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 \mbox { km s } ^ { -1 } ) } + \beta _ { 2 } [ % \log { ( \sigma / 200 \mbox { km s } ^ { -1 } ) } ] ^ { 2 } .
We perform a Bayesian analysis using the Local sample described in Tremaine et al .
( 2002 ) , and solve for \beta , \beta _ { 2 } and \alpha , in addition to the intrinsic scatter ( \delta ) .
We find unbiased parameter estimates of \beta = 4.2 \pm 0.37 , \beta _ { 2 } = 1.6 \pm 1.3 and \delta = 0.275 \pm 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 \la \sigma \la 380km/s the logarithmic slope d \log { M _ { bh } } / d \log { \sigma } of the best fit relation varies between 2.7 and 5.1 , which should be compared with a power-law estimate of 4.02 \pm 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 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 } \ga 10 ^ { 9 } M _ { \odot } , leading to densities of SMBHs with M _ { bh } \ga 10 ^ { 10 } M _ { \odot } 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 Haering & Rix ( 2004 ) .
With a parameterization \log { ( M _ { bh } ) } = \alpha _ { bulge } + \beta _ { bulge } \log { ( M _ { bulge } / 10 ^ % { 11 } M _ { \odot } ) } + \beta _ { 2 ,bulge } [ \log { ( M _ { bulge } / 10 ^ { 11 } M _ { \odot } ) } ] ^ { 2 } , we find \beta _ { bulge } = 1.15 \pm 0.18 and \beta _ { 2 ,bulge } = 0.12 \pm 0.14 .
We determined an intrinsic scatter \delta _ { bulge } = 0.41 \pm 0.07 which is \sim 50 % larger than the scatter in the M _ { bh } - \sigma relation .