The Sub-Eddington Boundary for the Quasar Mass–Luminosity Plane: a Theoretical Perspective

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Article The Sub-Eddington Boundary for the Quasar Mass–Luminosity Plane: A Theoretical Perspective

David Garofalo 1,* , Damian J. Christian 2 and Andrew M. Jones 3

1 Department of Physics, Kennesaw State University, Marietta, GA 30060, USA 2 Department of Physics & Astronomy, California State University, Northridge, CA 91330, USA; [email protected] 3 Department of Mathematics, Boise State University, Boise, ID 83725, USA; [email protected] * Correspondence: [email protected]

Received: 17 May 2019; Accepted: 6 June 2019; Published: 11 June 2019

Abstract: By exploring more than sixty thousand quasars from the Sloan Digital Sky Survey Data Release 5, Steinhardt & Elvis discovered a sub-Eddington boundary and a redshift-dependent drop-off at higher black hole mass, possible clues to the growth history of massive black holes. Our contribution to this special issue of Universe amounts to an application of a model for black hole accretion and jet formation to these observations. For illustrative purposes, we include ~100,000 data points from the Sloan Digital Sky Survey Data Release 7 where the sub-Eddington boundary is also visible and propose a theoretical picture that explains these features. By appealing to thin disk theory and both the lower accretion efficiency and the time evolution of jetted quasars compared to non-jetted quasars in our “gap paradigm”, we explain two features of the sub-Eddington boundary. First, we show that a drop-off on the quasar mass-luminosity plane for larger black hole mass occurs at all redshifts. But the fraction of jetted quasars is directly related to the merger function in this paradigm, which means the jetted quasar fraction drops with decrease in redshift, which allows us to explain a second feature of the sub-Eddington boundary, namely a redshift dependence of the slope of the quasar mass–luminosity boundary at high black hole mass stemming from a change in radiative efficiency with time. We are able to reproduce the mass dependence of, as well as the oscillating behavior in, the slope of the sub-Eddington boundary as a function of time. The basic physical idea involves retrograde accretion occurring only for a subset of the more massive black holes, which implies that most spinning black holes in our model are prograde accretors. In short, this paper amounts to a qualitative overview of how a sub-Eddington boundary naturally emerges in the gap paradigm.

Keywords: quasars; active galactic nuclei; supermassive black holes

1. Introduction Active Galactic Nuclei (AGN) have been found to obey their Eddington luminosity limit, and even supermassive black holes in quiescent galaxies can be actively accreting at rates of 0.1% that of ≈ Eddington (Peterson 2014). Building on the work of [1] showing that black holes grow predominantly by accreting close to their Eddington limit, [2] explored the quasar mass-luminosity plane for 62,185 quasars from the Sloan Digital Sky Survey (SDSS) and discovered an additional surprising and as yet unexplained ‘Sub-Eddington Boundary’ (see also [1] Figure 6 including objects at higher redshift where the sub-Eddington boundary appears). Their results are shown in Figure1 (taken from Figure 1 of [2]), showing that quasars tend to not only obey the Eddington limit, but seem to shy away from it in a mass-dependent way such that at higher black hole mass there is a “drop-off”, with the slope of the boundary line decreasing. In addition, they also find the drop-off to increase further beyond some redshift threshold, in the sense that its slope is smaller. We find similar behavior for a “drop-off” in

Universe 2019, 5, 145; doi:10.3390/universe5060145 www.mdpi.com/journal/universe Universe 2019, 5, 145 2 of 19

off” in luminosity for higher mass black holes and a sub-Eddington boundary in our analysis of the SDSS DR7 data [3] shown in our Figure 2. Understanding this behavior promises to shed light on the luminosityoff” for in luminosity higher mass for blackhigher holesmass black and aholes sub-Eddington and a sub-Eddington boundary boundary in our in analysis our analysis of the of SDSSthe DR7 SDSShistory DR7 of data black [3] holeshown growth. in our FigureThe existence 2. Understanding of a sub-Eddington this behavior boundary, promises tohowever, shed light is on quite the data [3] shown in our Figure2. Understanding this behavior promises to shed light on the history of historycontroversial of black [4–6]. hole growth. The existence of a sub-Eddington boundary, however, is quite black holecontroversial growth. The [4–6]. existence of a sub-Eddington boundary, however, is quite controversial [4–6].

Figure 1. The quasar mass-luminosity plane from Figure 1 of [2]. The dashed line represents the Figure 1. The quasar mass-luminosity plane from Figure 1 of [2]. The dashed line represents the FigureEddington 1. The boundary. quasar mass-luminosityRed indicates a redshift plane fromvalue Figure of 0.2 <1 zof < 0.8,[2]. yellowThe dashed indicates line 0.8 represents < z < 1.4 andthe Eddington boundary. Red indicates a redshift value of 0.2 < z < 0.8, yellow indicates 0.8 < z < 1.4 Eddingtongreen is for boundary. 1.4 < z

Figure 2. The plot of data from Sloan Digital Sky Survey (SDSS) Data Release 7 including average Figure 2.FigureluminosityThe plot2. The as of plota datafunction of data from of from mass Sloan Sloan ([3])—the Digital Digital top Sky Sky lin SurveySurve is theey (SDSS)Eddington (SDSS) Data Data boundary, Release Release 7 andincluding 7 two including additional average average lines at 1% and 0.1% of Eddington. Units are the same as in Figure 1. Note the absence of objects luminosityluminosity as a function as a function of mass of mass ([3])—the ([3])—the top top line line isis thethe Eddington Eddington boundary, boundary, and andtwo additional two additional lines at 1% and 0.1% of Eddington. Units are the same as in Figure 1. Note the absence of objects lines at 1% and 0.1% of Eddington. Units are the same as in Figure1. Note the absence of objects reaching or even approaching the Eddington limit at log M = 10.5 (i.e., just above Log L = 48) and beyond. Although the possible missing low luminosity points could change the slope of the black line, it would not change the fact that whereas some of the data for small values of log M exceed the Eddington boundary, at high log M no such data exceed it. In fact, the data at high log M actually drop oﬀ sharply from the Eddington boundary. Note how the number of points above the Eddington limit is greater here compared to SDSS 5 of Figure1. Of course, SDSS 7 almost doubles the number of objects. Universe 2019, 5, 145 3 of 19

Whether or not a sub-Eddington boundary exists, we show how it may emerge from our theoretical framework. We reproduce these features of the sub-Eddington boundary by adopting the theoretical framework known as the ‘gap paradigm’ for black hole accretion and jet formation ([7]), according to which retrograde black hole accretion served an important role in the high redshift Universe. Our goal is to describe a picture in which the sub-Eddington boundary emerges from a prescription for redshift z~1.5–2 quasar evolution focused on retrograde accreting black holes, which is both difficult to explore (e.g., [8]) and where a variety of processes appear to be at work (e.g., [9]), by applying the same ideas that allowed the model to shed light on the high redshift AGN phenomenon ([10,11]). In order to accomplish our goal, we appeal to the stabilizing effect of larger black hole mass on retrograde accretion in tandem with its implication for the redshift distribution of different types of active galactic nuclei ([11–14]), as we now explain. When the black hole is massive enough compared to the mass content of the accreting material, the post-merger cold gas funneled toward the supermassive black hole will eventually settle into a thin disk configuration that is equally likely to form in either prograde or retrograde configuration. But when the relative mass constraint between a black hole and accretion disk is not satisfied, retrograde accretion configurations are absent. This constraint comes to us from [12] which we will describe quantitatively. The essence of that work, however, is that the most massive black holes have equal probability of being surrounded by either a retrograde or prograde accretion disk while that probability for retrograde configurations drops as the black hole mass decreases. We will provide quantitative prescriptions as a function of redshift for the fraction of retrograde forming accreting black holes in post mergers that are informed by models of black hole spin formation. Because these accreting black holes are assumed to result from mergers, our mass constraint is modulated by the merger function, which evolves with time. We will use a merger function that is a combination of observations and simulations that ends up producing an almost linear decrease in the merger rate as the redshift decreases from about z = 2. As the merger rate drops, the fraction of accreting black holes that are the product of mergers decreases compared to the entire AGN population, which eventually becomes dominated or fed by secular processes. Consequently, the fraction of retrograde accreting black holes drops and this has an effect on the average efficiency of the Eddington-limited accreting black holes. In short, there is both a black hole mass as well as a redshift dependence on the fraction of retrograde accreting black holes which we will make quantitative. In making clear the spirit of our work, we emphasize that our explanation is contingent or dependent on a simple prescription concerning the nature of the black hole engine. Our goal is to show how much this simple picture can account for. The simplicity of the model, however, does not imply that the physics of the greater galaxy is irrelevant to the overall character of the quasar, but simply that the black hole engine leaves an indelible imprint. We will construct a mass–luminosity plane from theory and show how the aforementioned features of the sub-Eddington boundary emerge. In Section2 we construct a mass–luminosity plane by averaging over a range of luminosities for fixed values of black hole mass to show how the slope at higher mass drops to lower values. We then explore how the value of this slope depends on redshift. In Section3, we conclude.

