# Rapidly Spinning Black Holes in Quasars: An Open Question

###### Abstract

Wang et al. (2006) estimated an average radiative efficiency of 30%–35% for quasars at moderate redshift. We find that their method is not independent of quasar lifetimes and thus that quasars do not necessarily have such high efficiencies. Nonetheless, it is possible to derive interrelated constraints on quasar lifetimes, Eddington ratios, and radiative efficiencies of supermassive black holes. We derive such constraints using a statistically complete sample of quasars with black hole mass estimates from broad Mg ii, made both with and without the radiation pressure correction of Marconi et al. (2008). We conclude that for quasars with , lifetimes can range from 140 to 750 Myr for Schwarzschild black holes. Coupled with observed black hole masses, quasar lifetimes of 140 Myr would imply that radiatively inefficient accretion or BH mergers must be important in the accretion history of quasars. Given reasonable assumptions about the quasar population, if the average quasar lifetime is Gyr, and if radiatively inefficient accretion is unimportant, then not many BHs with Eddington ratio can be rapidly spinning.

## 1 Introduction

A quasar is powered by matter accreting onto a supermassive black hole (e.g., Rees 1984). Gas orbiting in the innermost stable circular orbit (ISCO) may be perturbed and fall into the black hole (BH), adding mass and angular momentum to it. To have reached the ISCO through a thin accretion disk, gas must have radiated away a fractional binding energy per unit rest mass which is the system’s radiative efficiency (Bardeen 1970). Since decreases from for Schwarzschild (nonrotating) BHs to for co-aligned accretion onto Extreme-Kerr (maximally rotating) BHs, more energy is produced by (co-aligned) thin disk accretion of a given mass onto a rotating BH than onto a non-rotating BH (Carter 1968).

If a quasar accretes gas with a fixed angular momentum vector at a fixed mass accretion rate , the BH spin will increase to a theoretical maximum and the luminosity of the quasar will increase along with it, since . In many realistic accretion models, the spin of a supermassive BH increases rapidly and then fluctuates around a maximum value, although spin-down is also possible (Volonteri 2007). Several such models, including numerical ones by Di Matteo et al. (2005), suggest that the accretion process stops when a BH becomes massive enough to support a kinetic and/or radiative luminosity capable of blowing away the gas fuelling it (Silk & Rees 1998; Fabian 1999; King 2003). However, the radiative luminosity for a given mass accretion rate depends upon the radiative efficiency. A rapidly rotating BH can help shut down the accretion process earlier than a nonrotating BH could. Knowledge of supermassive BH spins is therefore useful in constraining models of quasar development.

Observationally, the quasi-periodic variability detected in Sgr A* may be evidence of rapid spin of the Galactic BH (e.g., Genzel et al. 2003). Evidence for rotating supermassive BHs comes from studies of X-ray Fe K line profiles (Blandford et al. 1990) and by theoretical arguments that powerful radio jets are powered by the extraction of energy from rotating BHs (Miller 2007).

In a recent study, Wang et al. (2006; hereafter WCHM) estimated a high average radiative efficiency of 30%–35% for quasars at , implying that most supermassive black holes are rapidly rotating. However, the existence of rotating BHs with is not confirmed by magnetohydronamic (MHD) simulations: gas loses more angular momentum prior to accretion in an MHD disk than in a standard thin disk (Gammie & Shapiro 2004; Shapiro 2005). Observationally, Shankar et al. (2008) present and review evidence for . In this work we show that the WCHM method is not independent of quasar lifetimes. In § 2 we correct and extend the WCHM method for determining radiative efficiencies, in § 3 we apply it to a sample of Sloan Digital Sky Survey (SDSS) quasars and in § 4 we discuss our conclusions.

## 2 The Method:

We assume that quasar light derives only from accretion of matter onto a black hole, neglecting the effects of BH mergers. Suppose that mass propagates through a thin accretion disk around an accreting BH at a rate of during some time interval of . The BH mass growth rate is given by , where is the radiative efficiency. The quasar radiates at bolometric luminosity (Marconi et al. 2004) given by:

(1) |

We can rewrite the above equation as:

(2) |

where is the change over the time in the comoving radiative energy density and is the accompanying change in the comoving BH mass density, both due solely to this BH’s accretion.

