The Black Hole Mass–Galaxy Bulge Relationship for QSOs in the Sloan Digital Sky Survey Data Release 3

We investigate the relationship between black hole mass, M BH , and host galaxy velocity dispersion, σ * , for QSOs in Data Release 3 of the Sloan Digital Sky Survey. We derive M BH from the broad Hβ line width and continuum luminosity and the bulge stellar velocity dispersion from the narrow [O III] line width (σ [O III] ). At higher redshifts, we use Mg II and [O II] in place of Hβ and [O III]. For redshifts z < 0.5, our results agree with the M BH -σ * relationship for nearby galaxies. For 0.5 < z < 1.2, the M BH -σ * relationship appears to show evolution with redshift in the sense that the bulges are too small for their black holes. However, we find that part of this apparent trend can be attributed to observational biases, including a Malmquist bias involving the QSO luminosity. Accounting for these biases, we find ~0.2 dex evolution in the M BH -σ * relationship between now and redshift z ≈ 1.