Dynamical Evidence for a Billion Solar Mass Black Hole in Galaxy NGC 4061 from ALMA $^{12}$CO(2-1) Kinematics

We present the first robust dynamical measurement of the supermassive black hole (SMBH) mass in the massive early-type galaxy NGC 4061 using high-spatial-resolution ALMA observations of the $^{12}$CO(2-1) emission. By combining archival Cycle 6 data with new Cycle 7 observations, we achieve a synthesized beam of $0''.16 \times 0''.13$, comparable to the expected sphere of influence of the central black hole. The molecular gas forms a regularly rotating circumnuclear disk aligned with the prominent dust lane seen in HST imaging. We model the full three-dimensional ALMA data cube using the KinMS forward-modeling framework, exploring both data-driven and analytic prescriptions for the gas surface brightness distribution. Our Bayesian analysis yields a best-fitting SMBH mass of $M_{\rm BH} = (1.17^{+0.08}_{-0.10}\,[{\rm stat.}] \pm 0.43\,[{\rm syst.}]) \times 10^{9}$ M$_\odot$ and an $I$-band stellar mass-to-light ratio of $M/L_{\rm F814W} = 3.46^{+0.07}_{-0.06}\,[{\rm stat.}] \pm 0.10\,[{\rm syst.}]$ M$_\odot$/L$_\odot$. The inferred black hole mass is fully consistent across different modeling assumptions and remains insensitive to plausible radial variations in the $M/L_{\rm F814W}$ profile. Our results resolve the long-standing discrepancy between previous indirect mass estimates based on conflicting stellar velocity dispersion measurements and demonstrate that the exceptionally large dispersion reported in the literature is likely spurious. This study highlights the power of high-resolution ALMA molecular gas kinematics for precision SMBH mass measurements at the high-mass end of the local black hole mass function.