By analogy, an average radiative efficiency can be defined for any sample of quasars at redshift :

(3) |

where and are the estimated changes in the cumulative radiative energy density and the cumulative mass density of the sample in the redshift range . We now define these cumulative densities.

We require the quasar black hole mass function , defined such that is the comoving number density of black holes with masses in the range () in the redshift range . We also require the quasar bolometric luminosity function , where is the comoving number density of black holes with bolometric luminosity in the range () in the redshift range .

Over its lifetime , a single quasar accretes a mass = and radiates an energy . We need to express those quantities in terms of the observables and , which are the black hole’s mass and bolometric luminosity at the redshift of observation, . We do so by defining correction factors and such that

(4) |

where the averages are over all quasars in the sample.

First, consider the mass density that contributes to Eq. 3. Over their lifetimes, all black holes observed at redshift will accrete a comoving matter density of

(5) |

The cumulative, lifetime amount of matter accreted by all black holes observed at and above is the comoving lifetime mass density summed over all redshifts :

(6) |

which agrees with Eqs. 2 and 6 of WCHM if . Notice that units of are mass per comoving volume per redshift.

Now consider the radiative energy component of Eq. 3. Over their lifetimes, all black holes observed at redshift will radiate a comoving energy density of

(7) |

where is the average quasar lifetime.^{1}^{1}1
Time periods without accretion are not counted in .
While can be defined as the sum of all time periods
during which an average quasar is actively accreting mass, an acceptable
observational definition might be the time required to accrete,
e.g., 95% of the quasar’s final mass (see Hopkins et al. 2006).
The cumulative, lifetime amount of energy radiated by all
black holes observed at redshift and above is the comoving
lifetime energy density summed over all redshifts above
:^{2}^{2}2For convenience, in Equations 6 and 8 the redshift
sum appears in front of the mass or luminosity sum for that
redshift bin. That latter sum is computed in the th redshift
bin, multiplied by and then added to the mass or
luminosity sum from the (+1)th redshift bin times , and so on. The redshift bin size does not matter as long
as or does not change considerably
within a bin. For example, if the redshift bin width was halved,
each term in the redshift sum would be half as large but there
would be twice as many terms, yielding the same result.

(8) |

which differs from Eqs. 3 and 5 of WCHM. Their expression for is a factor of times the true value above (assuming ), where is the cosmological time spanned by the redshift interval . The units of above are energy per comoving volume per redshift, whereas in Eq. 5 of WCHM they are energy per comoving volume per (redshift), which is incorrect.

The first quantity of interest, the change in the cumulative comoving mass density of actively accreting black holes over the redshift range — — is times the sum of the masses of all individual quasar black holes in the sample in that range, divided by the comoving volume:

(9) |

In other words, the change in the cumulative comoving mass density in the redshift bin , which is , is the same as the comoving mass density in that bin. Similarly, the second quantity of interest, the change over the redshift range in the cumulative radiative energy density ever observed from all quasars — — is

(10) |

These expressions for and can be substituted into Eq. 3 to find for the sample of quasars under consideration. Doing so, and grouping , and together, we obtain:

(11) |

where the in the numerator and denominator have
cancelled out, making the calculation independent of even if is longer than the cosmic time interval corresponding to .
^{3}^{3}3For example, consider a population of quasars
observed at a redshift , and a population observed at a
redshift . Those quasars are different objects in
different regions of the universe, but the low-z population might
still be the descendents of the high-z population. (If we could
watch both regions of the universe over cosmic time, we might see
the high-z population evolve into the low-z population.) In that
case, the masses of the quasars would be counted twice, at
different cosmic times, but the radiative output would also be
counted twice, at those same times. The estimates of the comoving
mass and radiative energy densities would be systematically off,
but in such a way that the radiative efficiency calculation would
still be accurate.

Thus, the WCHM method for studying radiative efficiencies is not independent of the average quasar lifetime . A factor of enters because both the mass-energy growth and the radiative energy output of a quasar must be summed over its entire lifetime (or over the same portion of its lifetime). The mass-energy sum yields the final mass-energy of the black hole , while the radiative energy sum yields . However, this method remains independent of obscured sources and is also powerful because it can be implemented for any sample of quasars regardless of selection effects. It requires only that the changes in the cumulative mass-energy and radiative energy densities be computed using the same objects. Of course, selection effects will determine if the resulting radiative efficiency is relevant for quasars in general.

