Mon. Not. R. Astron. Soc. 000, 000–000 (0000)

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Comment on “Biases in the Quasar Mass-Luminosity
Plane”Charles L. Steinhardt and Martin Elvis
IPMU, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba-ken, Japan 277-8583;
Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138

22 November 2010

In their paper “Biases in the Quasar Mass-Luminosity
Plane”, Rafiee & Hall (2010) (RH10) recalibrated MgIIbased virial mass estimators for quasar black holes, andused these recalibrations to conclude that the sub-Eddington
boundary (SEB), described in Steinhardt & Elvis (2010a)(SE10) is an artifact due to biases in the Shen et al. (2008)(S08) virial mass catalog. The SEB is a bound restricting themost massive black holes over a wide range of redshifts toradiate significantly below their Eddington luminosity. Thisis a surprising feature of the quasar Mass-Luminosity plane,
likely with major implications, and so deserves to be tested
rigorously.
Here we note, however, that the biases claimed byRH10: (1) are not statistically significant; (2) imply nonvirial motions in the quasar broad line region so substantialthat it is unlikely that the masses derived have validity; (3)do not remove the SEB at most redshifts, but rather replacea bound restricting most quasars to radiate below Eddington at all redshifts with one restricting all quasars to radiate
below Eddington at most redshifts.
1. Statistical Significance:
RH10 use an empirical relation between reverberation
mapping masses and line widths. They fit a slope, γ, toMBH ∝ FWHMγ . RH10 find a best-fit value of γ =1.27 based on the MLINMIX ERR fitting method of Kelly(2007). The claim that the SEB disappears at z ∼ 2 rests
entirely upon this small value of γ.
RH10 argue that MLINMIX ERR is optimal. However
the 18 different statistical techniques they test produce values of γ ranging from 1.21 to 3.95. The value of γ is clearly
not robust but, rather, is highly sensitive to the chosen fitting technique. The error estimates provided by these different methods reflect this lack of robustness in that none
gives a & 2σ deviation below γ=2, the virial value⋆ , including the value of 1.27 ± 0.40 preferred by RH10. We conclude
that the current evidence, as presented both by RH10 and inprevious work, is consistent with the virial approximation.
In the absence of an underlying physical model, corrections for substantial non-virial motion are difficult to calibrate against the quite small (N = 29) reverberation mass
sample. If RH10 are correct that γ 6= 2, many more reverberation mass estimates will be required to produce compelling⋆ Two methods give values ∼3σ greater than 2. [OLS(X—Y) andBCES(X—Y).]

c 0000 RAS

evidence, robust with regard to fitting methodology, for a
new choice of mass estimation technique.
2. Non-Virial Motions:
A key difference between this analysis and previous
work is that RH10 do not assume that motions in the quasar
broad-line region are predominantly virial (i.e. MBH ∝2vvirial∝ FWHM2 ). While it is well known that broad emission lines have a more triangular ’log’ shape (Peterson 2008),
RH10 find that larger S08 BH masses are systematicallypoorer approximations to gaussians, which could lead to a
mass-dependent bias in the S08 BH masses.
Certainly, non-virial motions, mostly thought to be due
to outflows, have long been known in quasar spectra (Gaskell
1982; Wilkes 1984). However, these are strongest in the highionization broad emission lines, notably in CIV, rather thanin low ionization lines such as MgII. This ionization dependence is expected for known sources of non-virial motion
such as radiation pressure (Marconi et al. 2009), as CIV
comes from an inner region closer to the ionizing continuum
source and the central black hole, while MgII, comes from
more distant material. In agreement with this expectation,
the statistical uncertainty in virial mass estimation is smallest for MgII (Steinhardt & Elvis 2010b).
If the Mass-FWHM relationship is as strongly non-virial(γ = 1.27 rather than γ = 2) as RH10 claim, then this would
call into question the entire concept of virial mass estimation, which is predicated upon the assumption that broadline region motions are dominated by the gravitational attraction of the central black hole. Without a physical model
it is hard to know how to correct for these motions, particularly given the strong dependence of the fit on the chosen
statistical methodology.
Although there is no reason a priori that quasar luminosities must be able to reach Eddington, it is a boundarywith physical meaning. Using γ = 2, the most luminous
quasars at every black hole mass reach but do not exceed
Eddington at some redshift, and the highest-Eddington ratio
quasars at every redshift reach but do not exceed Eddington. Using γ = 1.27, as described below, for most choicesof redshift no quasars reach Eddington. That the virial approximation produces quasars reaching this boundary undera variety of conditions but never exceeding it implies thatvirial masses likely also have a physical meaning. Non-virial
masses do not share this property.
3. RH10 Do See a Below-Eddington Boundary:2Charles L. Steinhardt and Martin Elvis
strains quasars to lie below both a mass- and redshiftdependent luminosity limit. Thus, adjusting the boundary
to lie at Eddington also requires a combination of massdependent and redshift-dependent effects (SE10, § 4). RH10
give only a mass-dependent effect.

