van der Wel et al. (2012). Galaxies with peculiar or poorly measured structure are omitted, as indicated by the GALFIT quality flag PhotFlag = 0, which excludes 13% of the selected objects lying within our mass and redshift limits. Visual inspection shows that ~75% of these objects appear to suffer from contamination from nearby objects, some fraction of which are mergers and disturbed (images are shown in F18). The remaining ~25% appear normal, but many of these latter have small angular sizes, which may preclude reliable fits (van der Wel et al. 2012). In this paper, we are not in general using absolute counts of objects, and so the loss of these objects per se is not an issue.

Completeness as a function of mass is described in F18. To summarize, quiescent galaxies down to $10^{10} M_\odot$ are included at all redshifts, which captures nearly all of them. Star-forming galaxies are likewise complete at most redshifts but are only ~50% complete for $M_* \le 10^{9.5} M_\odot$ at $z \sim 2$. However, all of the missing galaxies are captured if the sample is extended to $H = 25.5$ mag, so we redid the *p*, *q* fitting formula and replotted Figure A1 to this lower limit. Neither the figure nor the formula changed perceptibly.

Figures A1(a) and (b) show that points are distributed in a plane, and fitted values are $p = 1.00$, $q = 1.10$, and $k_{\Sigma_1} = 1.38$. The formal error in both *p* and *q* is 0.01. There is no systematic shift in the zero point with time, and the rms scatter in both panels of Figure A1 is about 0.25 dex. However, some curvature of the plane is indicated by the curvature of the $R_e$ residuals at high redshift and by systematic errors as a function of $\Sigma_1$ at all redshifts. We rotated the space by hand to verify that the fitted values are optimum; further investigation is postponed to future work.
<figure><img src="image_11.png" /><figcaption>(a) Observed vs. model values of R<sub>e</sub> using the *p*,*q* mapping from Σ<sub>1</sub>-M<sub>*</sub> with *p* = 1.00 and *q* = 1.10, color-coded by Σ<sub>1</sub>. (SMA stands for effective half-light semi-major axis.)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Observed vs. model values of Σ₁ using the same p,q mapping from Rₑ-M*, color-coded by Rₑ.</figcaption></figure>

**Figure A1.** This figure shows the results of mapping the two spaces R_e-M_* and Σ_1-M_* onto one another using the *p*, *q* power-law expression Σ_1 = k_1 R_e^p M_*^q with *p* = 1.00 and *q* = 1.10. Predicted values are on the X-axes, and observed values are on the Y-axes. The sample consists of the same face-on star-forming galaxies shown in Figure 3. All five CANDELS fields are used, and “star-forming” is defined as main-sequence galaxies lying above the line that lies -0.45 dex below the main-sequence ridgeline. There is no systematic shift in zero point with time, and the rms scatter in both relations is about 0.25 dex. Some systematic residuals are seen, as discussed in the text.

**Appendix B**
The Evolving Zero Point of the Σ₁−M* Ridgeline

The ridgeline of quenched galaxies in Σ₁−M* determines
the location of the quenching boundary, which is offset
downwards by 0.2 dex in Σ₁ from the ridgeline at all redshifts
and masses. The zero point of the ridgeline was measured in

redshift bins by B17. Figure B1 fits a smooth curve to
the B17 zero points as a function of the normalized Hubble
constant h(z) = H(z)/H(0). It is seen that the zero point
evolves as h(z)0.66. This functional form is used to calculate
the evolving zero point in the quenching equation
Equation (20).
<figure><img src="image_13.png" /><figcaption>Figure B1. The zero-point evolution of the $\Sigma_1-M_*$ relation from B17, fitted as a function of $H(z)/H(0)$.</figcaption></figure>

## Appendix C

### Power-law Expressions in the Empirical Power-law Model

This section collects together all the exponents and zero points of the various power laws used in the empirical power-law model (Section 4) and its variants (Section 6). This model has a fixed zero point in the SMHM (Equation (1)) and an evolving zero point in the BH-halo energy quota equation (Equation (20)).

1. The $\Sigma_1-R_e-M_*$ relation (p, q mapping): From Appendix A, the $\Sigma_1-R_e-M_*$ relation is

$$ \frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} = k_{\Sigma_1} \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^q \times \left( \frac{R_e}{1 \text{ kpc}} \right)^{-p}, \quad (C1) $$

where $p = 1.00$, $q = 1.10$, and $k_{\Sigma_1} = 1.38$.

2. The SMHM relation: This is the power-law fit by eye to the average SMHM relation for star-forming galaxies from RP17. It is assumed to hold with constant zero point in the range $z = 0-4$. It also forms the lower branch of the broken-power-law model (Variant 1) and as such is plotted in Figure D1. The adopted relation is

$$ \frac{M_*}{10^{10} M_{\odot}} = k_m \times \left( \frac{M_{\text{vir}}}{10^{12} M_{\odot}} \right)^s, \quad (C2) $$

where $s = 1.75$ and $k_m = 2.51$.

