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

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Analysis and Discussion

#### 2.1. Slope Vs. Mass of the Sub-Eddington Boundary

_{isco}according to:

^{2}–9 GM/c

^{2}(with c the speed of light constant) while the smaller M objects tend to span the smaller range 1.23 GM/c

^{2}–6 GM/c

^{2}. 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 Table 1 with the ISCO value in gravitational radii and plot them in Figure 3.

^{7}M

_{◉}, where M

_{◉}is one solar mass. A more sophisticated prescription for retrograde formation at lower black hole mass would not qualitatively change any of our conclusions. This means that in the aftermath of mergers that funnel cold gas to the central merging black holes, an accretion disk forms whose angular momentum vector is aligned with that of the black hole, which constitutes prograde accretion. As discussed in the Introduction, the black hole is not massive enough to remain stable in a retrograde configuration so any retrograde forming disks simply flip to a prograde state. As we consider more massive black holes on the order of 10

^{8}M

_{◉}or more, however, we relax our constraint by allowing the fraction of retrograde objects to increase progressively toward 50% at high masses near 10

^{9}M

_{◉}, implying that both prograde and retrograde disks become equally stable for large black holes. Our prescription for the retrograde fraction is grounded in the work of [12] who showed that the relative total angular momentum of the accretion disk to that of the black hole determines the retrograde fraction according to:

_{d}/2J

_{h})

_{d}and J

_{h}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 retrograde occurrences can be translated into a mass ratio, with a higher fraction corresponding to lower accretion mass. Smaller total accreting mass compared to the black hole mass results in stability in either prograde or retrograde cases and the fraction tends to ½ (Figure 4). Whereas the range of accretion rates is assumed scale-invariant, the ratio J

_{d}/J

_{h}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 J

_{d}/J

_{h}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.

^{10}solar masses, where the retrograde fraction is closest to ½. 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 Figure 5 must converge. In Figure 5 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 (Figure 6), while the blue data represents the retrograde fraction at lower redshift when the merger rate has dropped and with it the retrograde fraction. Figure 4, Figure 5 and Figure 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.

^{8}M

_{◉}, we prescribe a cutoff for simplicity such that 100% of accreting black holes end up surrounded by prograde disks that are associated with larger efficiencies (i.e., smaller average ISCO value). This cutoff will not change our results qualitatively. When the black hole mass is large enough to ensure stability, the fraction of retrograde accreting black holes approaches the value that reflects the randomness of both the angular momentum vector of the black hole and that of the accretion disk, namely 50% ([12]), so both configurations become equally likely.

^{8}M

_{◉}decreases regardless of redshift. This is a direct consequence of the fact that accreting supermassive black holes span a wider range in efficiency (i.e., a larger range in ISCO values) due to the fact that retrograde orientations have increased occurrence in the high mass accreting black hole population. We have associated the highest prograde spin at 100% efficiency (see Table 1) with the Eddington limit as a normalization condition. If all accreting black holes were chosen to spin at their maximum prograde value, our results for Figure 7 would match the Eddington limit. But since prograde-only black holes are assumed to possess a range of prograde spin values, the average amounts to 0.4 of the Eddington limit. The focus should therefore be on the relative difference in the data in Figure 7 (i.e., on the drop for points associated with black hole masses above a threshold limit), and not on the actual Eddington value of each data point. The exact values for the average luminosity as a function of black hole mass plotted in Figure 7 for the two redshift values are reported in Table 2.

^{6}M

_{◉}to about 10

^{8}M

_{◉}, none of the accreting black holes are allowed to accrete in retrograde configurations (i.e., Figure 5). Therefore, the quasar population at this black hole mass is restricted to prograde accreting black holes, which means they span a narrower range in ISCO values that are no larger than 6 GM/c

^{2}. Furthermore, we assume that for a large range in spin magnitude, no black hole spin values are preferred or overrepresented with respect to others, so a straightforward average over the range of efficiencies for prograde accretion is determined. It is important to note that our constraints are not arbitrary as we are only imposing constraints on the retrograde versus prograde fraction and not on the individual spins which are determined by the physics of the merging black holes. This constraint ensures that average efficiencies must be above the efficiency for zero black hole spin. There are no free parameters that we can manipulate to obtain an average efficiency that is below that of a zero-spin accreting black hole. This constraint ensures that average efficiency at all redshifts be equal to an efficiency value corresponding to the efficiency of some intermediate spin prograde disk because no physical processes allow the retrograde regime 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, therefore, allows us to produce theoretical points in the mass–luminosity plane. As we consider larger black hole masses, the data in Figure 5 indicates to us that a progressively greater range of ISCO values, and thus disk efficiencies, are included as a result of the increased presence of retrograde systems. Around 4 × 10

^{8}M

_{◉}, the fraction of retrograde occurrences is no longer negligible, which means that to obtain a data point in our theoretical plot for this mass, we must average over most of the range from retrograde to prograde efficiencies under the constraint that 0.125 of all the objects are accreting in the retrograde mode (Figure 5). As done for the prograde-only case, we do not limit nor constrain black hole spin (except as previously discussed at very high spin as mandated by numerical work), which means a wide range in spin values from high to low are not only represented, but equally so. This then produces an average efficiency for the 4 × 10

^{8}M

_{◉}bin and another point on the mass–luminosity plane. The same analysis continues up to the highest black hole mass, where the average is over an equally represented range of both prograde and retrograde systems. It should be clear, therefore, that the average efficiency is decreasing with the increase in black hole mass.

#### 2.2. Slope Vs. Redshift of the Sub-Eddington Boundary

^{48}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).