2. Analysis and Discussion

2.1. Slope Vs. Mass of the Sub-Eddington Boundary From Figures1 and2, we recognize how accreting black holes are not restricted to radiate at their Eddington values, but span a range of about 1 or 2 orders of magnitude for the available data. More relevant to our work is the fact that the data also suggests that the Eddington limit tends to be obeyed, the degree to which that limit is obeyed being the subject of our paper. Surprisingly, the way the AGN population obeys the Eddington limit does not appear to be scale-invariant since Figures1 and2 indicate that smaller mass black holes seem more willing to approach or violate that limit compared to their high mass counterparts, albeit by a small amount. We see that although a variety of luminosities are possible for any black hole mass, at higher black hole mass, the Eddington Universe 2019, 5, 145 4 of 19 limit becomes an increasingly significant constraint, with fewer objects approaching or violating that limit. There seems to be an apparent scale-invariant violation in the way the luminosity range clusters about the Eddington limit. We will argue that this is a feature of the difference in the average type of accretion for lower versus higher black hole mass, with differences due to different accretion efficiencies, resulting directly from the above-mentioned accretion disk stability dependence on black hole mass (i.e., on the fraction of retrograde accreting black holes). We now highlight the elements of standard thin disk accretion that are needed to understand our explanation of the sub-Eddington boundary. While low angular momentum advection-dominated and super Eddington accretion flows have significantly lower efficiencies [15,16]), flows which radiate closest to the Eddington limit are standard thin disk accretors. Because we are interested in exploring the behavior of accretion close to the Eddington boundary so as to explain the sub-Eddington boundary, we will ignore less efficient accretion models for most of the discussion and focus on standard thin disk, [17] accretion, or more precisely, on the relativistic version of [18]. The reason is that less efficient accretion fails by definition to reach the Eddington limit. However, in order to explain the observations of [2] at redshift less than about 0.5, we will need to look at the time evolution of massive accreting black holes, which will force us to deal with low efficiency, ADAF accretion. For our initial exploration, however, we focus only on thin relativistic accretion disks, which are characterized by stable circular orbits that are truncated at the innermost stable circular orbit (ISCO), a value that depends on black hole spin as well as on the orientation of the accretion flow. The non-relativistic version of the dissipation as a function of radial distance in the disk depends on the location of the inner boundary risco according to:

3GM dM r D(r) = [1 isco )1/2] (1) 4πr3 dt − r with G Newton’s constant, M the black hole mass, and dM/dt the accretion rate. When integrated over the two faces of the thin disk, one obtains the luminosity as:

GM dM Ldisk = (2) 2risco dt

The disk luminosity is, therefore, ISCO dependent. The smaller the ISCO is, the larger the total luminosity. Because (as emphasized in the Introduction) the larger mass (M) objects less strongly constrain disk orientation and are thus progressively more likely in the paradigm to be surrounded by retrograde or prograde accretion, the larger M objects are also more likely to have ISCO values that span a wider range than those with smaller M values. The ISCO values for the larger M objects, therefore, tend to span much more of the range 1.23 GM/c2–9 GM/c2 (with c the speed of light constant) while the smaller M objects tend to span the smaller range 1.23 GM/c2–6 GM/c2. The former range is associated with ISCO values spanning the full retrograde through prograde accretion from high to low black hole spin. The latter range, on the other hand, is associated with prograde-only accretion from high to low black hole spin, which we employ to model lower mass accreting black holes (see, e.g., the AGN branching-tree diagram in [11]. In short, associated with the larger fraction of retrograde accreting black holes at larger black hole mass, there is also a greater range of ISCO values. And as Equation (2) illustrates, ISCO values determine the luminous eﬃciency. We show the relationship between the ISCO value and the disk eﬃciency in Table1 with the ISCO value in gravitational radii and plot them in Figure3. Universe 2019, 5, 145 5 of 19

Table 1. The left column shows the innermost stable circular orbit (ISCO) values in gravitational radii, while the right column reports the disk eﬃciency normalized to the highest prograde spin case.

ISCO (rg)H 9.0 0.111 8.7 0.115 8.5 0.118 8.2 0.122 7.9 0.127 7.5 0.133 7.2 0.139 6.9 0.145 6.6 0.152 6.3 0.159 6.0 0.167 5.8 0.172 5.33 0.188 5.0 0.200 4.61 0.217 4.3 0.233 3.83 0.261 3.3 0.303 2.9 0.345 2.2 0.455 1.8 0.555 1.5 0.667 1.237 0.808 1.00 1.00

FigureFigure 3. 3. ISCOISCO values values and and disk disk efficiency eﬃciency from from Table Table 1.1.

Since we are interested in scale-invariant explanations, we impose scale-invariance in the accretion rate, i.e., we assume a fixed average Eddington accretion rate across the black hole mass population. This is not unusual, nor restrictive, but a natural consequence of the fact that galactic processes that funnel gas toward the black hole are decoupled from the black hole sphere of influence (e.g., [19–22]). This leaves us with one free parameter, the ISCO value. Because of the absence of any physical constraints, we allow black hole spin to vary randomly over a large range in spin except at the very high end of the spin where simulations suggest the highest spins are more difficult to obtain in mergers ([23]). This means that most values of black hole spin are weighted equally. This does not mean, however, that prograde, as well as retrograde systems, are equally likely and this is a crucial distinction that

Figure 4. Retrograde fraction as a function of disk mass in terms of the black hole mass such that a value of 1 indicates equality between disk and black hole from the analysis of [12] via our Equation (3). As the total mass of the disk decreases compared to the black hole, its total angular momentum decreases as well, and the retrograde fraction approaches 0.5, reflecting the random nature of the disk configuration around the rotating black hole.

Since we are interested in scale-invariant explanations, we impose scale-invariance in the accretion rate, i.e., we assume a fixed average Eddington accretion rate across the black hole mass population. This is not unusual, nor restrictive, but a natural consequence of the fact that galactic processes that funnel gas toward the black hole are decoupled from the black hole sphere of influence (e.g., [19–22]). This leaves us with one free parameter, the ISCO value. Because of the absence of any physical constraints, we allow black hole spin to vary randomly over a large range in spin except at the very high end of the spin where simulations suggest the highest spins are more difficult to obtain in mergers ([23]). This means that most values of black hole spin are weighted equally. This does not mean, however, that prograde, as well as retrograde systems, are equally likely and this is a crucial distinction that imposes non-negligible constraints on the results. As discussed above, the retrograde fraction has both a mass as well as a redshift dependence. With that in mind, and with an eye toward constructing a theoretical version of Figure 1, we proceed to obtain an average luminosity for a given black hole mass. We average over all possible accretion disks that differ in ISCO values to determine an average disk luminosity as a function of black hole mass by imposing the following prescription for the retrograde accreting fraction. In attempting to impose the physical constraints from King et Universe 2019, 5, 145 6 of 19 imposes non-negligible constraints on the results. As discussed above, the retrograde fraction has both a mass as well as a redshift dependence. With that in mind, and with an eye toward constructing a theoretical version of Figure1, we proceed to obtain an average luminosity for a given black hole mass. We average over all possible accretion disks that differ in ISCO values to determine an average disk luminosity as a function of black hole mass by imposing the following prescription for the retrograde accreting fraction. In attempting to impose the physical constraintsal. from(2005) King as discussed et al. (2005) further as discussed below, we assume that no retrograde accreting black holes form at M further below, we assume that no retrograde accreting black holes< form 107 M at◉ M, where< 107 M◉ ,is where one solaris mass. one A more sophisticated prescription for retrograde formation at solar mass. A more sophisticated prescription for retrograde formationlower atblack lower hole black mass hole would mass not would qualitatively change any of our conclusions. This means that in the not qualitatively change any of our conclusions. This means thataftermath in the of aftermath mergers ofthat mergers funnel thatcold gas to the central merging black holes, an accretion disk forms funnel cold gas to the central merging black holes, an accretion diskwhose forms angular whose momentum angular momentum vector is aligned with that of the black hole, which constitutes prograde vector is aligned with that of the black hole, which constitutes progradeaccretion. accretion. As discussed As discussed in the Introduction, in the the black hole is not massive enough to remain stable in Introduction, the black hole is not massive enough to remain stablea retrograde in a retrograde configuration configuration so any soretrograde forming disks simply flip to a prograde state. As we any retrograde forming disks simply flip to a prograde state. As weconsider consider more more massive massive black black holes holes on the order of 108 M◉ or more, however, we relax our constraint on the order of 108 or more, however, we relax our constraintby by allowing allowing the fractionfraction ofof retrograde retrograde objects to increase progressively toward 50% at high masses objects to increase progressively toward 50% at high masses nearnear 109 109 ,M implying◉, implying that that both both prograde prograde and retrograde disks become equally stable for large black and retrograde disks become equally stable for large black holes.holes. Our Our prescription prescription for the for retrograde the retrograde fraction is grounded in the work of [12] who showed that fraction is grounded in the work of [12] who showed that the relativethe relative total angular total angular momentum momentum of the of the accretion disk to that of the black hole determines the accretion disk to that of the black hole determines the retrograderetrograde fraction according fraction according to: to:

