On the Inconsistency between the Black Hole Mass Function Inferred from M•-σ and M•-L Correlations

Black hole masses are tightly correlated with the stellar velocity dispersions of the bulges which surround them and slightly less well correlated with the bulge luminosity. It is common to use these correlations to estimate the expected abundance of massive black holes. This is usually done by starting from an observed distribution of velocity dispersions or luminosities and then changing variables. This procedure neglects the fact that there is intrinsic scatter in these black hole mass-observable correlations. Accounting for this scatter results in estimates of black hole abundances which are larger by almost an order of magnitude at masses >109 Msolar. Including this scatter is particularly important for models which seek to infer quasar lifetimes and duty cycles from the local black hole mass function. However, even when scatter has been accounted for, the M•-σ relation predicts fewer massive black holes than does the M•-L relation. This is because the σ-L relation in the black hole samples currently available is inconsistent with that in the SDSS sample from which the distributions of L or σ are based: the black hole samples have smaller L for a given σ, or larger σ for a given L. The σ-L relation in the black hole samples is similarly discrepant with that in other samples of nearby early-type galaxies. This suggests that current black hole samples are biased: if this is a selection rather than a physical effect, then the M•-σ and M•-L relations currently in the literature are also biased from their true values.