3. $R_{\text{vir}}$ as a function of $M_{\text{vir}}$: This is the relation between $R_{\text{vir}}$ and $M_{\text{vir}}$ based on the definition of halo overdensity $\Delta_{\text{vir}}$ from Bryan & Norman (1998). We solved Equation (17) for $R_{\text{vir}}$ and fitted $\Delta_{\text{vir}}^{-1/3} \rho_m^{-1/3}$ as a power of $h(z) \equiv H(z)/H(0)$ between $z = 0.5$ and $z = 3$ to find

$$ \frac{R_{\text{vir}}}{1 \text{ kpc}} = k_{R\text{vir}} \left( \frac{M_{\text{vir}}}{10^{12} M_{\odot}} \right)^{1/3} \left( \frac{h(z)}{2.3} \right)^{-0.74}, \quad (C3) $$

where the constant $k_{R\text{vir}} = 215$ is evaluated at $z = 1.5$ and $h(z)/2.3 = h(z)/h(1.5)$.

4. $R_e$ versus $R_{\text{vir}}$: This is the assumed relation between the effective radius of a typical star-forming galaxy, $\langle R_e \rangle$, and the

radius of its halo:

$$ \langle R_e \rangle = k_{R_e} \times R_{\text{vir}}, \quad (C4) $$

where $k_{R_e} = 0.02$.

5. The $\Sigma_1-\sigma_1$ relation: $\Sigma_1$ is the projected stellar surface density within a radius of 1 kpc from the center of the galaxy, and $\sigma_1$ is the projected velocity dispersion along the line of sight within the same aperture. The relation we use is from F13:

$$ \frac{\Sigma_1 \text{ kpc}}{10^9 M_{\odot} \text{ kpc}^{-2}} = k_{\Sigma_1, \sigma_1} \times \left( \frac{\sigma_1}{200 \text{ km s}^{-1}} \right)^{1.99}, \quad (C5) $$

where $k_{\Sigma_1} = 6.0$. It is assumed to hold for all galaxies at all redshifts.

6. The $M_{BH}-\Sigma_1$ relation for star-forming galaxies: This is the equation of the blue line in Figure 4. In Section 3.1, we motivated the use of the power law $M_{BH} \sim \Sigma_1^\nu$ to express BH mass in star-forming galaxies, and we found the value of the exponent $\nu = 1.76$ by taking the ratio of the quenched ridgeline slopes in $\Sigma_1-M_*$ and $M_{BH}-M_*$. Here we set the zero point of this relation by equating our strongly star-forming ridgeline galaxies to the pseudo-bulge galaxies in Figure 26 of KH13 and noting that $M_{BH}/M_* = 10^{-3.7}$ for such objects on average (with scatter). Normalizing $\Sigma_1$ at $10^{9.0} M_{\odot} \text{ kpc}^{-2}$ corresponds to $M_* = 10^{10.1} M_{\odot}$ for star-forming SDSS galaxies in F13. We therefore have

$$ \frac{M_{\text{BH}}}{10^8 M_{\odot}} = k_{\text{BH}} \times \left( \frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} \right)^{\nu}, \quad (C6) $$

where $\nu = 1.76$ and $k_{\text{BH}} = 0.025$. This relation is assumed to be valid for all star-forming galaxies at all redshifts. It says that BH mass is purely a function of central stellar density within 1 kpc in star-forming galaxies. Again, this is for the basic model, which says that star-forming galaxies evolve along the blue line in Figure 4. The alternative picture for BH growth that is presented at the end of Section 7.8 says that BHs in star-forming galaxies are smaller than this mass until their host galaxies near the green valley.

7. The $\Sigma_1-M_*$ quenched ridgeline relation: The equation of the quenched ridgeline in $\Sigma_1-M_*$ is (B17; F13)

$$ \frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} = 1.26 \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^{0.66} \times h(z)^{0.66}. \quad (C7) $$

The location of the quenching boundary is taken to be 0.2 dex below this ridgeline based on the location of the “structural valley” in $\Sigma_1-M_*$ identified by Luo et al. (2020) for SDSS galaxies today.

8. The $M_{BH}-M_*$ quenched ridgeline relation: The basic scaling relation for BHs in quenched galaxies is the red dashed line in Figure 4, which plots $M_{BH}$ versus $\Sigma_1$. The slope of this line comes from the quenched ridgeline in $M_{BH}-M_*$ today from KH13:

$$ \frac{M_{\text{BH}}}{10^8 M_{\odot}} = 0.36 \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^{1.16} \times h(z)^{0.66\nu}, \quad (C8) $$

which is converted to $\Sigma_1$ using $\Sigma_1 \sim M_*^{0.66}$ for quenched ridgeline galaxies (Item 7 above). However, the zero point of the red dashed line comes from the zero point for star-forming galaxies in Equation (C6) incremented by the length and slope of the green-valley vectors in Figure 4. These in turn come
from the growth of $M_{BH}$ in the green valley from Terrazas et al. (2017) and the growth of $\Sigma_1$ in the green valley from F13 and B17 and are (coincidentally) such that the zero point of the red line is offset from the blue line by almost precisely a factor of 10. Using this number and combining Equations (C6) and (C7) above, we can therefore perform a sanity check to see if the zero point for quenched galaxies in Figure 4 is consistent with the measured KH13 zero point for quenched galaxies in $M_{BH}-M_*$. The predicted zero point using this route is 0.37, which compares well with the observed zero point of 0.36 used in Equation (C8).