1 f = ½ (1 – Jd/2Jh) (3) f = (1 Jd/2Jh) (3) 2 − where Jd and Jh are the total angular momentum of the disk and black hole, respectively. Because both angular momenta are obtained as an integral that is linearly dependent on mass, the fraction of where Jd and Jh are the total angular momentum of the disk and black hole, respectively. Because both angular momenta are obtained as an integral that is linearlyretrograde dependent occurrences on mass,the can fraction be translated of into a mass ratio, with a higher fraction corresponding to retrograde occurrences can be translated into a mass ratio, withlower a higher accretion fraction mass. corresponding Smaller total accreting to mass compared to the black hole mass results in stability lower accretion mass. Smaller total accreting mass compared to thein either black prograde hole mass or results retrograde in stability cases and the fraction tends to ½ (Figure 4). Whereas the range of accretion1 rates is assumed scale-invariant, the ratio Jd/Jh is not. This comes from the fact that the in either prograde or retrograde cases and the fraction tends to 2 (Figure4). Whereas the range of amount of cold gas that makes it into the central region to form the accretion disk is decoupled from accretion rates is assumed scale-invariant, the ratio Jd/Jh is not. This comes from the fact that the amount of cold gas that makes it into the central region to formblack the accretion hole gravity. disk isFor decoupled mergers of from larger galaxies, the scale invariance in the range of accretion means black hole gravity. For mergers of larger galaxies, the scale invariancethere is in athe larger range range of accretion in that rate. means Black hole gravity cannot regulate or determine the amount of cold there is a larger range in that rate. Black hole gravity cannot regulategas that or determineis funneled the into amount its accretion of cold disk because black hole feedback has yet to occur. Bigger black gas that is funneled into its accretion disk because black hole feedbackholes live has in yetbigger to occur. galaxies Bigger and blacktheir mergers will produce a larger range of accretion rates that reach holes live in bigger galaxies and their mergers will produce a largerhigher range values of accretion that scale rates with that the reach galaxy size and thus black hole mass. But because the formation of higher values that scale with the galaxy size and thus black holethe mass. disk Butis uncoupled because the from formation the black of hole, that larger range means that accretion disks with a wider the disk is uncoupled from the black hole, that larger range meansrange that of accretiontotal mass disks and th withus total a wider angular momentum are possible, which means the existence of an range of total mass and thus total angular momentum are possible,increasing which subset means (compared the existence to smaller of black holes) of the population with small(er) total disk an increasing subset (compared to smaller black holes) of theangular population momentum with small(er) compared total to disk that of the black hole. This means that on average for galaxies with angular momentum compared to that of the black hole. This meanslarger that black on averageholes, the for angular galaxies momentum with fraction Jd/Jh tends to be smaller. Scale invariance does not ensure that larger black holes become surrounded by comparatively larger disks with larger total larger black holes, the angular momentum fraction Jd/Jh tends to be smaller. Scale invariance does not ensure that larger black holes become surrounded by comparativelyangular momentum. larger disks withIt, instead, larger makes total it possible for larger black holes to be fed by a wider range of angular momentum. It, instead, makes it possible for larger blackmass holes for tothe be accretion fed by a widerdisk, some range of of which have small mass compared to the black hole. This is true mass for the accretion disk, some of which have small mass comparedat the formation to the black of the hole. disk This as iswell true as at later times. In fact, the ratio of angular momenta becomes at the formation of the disk as well as at later times. In fact, theprogressively ratio of angular skewed momenta toward becomes the black hole once accretion settles into a steady state since accretion progressively skewed toward the black hole once accretion settlesgrows into the a steady black hole state mass since and accretion its total angular momentum but does not grow the disk mass or its grows the black hole mass and its total angular momentum butangular does not momentum. grow the disk mass or its angular momentum. Equation (3) is not the end of the story, however. Models of black hole spin formation and evolution suggest that high spinning black holes formed from mergers are a small subset of the total population of accreting black holes ([23,24]). Therefore, on top of the stability issue explored in Equation (3), we need to lower the occurrence of high-spinning retrograde accretion configurations relative to medium and low spin configurations. To capture this, we impose an upper boundary to the spin of 0.9 which limits the disk efficiency in retrograde mode to no less than about 0.115 which corresponds to an ISCO value of 8.7 gravitational radii (see Table 1). In addition to the mass dependence, the assumption that only mergers can lead to retrograde configurations implies that the retrograde fraction also depends on the behavior of the merger function which is changing with time. Because we are using the merger function in this Section, we now discuss the time evolution of the retrograde fraction that is affected by the merger rate. In Section IIb we will discuss more general time evolution of the retrograde fraction that comes about as a result of both the evolution of the merger rate as well as the evolution of originally retrograde accreting black holes as determined by the accretion process itself. In order to quantitatively deal with the effect

Universe 2019, 5, 145 Figure 3. ISCO values and disk efficiency from Table 1. 7 of 19

FigureFigure 4. Retrograde 4. Retrograde fraction fraction as a functionas a function of disk of mass disk inmass terms in ofterms the black of the hole black mass hole such mass that such a value that a of 1value indicates of 1 equality indicates between equality disk between and black disk hole and from black the hole analysis from ofthe [12 analysis] via our of Equation [12] via (3).our AsEquation the total(3). mass As ofthe the total disk mass decreases of the compareddisk decreases to the compar black hole,ed to itsthe total black angular hole, its momentum total angular decreases momentum as well,decreases and the retrogradeas well, and fraction the retrograde approaches fraction 0.5, reﬂecting approach thees 0.5, random reflecting nature the of random the disk nature conﬁguration of the disk aroundconfiguration the rotating around black the hole. rotating black hole.

Equation (3) is not the end of the story, however. Models of black hole spin formation and Since we are interested in scale-invariant explanations, we impose scale-invariance in the evolution suggest that high spinning black holes formed from mergers are a small subset of the accretion rate, i.e., we assume a fixed average Eddington accretion rate across the black hole mass total population of accreting black holes ([23,24]). Therefore, on top of the stability issue explored in population. This is not unusual, nor restrictive, but a natural consequence of the fact that galactic Equation (3), we need to lower the occurrence of high-spinning retrograde accretion configurations processes that funnel gas toward the black hole are decoupled from the black hole sphere of influence relative to medium and low spin configurations. To capture this, we impose an upper boundary to (e.g., [19–22]). This leaves us with one free parameter, the ISCO value. Because of the absence of any the spin of 0.9 which limits the disk efficiency in retrograde mode to no less than about 0.115 which physical constraints, we allow black hole spin to vary randomly over a large range in spin except at corresponds to an ISCO value of 8.7 gravitational radii (see Table1). the very high end of the spin where simulations suggest the highest spins are more difficult to obtain In addition to the mass dependence, the assumption that only mergers can lead to retrograde in mergers ([23]). This means that most values of black hole spin are weighted equally. This does not configurations implies that the retrograde fraction also depends on the behavior of the merger function mean, however, that prograde, as well as retrograde systems, are equally likely and this is a crucial which is changing with time. Because we are using the merger function in this Section, we now discuss distinction that imposes non-negligible constraints on the results. As discussed above, the retrograde the time evolution of the retrograde fraction that is affected by the merger rate. In Section 2.2 we fraction has both a mass as well as a redshift dependence. With that in mind, and with an eye toward will discuss more general time evolution of the retrograde fraction that comes about as a result of constructing a theoretical version of Figure 1, we proceed to obtain an average luminosity for a given both the evolution of the merger rate as well as the evolution of originally retrograde accreting black black hole mass. We average over all possible accretion disks that differ in ISCO values to determine holes as determined by the accretion process itself. In order to quantitatively deal with the effect on an average disk luminosity as a function of black hole mass by imposing the following prescription the retrograde fraction as determined by the evolution in time of the merger rate, we use the merger for the retrograde accreting fraction. In attempting to impose the physical constraints from King et function results of [25] to explain our results for Figure5. We determine an average merger rate from Figure6 values for the redshift range 1.5–2 and an average merger rate for the redshift range 0.5–1, obtaining 0.11 and 0.0625 respectively. The merger rate has, therefore, dropped by a factor of 1.76 over this redshift range. Because retrograde accreting thin disks are assumed only to result from mergers, the fraction of possible retrograde accreting thin disks drops to about a factor of 0.6 (the inverse of 1.76) in the redshift range 0.5–1 compared to the redshift range 1.5–2. Therefore, a factor of about 0.6 is the ratio between the retrograde fraction at these two redshift values. However, this is the ratio for black holes at the highest mass, equal to or above 1010 solar masses, where the retrograde fraction is 1 closest to 2 . As the mass of the black hole decreases, so does the retrograde fraction but such that at a threshold mass value, the retrograde fraction becomes zero. Hence, the two curves in Figure5 must converge. In Figure5 we show this redshift dependence with the red data representing the fraction of post-merger systems accreting in retrograde mode at higher redshift close to 2 when the merger function is at its peak (Figure6), while the blue data represents the retrograde fraction at lower redshift when the merger rate has dropped and with it the retrograde fraction. Figures4–6 are combined to produce the time dependence of the retrograde fraction that is needed to determine the evolution of the quasar mass–luminosity function. Universe 2019, 5, 145 8 of 19

Figure 5. Theoretical prescription for the fraction of accreting black holes that form retrograde thin FigureFigure 5.5. TheoreticalTheoretical prescriptionprescription forfor thethe fractionfraction ofof accretingaccreting blackblack holesholes thatthat formform retrograderetrograde thinthin disks as a function of black hole mass. Because accretion around massive black holes is stable, the disksdisks asas a a function function of of black black hole hole mass. mass. Because Because accretion accretion around around massive massive black black holes holes is stable, is stable, the high the high mass black holes are progressively more likely to be in either prograde or retrograde masshigh blackmass holesblack are holes progressively are progressively more likely more to be likel in eithery to progradebe in either or retrograde prograde conﬁgurations, or retrograde configurations, which is why the fraction approaches 0.5 at high mass. The red represents the behavior whichconfigurations, is why the which fraction is why approaches the fraction 0.5 atapproaches high mass. 0.5 The at high red represents mass. The theredbehavior represents in the the behavior redshift in the redshift range of 1.5–2 when the merger function is larger (Figure 6) while the blue represents rangein the ofredshift 1.5–2 whenrange theof 1.5–2 merger when function the merger is larger functi (Figureon is6 larger) while (Figure the blue 6) while represents the blue the represents behavior the behavior when the redshift is closer to 1 and the merger function has dropped (Figure 6). whenthe behavior the redshift when is the closer redshift to 1 and is closer the merger to 1 and function the merger has dropped function (Figure has dropped6). (Figure 6).

