On the correlations of massive black holes with their host galaxies

We address the correlations of black hole (BH) mass with four different host-galaxy properties from 11 existing data sets. For the purpose of guiding theoretical understanding, we first try to quantify the tightness of the intrinsic correlations. We assume that all of the relations are power laws and perform linear regressions that are symmetric in the two variables on the logarithms of the data points. Given the estimated measurement errors, we evaluate the probability distribution of the residual variance in excess of that expected from the measurement errors. Our central result is that the current data sets do not allow definite conclusions regarding the quality of the true correlations because the obtained probability distributions for the residual variance overlap for most quantities. Velocity dispersion as collected by Merritt & Ferrarese (sigma(MF)) and galaxy light concentration as measured by Graham and coworkers (C-Re) are consistent with zero residual variance. Taken at face value, this means that these two correlations are better than the others, but this conclusion is highly sensitive to the assumed measurement errors and would be undone if the present estimated errors were too large. We then consider which of the relations offer the best inferences of BH mass when there is no direct measurement available. As with the residual variances, we find that the probability distribution of expected uncertainty in inferred BH masses overlaps significantly for most of the relations. Photometric methods would then be preferred because the data are easier to obtain, as long as bulge-disk decomposition or detailed modeling of the photometric profile (as studied by Graham and coworkers) do not present problems. Determining which correlation offers the best inferences requires reducing the uncertainty in the expected error in the inferred BH masses (the "error on the error''). This uncertainty is currently limited by uncertainty in the residual variance for all of the relations. The only quantities for which BH mass inferences are limited by measurement error are sigma(MF) and C-Re. Therefore, if these relations are truly better than the others, then new, improved measurements should allow improved inferences of BH masses. If they do not, the conclusion must be that the present low residual variances for these two relations result from overestimated error bars.