Also, WCHM in effect assumed . We adopt more realistic estimates by examining Fig. 14 of Springel et al. (2005) and Fig. 1 of Hopkins et al. (2005), from which we respectively estimate and for those models. Adopting =1 means that a quasar is observed when its black hole has accumulated essentially all of its mass, while =10 means that the observed bolometric luminosity of a quasar is around 10 times larger than the average luminosity of a quasar throughout its entire life. Our value of is estimated by averaging over the period of time when quasar shows its final burst of activity (between 1.2 to 1.4 Gyr in Fig. 1 of Hopkins et al. 2005).

If WCHM had assumed and , we estimate that their (incorrect) calculation would have yielded instead of . However, Eq. 11 shows that even with realistic and values the average quasar radiative efficiency is dependent on the average quasar lifetime. We now explore the implications of this dependency.

## 3 Application to SDSS Quasars

### 3.1 Estimation of Black Hole Masses and Bolometric Luminosity

Based on reverberation mapping studies of local active galaxies, an empirical scaling relationship has been developed to estimate black hole masses by (e.g.) Kaspi et al. (2000), Vestergaard (2002) and McLure et al. (2002). The Sloan Digital Sky Survey (York et al. 2000) Data Release 3 quasar catalog (Schneider et al. 2005) provides us with a large sample of quasars at redshifts which have their Mg ii 2800 emission line redshifted into the SDSS spectral range. We have estimated black hole masses for 27728 such quasars from the dispersion of the Mg ii emission line and the continuum luminosity at =3000 Å and for two scenarios. Scenario A: we assume that a black hole mass can be estimated from a conventional virial relationship as: where has units of erg s and km s. Scenario B: we follow Marconi et al. (2008) in considering the effect of the radiation pressure of the quasar’s radiation on black hole mass estimates, namely that it reduces the effective gravity on the broad emission line region, yielding narrower lines at a given mass. We adopt the BH mass relationship with this radiation correction to be: where and log (Rafiee et al. 2008, in prep.).

Furthermore a Malmquist-like bias has been estimated following § 4 of Shen et al. (2008). Our black hole mass estimates have been adjusted downward by a mean bias of dex for scenario A and dex for scenario B, arising from the steepness of the BH mass function and the scatter in BH mass estimates.

We have matched our sample with that of Richards et al. (2006a), a homogeneously selected and statistically complete sample of 15343 DR3 quasars with redshifts drawn from an effective area of 1622 . This procedure yields a subsample of 6704 quasars. The bolometric luminosities of these quasars have been estimated from with for =3000 Å following Shen et al. 2008.

Comoving volumes and luminosity distances have been calculated using a CDM cosmology with , and (Spergel at al. 2007). Corrections have been made for the limited areal coverage of the Richards et al. (2006a) sample and for the 5% incompleteness of the SDSS at . For more details, see Fig. 8 of Richards et al. (2006a).

### 3.2 Radiative Efficiency and Redshift Binning

For thin accretion disks, the only disks we consider in this paper, varies from for a
Schwarzschild BH (with , where is the
angular momentum) to for an Extreme-Kerr BH ().
When one takes into account the effect of radiation energy
captured by the BH (Thorne 1974), the radiative efficiency reaches
a maximum of (); we refer to
that case as a Thorne BH.
It is more realistic to assume an MHD disk wherein magnetic turbulence
provides a torque to remove angular momentum from the inflowing gas
(Shapiro 2005); in that case, the maximum radiative efficiency is
().^{4}^{4}4We relate
to assuming co-aligned accretion on to rotating BHs. King,
Pringle & Hofmann (2008) have pointed out that the effective
will be different for randomly-aligned accretion.
The conversion from to will differ
for each combination of co- and randomly-aligned accretion,
but high will always require high .