FigureFigure 6. Observed 6. Observed (red (red circles) circles) and and simulated simulated (blue (blue diamonds) diamonds) merger merger rates rates for thefor mostthe most massive massive Figure 6. Observed (red circles) and simulated (blue diamonds) merger rates for the most massive mergersmergers above above 100 100 billion billion solar solar masses masses from from [25] [25] For For simplicity, simplicity, we havewe have produced produced an average an average (green (green mergers above 100 billion solar masses from [25] For simplicity, we have produced an average (green squares)squares) of the of twothe two functions. functions. Based Based on thison this we obtainwe obtain the retrogradethe retrograde fraction fraction as a as function a function of redshift of redshift squares) of the two functions. Based on this we obtain the retrograde fraction as a function of redshift displayeddisplayed in Figure in Figure5. We 5. couldWe could have have used used the redthe circlesred circles or the or blue the blue diamonds diamonds and ourand conclusionsour conclusions displayed in Figure 5. We could have used the red circles or the blue diamonds and our conclusions wouldwould remain remain qualitatively qualitatively the same.the same. would remain qualitatively the same. Our basic result for the mass dependence is that the average dimensionless ISCO value (i.e., dim risco risco = GM ) increases with M due to the greater fraction of retrograde accreting black holes. Because c2 the accretion rate is assumed scale-invariant, the increased fraction of retrograde accreting objects will result in an average luminosity that is less than the Eddington luminosity as the black hole mass increases. The slope of average luminosity versus mass will drop away from the Eddington boundary as the black hole mass increases. Another way of restating this is to recognize that average accretion efficiency drops as the black hole mass increases due to a greater fraction of less efficient retrograde objects. In order to show this, we produce an average luminosity for each black hole mass that reflects this one parameter difference in the accreting black hole population, i.e., an increase in Universe 2019, 5, 145 9 of 19 retrograde accreting objects with larger black hole mass. And this prescription (Figure5) increases with black hole mass for both redshifts as a result of their connectional. (2005) as to discussed the merger further fraction below, obtained we assume that no retrograde accreting black holes form at M from [25]—Figure6. When the post-merger black hole is less< than107 M about◉, where 10 8 M◉ ,is we one prescribe solar mass. a cuto Aff more sophisticated prescription for retrograde formation at for simplicity such that 100% of accreting black holes endlower up surrounded black hole mass by prograde would not disks qualitatively that change any of our conclusions. This means that in the are associated with larger efficiencies (i.e., smaller averageaftermath ISCO value). of mergers This cuto thatff willfunnel not cold change gas to the central merging black holes, an accretion disk forms our results qualitatively. When the black hole mass is largewhose enough angular to ensure momentum stability, vector the fraction is aligned with that of the black hole, which constitutes prograde of retrograde accreting black holes approaches the valueaccretion. that reflects As thediscussed randomness in the ofIntroduction, both the the black hole is not massive enough to remain stable in angular momentum vector of the black hole and that of thea accretion retrograde disk, configuration namely 50% so ([12 any]), soretrograde both forming disks simply flip to a prograde state. As we configurations become equally likely. consider more massive black holes on the order of 108 M◉ or more, however, we relax our constraint Our results are displayed in Figure7. Note the drop-obyff allowingat higher the black fraction hole mass,of retrograde where theobjects to increase progressively toward 50% at high masses slope above 4 108 decreases regardless of redshift. Thisnear is 10 a9 directM◉, implying consequence that both of the prograde fact that and retrograde disks become equally stable for large black ≈ × accreting supermassive black holes span a wider range in effiholes.ciency Our (i.e., prescription a larger range for in the ISCO retrograde values) fraction is grounded in the work of [12] who showed that due to the fact that retrograde orientations have increased occurrencethe relative in total the high angular mass momentum accreting black of the accretion disk to that of the black hole determines the hole population. We have associated the highest prograderetrograde spin at 100% fraction efficiency according (see Table to: 1) with the Eddington limit as a normalization condition. If all accreting black holes were chosen to spin at f = ½ (1 – Jd/2Jh) (3) their maximum prograde value, our results for Figure7 would match the Eddington limit. But since prograde-only black holes are assumed to possess a range of progradewhere Jd and spin Jh values, are the the total average angular amounts momentum of the disk and black hole, respectively. Because both to 0.4 of the Eddington limit. The focus should therefore beangular on the momenta relative di arefference obtained in the as data an integral in that is linearly dependent on mass, the fraction of Figure7 (i.e., on the drop for points associated with black holeretrograde masses above occurrences a threshold can limit), be translated and not into a mass ratio, with a higher fraction corresponding to on the actual Eddington value of each data point. The exactlower values accretion for the mass. average Smaller luminosity total accreting as a mass compared to the black hole mass results in stability function of black hole mass plotted in Figure7 for the two redshiftin either values prograde are reported or retrograde in Table cases2. and the fraction tends to ½ (Figure 4). Whereas the range of accretion rates is assumed scale-invariant, the ratio Jd/Jh is not. This comes from the fact that the amount of cold gas that makes it into the central region to form the accretion disk is decoupled from black hole gravity. For mergers of larger galaxies, the scale invariance in the range of accretion means there is a larger range in that rate. Black hole gravity cannot regulate or determine the amount of cold gas that is funneled into its accretion disk because black hole feedback has yet to occur. Bigger black holes live in bigger galaxies and their mergers will produce a larger range of accretion rates that reach higher values that scale with the galaxy size and thus black hole mass. But because the formation of the disk is uncoupled from the black hole, that larger range means that accretion disks with a wider range of total mass and thus total angular momentum are possible, which means the existence of an increasing subset (compared to smaller black holes) of the population with small(er) total disk angular momentum compared to that of the black hole. This means that on average for galaxies with larger black holes, the angular momentum fraction Jd/Jh tends to be smaller. Scale invariance does not ensure that larger black holes become surrounded by comparatively larger disks with larger total angular momentum. It, instead, makes it possible for larger black holes to be fed by a wider range of mass for the accretion disk, some of which have small mass compared to the black hole. This is true at the formation of the disk as well as at later times. In fact, the ratio of angular momenta becomes progressively skewed toward the black hole once accretion settles into a steady state since accretion grows the black hole mass and its total angular momentum but does not grow the disk mass or its angular momentum. Equation (3) is not the end of the story, however. Models of black hole spin formation and

Figure 7. Average Eddington luminosity as a function of blackevolution hole mass suggest (in billions that ofhigh solar spinning masses) black holes formed from mergers are a small subset of the total Figure 7. Average Eddington luminosity as a function of black hole mass (in billions of solar masses) from theory. The red line represents the Eddington limitpopulation while the blue of and accreting green represent black holes the ([23,24]). Therefore, on top of the stability issue explored in from theory. The red line represents the Eddington limit while the blue and green represent the behavior at redshift below 1 and 1.5, respectively. Note theE turnoquationff at higher(3), we black need hole to masslower (above the occurrence of high-spinning retrograde accretion configurations behavior at redshift below 1 and 1.5, respectively. Note the turnoff at higher black hole mass (above 4 108 ) resulting from the greater presence of lessrelative efficient to Shakura medium & and Sunyaev low spin disks configurations. in To capture this, we impose an upper boundary to ≈ ×≈ 4 × 108 M◉) resulting from the greater presence of less efficient Shakura & Sunyaev disks in retrograde configurations. the spin of 0.9 which limits the disk efficiency in retrograde mode to no less than about 0.115 which retrograde configurations. corresponds to an ISCO value of 8.7 gravitational radii (see Table 1). Table 2. Data for Figure 7. The left column is mass in solar masses.In addition The right to columnthe mass is the dependence, Eddington the assumption that only mergers can lead to retrograde luminosity or efficiency. Blue and green refer to redshiftsconfigurations 0.8 and 1.5, respectively. implies that Because the retrogradewe have fraction also depends on the behavior of the merger allowed a slightly larger fraction of retrograde black functionholes in the which green is data,changing the efficiency with time. drops Because we are using the merger function in this Section, we below that for the blue data. now discuss the time evolution of the retrograde fraction that is affected by the merger rate. In Section IIb we will discuss more general time evolution of the retrograde fraction that comes about as a result

Mass (In Solar Masses) L/LEdd (Blue)of both the evolution L/LEdd of (Green) the merger rate as well as the evolution of originally retrograde accreting black holes as determined by the accretion process itself. In order to quantitatively deal with the effect 107 0.398 0.398

2 × 107 0.398 0.398

3 × 107 0.398 0.398

4 × 107 0.398 0.398

5 × 107 0.398 0.398

6 × 107 0.398 0.398

7 × 107 0.398 0.398

8 × 107 0.398 0.398

9 × 107 0.398 0.398

108 0.398 0.398

2 × 108 0.398 0.398

3 × 108 0.398 0.398 Universe 2019, 5, 145 10 of 19

Table 2. Data for Figure7. The left column is mass in solar masses. The right column is the Eddington luminosity or eﬃciency. Blue and green refer to redshifts 0.8 and 1.5, respectively. Because we have allowed a slightly larger fraction of retrograde black holes in the green data, the eﬃciency drops below that for the blue data.

