A novel approach to understanding the link between supermassive black holes and host galaxies

The strongest and most universal scaling relation between a supermassive black hole and its host galaxy is known as the $M_\bullet-\sigma$ relation, where $M_\bullet$ is the mass of the central black hole and $\sigma$ is the stellar velocity dispersion of the host galaxy. This relation has been studied for decades and is crucial for estimating black hole masses of distant galaxies. However, recent studies suggest the potential absence of central black holes in some galaxies, and a significant portion of current data only provides upper limits for the mass. Here, we introduce a novel approach using a Bayesian hurdle model to analyze the $M_\bullet-\sigma$ relation across 244 galaxies. This model integrates upper mass limits and the likelihood of hosting a central black hole, combining logistic regression for black hole hosting probability with a linear regression of mass on $\sigma$. From the logistic regression, we find that galaxies with a velocity dispersion of $11$, $34$ and $126$ km/s have a $50$%, $90$% and $99$% probability of hosting a central black hole, respectively. Furthermore, from the linear regression portion of the model, we find that $M_\bullet \propto \sigma^{5.8}$, which is significantly steeper than the slope reported in earlier studies. Our model also predicts a population of under-massive black holes ($M_\bullet=10-10^5 M_\odot$) in galaxies with $\sigma \lesssim 127$ km/s and over-massive black holes ($M_\bullet \geq 1.8 \times 10^7$) above this threshold. This reveals an unexpected abundance of galaxies with intermediate-mass and ultramassive black holes, accessible to next-generation telescopes like the Extremely Large Telescope.