We divide our sample into twelve redshift bins. In each bin we can compute for any given value of . The results are shown in Figure 1 as tracks of for eight different values of chosen to match the of a Schwarzschild, MHD, Thorne or Extreme-Kerr black hole at either the high or low redshift limit of our sample. Figure 1 is the corrected version of Figure 2 of WCHM. Both figures show the evolution of with for a flux-limited quasar sample, assuming constant . However, this figure does not give a complete picture since there is another degree of freedom not being considered; namely, the Eddington ratios of the quasars. For example, taken at face value, Figure 1 suggests that a quasar with a lifetime of 0.68 Gyr can be powered by a Extreme-Kerr BH at while the same quasar would violate the maximum possible radiative efficiency at . Since quasars in our sample have larger values of the Eddington ratio at than at , the trend in Figure 1 might be explained by the importance of the Eddington ratio rather than the redshift.

### 3.3 Eddington Ratio Binning

We assume the radiative efficiency may be a function of the Eddington ratio , where erg s. In that case, the changes in the cumulative comoving mass and energy densities in the Eddington ratio bin () are:

(12) |

(13) |

where and now depend on as well as . The bullet subscript denotes quantities binned in as well as .

The average radiative efficiency for a given and is

(14) |

For the same sub-sample used in section 3.2 and for twelve redshift bins, ten mass bins, and ten luminosity bins, the radiative efficiency has been estimated for four different Eddington ratio bins (Figure 2), each containing one quartile of the objects. Figure 2 shows that radiative efficiency is not a function of redshift but rather of quasar lifetime and Eddington ratio.

## 4 Discussion and Summary

Determinations of quasar lifetimes, Eddington ratios and radiative efficiencies are interrelated. Given constraints on (or assumptions about) quasar lifetimes, the WCHM method can be used to constrain quasar radiative efficiencies and BH spins. (Without such constraints, the average quasar cannot be estimated by this method.) Conversely, the range of radiative efficiencies possible for the full range of BH spins can be used to constrain average quasar lifetimes, as long as luminous quasars are not powered by radiatively inefficient accretion flows (RIAFs; see, e.g., Blandford & Begelman 1999). For example, for the model of Shankar et al. (2008), we predict million years in our scenario A or million years in scenario B, which could be used as a further test of those models in comparison to others.

Assuming and (see the end of § 2), quasar lifetimes can be constrained according to the Eddington ratio of the quasar. Lifetimes estimated this way are within a factor of a few of literature lifetime estimates. For example, for BHs in the mass range of our sample (), a lower limit lifetime of 530 million years can be established for black holes with (Panel 1 of Figure 2a, scenario A) or around 750 million years in scenario B for . This lower limit corresponds to the Schwarzschild case, since a rotating black hole at the same will require a longer lifetime to build up its observed mass. This lower limit lifetime is less than a factor of two lower than the mean lifetime of one billion years estimated by Marconi et al. (2004) for and in the same range of . As another example, a luminous quasar powered by a relatively low-mass black hole — which may be a typical early stage in a quasar’s evolution — will have and can have a typical lifetime of 140 to 510 million years (Panel 4 of Figure 2a). This range is only a factor of larger than the mean lifetime of million years estimated by Yu & Tremaine (2002) for luminous quasars, and is consistent with the mean lifetime of million years estimated by Marconi et al. (2004) for super-Eddington accretors.

In principle, given constraints on , and for quasar samples, one could estimate the historical frequency of RIAF episodes in those quasars by plotting quasar lifetimes versus Eddington ratios. For example, consider quasars lying below the Schwarzschild curves in Figure 3 (the normalization of which is a function of , as seen by comparing the two axis in either scenario A or B). Above our lower mass limit of , such quasars must either have had a RIAF phase in order to explain their observed masses, or they must have observed Eddington ratios lower than their historical average: . (In the latter case, the quasars historically would have been located horizontally to the right in the diagram, lying between the Schwarzschild and Thorne curves at a value of sufficient to yield the observed in the observed .) Conversely, quasars lying above the Thorne curves in Figure 3 require . A low might result if the BH spin does not increase as fast as the BH mass does, perhaps due to counter-rotating gas accretion phases. If and and Gyr, and if RIAFs are unimportant, then not many BHs with can be rapidly spinning. On the other hand, if and and Myr, then RIAFs or BH mergers must be important for quasars regardless of , since only then could the observed masses be reached in the inferred lifetimes.

