Discovering Black Hole Mass Scaling Relations with Symbolic Regression

Our knowledge of supermassive black holes (SMBHs) and their relation to their host galaxies is still limited, and there are only around 150 SMBHs that have their masses directly measured and confirmed. Better black hole mass scaling relations will help us reveal the physics of black holes, as well as predict black hole masses that are not yet measured. Here, we apply symbolic regression, combined with random forest to those directly-measured black hole masses and host galaxy properties, and find a collection of higher-dimensional (N-D) black hole mass scaling relations. These N-D black hole mass scaling relations have scatter smaller than any of the existing black hole mass scaling relations. One of the best among them involves the parameters of central stellar velocity dispersion, bulge-to-total ratio, and density at the black hole's sphere-of-influence with an intrinsic scatter of $\epsilon=0.083\,\ \text{dex}$, significantly lower than $\epsilon \sim 0.3\,\ \text{dex}$ for the M-$\sigma$ relation. These relations will inspire black hole physics, test black hole models implemented in simulations, and estimate unknown black hole masses on an unprecedented precision.