9. The empirical power-law expression for $E_{\text{quench}}$: $E_{\text{quench}}$ is the empirically derived value for the halo quenching-energy quota defined in Equation (20). By fitting to the red lines in Figure 1, we find the evolving constant in the following equation:

$$E_{\text{BH}} = E_{\text{quench}} = \eta \frac{M_{\text{BH}}}{10^8 M_{\odot}} c^2 = k_e(z) \left( \frac{M_{\text{vir}}}{10^{12} M_{\odot}} \right)^t, \quad (\text{C9})$$

where

$$k_e(z) = k_e(0) \times h(z)^{0.66\nu}, \quad (\text{C10})$$

the term $h(z)^{0.66\nu}$ is based on the zero-point fit in Figure B1, and $k_e(0) = 10^{17.66}$.

10. The zero point $k_{\text{SMHM}}(0)$ of the SMHM relation in Equation (D1): This is $k_{\text{SMHM}}(0)$ in the Variant 2 model:

$$k_{\text{SMHM}}(z) = k_{\text{SMHM}}(0) \times h(z)^{-1.00}, \quad (\text{C11})$$

where $k_{\text{SMHM}}(0) = 2.51$.

11. The $M_{BH}-R_e-M_*$ relation: For convenience, we include the power law that makes it possible to read off the BH mass for star-forming galaxies from $R_e$ and $M_*$. From Equations (C1) and (C6), the relation is

$$\frac{M_{\text{BH}}}{10^8 M_{\odot}} = k_{\text{BH1}} \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^{q\nu} \times \left( \frac{R_e}{1 \text{ kpc}} \right)^{-pv}, \quad (\text{C12})$$

where $q\nu = 1.94$, $pv = 1.76$, and $k_{\text{BH1}} = 0.044$. This is the value of BH mass predicted by the blue line in Figure 4. Note that the alternative picture of BH growth in star-forming galaxies that is presented in Section 7.8 says that BHs in star-forming galaxies actually spend most of their lives below this value, growing only later as galaxies near the green valley.

We conclude this appendix with some useful power laws that approximate the evolutionary tracks of three quantities versus $M_*$ for galaxies with $R_e/\langle R_e \rangle = 1$ during the star-forming phase. These are evolutionary tracks, not quenched ridgelines. The slopes are taken from Figure 7, and the zero points are set by eye to fit the middle curves in the redshift range $z = 0.5-3$. The actual model calculations use the real, curved tracks in Figure 7, not these power laws.

12. The average radius-mass evolutionary track:

$$\frac{R_e}{1 \text{ kpc}} = \kappa_R \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^R, \quad (\text{C13})$$

where $R = 0.40$ and $\kappa_R \sim 3$.

13. The average $\Sigma_1$-mass evolutionary track:

$$\frac{\Sigma_1}{10^9 M_{\odot} \text{ kpc}^{-2}} = \kappa \Sigma_1 \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^S, \quad (\text{C14})$$

where $S = 0.70$ and $\kappa_{\Sigma_1} \sim 0.6$.

14. The average $M_{BH}$-mass evolutionary track:

$$\frac{M_{\text{BH}}}{10^8 M_{\odot}} = \kappa M_{\text{BH}} \times \left( \frac{M_*}{10^{10} M_{\odot}} \right)^B, \quad (\text{C15})$$

where $B = 1.23$ and $\kappa_{M_{BH}} \sim 0.008$. This assumes that star-forming galaxies evolve along the blue line in Figure 4. The alternative picture for BH growth that is presented at the end of Section 7.8 says that BHs in star-forming galaxies are smaller than this mass until their host galaxies near the green valley.

## Appendix D

### Four Variants on the Empirical Power-law Model

This section provides more details on the four variants of the empirical power-law model that are discussed in Section 6.

#### D.1. Variant 1: A Broken-power-law Model

A feature of the empirical power-law model that does not quite match observed data is the long “tongue” of star-forming galaxies extending up to $M_* > 10^{12} M_{\odot}$ at late redshifts in Figure 10. This tongue is due to the fact that we have used the curved HMSM relation from RP17 at $z=0$ to find the starting redshifts, $z_s$, but have calculated stellar masses going forward using the power-law SMHM relation of Equation (1), $M_* = k_m M_{\text{vir}}^\nu$. Because the latter keeps climbing with $M_{\text{vir}}$, massive galaxies that start near the knee of the real SMHM relation today can wind up with higher stellar mass than their assumed initial mass. Galaxies with $M_* > 10^{10.7} M_{\odot}$ and large $R_e$ (and small BHs) are the objects that populate the massive tongue. We tried to limit this by keeping starting stellar masses close to the knee, which is at approximately $10^{10.7} M_{\odot}$ in RP17. However, that is not high enough to populate the full mass range of observed star-forming galaxies, which reaches $\sim 10^{11.2} M_{\odot}$ in quantity at all redshifts (F13; vdW14; B17).

To attempt to solve this problem, the Variant 1 broken-power-law model uses the same power-law SMHM relation for star-forming galaxies up to $10^{10.7} M_{\odot}$ but switches to a flatter power-law for SMHM above that. This retards stellar mass growth and also BH growth, as $\Sigma_1$ is not increasing. We also take into account the related fact that $\langle R_e \rangle$ now also lags behind $R_{\text{vir}}$ above the knee and the ratio $\langle R_e \rangle/R_{\text{vir}}$ therefore declines. The latter effect has been estimated using the abundance-matching results of RP17, which give $\langle R_e \rangle/R_{\text{vir}}$ versus $R_{\text{vir}}$ for massive halos. The adopted new broken-power-law relations for $M_*$ and $R_e$ are compared to the RP17 SMHM relations in Figure D1.