Mass (In Solar Masses) L/LEdd (Blue) L/LEdd (Green) 107 0.398 0.398 2 107 0.398 0.398 × 3 107 0.398 0.398 × 4 107 0.398 0.398 × 5 107 0.398 0.398 × 6 107 0.398 0.398 × 7 107 0.398 0.398 × 8 107 0.398 0.398 × 9 107 0.398 0.398 × 108 0.398 0.398 2 108 0.398 0.398 × 3 108 0.398 0.398 × 4 108 0.382 0.382 × 5 108 0.354 0.342 × 6 108 0.342 0.321 × 7 108 0.331 0.312 × 8 108 0.321 0.303 × 9 108 0.312 0.295 × 109 0.303 0.287 al. (2005) as discussed further below, we assume that no retrograde accreting black holes form at M In greater detail, for black hole masses ranging< 107 M◉ from, where 106 M◉ tois one about solar 108 mass., none A more of the sophisticated accreting prescription for retrograde formation at black holes are allowed to accrete in retrogradelower configurations black hole mass (i.e., would Figure not5 ).qualitatively Therefore, thechange quasar any of our conclusions. This means that in the population at this black hole mass is restrictedaftermath to prograde of mergers accreting that black funnel holes, cold which gas means to the theycentral span merging black holes, an accretion disk forms a narrower range in ISCO values that are nowhose larger angular than 6 GMmomentum/c2. Furthermore, vector is wealigned assume with that that for of a the black hole, which constitutes prograde large range in spin magnitude, no black holeaccretion. spin values As are discussed preferred in or the overrepresented Introduction, with the black respect hole to is not massive enough to remain stable in others, so a straightforward average over thea rangeretrograde of effi cienciesconfiguration for prograde so any accretionretrograde is determined.forming disks simply flip to a prograde state. As we It is important to note that our constraintsconsider are not arbitrary more massive as we black are only holes imposing on the order constraints of 108 M on◉ or more, however, we relax our constraint the retrograde versus prograde fraction andby not allowing on the individualthe fraction spins of retrograde which are determinedobjects to increase by the progressively toward 50% at high masses physics of the merging black holes. This constraintnear 109 M ensures◉, implying that averagethat both e ffiprogradeciencies and must retrograde be above disks become equally stable for large black the efficiency for zero black hole spin. Thereholes. are no Our free prescription parameters for that the we retrograde can manipulate fraction to is obtain grounded in the work of [12] who showed that an average efficiency that is below that ofthe a zero-spin relative accretingtotal angular black momentum hole. This of constraint the accretion ensures disk to that of the black hole determines the that average efficiency at all redshifts be equalretrograde to an e ffifractionciency according value corresponding to: to the efficiency of some intermediate spin prograde disk because no physical processes allow the retrograde regime f = ½ (1 – Jd/2Jh) (3) to be over-represented at any time. In fact, it is precisely the opposite, with the prograde regime over-represented in the population. This analysis,where therefore,Jd and Jh are allows the total us toangular produce momentum theoretical of points the disk in and black hole, respectively. Because both the mass–luminosity plane. As we considerangular larger blackmomenta hole are masses, obtained the data as an in integralFigure5 indicates that is linearly dependent on mass, the fraction of to us that a progressively greater range of ISCOretrograde values, occurrences and thus diskcan ebeffi translatedciencies, are into included a mass asratio, a with a higher fraction corresponding to result of the increased presence of retrogradelower systems. accretion Around mass.4 Smaller108 to,tal the accreting fraction ofmass retrograde compared to the black hole mass results in stability × occurrences is no longer negligible, whichin means either that prograde to obtain or retrograde a data point cases in our and theoretical the fraction plot tends to ½ (Figure 4). Whereas the range of for this mass, we must average over most ofaccretion the range rates from is retrograde assumed to scale prograde-invariant, efficiencies the ratio under Jd/Jh is not. This comes from the fact that the the constraint that 0.125 of all the objects areamount accreting of cold in the gas retrograde that makes mode it into (Figure the central5). As doneregion for to form the accretion disk is decoupled from the prograde-only case, we do not limit norblack constrain hole blackgravity. hole For spin mergers (except of aslarger previously galaxies, discussed the scale invariance in the range of accretion means at very high spin as mandated by numericalthere work), is a larger which range means in athat wide rate. range Black in hole spin gravity values cannot from regulate or determine the amount of cold high to low are not only represented, but equallygas that so. is This funneled then producesinto its accretion an average disk e ffibecauseciency black for the hole feedback has yet to occur. Bigger black 4 108 bin and another point on the mass–luminosityholes live in bigger plane. galaxies The sameand their analysis mergers continues will produce up to a larger range of accretion rates that reach × the highest black hole mass, where the averagehigher is over values an that equally scale represented with the galaxy range size of both and prograde thus black hole mass. But because the formation of the disk is uncoupled from the black hole, that larger range means that accretion disks with a wider range of total mass and thus total angular momentum are possible, which means the existence of an increasing subset (compared to smaller black holes) of the population with small(er) total disk angular momentum compared to that of the black hole. This means that on average for galaxies with larger black holes, the angular momentum fraction Jd/Jh tends to be smaller. Scale invariance does not ensure that larger black holes become surrounded by comparatively larger disks with larger total angular momentum. It, instead, makes it possible for larger black holes to be fed by a wider range of mass for the accretion disk, some of which have small mass compared to the black hole. This is true at the formation of the disk as well as at later times. In fact, the ratio of angular momenta becomes progressively skewed toward the black hole once accretion settles into a steady state since accretion grows the black hole mass and its total angular momentum but does not grow the disk mass or its angular momentum. Equation (3) is not the end of the story, however. Models of black hole spin formation and evolution suggest that high spinning black holes formed from mergers are a small subset of the total population of accreting black holes ([23,24]). Therefore, on top of the stability issue explored in Equation (3), we need to lower the occurrence of high-spinning retrograde accretion configurations relative to medium and low spin configurations. To capture this, we impose an upper boundary to the spin of 0.9 which limits the disk efficiency in retrograde mode to no less than about 0.115 which corresponds to an ISCO value of 8.7 gravitational radii (see Table 1). In addition to the mass dependence, the assumption that only mergers can lead to retrograde configurations implies that the retrograde fraction also depends on the behavior of the merger function which is changing with time. Because we are using the merger function in this Section, we now discuss the time evolution of the retrograde fraction that is affected by the merger rate. In Section IIb we will discuss more general time evolution of the retrograde fraction that comes about as a result of both the evolution of the merger rate as well as the evolution of originally retrograde accreting black holes as determined by the accretion process itself. In order to quantitatively deal with the effect Universe 2019, 5, 145 11 of 19 and retrograde systems. It should be clear, therefore, that the average efficiency is decreasing with the increase in black hole mass. To summarize the results, our explanation for the sub-Eddington boundary is that the most massive black holes produce less restrictive constraints on their accretion configurations, which determines, therefore, an average efficiency that is less than the average efficiency of less massive black holes that do, instead, place constraints on the formation of retrograde systems. While Figure7 emerges from the precise prescription we use for the retrograde fraction as a function of mass, the uncertainties in this function can only result in a shift in the green line either toward the Eddington limit or further removed from it so the qualitative conclusion remains.

2.2. Slope Vs. Redshift of the Sub-Eddington Boundary In addition to the drop-off at higher black hole mass, [2] evaluate the slope of the fit at higher black hole mass as a function of redshift and find the slope to increase initially with redshift, peak, andwill subsequently incorporate decreasethis feature with in higher our explanation. redshift. They First, report however, the slopes we want in what to isobservationally our Figure8. We explore will incorporatethe [3] data this as a feature function in ourof redshift. explanation. We produce First, however, the equivalent we want of to Figure observationally 2 but at z explore> 4 in Figure the [3 ]9. dataBy looking as a function at the of observational redshift. We producedata, one the can equivalent identify ofwhether Figure 2a but sub-Eddington at z > 4 in Figure boundary9. By lookingappears. atWe the have observational explored the data, [3] onedata canwhere identify the existence whether of a a sub-Eddington high mass sub-Eddington boundary appears. boundary We is visible have exploredat the high the [mass3] data limit where around the existence 10 billion of solar a high masse masss, sub-Eddington with the luminosities boundary there is visible falling at shy the of high the 48 mass1048 limiterg/s aroundof the Eddington 10 billion solarlimit masses,(Figure with2). But the we luminosities now can compare there falling that shyto the of thedata 10 as aerg function/s of the of Eddingtonredshift. If limit we focus (Figure on2 the). But z > we4 data, now the can high compare mass sub-Eddington that to the data boundary as a function is even of redshift. more visible If we as focusa result on theof a z flat> 4 distribution data, the high across mass the sub-Eddington mass scale (Figure boundary 9). is even more visible as a result of a flat distribution across the mass scale (Figure9).

FigureFigure 8. 8.Redshift Redshift dependence dependence of of the the slope slope of of the the Eddington Eddington boundary boundary from from H Hββ(green), (green), MgII MgII (blue), (blue), andand corrected corrected MgII MgII values values (magenta). (magenta). From From [2 [2]] (Figure (Figure9). 9).

Figure 9. Luminosity versus Mass for redshift z > 4 from SDSS DR7 ([3]) with the average luminosity over-plotted as the black points. will incorporate this feature in our explanation. First, however, we want to observationally explore the [3] data as a function of redshift. We produce the equivalent of Figure 2 but at z > 4 in Figure 9. By looking at the observational data, one can identify whether a sub-Eddington boundary appears. We have explored the [3] data where the existence of a high mass sub-Eddington boundary is visible at the high mass limit around 10 billion solar masses, with the luminosities there falling shy of the 1048 erg/s of the Eddington limit (Figure 2). But we now can compare that to the data as a function of redshift. If we focus on the z > 4 data, the high mass sub-Eddington boundary is even more visible as a result of a flat distribution across the mass scale (Figure 9).