What can we conclude if we assume that the Marconi et al. (2008) correction for the effect of radiation pressure on quasar BH masses is valid, and that Gyr (Martini et al. 2004; Marconi et al. 2004)? First of all, most quasars can not be rapidly spinning in that case. Second, most of the quasars in our sample have a radiative efficiency of consistent with the results of Yu & Tremaine (2002). This being lower than the MHD prediction of Shapiro (2005) might be explained by the effects of BH mergers or by the fraction of maximally spinning BHs being low, at least in our sample.

Alternatively, if one assumes thick disk accretion, where the relation between the BH spin and radiative efficiency differs from thin disk accretion, then radiatively inefficient accretion becomes more important even for MHD accretion and spinning BHs, making a lower more plausible.

In conclusion, the Wang et al. (2006) method, despite its advantages, can only estimate the
radiative efficiency of quasars and ultimately the spin of black holes if we know enough about the accretion process
and the evolutionary history of black holes. Better estimates of , and from ultimately, however, more detailed evolutionary models,
might improve the reliability of the results. Lack of knowledge about the geometry and dynamics of the accretion
disk limits the level of reliability of this method.

PBH and AR are supported in part by NSERC.

## References

- (1) Bardeen, J. M. 1970, Nature, 226, 64.
- (2) Blandford, R. D., Netzer, H., Woltjer, L. Courvoisier, T. J. L., & Mayor, M. 1990, Active Galactic Nuclei (Berlin: Springer-Verlag)
- (3) Blandford, R. D. & Begelman, M. C. 1999, MNRAS, 303, L1
- (4) Carter, B. 1968, Phys. Rev., 1559
- (5) Di Matteo, T., Springel, V. & Hernquist, L. 2005, Nature, 433, 604
- (6) Fabian, A. C. 1999, MNRAS, 308, L39
- (7) Gammie, C., Shapiro, S. & McKinney, J. 2004, ApJ, 602, 312
- (8) Genzel, R., et al. 2003, Nature, 425, 934
- (9) Hopkins, P. F., et al. 2005, ApJ, 625, L71
- (10) Hopkins, P. F., et al. 2006, ApJ, 643, 641
- (11) Kaspi, S., Smith, P. S., Netzer, H., et al. 2000, ApJ, 533, 631
- (12) King, A. 2003, ApJ, 596, L27
- (13) King, A., Pringle, J., Hofmann, J. 2008, MNRAS, 385, 1621
- (14) Marconi, A., Risaliti, G., et al. 2004, MNRAS, 351, 169
- (15) Marconi, A., Axon, J. D., et al. 2008, ApJ, 678, 693
- (16) Martini, P., et al. 2003, ApJ, 597, L109
- (17) McLure, R. J. & Jarvis, M. J. 2002, MNRAS, 337, 109
- (18) Miller, J. M. 2007, ARA&A, 45:441-79
- (19) Rees, M. J. 1984, ARA&A, 22, 471
- (20) Richards, G. T., et al. 2006a, AJ, 131, 2766
- (21) Richards, G. T., et al. 2006b, ApJS, 166, 470
- (22) Schneider, D. P., et al. 2005 ApJ, 130, 367
- (23) Shankar, F., Weinberg, D. & Miraldi-Escude, J. 2008, submitted
- (24) Shapiro, S. L. 2005 ApJ, 620, 59
- (25) Shen, Yue., et al. 2008, ApJ, 680, 169
- (26) Silk, J. & Rees, M. J. 1998, A&A 331, L1
- (27) Spergel, D. N., et al. 2007, ApJS, 170, 377
- (28) Springel, V., et al. 2005, MNRAS, 361, 776
- (29) Thorne, K. S. 1974, ApJ, 191, 507
- (30) Vestergaard, M. 2002, ApJ, 571, 733
- (31) Volonteri, M. 2008, MNRAS, 383, 1079
- (32) Wang, J., Chen, Y., Ho, L. & McLure, R. 2006, 642, L111 [WCHM]
- (33) York, D. G., et al. 2000, AJ, 120, 1579
- (34) Yu, Q. & Tremaine, S. 2002, MNRAS, 335, 965