THE M_•–σ_* RELATION DERIVED FROM SPHERE OF INFLUENCE ARGUMENTS

There has been a growing database of direct supermassive black hole (SMBH) mass (M_•) estimates from the centers of nearby galactic bulges (e.g., Graham 2008b). While the limits of our current abilities to significantly expand this database may have been reached (Batcheldor & Koekemoer 2009), the last decade has seen a wealth of M_• estimates that have increased the SMBH catalog from 13 or 26 (Ferrarese & Merritt 2000; Gebhardt et al. 2000) to ∼70 (Graham 2008b; Hu 2008; Gültekin et al. 2009). An intense interest in populating the SMBH database was sparked by observed correlations between M_• and fundamental properties of their host bulges (e.g., Kormendy & Richstone 1995; Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Graham et al. 2001; Ferrarese 2002; Marconi & Hunt 2003; Baes et al. 2003; Häring & Rix 2004; Pizzella et al. 2005). These M_• scaling relations have generated numerous theoretical investigations (e.g., Ciotti & van Albada 2001; Adams et al. 2003; Cattaneo et al. 2005; Robertson et al. 2006) and have possibly added valuable limits to evolutionary models (e.g., Heckman et al. 2004; Wyithe & Loeb 2005; Treu et al. 2007; Ciotti 2009).

The degree to which an SMBH's sphere of influence, r_i, is resolved has been used as a quality measure for M_• estimates (Ferrarese 2002; Marconi & Hunt 2003; Valluri et al. 2004). The only method available to a priori determine if a M_• estimate can be made is to assume r_i = GM_•/σ²_* (Peebles 1972). However, to calculate r_i in galaxies with known σ_*, M_• is estimated using the M_•–σ_* relation given by log M_• = α + βlog(σ_*/200 km s^-1), where α = 8.1–8.2 and β = 3.7–4.9. The observed scatter, , is 0.4 dex (Novak et al. 2006; Graham & Li 2009). Following this, Figure 1 demonstrates where r_i will be resolved, given a spatial resolution of ℜ = 01. A value of ℜ = 01 is used here as that is the typical FWHM of the Hubble Space Telescope (HST) point-spread function (PSF). To date, HST has been responsible for most M_• estimates.

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Ferrarese & Ford (2005, FF05) discussed the limited abilities of HST to resolve r_i, and Gültekin et al. (2009, G09) found the M_•–σ_* relation to be biased when applying the r_i argument. The influence of r_i cuts on the M_•–σ_* relation is continued in this Letter with two significant advances. First, a sample of all galaxies with a known σ_* (<100 Mpc) is used. It is important to only use these galaxies as there are no σ_* = 400 km s⁻¹ galaxies at 1 Mpc, for example. Second, random M_• estimates are applied to each galaxy, i.e., no galaxy is assumed to intrinsically fall on the M_•–σ_* relation. This ensures no pre-selection of galaxies that lie on the M_•–σ_* relation. Throughout, a distinction is made between the observed M_•–σ_* relation (published values) and the observable M_•–σ_* relation (the relation that can be fitted using the data simulated here).

In short, these simulations take a galaxy with a known σ_* and distance, assign a M_•, then calculate r_i. The r_i argument is then applied; if r_i is unresolved the galaxy is removed from the sample (the low-mass cut). If the assigned M_• generates a galaxy that lies above the observed M_•–σ_* relation, it is also removed from the sample (the high-mass cut). The observable M_•–σ_* relation is then fit to the remaining galaxies using a Levenberg–Marquardt least-squares add-on to IDL.

A SQL search of the Hyperleda¹ catalog was performed (Paturel et al. 2003). All galaxies with a kinematical distance modulus less than 35.0 (100 Mpc, assuming H₀ = 70 km s^-1 Mpc^-1) were found. Of the 49,740 galaxies returned, 2518 have published values of σ_*. The incompleteness of this sample, due to difficulties in measuring σ_* in faint galaxies, is noted. Values of r_i (in arcsec) were then calculated for each galaxy by assigning a M_•.