Inserting these new relations produces a difference between high-mass and low-mass galaxies. The evolution is identical for galaxies whose stellar masses stay below the knee today, and so the $R_e-M_*$ and $\Sigma_1-M_*$ diagrams remain the same below $10^{10.7} M_{\odot}$. Above this mass, $M_*$ tends to saturate, and BH growth is retarded. This is evident in Figure D2, which replots $E_{\text{BH}}$ and $E_{\text{quench}}$ as in Figure 9 but with the new broken power laws. With the new relation, high-mass galaxies now stop growing but also never quench. This solves the too-massive galaxy problem, but these galaxies are now permanently stuck in a star-forming state because their BHs never grow large enough. We were not able to find a broken-power-law scaling relation that both prevents the formation of too-massive
<figure><img src="image_14.png" /><figcaption>**Figure D1.** Broken power laws used for the Variant 1 evolving model in Section 6.1. Left panel: $M_*$ vs. $M_{vir}$. The power-law slope of the dashed line below $M_{vir} = 10^{12.2} M_{\odot}$ is 1.75, which is also the value of $s$ used in the empirical power-law model. Above $M_{vir} = 10^{12.2} M_{\odot}$, the slope is 0.35. The overplotted colored curves for comparison are the RP17 SMHM relations from abundance matching. Right panel: $(R_c/R_{vir})$ vs. $M_{vir}$ for the corresponding lines in the left panel. The colored curves are calculated from equations in RP17. The constant ratio for the dashed line below $M_{vir} = 10^{12.2} M_{\odot}$ is 0.02, which is used in the empirical power-law model and agrees fairly well in the mean with RP17 at those halo masses. Above $M_{vir} = 10^{12.2} M_{\odot}$, the broken-power-law slope is -0.2.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure D2.** Evolutionary plots of $E_{BH}$ and $E_{quench}$ for the Variant 1 broken-power-law model in the style of Figure 9. The single power-law SMHM and $\langle R_c \rangle/R_{vir}$ relations of the empirical power-law model are replaced with the broken power laws shown in Figure D1. The evolution of galaxies below the break point at $10^{10.7} M_{\odot}$ is unaffected. Above this, BHs fail to grow fast enough to keep up with the energy requirements of their halos, and the majority of the most massive galaxies never quench.</figcaption></figure>

star-forming galaxies and at the same time allows typical slightly less-massive star-forming galaxies to quench.

On the other hand, more recent information about the shape of the quenching boundary from a large sample of SDSS galaxies (Luo et al. 2020) offers a different solution. The quenched ridgeline in $\Sigma_1-M_*$ in this new sample is seen to bend over to lower $\Sigma_1$ at high stellar mass, which would create a matching bend in the quenching boundary using our assumptions. Our assumed straight power law lies about 0.2 dex above the new SDSS ridgeline at $10^{11.4} M_{\odot}$, and it is seen from Figure 10 that bending the ridgeline down by this amount would remove all of the problem massive galaxies.

We conclude that the existence of too-massive star-forming galaxies in the model may be an artifact of assuming a perfectly linear power-law quenching boundary, whereas the real boundary may curve slightly down in $\Sigma_1$ at high mass. This

curvature could be due either to larger BH mass growth per galaxy than assumed in the model or to a change in the structure of massive halos that makes them easier to quench. Either way, the lesson of Variant 1 is that the fraction of quenched versus star-forming galaxies at high masses is extremely sensitive to the shape of the quenching boundary, and a detailed physical understanding is needed to predict it accurately. A related consequence is that the shape of the mass function of quenched galaxies at high values of $M_*$ is also very difficult to predict.

## D.2. Variant 2: Move the Evolving Zero Point to the SMHM Relation

This variant moves the evolving zero point from the halo energy quota in Equation (20) to the SMHM relation. The
<figure><img src="image_16.png" /><figcaption>**Figure D3.** The BH energy output and the halo quenching-energy quota are plotted in the style of Figure 9 for the Variant 2 model, which puts the evolving zero point into the SMHM relation rather than $E_{quench}$. The red dashed lines are the new halo quenching energy $E_{quench}$, and the blue solid curves repeat the energy content of the halo gas, $E_{bind}$, from Figure 9. $E_{quench}$ no longer follows $E_{bind}$, which means that the energy needed to quench a halo is not closely equal to the gas-binding energy in this model (see Section 6.2). Also, quenching for both curves begins too late at $z \sim 1$ rather than at $z \sim 2.5$ as observed.</figcaption></figure>

expression used for $M_*$ is then

$$ \frac{M_*}{10^{10} M_{\odot}} = k_{\text{SMHM}}(0) \times \left( \frac{M_{\text{vir}}}{10^{12} M_{\odot}} \right)^s \times h(z)^{-1.00}, \quad (\text{D1}) $$

where $k_{\text{SMHM}}(0) = 2.51$.

This equation was derived by substituting $M_*$ for $M_{\text{vir}}$ in Equation (20) and demanding that galaxies quench with their previous values of $M_{\text{BH}}$ and $M_*$ at the same $z$. The exponent is fortuitously $-1.00$ owing to the zero-point evolution of the $\Sigma_1-M_*$ ridgeline as $h(z)^{-0.66}$ in Figure B1 and the relationship among $s$, $t$, and $v$ from Equation (14).

