New Black Hole Mass Calibrations and the Fundamental Plane of the Broad-line Region Size, Luminosity, and Velocity

We present a new calibration of the broad-line region (BLR) size–luminosity–velocity relation using a sample of 157 active galactic nuclei (AGNs) with reliable measurements of H β time-delay ( τ H β ) from S. Wang & J.-H. Woo. By incorporating the Eddington ratio as a third parameter, we effectively correct the systematic offset of high Eddington AGNs in the traditional BLR size–luminosity relation. The resulting three-parameter fit defines a fundamental plane in the 3D space of τ H β , optical luminosity, and H β velocity, with an intrinsic scatter of 0.21 dex. This tight correlation reflects the coupled effects of gas kinematics, photoionization, and BLR geometry. In turn, we develop a new method to infer τ H β from the combination of optical luminosity and H β velocity, and derive single-epoch black hole mass estimators by adopting either the FWHM or the line dispersion ( σ ) of the H β line profile as the velocity indicator. The derived τ H β shows a ∼0.1 dex scatter, depending on the choice of calibrations. We show that the previous mass estimates based on the two-parameter size–luminosity relation with a slope of 0.5 can be overestimated by up to 0.5 dex, demonstrating that the new mass estimator substantially changes the cosmic black hole mass density and the growth of black hole seeds in the early Universe.