Selection bias in the M.-σ and M.-L correlations and its consequences

It is common to estimate black hole abundances by using a measured correlation between black hole mass and another more easily measured observable, such as the velocity dispersion or luminosity of the surrounding bulge. The correlation is used to transform the distribution of the observable into an estimate of the distribution of black hole masses. However, different observables provide different estimates; the M-center dot -sigma relation predicts fewer massive black holes than does the M-center dot -L relation. This is because the sigma-L relation in black hole samples currently available is inconsistent with that in the SDSS sample, on which the distributions of L or sigma are based; the black hole samples have smaller L for a given sigma or have larger sigma for a given L. This is true whether L is estimated in the optical or in the NIR. If this is a selection rather than physical effect, then the M-center dot-sigma similar to and M-center dot-L relations currently in the literature are also biased from their true values. We provide a framework for describing the effect of this bias. We then combine it with a model of the bias to make an estimate of the true intrinsic relations. If we have correctly modeled the selection effect, then our analysis suggests that the bias in the < M-center dot vertical bar sigma > relation is likely to be small, whereas the < M-center dot-L > relation is biased toward predicting more massive black holes for a given luminosity, and the M-center dot-L relation is entirely a consequence of more fundamental relations between M-center dot and sigma and between sigma and L. The intrinsic relation we find suggests that at fixed luminosity, older galaxies tend to host more massive black holes.