Figure 8. Redshift dependence of the slope of the Eddington boundary from Hβ (green), MgII (blue), and corrected MgII values (magenta). From [2] (Figure 9).

Universe 2019, 5, 145 12 of 19

FigureFigure 9. Luminosity 9. Luminosity versus versus Mass Mass for redshiftfor redshift z > z4 > from 4 from SDSS SDSS DR7 DR7 ([3]) ([3]) with with the the average average luminosity luminosity over-plottedover-plotted as the as blackthe black points. points.

In order to construct a redshift dependence in the slope, we must determine from theory the redshift dependence of the average luminosity at high black hole mass, which in turn requires that we evaluate how retrograde black hole formation depends on time. This, as we have already explored in Section IIa, forces us to consider the time dependence of the merger function (e.g., [25] as shown in Figure6. Because the merger function peaks at z~1.5–2, this is the redshift corresponding to the greatest fraction of retrograde accreting black holes. In addition to the mass dependence of retrograde accretion, there is a time dependence as well that is directly connected to the time dependence of the physical mechanism that leads to retrograde accretion formation, namely galaxy mergers. As the redshift drops from z~1.5–2, the merger function drops and the fraction of retrograde accreting black holes drops in response to that. This connection between the fraction of retrograde accreting black holes and redshift was used to explain the observations of the shift in radio versus X-ray/optical peak in [26] (Figure 12), as well as the compatibility with the Soltan argument in [11] where the branching-tree diagrams in [11] show how the FRI LERGs described in this paper’s Figures 10 and 11 are connected to the most massive black holes that began in retrograde mode. Here, we appeal to the same physics to evaluate the average luminosity as a function of redshift and to explain the time dependence of the slope on the mass–luminosity plane. Because the fraction of retrograde accreting black holes is largest, and thus the average efficiency is smallest, at z~1.5–2, this is also a time when the average luminosity struggles to reach Eddington values. According to Steinhardt & Elvis (2010), the slope of the sub-Eddington boundary at z~2 ranges from 0.2–0.4 depending on how it is measured (Figure8). The slope at z~1, on the other hand, ranges from 0.65–0.35 for the blue and magenta points, respectively. In other words, the slope of the Eddington boundary at z~2 is about 60% of its value at z = 1 for both measurements. We construct the slope at both z~1.5 and z~0.8 by referring to the data in Table2, but since we wish to explore the slope for the high mass population, we include Table2 data only for black hole masses above 108 solar masses. We include this data in the last two columns of Table3. We plot these values in Figure 12. In addition to the behavior of the slope of the sub-Eddington boundary at higher redshift (i.e., between z~1 and z~2), [2] also find a drop at redshifts below z~0.6. In other words, they find a peak in the slope of the sub-Eddington boundary for the highest black hole mass population at z~1, implying a decrease not only at higher z but also at lower z. At first glance, there seems to be tension between this observational fact and our explanation. Since the merger function continues to drop, in fact, the fraction of retrograde-forming objects continues to decrease. From the further drop postulated in our theoretical framework for the retrograde fraction as redshift decreases, we would expect the slope of the Universe 2019, 5, 145 13 of 19 sub-Eddington boundary to continue increasing below z~1. However, this ignores the longer-term time evolution of the originally retrograde accreting objects that begins to have a dominant effect (Figures 10 and 11) by shifting the average to much lower disk efficiencies, as we now describe. The time evolution of powerful radio quasars involves a transition not only in black hole spin as the disk evolves into a prograde configuration from the originally retrograde one, but also in the type of accretion, which sees the radiatively efficient disk evolve into a hot, advection dominated (ADAF) flow on timescales of 107 years or less ([27]; [28]; [7]; [11]). In other words, while the slope of the sub-Eddington boundary drops at higher redshift due to the lower efficiency of the average accreting high-mass black hole, the time evolution at lower redshifts of the efficiency involves competing effects. To appreciate the nature of this competition, see Figures 10 and 11. On the one hand, the ISCO decreases because continued accretion turns retrograde objects into prograde ones, which means that as long as the disk is radiatively efficient and thin, the efficiency increases. However, on a slower timescale, the state of accretion is transitioning toward advection-dominated, which means that despite the initial increase, the efficiency eventually not only drops but does so dramatically. This is true of all the Fanaroff-Riley II ([29]—FRII) quasars as can be seen in Figures 10 and 11. The ones with lower jet powers, in fact, also transition from a radiatively efficient thin disk to an advection-dominated disk. However, this takes longer than for the objects described in Figure 10. At the Eddington rate, the objects of Figure 10 approach the higher prograde regime after about 5 107–108 years and eventually transition to lower × efficiency ADAFs. Therefore, for the fraction of the most massive black holes that find themselves in retrograde accreting states, the efficiency of accretion begins low and starts to increase as ISCO values drop.millions But of on years, the order their ofefficiencies a few tens not of millionsonly drop of ag years,ain, but their do effi sociencies dramatically. not only We drop argue again, this butas the do soexplanation dramatically. for the We arguedata observed this as the in explanation Figure 8 an ford produce the data observedour own estimate in Figure 8of and the produceslope of our the ownEddington estimate boundary of the slope at lower of the redshift Eddington based boundary on the above at lower explanation. redshift based The on ob thejects above we are explanation. referring Theto (FRI objects LERGs we on are the referring diagrams to (FRI of Figures LERGs 10 on and the diagrams11) are the of massive, Figures 10FRI and radio 11) galaxies, are the massive, that accrete FRI radioin low galaxies, luminosity that systems accrete and in low that luminosity are thought systems to be responsible and that are for thought the quen toching be responsible of star formation for the quenching[30]. of star formation [30].

Figure 10. Time evolution of an initially retrograde accreting black hole with powerful jets (from [7]; [7]; where LERG is is “low-excitation “low-excitation radio radio galaxies” galaxies” and and HERG HERG is “high-excitation is “high-excitation radio radio galaxies”). galaxies”). The Theradiative radiative efficiency efficiency of the of initially the initially radiatively radiatively efficient effi thincient disk thin (lower disk (lowerpanel) panel)evolves evolves quickly quicklyinto an intoadvection-dominated an advection-dominated disk, which disk, has which a radiative has a radiativeefficiency ethatfficiency is at least that two is at orders least twoof magnitude orders of magnitudesmaller than smaller the Eddington than the value. Eddington BZ refers value. to BZthe refers Blandford-Znaj to the Blandford-Znajekek jet ([31] and jet BP ([ to31 ]the and Blandford- BP to the Blandford-PaynePayne jet ([32]). jet ([32]).

Figure 11. Time evolution of an initially retrograde accreting black hole with less powerful jets (from [7]; details same as Figure 10). The radiative efficiency of the initially radiatively-efficient thin disk (lower two panels) evolves less quickly (compared to the object in Figure 10) into an advection- dominated disk, which has a radiative efficiency that is at least two orders of magnitude smaller than the Eddington value.

The objects that remain radiatively efficient are generally part of the less massive of the massive black holes so they are less represented in this group. It is also worth noting that unlike in the discussion of [2] the drop-off at higher mass derived here is not a turn-off associated with cosmic downsizing, quite the contrary. It represents the early stage of evolution for some of the massive systems that accrete in retrograde mode. We have calculated the redshift dependence of the slope of the sub-Eddington boundary for three redshift values and report them in Figure 12. In comparing our results to those of [2] it is necessary to appreciate that for the slopes at redshift of 0.8 and 1.5 our results involve averaging only over the radiatively efficient class of accretors, and will therefore lead to a higher slope compared to [2] who, instead, deal with a distribution of objects that vary in Eddington luminosity by an order of magnitude. Our theoretical calculation is limited to accretion rates near the Eddington limit. Therefore, our theoretical slope must be larger by construction. For the data point at z~0.2, on the other hand, we explicitly include efficiencies that are much closer to the ADAF/thin disk efficiency at 0.01 Eddington. While FRII quasars are forming at z~0.2 at 1/4 the rate compared to z~1.5 (Figure 6), the distribution of accretion among the most massive black holes at z~0.2 is strongly influenced by the time evolution of their progenitors that have undergone the millions of years, their efficiencies not only drop again, but do so dramatically. We argue this as the explanation for the data observed in Figure 8 and produce our own estimate of the slope of the Eddington boundary at lower redshift based on the above explanation. The objects we are referring to (FRI LERGs on the diagrams of Figures 10 and 11) are the massive, FRI radio galaxies, that accrete in low luminosity systems and that are thought to be responsible for the quenching of star formation [30].

Figure 10. Time evolution of an initially retrograde accreting black hole with powerful jets (from [7]; where LERG is “low-excitation radio galaxies” and HERG is “high-excitation radio galaxies”). The radiative efficiency of the initially radiatively efficient thin disk (lower panel) evolves quickly into an advection-dominated disk, which has a radiative efficiency that is at least two orders of magnitude Universesmaller2019, 5 than, 145 the Eddington value. BZ refers to the Blandford-Znajek jet ([31] and BP to the Blandford-14 of 19 Payne jet ([32]).