The new law makes the zero point of $M_*$ larger at late times, by about 0.5 dex at $z=0$ compared to $z=2$. The growth of $R_e$ versus $M_{\text{vir}}$ is unaffected, but $M_*$ grows more steeply versus $M_{\text{vir}}$ than before. This change effectively grows BHs more strongly at late times, allowing them to overcome halos without having to reduce $E_{\text{quench}}$. The evolutionary tracks for this model are shown as the dashed lines in Figure 7. The main change is a reduction of the log-log slope in $R_e$ versus $M_*$ from 0.40 to 0.33, which actually agrees better with the slope of 0.3 estimated by vD15.

The shift of the evolving zero point affects stellar mass, BH growth, and $E_{\text{quench}}$, but $E_{\text{bind}}$ remains unchanged. Figure D3 plots these quantities for the Variant 2 model in the style of Figure 9. $E_{\text{quench}}$ is now simply proportional to $M_{\text{vir}}^t = M_{\text{vir}}^2$, which climbs strongly with time (red dashed lines). However, it still roughly parallels $E_{\text{BH}}$, which also climbs rapidly, and this maintains a wide green valley. On the other hand, close equality between $E_{\text{quench}}$ and halo gas-binding energy, $E_{\text{bind}}$, has been lost. This is shown by the blue curves, which are now much shallower than $E_{\text{quench}}$. This has the consequence that quenching using $E_{\text{bind}}$, which began for massive galaxies at the plausible redshift $z \sim 2.5$ in Figure 9, does not start here until $z=1$. Quenching using $E_{\text{quench}}$ also starts very late. Thus, even though Variant 2 can match the moving quenching boundary, its $E_{\text{quench}}$ no longer has physical justification, and the late growth of BH masses makes quenching too recent under all scenarios.

A related question is whether the measured changes in the zero point of the SMHM relation back in time are as large as needed by this variant. Historically, measurements of SMHM back in time have scattered. Several authors have reported finding similar values of SMHM at $M_{\text{vir}} \sim 10^{10} M_{\odot}$ but lower

values back in time near the knee at $M_{\text{vir}} \sim 10^{12} M_{\odot}$ (e.g., Moster et al. 2013; Skibba et al. 2015; RP17), implying shallower slopes at higher $z$. However, some works (Leauthaud et al. 2012; Yang et al. 2012; Behroozi et al. 2013, 2019) find roughly parallel slopes at all redshifts but a shift to a lower zero point in $M_*$ at early times. This is the right direction needed for Variant 2. However, the zero-point shift from $z=0$ to $z=1$ in Behroozi et al. (2019) is 0.25 dex, with little change beyond that to $z=3$. This is only half the value of 0.5 dex needed by this model.

### D.3. Variants 3 and 4: Energy Rates or Energy Totals?

Variants 3 and 4 replace the total energies on both sides of Equation (20) with mass rates. Entry into the green valley then occurs when the halo mass-heating rate from the BH equals the rate at which mass cools in the halo. For both, the halo energy term $E_{\text{quench}}$ on the right side is replaced by the standard halo mass-cooling rate, $\dot{m}_{\text{cool}}$, from Croton et al. (2006; see also Somerville et al. 2008; Henriques et al. 2015):

$$ \dot{m}_{\text{cool}} = M_{\text{hot}} \frac{r_{\text{cool}}}{R_{\text{vir}}} \frac{1}{t_{\text{dyn,h}}}, \quad (\text{D2}) $$

where

$$ r_{\text{cool}} = \left[ \frac{t_{\text{dyn,h}} M_{\text{hot}} \Lambda(T_{\text{hot}}, Z_{\text{hot}})}{6\pi \mu m_{\text{H}} \kappa T_{\text{hot}} R_{\text{vir}}} \right]^{\frac{1}{2}}, \quad (\text{D3}) $$

and where $\mu$ is the mean molecular weight, set here to 0.59, $\kappa$ is Bolzmann's constant, and $t_{\text{dyn,h}}$ is the halo dynamical time $=R_{\text{vir}}/V_{\text{vir}}$ (Henriques et al. 2015). $M_{\text{hot}}$ is the mass of hot gas defined as

$$ M_{\text{hot}} = f_{\text{hot}} \times M_{\text{vir}}, \quad (\text{D4}) $$

where $f_{\text{hot}}$ is the hot-gas fraction, set here to 0.1. $\Lambda$ is the equilibrium cooling function for collisional processes, which depends both on the metallicity and temperature but ignores radiative ionization effects (Sutherland & Dopita 1993). For $Z_{\text{hot}}$, we use one-third solar metallicity at all redshifts. $T_{\text{hot}}$ is the postshock temperature of the infalling gas, which is assumed to be the virial temperature of the dark halo
<figure><img src="image_17.png" /><figcaption>**Figure D4.** BH-related quantities used in Variants 3 and 4. The three lines correspond to the final halo masses of Figure 6. Panels (a) and (b): the original BH masses from the empirical power-law model, $M_{BH} \sim \Sigma_1^{\gamma}$, are plotted at upper left, and the BH accretion rates obtained by differentiating those masses vs. time are plotted at upper right. Panels (a) and (b) are used in Variant 3. Panel (c): BH masses in the Croton et al. (2006) radio-mode model, obtained by integrating the BH accretion rates in that model, which are shown in panel (d). They have been normalized by forcing them to match, on average, the BH masses from the empirical model in panel (a) at $z=0$. Because the ranges of these masses are roughly similar, this proves easy to do just by adjusting the zero point $k_{AGN}$ in Equation (D6). Panels (c) and (d) are used in Variant 4. The different behaviors of the accretion rates vs. time predict very different shapes and evolution of the quenching boundaries, as shown in Figures D5 and D7.</figcaption></figure>