^{8}solar masses. We include this data in the last two columns of Table 3. We plot these values in Figure 12.

^{7}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 Figure 10 and Figure 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 Figure 10 and Figure 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 × 10

^{7}–10

^{8}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. But on the order of a few tens of 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 Figure 10 and Figure 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].

^{8}solar masses to about 0.2 at the higher end near 10

^{9}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 Figure 4, a change in that prescription will necessarily affect 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 Figure 8.

_{Edd}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/L

_{Edd}distribution to be relatively independent of black hole mass. However, at redshifts 0.8 < z < 2.65 the distribution of L/L

_{Edd}at fixed black hole mass shifts to larger Eddington ratios from black hole masses of ~5 × 10

^{8}M

_{☉}to M

_{BH}~5 × 10

^{9}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/L

_{Edd}. 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—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 Figure 10 and Figure 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.

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The quasar mass-luminosity plane from Figure 1 of [2]. The dashed line represents the Eddington boundary. Red indicates a redshift value of 0.2 < z < 0.8, yellow indicates 0.8 < z < 1.4 and green is for 1.4 < z < 2. Black hole mass is estimated from broad line region emission lines in combination with the virial velocity (details in [2]).

**Figure 2.**The plot of data from Sloan Digital Sky Survey (SDSS) Data Release 7 including average luminosity as a function of mass ([3])—the top line is the Eddington boundary, and two additional lines at 1% and 0.1% of Eddington. Units are the same as in Figure 1. 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 Figure 1. Of course, SDSS 7 almost doubles the number of objects.

**Figure 3.**ISCO values and disk efficiency from Table 1.

**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.

**Figure 5.**Theoretical prescription for the fraction of accreting black holes that form retrograde thin disks as a function of black hole mass. Because accretion around massive black holes is stable, the high mass black holes are progressively more likely to be in either prograde or retrograde configurations, which is why the fraction approaches 0.5 at high mass. The red represents the behavior in the redshift range of 1.5–2 when the merger function is larger (Figure 6) while the blue represents the behavior when the redshift is closer to 1 and the merger function has dropped (Figure 6).

**Figure 6.**Observed (red circles) and simulated (blue diamonds) merger rates for the most massive mergers above 100 billion solar masses from [25] For simplicity, we have produced an average (green squares) of the two functions. Based on this we obtain the retrograde fraction as a function of redshift displayed in Figure 5. We could have used the red circles or the blue diamonds and our conclusions would remain qualitatively the same.

**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 limit while the blue and green represent the behavior at redshift below 1 and 1.5, respectively. Note the turnoff at higher black hole mass (above ≈ 4 × 10

^{8}M

_{◉}) resulting from the greater presence of less efficient Shakura & Sunyaev disks in retrograde configurations.

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

**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 smaller than the Eddington value. BZ refers to the Blandford-Znajek jet ([31] and BP to the Blandford-Payne 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.

**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 evolution of efficient accretors into advection dominated flows (ADAFs).

**Figure 13.**Distribution of quasars as a function of redshift from SDSS DR7 ([3]). Z = 1.5 is indicated with the vertical dashed line.

**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 (r_{g}) | Η |
---|---|

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 |

**Table 2.**Data for Figure 7. 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/L_{Edd} (Blue) | L/L_{Edd} (Green) |
---|---|---|

10^{7} | 0.398 | 0.398 |

2 × 10^{7} | 0.398 | 0.398 |

3 × 10^{7} | 0.398 | 0.398 |

4 × 10^{7} | 0.398 | 0.398 |

5 × 10^{7} | 0.398 | 0.398 |

6 × 10^{7} | 0.398 | 0.398 |

7 × 10^{7} | 0.398 | 0.398 |

8 × 10^{7} | 0.398 | 0.398 |

9 × 10^{7} | 0.398 | 0.398 |

10^{8} | 0.398 | 0.398 |

2 × 10^{8} | 0.398 | 0.398 |

3 × 10^{8} | 0.398 | 0.398 |

4 × 10^{8} | 0.382 | 0.382 |

5 × 10^{8} | 0.354 | 0.342 |

6 × 10^{8} | 0.342 | 0.321 |

7 × 10^{8} | 0.331 | 0.312 |

8 × 10^{8} | 0.321 | 0.303 |

9 × 10^{8} | 0.312 | 0.295 |

10^{9} | 0.303 | 0.287 |

**Table 3.**Log of L/L

_{E}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 Table 2. However, we only used the data for black hole mass above 10

^{8}solar masses.

Log M_{9} | Log (L/L_{Edd})Z = 0.2 | Log (L/L_{Edd})Z = 0.8 | Log (L/L_{Edd})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 |

8.845 | 46.0 | 46.4 | 46.3 |

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 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Garofalo, D.; Christian, D.J.; Jones, A.M.
The Sub-Eddington Boundary for the Quasar Mass–Luminosity Plane: A Theoretical Perspective. *Universe* **2019**, *5*, 145.
https://doi.org/10.3390/universe5060145

**AMA Style**

Garofalo D, Christian DJ, Jones AM.
The Sub-Eddington Boundary for the Quasar Mass–Luminosity Plane: A Theoretical Perspective. *Universe*. 2019; 5(6):145.
https://doi.org/10.3390/universe5060145

**Chicago/Turabian Style**

Garofalo, David, Damian J. Christian, and Andrew M. Jones.
2019. "The Sub-Eddington Boundary for the Quasar Mass–Luminosity Plane: A Theoretical Perspective" *Universe* 5, no. 6: 145.
https://doi.org/10.3390/universe5060145