47.5
0.4<z<0.6 (Shen08)

0.4<z<0.6 (RH10)

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4. Further Problems
RH10 includes some misleading figures, captions, andtext. Individually these are not major, but the cumulative
effect is unfortunate:

45.5

47.5
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0.6<z<0.8 (Shen08)

0.6<z<0.8 (RH10)

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46.5

46

45.5

457

7.5

88.5

99.5
77.5

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Figure 1. The quasar mass-luminosity plane using S08 masses(left) and RH10 masses (right) at 0.4 < z < 0.6 (top) and
0.6 < z < 0.8 (bottom). Using S08 masses, the highest-massquasars are all limited to lie below their Eddington luminosities (black, short-dashed), while using RH10 all quasars are subEddington. Using RH10 masses, quasars are increasingly subEddington at lower redshifts; quasars at 0.6 < z < 0.8 approach closer to log(L/LE ) = 0.5 (red, long-dashed) than at
0.4 < z < 0.6.

The sub-Eddington boundary described in SE10 aboundary constraining quasars at most combinations ofmass and redshift to lie below their Eddington luminosity,with the highest-mass quasars at each redshift being themost sub-Eddington and the lowest-mass quasars approaching LEdd . RH10 also see a below-Eddington boundary constraining quasars at most combinations of mass and redshift
to lie below their Eddington luminosity, but with a different
mass-dependence.
Even using the RH10 adjusted masses, their Figure 7
shows that the highest-luminosity quasars at every mass atz < 1.68 fall below Eddington, with the shortfall monotonically increasing larger towards lower redshift. The tilt withmass, however, is largely removed, although a few redshiftbins (e.g. 1.06<z<1.22) show a weak inversion of the SE10
result, with lower mass quasars being more sub-Eddington
than higher mass quasars. The highest-redshift bin shown,
1.83 < z < 1.98, again shows the lowest-mass quasars reaching Eddington but the highest-mass quasars falling short.
The shortfall in RH10 Fig. 7 increases towards lower
redshift down to the 0.76<z<0.912 bin, the lowest range
plotted. However, MgII is visible in SDSS spectra down toz∼0.4. Fig. 1 shows the quasar M − L plane for 2 lower redshift bins, using both S08 (left) and RH10 (right) masses,down to this limit. The additional shortfall using RH10
masses is more apparent in these bins, lying at least 0.5 dex
below Eddington. A clear slope of increasing deviation fromEddington at high mass is present in the lowest redshift bin(0.4<z<0.6). In SE10, we showed the entire redshift range(0.2 < z < 4.1) allowed by our sample and techniques. RH10
should do the same.
The sub-Eddington boundary described in SE10 con-• In RH10 Fig. 1, the authors show quasars at all redshifts
in their sample, ranging from 0.76 < z < 1.98, yet claim todraw the SEB described in SE10. As we demonstrate (SE10Figs. 1, 3), the SEB is present in the quasar M − L plane
only in narrow redshift intervals, and the location of the SEB
moves upwards with redshift. Hence the SEB is obscured bycombining quasars spanning a wide range of redshift (seee.g. SE10, Figure 1, and Kollmeier et al. (2006)).
• In RH10 Fig. 7, Rafiee & Hall note, as mentioned above,that their SEB has disappeared “at high redshift”, not in allpanels. Yet, in their abstract, they claim to have explained
the entire effect as due to biases in estimating masses. Similarly, RH10 §4.1 claims that a change from γ = 2 to γ < 2has eliminated the sub-Eddington boundary; this is not what
they have found.
• In §4.1, RH10 quote SE10 as suggesting that any change
in mass-scaling relations should only shift the quasar locus
in the M − L plane. In §4.5.2, SE10 includes a full discussion of the topic, including allowing for the possibility ofa mass-dependent change in virial mass estimation. SE10also considers previously proposed mass-dependent changes(Onken & Kollmeier 2008; Risaliti et al. 2009).
5. Discussion
As discussed in a followup paper (Steinhardt & Elvis2009), the most puzzling behavior in the mass-luminosity
plane is that: (1) quasar accretion at a given mass andredshift is apparently only possible within a narrow (< 1dex); luminosity range around a central, characteristic, subEddington accretion rate and (2) that characteristic accretion rate is time-dependent, decreasing towards lower redshift.
The key conclusion that quasar accretion rates are timedependent yet synchronized in host galaxies with quite different properties cannot be explained merely by a massdependent bias in black hole mass estimation. Instead a
concidence between mass- and time-dependent biases is required.
The detailed quasar M − L distribution, and our interpretation of that distribution in an effort to build physicalmodels, would be affected by solely mass-dependent biases,
so it is important to continue to improve our understandingof virial mass estimation. If motion in the broad-line region
is not predominantly virial, the technique may well need tobe discarded rather than patched. However, Rafiee & Hall
do not claim statistically significant deviations from virial
masses, nor do their new calibrations remove either this subEddington behavior or other puzzling limits on quasar accretion.
The authors would like to thank John Silverman andMichael Strauss for valuable comments.
c 0000 RAS, MNRAS 000, 000–000Comment on “Biases in the Quasar Mass-Luminosity Plane”
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