(Henriques et al. 2015), that is,

$$T_{\text{hot}} = T_{\text{vir}} = 35.9 \left( \frac{V_{\text{vir}}}{\text{km s}^{-1}} \right)^2 K. \qquad (\text{D5})$$

(Note that this $T_{\text{vir}}$ is half the virial temperature of Bryan & Norman 1998.)

To replace the total energy from the BH on the left side of Equation (20) by an energy input rate, two formulations for the BH mass-accretion rate are used. Variant 3 uses the rate obtained by differentiating the instantaneous $M_{BH}$ values given in our empirical power-law model, while Variant 4 uses the BH accretion expression introduced by Croton et al. (2006; and modified by Henriques et al. 2015) to supply the heating for radio mode:

$$\dot{M}_{\text{BH}} = k_{\text{AGN}} \left( \frac{M_{\text{BH}}}{10^8 M_{\odot}} \right) \left( \frac{f_{\text{hot}}}{0.1} \right) \left( \frac{V_{\text{vir}}}{200 \text{ km s}^{-1}} \right)^3, \qquad (\text{D6})$$

where $M_{BH}$ is the BH mass, $f_{\text{hot}}$ is the hot-gas mass fraction, and $k_{\text{AGN}} = 6 \times 10^{-6}$ is a free parameter with units of $M_{\odot} \text{yr}^{-1}$. We note that $V_{\text{vir}}^3$ is approximately $M_{\text{vir}}$ and thus the BH grows super-exponentially with time in this formulation. This has important consequences for the shape and location of the quenching boundary, as discussed below.

Both mass-accretion rates are converted into a halo mass-heating rate, $\dot{m}_{\text{heat}}$, in the standard way:

$$\dot{m}_{\text{heat}} = \frac{2\dot{E}_{\text{radio}}}{V_{\text{vir}}^2} = \frac{2\eta\dot{M}_{\text{BH}}c^2}{V_{\text{vir}}^2}, \qquad (\text{D7})$$

where $\eta = 0.1$ is an efficiency parameter and $c$ is the speed of light.

To use the Croton et al. (2006) expression in Equation (D6) for Variant 4 requires two more inputs. The initial values of $M_{BH}$ in Croton et al. (2006; and in Somerville et al. 2008; Henriques et al. 2015) come from growing the BH masses previously in quasar mode. We do not have access to those masses, so we start each galaxy at $z_s$ with $M_{BH}$ given by the empirical power-law model using $\Sigma_1$ from $R_e$ and $M_*$, and grow the BH thereafter using Croton's $\dot{M}_{BH}$. Meanwhile, host galaxy properties $\Sigma_1$, $R_e$, and $M_*$ are tracked using the empirical power-law model. The coefficient $k_{\text{AGN}}$ in front of the accretion rate in Equation (D6) also needs to be adjusted. This was originally set low in Croton et al. (2006) because the purpose was to keep an already-hot halo hot, but we are using it to stop accretion in a cold halo and need to maximize it within reason. We therefore adjust it so that the total BH mass growth in radio mode approximately equals the total mass growth in the empirical model (Equation (7)), which has the benefit of making the final BH masses satisfy the $M_{BH}-M_*$ relation today. It turns out that a wide range of BH masses can be approximately matched today by adjusting this one zero point (compare the BH masses in panel (c) with those in panel (a) at $z=0$ in Figure D4 below).

In order to compute Variants 3 and 4 in a fashion that is exactly parallel to the empirical power-law model, we would need to take into account any effect that radius variations, $R_e/\langle R_e \rangle$, would have on $\dot{M}_{BH}$. This is automatically taken care of for Variant 3 because $\dot{M}_{BH}$ is based on the time derivative of $M_{BH}$ from the empirical model, which itself is scattered by
<figure><img src="image_18.png" /><figcaption>**Figure D5.** $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ vs. redshift for the Variant 3 model. This figure is analogous to Figure 9 for the empirical power-law model but uses heating and cooling rates instead of total energies. (Note that, as in Figure 9, galaxies are plotted ignoring the fact that objects above the dashed lines would have quenched, so that actual values of $M_{\text{BH}}$ and $E_{\text{BH}}$ for these objects would be smaller than shown.) $\dot{m}_{\text{cool}}$ for the halo cooling rate is given by Equation (D2), while BH $\dot{m}_{\text{heat}}$ uses the BH mass-accretion rates from the empirical model, shown in panel (b) of Figure D4. Note how the BH energy output rate turns down at late times and also how the halo $\dot{m}_{\text{cool}}$ stays very constant with mass while $\dot{m}_{\text{heat}}$ trends sharply down with time. These features generate very different quenching boundaries with time from the empirical model, as shown in Figure D6.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure D6.** $\Sigma_I$ vs. $M_*$ for the Variant 3 model. This figure is analogous to Figure 10 for the empirical power-law model but uses the $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ rates from Figure D5. The observed quenching boundary that we want to match at each redshift is given by the straight red power law. Model galaxies that are actually quenching at that redshift are the pale green points. The colored regions represent objects that are still star-forming. No galaxies quench at late times in this model, which stems from the turndown in $\dot{m}_{\text{heat}}$ at late times.</figcaption></figure>