FigureFigure 11.11. Time evolutionevolution of of an an initially initially retrograde retrograde accreting accreting black black hole hole with with less less powerful powerful jets jets (from (from [7]; details[7]; details same same as Figure as Figure 10). The10). radiativeThe radiative efficiency efficien of thecy of initially the initially radiatively-e radiatively-efficientfficient thin disk thin (lower disk two(lower panels) two evolvespanels) lessevolves quickly less (compared quickly (compared to the object to in the Figure object 10) intoin Figure an advection-dominated 10) into an advection- disk, whichdominated has a radiativedisk, which effi hasciency a radiative that is at leastefficiency two orders that is of at magnitude least two orders smaller of than magnitude the Eddington smaller value. than the Eddington value. The objects that remain radiatively efficient are generally part of the less massive of the massive blackThe holes objects so they that are remain less represented radiatively in efficient this group. are It ge isnerally also worth part noting of the thatless unlikemassive in of the the discussion massive ofblack [2] theholes drop-o so theyff at higherare less mass represented derived herein this is notgroup. a turn-o It is ffalsoassociated worth noting with cosmic that unlike downsizing, in the quitediscussion the contrary. of [2] the It represents drop-off at the higher early stagemass of derived evolution here for is some not ofa turn-off the massive associated systems with that accretecosmic indownsizing, retrograde quite mode. the We contrary. have calculated It represents the redshift the early dependence stage of evolution of the slope for ofsome the sub-Eddingtonof the massive boundarysystems that for accrete three redshift in retrograde values mode. and report We have them calculated in Figure the 12 .redshift In comparing dependence our results of the toslope those of ofthe [ 2sub-Eddington] it is necessary boundary to appreciate for three that redshift for the slopesvalues atand redshift report of them 0.8 andin Figure 1.5 our 12. results In comparing involve averagingour results only to those over of the [2] radiatively it is necessary efficient to appreciate class of accretors, that for andthe willslopes therefore at redshift lead of to 0.8 a higher and 1.5 slope our comparedresults involve to [2] averaging who, instead, only deal over with the aradiatively distribution efficient of objects class that of accretors, vary in Eddington and will luminositytherefore lead by anto ordera higher of magnitude.slope compared Our theoreticalto [2] who, calculation instead, de isal limited with a to distribution accretion rates of objects near the that Eddington vary in limit.Eddington Therefore, luminosity our theoretical by an order slope of mustmagnitude. be larger Ou byr theoretical construction. calculation For the datais limited point atto z~0.2,accretion on therates other near hand, the Eddington we explicitly limit. include Therefore, efficiencies our theore that aretical much slope closer must to be the larger ADAF by/thin construction. disk efficiency For atthe 0.01 data Eddington. point at z~0.2, While on FRII the quasars other hand, are forming we explic at z~0.2itly include at 1/4 the efficiencies rate compared that are toz~1.5 much (Figure closer6 to), thethe distributionADAF/thin ofdisk accretion efficiency among at 0.01 the Eddington. most massive While black FRII holes quasars at z~0.2 are is forming strongly at influenced z~0.2 at 1/4 by thethe timerate compared evolution ofto theirz~1.5 progenitors (Figure 6), the that distribution have undergone of accretion the evolution among described the most inmassive Figures black 10 and holes 11. Thisat z~0.2 evolution is strongly involves influenced both a transition by the time toward evolution the prograde of their accretion progenitors regime that (which have isundergone what initially the increases the efficiency and explains the data point at z~0.8) and a transition from radiatively efficient thin disk accretion into hot, advection-dominated, low-efficiency accretion. The efficiency for such objects spans a large range (orders of magnitude), but we are not interested in the lower or lowest efficiency ADAFs. We are still focused on objects that radiate efficiently or as efficiently as possible, so as to affect the Eddington boundary. Whereas at z~1.5 when thin disk accretion dominated the most massive black hole population and we averaged over the range of thin disk efficiencies above 0.1 Eddington as a result, here we also average over the most efficient accretors, but need to take into account the time evolution of the most massive black holes that were efficiently accreting in the past but are now evolving away from thin disk accretion. The effect of this time evolution for z~0.2 is the presence of thin, radiatively efficient accretion, that spans a wider range of efficiencies because the accreted material from the larger galaxy is hotter on average for the most massive black hole population compared to larger z. This forces us to average over all efficiencies that are associated with radiatively efficient accretion, namely accretion at or above 0.01 Eddington which is the theoretical boundary between ADAFs and thin disks. In order to account for the entire population of massive black holes accreting in radiatively efficient mode, we need to include the subgroup of AGN that were never strong jet producers and did not evolve from radio-loud quasar progenitors. A good fraction of the massive black hole population (approximately 1–0.43 and 1–0.35 at z~1.5–2 and z~1, respectively, as seen in Figure5) were prograde accreting supermassive black holes and therefore highly-e fficient radio-quiet quasars that, as long as they continued to be fed, remained prograde accreting systems. Hence, the distribution at z~0.2 is over a range of efficiencies from 0.8–0.01 to account for the radio-quiet quasar Universe 2019, 5, 145 15 of 19 class. We produce an estimate that ranges from about 0.5 near the lower end of the massive black hole population at 3 x 108 solar masses to about 0.2 at the higher end near 109 solar masses. The more massive black holes, in fact, are more likely to come from progenitor FRII quasars that were more effective in heating the galactic and intergalactic medium leading to the ADAF phase, and we try to capture that effect by lowering the efficiency slightly with an increase in black hole mass. Hence, the average efficiency for thin disk accretion at z~0.2 drops as a result of the presence of a large range of efficiencies that are close(r) to the efficiency that serves as the boundary between thin disk and ADAF. This explains the lowest point in Figure 10. While ours is an exact calculation given the postulated retrograde fraction in Figure4, a change in that prescription will necessarily a ffect the z~0.2 point in Figure 12, further lowering that data point if we allow the fraction of retrograde objects to increase at lower black hole mass. Given this, the takeaway message should be that Figure 12 is qualitatively compatible with Figure8.

Table 3. Log of L/LE vs. Log of black hole mass in units of 1 billion solar masses for z = 1.5, z = 0.8, and z = 0.2. The data for z = 0.8 and z = 1.5 was available in Table2. However, we only used the data for black hole mass above 108 solar masses.

Log (L/L ) Log (L/L ) Log (L/L ) Log M Edd Edd Edd 9 Z = 0.2 Z = 0.8 Z = 1.5 8.301 45.6 45.9 45.9 8.477 45.8 46.1 46.1 8.602 45.9 46.2 46.2 8.699 45.9 46.2 46.2 8.778 45.9 46.3 46.3 8.903 45.9 46.4 46.4 8.845 46.0 46.4 46.3 8.9548.903 45.945.7 46.446.4 46.4 46.4 9.008.954 45.745.8 46.446.5 46.4 46.5 9.00 45.8 46.5 46.5 Slope 0.24 0.81 0.77 Slope 0.24 0.81 0.77

Figure 12. Redshift dependence of the slope of the Eddington boundary from theory. While the redshift Figure 12. Redshift dependence of the slope of the Eddington boundary from theory. While the 1.5 data pointredshift drops 1.5 withdata respectpoint drops to the with redshift respect 0.8 to point the redshift due to 0.8 the point larger due fraction to the oflarger radiatively fraction eofffi cient thin disksradiatively in retrograde efficient configurations, thin disks in retrograde the 0.2 redshiftconfigurations, point the drops 0.2 redshift due to point the evolution drops due ofto etheffi cient accretors intoevolution advection of efficient dominated accretors flowsinto advection (ADAFs). dominated flows (ADAFs).

In closing, we point out that [5] have explored the L/LEdd ratios of ~58,000 type 1 quasars from the SDSS Data Release 7 and fail to find a sub-Eddington boundary, and [33] argues there is no sub- Eddington boundary in the [4] sample of ~28,000 SDSS quasars. At both z~3.2 and z~0.6, [5] find the L/LEdd distribution to be relatively independent of black hole mass. However, at redshifts 0.8 < z < 2.65 the distribution of L/LEdd at fixed black hole mass shifts to larger Eddington ratios from black

hole masses of ~5 × 108 M☉ to MBH ~5 × 109 M☉. This therefore implies that at 0.8 < z < 2.65 type 1 quasars with more massive black holes are more likely to be radiating near the Eddington limit. It is important to emphasize that our conclusions are based on the presence of retrograde accreting black holes, which the model associates with broad line radio galaxies and FRII quasars. Absence of such objects in the population of radiatively efficient accretors will indeed increase the average L/LEdd. We therefore caution observers to pay particular attention to an important difference for high black hole mass accretors from the perspective of the theoretical framework. If the higher black hole mass sample involves objects that have enough FRII quasars, the theoretical framework predicts that a sub- Eddington boundary will show up at higher mass. From the perspective of the paradigm, observers should verify whether the drop off on the mass luminosity plane exists for higher black hole mass quasars that involve FRII quasars, not simply for higher black hole mass quasars. If, in fact, the sample is primarily characterized by massive black holes in radio-quiet or non-jetted quasars, the paradigm prescribes the absence of a sub-Eddington boundary. A practical question addressing this issue, therefore, would be to ask if the fraction of radio loud or jetted quasars to the total quasar data is smaller in any subset of the SDSS data release 7 that is chosen for analysis compared to any subset of the SDSS data release 5. Because this work attempts to explain objects near the Eddington boundary— 8.903 45.9 46.4 46.4

8.954 45.7 46.4 46.4

9.00 45.8 46.5 46.5

Slope 0.24 0.81 0.77

Figure 12. Redshift dependence of the slope of the Eddington boundary from theory. While the redshift 1.5 data point drops with respect to the redshift 0.8 point due to the larger fraction of radiatively efficient thin disks in retrograde configurations, the 0.2 redshift point drops due to the Universe 2019, 5, 145 16 of 19 evolution of efficient accretors into advection dominated flows (ADAFs).

