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 (SMBHs, M-bh) and the velocity dispersions of their host spheroid (sigma). An unknown relation between log M-bh and log sigma may be expanded to second order to obtain a log-quadratic relation of the form log (M-bh) =alpha+beta log (sigma/200 km s(-1)) +beta(2)[log (sigma/200 km s(-1))](2). We perform a Bayesian analysis using the local sample described in Tremaine et al., and solve for beta, beta(2) and alpha, in addition to the intrinsic scatter (delta). We find unbiased parameter estimates of beta= 4.2 +/- 0.37, beta(2)= 1.6 +/- 1.3 and delta= 0.275 +/- 0.05. At the 90 per cent level the M-bh-sigma relation does not follow a uniform power law. Indeed, over the velocity range 70 less than or similar to sigma less than or similar to 380 km s(-1) the logarithmic slope d log M-bh/d log sigma of the best-fitting 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 to the local SMBH sample leads to a log-quadratic relation with the same best fit as the local sample. However, the addition of the reverberation masses increases the significance of the log-quadratic contribution, yielding a value of beta(2) that is non-zero at the 5 sigma level. Furthermore, assuming no systematic offset, single epoch virial SMBH masses estimated for active galactic nuclei (AGNs) follow the same log-quadratic M-bh-sigma relation as the local sample, but extend it downward in mass by an order of magnitude. The log-quadratic term in the M-bh-sigma relation has a significant effect on estimates of the local SMBH mass function at M-bh greater than or similar to 10(9) M-circle dot, leading to densities of SMBHs with M-bh greater than or similar to 10(10) M-circle dot that are several orders of magnitude larger than inferred from a log-linear M-bh-sigma relation. We also estimate unbiased parameters for the SMBH-bulge mass relation using the sample assembled by Haring and Rix. With a parametrization log(M-bh) =alpha(bulge)+beta(bulge) log(M-bulge/10(11) M-circle dot) +beta(2,bulge)[log(M-bulge/10(11) M-circle dot)](2), we find beta(bulge)= 1.15 +/- 0.18 and beta(2,bulge)= 0.12 +/- 0.14. We determined an intrinsic scatter delta(bulge)= 0.41 +/- 0.07 which is similar to 50 per cent larger than the scatter in the M-bh-sigma relation.