$R_e/\langle R_e \rangle$. In contrast, as written, $\dot{M}_{\text{BH}}$ in Variant 4 depends only on $M_{\text{BH}}$ and halo $V_{\text{vir}}$, neither of which explicitly has a theory to relate it to $R_e/\langle R_e \rangle$ in this alternate picture. For Variant 4, we have therefore simply used the average galaxy with $R_e = \langle R_e \rangle$ in the empirical power-law model as a starting point and evolved its BH mass from there. Variant 4 is therefore effectively a one-parameter model at each epoch.

The resulting $M_{\text{BH}}$ and $\dot{M}_{\text{BH}}$ curves for Variants 3 and 4 are shown in Figure D4. The $M_{\text{BH}}$ growth curve from the empirical power-law model is shown at upper left, and its time derivative, $\dot{M}_{\text{BH}}$, is shown at upper right. This latter is the mass-accretion rate used in Variant 3. The lower panels show $M_{\text{BH}}$ and $\dot{M}_{\text{BH}}$ from the Croton et al. (2006) radio-mode prescription. These have been normalized by forcing the final BH masses today to roughly equal the values in the empirical model by adjusting $k_{\text{AGN}}$ in Equation (D6).

Comparing the four panels, it is seen that the $M_{\text{BH}}$ curves from the empirical power-law model (panel (a)) curve gently down and are very different from the exponentially growing, upwardly curved $M_{\text{BH}}$ curves in the radio-mode formulation (panel (c)). The mass-accretion rates on the right side also differ. $\dot{M}_{\text{BH}}$ trends down with time using the empirical-model accretion rate (panel (b)). This is because $M_{\text{BH}} \sim M_*^{1.23}$ in this model (Figure 7), and thus $\dot{M}_{\text{BH}} \sim \dot{M}_*$ very approximately. Because the star formation rate is falling at late times, $\dot{M}_*$ is

falling and $\dot{M}_{\text{BH}}$ falls, too. In contrast, $\dot{M}_{\text{BH}}$ in the exponentially growing radio-mode trends up super-exponentially (panel (d)).

To summarize, Figure D4 compares three separate models for the energy coming from the BH: (1) empirical power-law model, where the relevant energy quantity is the total BH mass (panel (a)); (2) BH accretion rates from the empirical model (Variant 3; panel (b)); and (3) the radio-mode accretion rates from Croton et al. (2006; Variant 4; panel (d)). The first is rather log-linear versus redshift, the second turns steeply down, and the third turns steeply up. The three models therefore quite fortuitously exemplify three very different BH heating histories.

Figure D5 plots the resulting values of $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ versus redshift for Variant 3. Compared to the analogous plot for the total-energy-based empirical model in Figure 9, the halo term $\dot{m}_{\text{cool}}$ is much more constant with redshift and mass than the halo term $E_{\text{quench}}$ is in Figure 9, whereas $\dot{m}_{\text{heat}}$ from the BH varies more. (Note that values of $M_{\text{BH}}$ and $E_{\text{BH}}$ are again plotted as though quenching did not occur, so that actual values for galaxies above the dashed line would be smaller than shown. See description for Figure 9.) The two terms move roughly together in Figure 9, which mean that all masses are never far from quenching. In Variant 3 by contrast, the highest halo mass is quenched early at high redshifts, while the lowest halo mass never quenches at all. The rate of BH heating, $\dot{m}_{\text{heat}}$,
<figure><img src="image_20.png" /><figcaption>**Figure D7.** $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ vs. redshift for the Variant 4 model. This figure is analogous to Figure D5 but uses BH mass-accretion rates based on the radio-mode model of Croton et al. (2006) (Figure D4, panel (d)). No scatter is applied because there is no explicit dependence of BH accretion rate on $R_e$ (Equation (D6)); tracks are widened slightly to show $M_*$. Note how the BH energy output rate turns up sharply at late times due to super-exponential growth of the BH, while $\dot{m}_{\text{cool}}$ stays roughly constant with mass. This together with zero scatter generate yet different quenching boundaries with time, as shown in Figure D8.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure D8.** $\Sigma_1$ vs. $M_*$ for the Variant 4 model. Similar to Figure D6 but uses the $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ rates from Figure D7. The observed quenching boundary at each redshift is given by the straight red power law. Model galaxies that are actually quenching at that redshift are the pale green points, and the colored regions indicate galaxies that are still star-forming. No scatter is applied because there is no explicit dependence of BH accretion rate on $R_e$ (Equation (D6)); tracks are widened slightly simply to show $M_{\text{BH}}$. This model matches the location of the quenching boundary fairly well but predicts that only a single mass should be quenching at any epoch.</figcaption></figure>

also tends to turn down at late times. That is because BH masses grow as $M_*^{1.23} \sim M_*$ (Figure 7), and $\dot{M}_*$, and therefore also $\dot{M}_{\text{BH}}$, is falling. Because $\dot{m}_{\text{cool}}$ stays flat, this means that BHs are generally less able to quench at late times in this model.<fnref n="51" /> Finally, $\dot{m}_{\text{heat}}$ and $\dot{m}_{\text{cool}}$ are roughly parallel at middle halo masses. This means that some galaxies at that mass will quench very early while others never will.

