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, 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) + beta_2[log(sigma/200)]^2. We perform a Bayesian analysis using the Nuker sample, 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 80% level the M_bh-sigma relation does not follow a uniform power-law. Indeed, over the velocity range 70km/s<sigma<380km/s the logarithmic slope 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. Assuming no systematic offset, single epoch virial SMBH masses estimated for AGN follow the same log-quadratic M_bh-sigma relation as the Nuker 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>10^9 solar masses, leading to densities of SMBHs with M_bh>10^10 solar masses that are several orders of magnitude larger than inferred from a log-linear relation. We also estimate unbiased parameters for the SMBH-bulge mass relation. With a parameterisation log(M_bh)=alpha_b + beta_b log(M_b/10^{11}) + beta_2b[log(M_b/10^{11})]^2, we find beta_b=1.15+/-0.18 and beta_2b=0.12+/-0.14. We determined an intrinsic scatter delta_b=0.41+/-0.07 which is ~50% larger than the scatter in the M_bh-sigma relation. 
Stuart Wyithe