In the first cases, the low-mass cut was made by applying ℜ = 01 to represent the HST PSF. However, several values of ℜ are applied later. The high-mass cuts were then made by considering the observed distribution of galaxies in the M_•–σ_* plane; there is a clear absence of over-massive SMBHs, i.e., there are no σ_* = 50 km s⁻¹ dwarf spheroids in the Local Group with M_• ∼10⁹M_☉. Such a nearby over-massive SMBH population would have been detected, and therefore it is assumed that these objects do not exist. Galaxies selected for the high-mass cuts were determined by assigning a scatter ( = 0.4 dex) and upper limits to the M_•–σ_* relation, (M_•–σ_*)_u. Galaxies with r_i > r_max(M_max) were removed from the sample, where $M_{\rm max} = [\epsilon + 10^{\alpha _u}(\sigma /200{\rm \ km\ s^{-1}})^{\beta _u}]M_\odot$.

Each galaxy was first assigned 90 separate values of M_• using a step size of 10^0.1 from 10^1.0M_☉ to 10^10.0M_☉. In this case, the high-mass cut was made by applying the α_u = 8.1, β_u = 4.2 relation of G09 and the α_u = 8.2, β_u = 4.9 relation of FF05. Figure 2 shows the distribution of observable galaxies in the M_•–σ_* plane assuming every galaxy within 100 Mpc can host any M_•  and that ℜ = 01. The fit to this sample (red line) is α = 8.4, β = 3.5.

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However, galaxies host single (or binary) SMBHs, therefore it is shrewd to assign each galaxy a single (uniformly sampled) random value for M_• in the range 10¹–10¹⁰M_☉. The distribution of observable galaxies in the M_•–σ_* plane using a random sample of M_• is shown in Figure 3 using the red open circles. In this case, the low-mass cut was made using ℜ = 01, and the high-mass cut was made using the (M_•–σ_*)_u relation of α_u = 8.1, β_u = 4.2, = 0.4 (G09). The M_•–σ_* relation fit to these observable galaxies (red line) is α_u = 8.3, β_u = 4.1, = 0.2 dex. An observable M_•–σ_* relation of α_u = 8.3, β_u = 4.6, = 0.4 dex is found using the high-mass (M_•–σ_*)_u relation of FF05. Figure 3 also shows all observed galaxies based on the combined catalogs of Graham (2008b), Hu (2008), and G09. No distinction is made between "good" and "bad" M_• estimates, or differences in quoted σ_*; all estimates are plotted (143 total).

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The similarity between the M_•–σ_* distribution of observable random mass SMBHs and observed SMBHs masses is striking. However, as this result derives from a single random sampling of M_•, it represents a single possible observable M_•–σ_* relation. If galaxies intrinsically have random M_• values, then the range of observable SMBHs could have a distribution given by the distribution in Figure 2. Therefore, 50 separate random M_• samplings were then made for each galaxy, and in each case a fit to the observable M_•–σ_* relation was performed. In addition, as the previously used value of ℜ = 01 only applies to HST observations, the analysis was repeated for a range of r_i lower cutoffs. Finally, as the high-mass cuts may not actually be defined by the G09 and FF05 fits, values of α_u from 7.8 to 8.7 (in 0.1 steps) and β_u from 3.6 to 5.4 (in 0.2 steps) were used to create an addition 100 individual (M_•–σ_*)_u cutoffs. The mean values of α, β and , derived by applying these different ℜ criteria, and from using these variable upper limits, are presented in Table 1.

Table 1. Effects of the r_i Cutoff from Averaged Random M_• Distributions

ℜ	Observed α	Observed β	Observed
	(Zero Point)	(Slope)	(Scatter)
Variable Upper Cutoffs
005	8.25 ± 0.17	3.80 ± 0.33	0.41 dex
010	8.25 ± 0.21	3.90 ± 0.34	0.27 dex
015	8.25 ± 0.23	3.93 ± 0.34	0.22 dex
020	8.25 ± 0.27	4.01 ± 0.38	0.18 dex
Fixed Upper Cutoffs
005	8.44 ± 0.03	3.93 ± 0.23	1.82 dex
010	8.57 ± 0.02	4.02 ± 0.21	1.22 dex
015	8.64 ± 0.02	4.03 ± 0.25	0.97 dex
020	8.68 ± 0.03	4.07 ± 0.32	0.85 dex

Notes. Average results from applying variable and fixed M_•–σ_* upper limits to 50 random M_• samples. A fixed upper limit of α_u = 8.7 and β_u = 5.0 was used based on the upper-limit fit in Figure 4. If there was no intrinsic M_•–σ_* relation, these are the values of the M_•–σ_* relation that would be observed using the different spatial resolutions listed.