In closing,In closing, we point we outpoint that out [5 ]that have [5] explored have explored the L/L theEdd L/LratiosEdd ratios of ~58,000 of ~58 type,000 1 type quasars 1 quasars from from the SDSSthe SDSS Data ReleaseData Release 7 and 7 failand tofail find to find a sub-Eddington a sub-Eddington boundary, boundary, and and [33 [33]] argues argues there there is is no no sub- sub-EddingtonEddington boundary boundary in in the the [ 4[4]] sample sample ofof ~28,000~28,000 SDSS quasars. At At both both z~3.2 z~3.2 and and z~0.6, z~0.6, [5] [ 5find] the find theL/L LEdd/LEdd distributiondistribution to be to berelativ relativelyely independent independent of black of black hole hole mass. mass. However, However, at redshifts at redshifts 0.8 < z < 0.8 < z2.65< 2.65 thethe distribution distribution of ofL/L L/EddLEdd at atfixed fixed black black hole hole mass mass shifts shifts to to larger larger Eddington Eddington ratios from black black holehole massesmasses ofof ~5~5 × 108 M☉ to MMBH ~5 × 101099 M☉. .This This therefore therefore implies implies that that at at 0.8 0.8 << zz << 2.652.65 type 1 × BH × type 1quasars quasars with more massive blackblack holesholes areare moremore likelylikely to to be be radiating radiating near near the the Eddington Eddington limit. limit. It is It is importantimportant to to emphasize emphasize that that our our conclusions conclusions are are based based on on the the presence presenceof of retrograderetrograde accretingaccreting black black holes, which the model associates with with b broadroad line line radio radio galaxies galaxies and and FRII FRII quasars. quasars. Absence Absence of such of suchobjects objects in inthe the population population of radiatively of radiatively efficient efficient accretors accretors will willindeed indeed increase increase the average the average L/LEdd. We L/LEddtherefore. We therefore caution caution observers observers to pay particular to pay particular attention attention to an important to an important difference di forfference high black for hole high blackmass hole accretors mass accretors from the from perspective the perspective of the theoretical of the theoretical framework. framework. If the If higher the higher black black hole mass hole masssample sample involves involves objects objects that thathave have enough enough FRII FRIIquasars, quasars, the theoretical the theoretical framework framework predicts predicts that a sub- that a sub-EddingtonEddington boundary boundary will willshow show up at up higher at higher mass. mass. From From the perspective the perspective of the of paradigm, the paradigm, observers observersshould should verify verify whether whether the drop the drop off on off theon mass the mass luminosity luminosity plane plane exists exists for higher for higher black black hole mass hole massquasars quasars that involve that involve FRII quasars, FRII quasars, not simply not simply for higher for black higher hole black mass hole quasars. mass quasars.If, in fact, If,the in sample fact, theis sampleprimarily is primarilycharacterized characterized by massive by black massive holes black in radio holes-quiet in radio-quiet or non-jetted or non-jetted quasars, quasars,the paradigm the paradigmprescribes prescribes the absence the absence of a sub of-Eddington a sub-Eddington boundary. boundary. A practical A practical question question addressing addressing this issue, this issue,therefore, therefore, would would be to be ask to askif the if thefraction fraction of radio of radio loud loud or orjet jettedted quasars quasars to to the the total total quasar quasar data is data issmaller smaller in in any any subset subset of of the the SDSS SDSS data data release release 7 that 7 that is chosen is chosen for for analysis analysis compared compared to any to any subset of subsetthe of theSDSS SDSS data data release release 5. Because 5. Because this work this work attempts attempts to explain to explain objects objects near the near Eddington the Eddington boundary— boundary—i.e., objects that are most luminous for a given black hole mass —Malmquist bias does not affect our results. It should be clear, i.e., that our analysis does not shed light on the range of luminosities per black hole mass but deals only with the largest luminosities at or near the Eddington boundary. We also emphasize that ours is not an attempt to strengthen the observational support in favor of a sub-Eddington boundary. We are, instead, focused on showing how such features emerge in a straightforward way from theoretical ideas that have been developed to explain a host of observations that are not directly related to the sub-Eddington boundary. In turn, even if it turns out that the observed sub-Eddington boundary is not physical, the theoretical ideas presented still incorporate it. It is important to point out that the ideas behind the sub-Eddington boundary as illustrated in Figures 10 and 11 are anchored to a prescription that makes many predictions but that fundamentally is grounded in the time evolution from retrograde to prograde accretion. In terms of jet morphology and excitation class, it is interesting to note that recent observations indeed fit within this prescription [34]. And, finally, in Figure 13 we show the number of quasars as a function of redshift, illustrating how a sub-Eddington boundary at a redshift of about 1.5 is even more significant than if the distribution were flat in the sense that more objects are available to reach their Eddington limit. i.e., objects that are most luminous for a given black hole mass —Malmquist bias does not affect our results. It should be clear, i.e., that our analysis does not shed light on the range of luminosities per black hole mass but deals only with the largest luminosities at or near the Eddington boundary. We also emphasize that ours is not an attempt to strengthen the observational support in favor of a sub- Eddington boundary. We are, instead, focused on showing how such features emerge in a straightforward way from theoretical ideas that have been developed to explain a host of observations that are not directly related to the sub-Eddington boundary. In turn, even if it turns out that the observed sub-Eddington boundary is not physical, the theoretical ideas presented still incorporate it. It is important to point out that the ideas behind the sub-Eddington boundary as illustrated in Figures 10 and 11 are anchored to a prescription that makes many predictions but that fundamentally is grounded in the time evolution from retrograde to prograde accretion. In terms of jet morphology and excitation class, it is interesting to note that recent observations indeed fit within this prescription [34]. And, finally, in Figure 13 we show the number of quasars as a function of redshift, illustrating how a sub-Eddington boundary at a redshift of about 1.5 is even more significant than if the distribution were flat in the sense that more objects are available to reach their Eddington Universelimit. 2019, 5, 145 17 of 19

Figure 13. Distribution of quasars as a function of redshift from SDSS DR7 ([3]). Z = 1.5 is indicated Figure 13. Distribution of quasars as a function of redshift from SDSS DR7 ([3]). Z = 1.5 is indicated with the vertical dashed line. with the vertical dashed line. 3. Conclusions 3. Conclusions In this paper, we have attempted to unify various features of the sub Eddington boundary discoveredIn this by paper, [2] within we have the context attempted of the to gap unify paradigm various for features black hole of the accretion sub Eddington and jet formation, boundary a phenomenologicaldiscovered by [2] frameworkwithin the context anchored of to the the gap notion paradigm that retrograde for black accretion hole accretion had an and important jet formation, role in thea phenomenological history of the most framework massive black anchored hole mass to the buildup. notion While that weretrograde have not accretion produced had any an explanation important asrole to whyin the the history Eddington of the boundary most massive should black be respectedhole mass (which buildup. the While observations, we have especially not produced at lower any blackexplanation hole mass, as doto notwhy require), the Eddington we have shownboundary how should the range be inrespected Eddington (which ratio tendsthe observations, to vary as a functionespecially of at black lower hole black mass. hole We mass, have do shown, not requir in othere), we words, have shown that if how the basicsthe range of standard in Eddington thin diskratio theorytends to apply, vary the as diska function efficiency, of black which hole depends mass. We on blackhave shown, hole spin, in mustother alsowords, depend that if on the black basics hole of mass.standard And thin the reasondisk theory has to apply, do with the the disk post-merger efficiency, fraction which ofdepends retrograde on black holehole formationspin, must from also whichdepend we on find black larger hole average mass. ISCOAnd the values reason for thehas mostto do massive with the accreting post-merger black fraction holes. In of addition retrograde to thisblack effi holeciency formation versus mass from dependence, which we find we larger have shownaverage that ISCO a redshift values dependencefor the most emergesmassive fromaccreting the basicblack theory holes. for In theaddition slope ofto thethis sub-Eddington efficiency vers boundary.us mass dependence, Everything we we have have addressed shown that in this a redshift paper comesdependence from the emerges connection from the between basic theory mergers for and the theirslope ability of the tosub-Eddington produce retrograde boundary. accreting Everything thin diskswe have around addressed spinning in black this holes.paper Whilecomes a from number the of connection assumptions between are made mergers for simplicity, and their they ability do not to haveproduce an e ffretrogradeect on the accreting basic qualitative thin disks conclusion around spinning concerning black the holes. sub-Eddington While a number boundary: of assumptions Accreting blackare made holes for in simplicity, retrograde configurationsthey do not have are an less effect effi cienton the than basic prograde qualitative ones conclusion which will concerning contribute the to loweringsub-Eddington the average boundary: efficiency Accreting of the population.black holes Wein retrograde have also pointed configurations out that ifare observations less efficient are than to constrain theory, an important distinction must be made between broad-line radio galaxies and FRII quasars and what are generally referred to as type 1 or 2 quasars.

Author Contributions: D.G. came up with the idea and wrote most of the paper. D.J.C. made many of the plots and contributed to the ideas as well as adding an analysis of the SDSS DR7 data. A.M.J. performed some of the calculations for his research project. Funding: This research received no external funding. Acknowledgments: We are grateful for the impressive and thorough review by the referees. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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