A Log-Quadratic Relation Between the Nuclear Black-Hole Masses and Velocity Dispersions of Galaxies

We demonstrate that a log-linear relation does not provide an adequate description of the correlation between the masses of Super-Massive Black-Holes (SMBH, Mbh) and the velocity dispersions of their host spheroid (σ). An unknown relation between logMbh and log σ may be expanded to second order to obtain a log-quadratic relation of the form log (Mbh) = α+β log (σ/200kms )+β2[log (σ/200kms )]2. We perform a Bayesian analysis using the Local sample described in Tremaine et al. (2002), and solve for β, β2 and α, in addition to the intrinsic scatter (δ). We find unbiased parameter estimates of β = 4.2± 0.37, β2 = 1.6± 1.3 and δ = 0.275± 0.05. At the 80% level the Mbh − σ relation does not follow a uniform power-law. Indeed, over the velocity range 70km/s<∼ σ <∼380km/s the logarithmic slope d logMbh/d logσ of the best fit relation varies between 2.7 and 5.1, which should be compared with a power-law estimate of 4.02± 0.33. The addition of the 14 galaxies with reverberation SMBH masses and measured velocity dispersions (Onken et al. 2004) to the Local SMBH sample leads to a log-quadratic relation with the same best fit as the Local sample. Furthermore, assuming no systematic offset, single epoch virial SMBH masses estimated for AGN (Barth et al. 2005) follow the same log-quadratic Mbh − σ relation as the Local sample, but extend it downward in mass by an order of magnitude. The log-quadratic term in the Mbh − σ relation has a significant effect on estimates of the local SMBH mass function at Mbh >∼ 10 M⊙, leading to densities of SMBHs with Mbh >∼ 10 M⊙ that are several orders of magnitude larger than inferred from a log-linear Mbh − σ relation. We also estimate unbiased parameters for the SMBH-bulge mass relation using the sample assembled by Haering & Rix (2004). With a parameterization log (Mbh) = αbulge + βbulge log (Mbulge/10 M⊙) + β2,bulge[log (Mbulge/10 M⊙)] , we find βbulge = 1.15±0.18 and β2,bulge = 0.12±0.14. We determined an intrinsic scatter δbulge = 0.41± 0.07 which is ∼50% larger than the scatter in the Mbh − σ relation.