Sub-Eddington Boundary 1

Here are my thoughts on arXiv:1011.6381 'Comment on "Biases in the Quasar Mass-Luminosity Plane' by Steinhardt & Elvis, which is a response to Rafiee & Hall arXiv:1011.1268. The original sub-Eddington boundary (SEB) paper is Steinhardt & Elvis 2010a.

These are my thoughts, not those of Alireza Rafiee (first author on our paper, who recently completed his PhD with me). I am posting them publicly as a personal experiment in 'open science'; feel free to send the URLs of this or any subsequent post around to all and sundry. Comments may be posted publicly or sent to me and Ali privately.

SUMMARY: Steinhardt & Elvis make some good points about where we could be clearer, but also make two mistakes (in section 1 and section 2). However, they do raise a valid issue regarding the definition of the SEB: I was thinking of the SEB as having a slope different from unity, but they have a valid point that a SEB with a slope of unity is still a SEB. By that definition, we do find a SEB at low redshifts, but nowhere does the slope appear significantly different from unity (see section 3 discussion below).

Going through the posting by S&E:

Section 1. "Statistical Significance:"

S&E say "The claim that the SEB disappears at z ∼ 2 rests entirely upon this small value of gamma." That is incorrect. In section 2 of our paper, specifically Figure 3, we show that the Rafiee & Hall (ApJS, submitted) BH masses measured using the line dispersion of MgII do not show a SEB with slope different from unity (see further discussion in Section 3 below). That result does not depend on gamma being less than 2; in fact, gamma = 2 for Fig. 3. Where gamma comes in is in section 4 & Figure 7, where we show that recalibration of the Shen et al. mass scale can eliminate the SEB.

S&E are quite correct that there is a large dispersion in gamma from different fitting techniques, but MLINMIX_ERR is the only method which considers intrinsic scatter and errors on both parameters, making it the optimal method and the one on which we base our conclusions. (e.g., we could have included least-squares fitting ignoring individual error bars in Table 2, but its inclusion wouldn't make its results worthy of consideration.)

That said, S&E are also correct that the MLINMIX_ERR gamma=1.27+-0.40 is not different from the canonical gamma=2 at a statistically significant level. We say this right before section 4.1, and our conclusions stand. (Though I would consider changing the statement in the conclusion that the SEB "is likely not a real boundary" to "is quite possibly not a real boundary". The point being that until we can constrain gamma much more tightly, we won't know for sure.)

Section 2. "Non-Virial Motions:"

S&E say that we "do not assume that motions in the quasar broad-line region are predominantly virial (i.e. M_BH \propto (v_virial)^2 \propto FWHM^2)". That's a misinterpretation of Wang et al. 2009, whose analysis we follow. Basically, Wang et al. point out that if Hbeta and MgII don't come from the same part of the BLR, then you can have M_BH(RM) \propto (sigma_Hbeta,rms)^2 \propto (FWHM_MgII)^gamma, where gamma != 2 (and where "RM" stands for reverberation mapping); see our Eqs. 1 and 2. It's true that if there are nonvirial motions, then allowing gamma != 2 helps account for them, but they don't need to exist for gamma != 2 to be correct. Note that Wang et al. find gamma=1.70+-0.42 whereas we find gamma=1.27+-0.40. Luis Ho (personal communication) has found that the SEB slope is not significantly different from unity using the Wang et al. mass calibration.

Section 3

S&E raise a valid issue regarding the definition of the SEB: I was thinking of the SEB as a boundary below Eddington _with slope different from unity_, but they have a point that a SEB with a slope of unity is still a SEB. We will need to rephrase our paper to be in line with this proper definition of a SEB.

By this definition, we do find a SEB at low redshifts, but nowhere does the slope appear significantly different from unity. That is, we find that the 95th percentile value of the Eddington ratio appears to be a function of redshift but independent of mass. (I'm saying "appears" until we measure the 95% values and compute the slopes.)

S&E consider this result by itself to be a puzzling limit on quasar accretion, and they could well be right. My gut reaction is that a mass dependence is more puzzling than a redshift dependence, but my gut could well be wrong. One natural idea is that a redshift dependence could be a result of downsizing, with BHs in general being fueled less at lower redshift. But while it's conceptually easy to understand the Eddington limit boundary (quasar luminosity saturates above a M_BH-dependent mass accretion rate) and a SEB with slope zero (maximum mass accretion rate independent of M_BH leads to maximum luminosity), other boundaries are harder to understand. A M_BH-independent 95th percentile Eddington ratio away from the Eddington limit corresponds to different physical mass accretion rates, so what prevents accretion above that rate for a given BH from occurring? The same goes for a SEB with nonunity slope.

On another note, near the end of the section S&E say "In SE10 we showed the entire redshift range (z = 0.2 to 4.1) allowed by our sample and techniques. RH10 should do the same." In fact we do; our use of only MgII-based masses and our fitting requirement of modeling the FeII emission from 2200-3000 Ang means that we cover a more limited redshift range than SE10.

Section 4

* Our Fig. 1 was intended to serve only as an introduction to the SEB, like Fig. 3 of SE10, not to illustrate the full redshift-dependent extent of the SEB (for which we use Figs 2, 3 and 7). To clarify this, we need to either edit the caption or change the figure to show only quasars in one redshift bin.

* The captions to our Figs 3 and 7 mention that the differences in the M-L plane are particularly noticeable at high redshift. We should rewrite these captions accounting for the discussion of the SEB definition in the Section 3 comments above.

* S&E are quite correct here; they do indeed discuss the possibility of a mass-dependent BH mass correction in their section 4.5.2, and we will need to rewrite our discussion accordingly.

Section 5

I agree that "it is important to continue to improve our understanding of virial mass estimation" and that "Rafiee & Hall do not claim statistically significant deviations from virial masses". However, our new calibrations do show a SEB slope which is consistent with unity, unlike what was found by SE10.


What Next?

I'm making my grad student Jesse Rogerson think about next steps following from his Master's Thesis, so I feel obligated to do the same. What do we need to make progress on these issues? Roughly in order of perceived usefulness:

* Reverberation mapping in MgII and Hbeta simultaneously, to put MgII masses on a more empirical footing and to better constrain gamma (targets need to be at redshift high enough that MgII is visible from ground)
* Next best way to constrain gamma would be simultaneous snapshot spectra of MgII and Hbeta in previously reverberation-mapped AGN (e.g. with STIS and COS in next HST cycle?), to compare single-epoch sigma_MgII and single-epoch sigma_Hbeta; the scatter between single-epoch sigma_Hbeta and sigma_Hbeta,rms has been characterized (Denney et al. 2009)
* Apply R&H10 PCA-reconstruction MgII BH mass technique to full DR7 (about to start)
* extend R&H10 PCA-reconstruction BH mass technique to Hbeta and CIV, intercalibrating masses where Hbeta+MgII or MgII+CIV are both available