These circumstances produce a very uneven feeding of galaxies through the quenching boundary versus time for Variant 3. This is shown in Figure D6, which is similar to Figure 10, showing plots of $\Sigma_1$ versus $M_*$ in four redshift bins. The locations of actually quenching galaxies are shown as the pale green points, and the target quenching boundary at that redshift is the red line. Galaxies that are still star-forming fill the colored regions. The quenching loci fall very far from the target locations in this model. Galaxies that quench do so too soon and at too low a mass. They come primarily from the easily quenched high-mass halos. In addition, there is a general lack of quenching at late times due to the late turndown in $\dot{m}_{\text{heat}}$. This is reflected in the complete disappearance of the pale green points in the lowest redshift panels, which does not match the observed continued growth in the number of quenched galaxies after $z \sim 1$ (e.g., Bell et al. 2004; Faber et al. 2007; Muzzin et al. 2013; Tomczak et al. 2014;

Straatman et al. 2015). Finally, the zero point of the predicted quenching boundary hardly falls with time, unlike the observations, and that is because $\dot{m}_{\text{cool}}$ is nearly constant with time.

Variant 3 illustrates the point that the BH and halo heating and cooling terms need to march closely together in zero point versus time in order to feed galaxies smoothly across the entire width of the green valley with the same average mass at all redshifts and at a uniform rate. BH energy rates vary a lot at late times, whereas $\dot{m}_{\text{cool}}$ for halos (Equation (D2)) hardly varies, and this produces a mismatch. More generally, matching a power-law-shaped quenching boundary is most naturally accomplished using BH and halo expressions that are logarithmically featureless. This favors the use of galaxy and halo structural parameters that vary smoothly with time and mass and disfavors the use of rates, which can vary irregularly (see Figure D4).

We conclude that using the mass-rate formulation for halo cooling together with BH accretion rates based on the empirical power-law model, as in Variant 3, fails to match the observed quenching boundary and its evolution with time.

Figures D7 and D8 are the analogous plots for the Variant 4 model, which uses the BH accretion rate $\dot{M}_{\text{BH}}$ from radio mode, as shown in Figure D4(d). The halo term $\dot{m}_{\text{cool}}$ is again rather constant whereas $\dot{m}_{\text{heat}}$ turns up steeply with time on account of the super-exponential BH mass growth in Equation (D6). This model does a better job of matching the observed quenching boundary locations, but its characteristic quenching mass drifts

<footnote n="51">⁵¹ Once a galaxy is quenched, it is counted as quenched forever even if its $\dot{m}_{\text{heat}}$ falls below $\dot{m}_{\text{cool}}$ later.</footnote>
down by 1.1 dex from $z = 1.4$ to now, in disagreement with the observed constant green-valley mass. This is because the super-exponential growth sets in earlier for more massive BHs (Figure D4(d)), which makes them quench at higher redshift. A second deficit is the lack of any theory to predict how BH mass would scale with galaxy radius (or any other potential second parameter), which means that neither the finite width of the green valley nor the slopes of the quenching boundaries through it (in $\Sigma_I-M_*$, $R_e-M_*$, and $M_{BH}-M_*$) are explained.

Variant 4 proves the inverse of Variant 3, that accretion rates cannot rise too rapidly with time. It also illustrates the point made in the Introduction that the observed properties of quenching galaxies are at least a two-parameter family at each epoch whereas models based just on halo cooling (like Variant 4) are only a one-parameter family.

## ORCID iDs

Zhu Chen https://orcid.org/0000-0002-2326-0476

David C. Koo https://orcid.org/0000-0003-3385-6799

Joel R. Primack https://orcid.org/0000-0001-5091-5098

Avishai Dekel https://orcid.org/0000-0003-4174-0374

Yicheng Guo https://orcid.org/0000-0003-2775-2002

Guillermo Barro https://orcid.org/0000-0001-6813-875X

A. van der Wel https://orcid.org/0000-0002-5027-0135

Eric F. Bell https://orcid.org/0000-0002-5564-9873

Jerome J. Fang https://orcid.org/0000-0002-7254-7174

Henry C. Ferguson https://orcid.org/0000-0001-7113-2738

Mauro Giavalisco https://orcid.org/0000-0002-7831-8751

Susan Kassin https://orcid.org/0000-0002-3838-8093

Lin Lin https://orcid.org/0000-0003-1138-8146

Yifei Luo https://orcid.org/0000-0001-7729-6629

Camilla Pacifici https://orcid.org/0000-0003-4196-0617

Viraj Pandya https://orcid.org/0000-0002-2499-9205

Samir Salim https://orcid.org/0000-0003-2342-7501

Sandro Tacchella https://orcid.org/0000-0002-8224-4505

Bryan A. Terrazas https://orcid.org/0000-0001-5529-7305

Hassen M. Yesuf https://orcid.org/0000-0002-4176-9145

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