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Table 1 has several notable features. First, as ℜ rises falls. This is expected because the range of observable SMBHs decreases with larger values of ℜ. Second, the slope (β) remains consistent. Finally, in the case of the variable upper limits, the zero point (α) remains constant and is consistent with the mid-value of the (M_•–σ_*)_u limits imposed on the sample. To test whether these values of α are a consequence of the high-mass cut conditions, the analysis was repeated using α_u limits of 7.0 and 8.8 (in 0.2 steps). Applying a ℜ = 01 low-mass cut, an observable relation with α = 8.0 ± 0.5 was found. Again, the values of α are consistent with the mid-value of the imposed limits. However, an estimate of the observed (M_•–σ_*)_u relation can be made. The observed SMBH sample in Figure 3 was used to define an upper limit by fitting all observed SMBHs that fall above the FF05 and G09 relations, and above = 0.4. The fit to these over-massive SMBHs produce an upper limit of α_u = 8.7, β_u = 5.0 (Figure 4). This (M_•–σ_*)_u fit was then applied to 50 random values of M_• in each galaxy, and the observable M_•–σ_* relations were then fitted in each case. The average of these fits are also presented in Table 1. In this case, the fitted values of α and β are consistent with the variable upper limits, but the values of are significantly larger. This is expected, as these fits were applied to data that had the maximum range in the M_•–σ_* plane due to the high observed values of α_u and β_u.

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In summary, as a result of applying the r_i argument, a M_•–σ_* relation consistent with observed values (α ≈ 8, β ≈ 4, ≈ 0.3 dex) can be fitted to a sample of galaxies that contain random mass SMBHs, and as a consequence do not follow a M_•–σ_* relation. The r_i argument removes low-mass SMBHs where r_i would not be resolved, and high-mass SMBHs where r_i should have been resolved if such a population were present. Therefore, for scaling relations to be of value in constraining galaxy evolution models, M_• estimates must not be solely proposed on the basis of critically resolving values of r_i derived from σ_*.

It is unlikely that an intrinsic M_•–σ_* relation has been observed as a result of critically sampling r_i in the M_•–σ_* plane, and it will be some time until we are able to sample a significant area rightward of the observed M_•–σ_* relation (e.g., FF05; Batcheldor & Koekemoer 2009). Therefore, it is clear that caution must be used in the application of the observed M_•–σ_* relation until this issue can be resolved.

The possibility of there being no tight intrinsic M_•–σ_* relation, and that the distribution of observed galaxies in the M_•–σ_* plane determine an upper limit only, is now discussed. As the same M_• estimates are used to define the other M_• scaling relations, this also implies that all observed scaling relations are upper limits. There are a number of questions arising from this possibility, beginning with the most fundamental. Why do observed SMBHs fall on the M_•–σ_* relation? There are several SMBHs with such high quality M_• estimates (Sgr A*, NGC 4258, M87), that there clearly is a relation between M_• and σ_* in some galaxies. However, the M_•–σ_* plane has been increasingly populated with less certain M_• estimates that have potentially been included on the assumption that r_i is spatially resolved, i.e., that they follow a previously estimated M_•–σ_* relation defined by higher quality M_• estimates. In these cases, as the simulations presented here show, an observed M_•–σ_* relation will arise simply as a result of the r_i selection effect, even if M_• is randomly distributed within galaxies.

Does a population of under-massive SMBHs exist, and have they been detected? If the M_•–σ_* relation is an upper limit, then there should be galaxies that host low-mass SMBHs, as suggested by simulations (Vittorini et al. 2005; Volonteri 2007). Indeed, both the simulations presented here, and the observations plotted in Figure 3, show that under-massive SMBHs can be, and have been, detected. Some notable cases are that of NGC 4435 (Coccato et al. 2006), a sample of barred galaxies (Graham 2008a), and narrow-line Seyfert 1 galaxies (Mathur & Grupe 2005). In fact, the evidence to suggest the presence of under-massive SMBHs is not matched by any evidence for a significant population of over-massive SMBHs leftward of the observed M_•–σ_* relation.

It is important to note some intricacies with the two dominant techniques for measuring M_• (stellar and gas dynamics). In the case of stellar dynamics, there are systematics to the models that may allow a large range of M_• in a given bulge (Valluri et al. 2004). In the case of gas dynamics, it is unclear what the inclination of the nuclear gas disk may be (e.g., Marconi et al. 2003). Both methods allow the potential for many of the current M_• estimates to in fact be upper limits, i.e., the true M_•–σ_* plane may have a large distribution of under-massive SMBHs. Therefore, under-massive SMBHs may have been observed, their mass overestimated, and their impact overlooked.

All SMBH models allow stringent upper limits to be placed on M_• (e.g., Sarzi et al. 2002; Beifiori et al. 2009). However, these M_• upper limits will also be dependent on the available spatial resolution. Any kinematical data will have two velocity points spatially separated on a scale of ℜ. An upper limit to M_• is estimated by including an increasing dark mass until the derived model becomes inconsistent with the data, i.e., a higher upper limit to M_• will be estimated using a lower ℜ. There are still important constraints that can be added to the M_•–σ_* plane from estimates of M_• upper limits, however, as upper limits that fall below the observed M_•–σ_* relation provide the same evidence for under-massive SMBHs as would a tightly constrained low-mass M_•. At present, most M_• upper limits are based on data derived from gas dynamics. Consequently, these limits generally fall above the observed M_•–σ_* relation due to unknown amounts of line broadening from non-gravitational processes, and uncertainty in the inclination of the nuclear gas disks.

If the M_•–σ_* relation represents an upper limit in the M_•–σ_* plane, then what is this limit? In Section 2, an upper limit of α_u = 8.7, β_u = 5.0 was found based on the distribution of observed SMBHs leftward of the observed M_•–σ_* relation. This is likely a good approximation to the (M_•–σ_*)_u limit as there are no reports of a steeper relation. However, this estimate does not include the potential over-massive SMBHs from the upper limits calculated by Beifiori et al. (2009). Including these limits to the upper limit sample gives values of α_u = 8.8, β_u = 3.9 to (M_•–σ_*)_u. As expected, due to the addition of SMBH limits at lower M_•, this (M_•–σ_*)_u limit is more shallow with a higher zero point. Including these limits at the lower M_• end of the M_•–σ_* plane addresses, in part, a limitation of the sample used here. As already noted, the σ_* catalog used here is likely incomplete due to the difficulty of measuring σ_* in faint galaxies at greater distances. In addition, the σ_* catalog likely contains inhomogeneous measurements that may not translate from bulge to bulge.

What are the consequences to galaxy evolution models if there is only a (M_•–σ_*)_u relation? First, galaxies will no longer be required to obey the M_•–σ_* relation, and could host a SMBH with any M_• below (M_•–σ_*)_u. Models that include feedback from the SMBH to the galaxy will then need to be carefully reconsidered. While the SMBH will undoubtedly have some influence on a portion of the host galaxy, it would not need to affect large-scale properties; evolution of the SMBH would be a result of host galaxy evolution. An upper limit in the M_•–σ_* plane would also represent the pinnacle of SMBH evolution as a function of σ_*, in which galaxies evolve up to the (M_•–σ_*)_u limit. A signature of such a scenario could be a cosmic variation in (the scatter would increase with redshift) and an observed M_•–σ_* relation that does not exceed (M_•–σ_*)_u. If the distribution of M_• is random within bulges, then when compared with the local observed M_•–σ_* relation, M_• estimates from higher redshift could fall to the left or the right. Treu et al. (2007) find a population of z = 0.36 Seyfert 1 galaxies that lie above the local observed M_•–σ_* relation by Δlog σ = 0.13, Δlog  M_• = 0.54, but this population still lies below the (M_•–σ_*)_u limit estimated here. Finally, if SMBHs can reside anywhere below the (M_•–σ_*)_u limit, then the local black hole mass function may have been overestimated. This would relax the observation that merging is not important and that SMBH growth is dominated by accretion (e.g., Marconi et al. 2004). This potentially allows anti-hierarchical SMBH growth to no longer present problems for hierarchical galaxy formation models.

The use of the HyperLeda database (http://leda.univ-lyon1.fr) is acknowledged. D.B. thanks Alessandro Marconi, David Merritt, and Andy Robinson for useful discussions. Support for this work was provided by proposal number HST-AR-10935.01 awarded by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.