Probing the merger rates of supermassive black holes and galaxies with gravitational waves

ABSTRACT The mergers of galaxies and supermassive black holes (SMBHs) are key drivers of galaxy evolution, contributing to the growth of both galaxies and their central black holes. Current and upcoming gravitational wave (GW) detectors – pulsar timing arrays (PTAs), Laser Interferometer Space Antenna (LISA), Taiji, and Tianqin – offer unique access to these processes by observing GW signals from SMBH binaries. We present a framework to infer galaxy and SMBH merger rates by combining mock LISA detections of SMBH mergers with PTA constraints on the stochastic GW background, while incorporating observational uncertainties in stellar mass functions and $M_\bullet$–$M_*$ relations. We find that the number of LISA-detected events and their joint distribution in mass and redshift are key to constraining merger rates – data sets with around 40 events yield results consistent with galaxy pair observations, whereas limited event counts lead to biases at high redshift. Including PTA data further reduces parameter uncertainties. Our method also effectively constrains the delay time between galaxy and SMBH mergers, with longer delays suppressing high-redshift SMBH merger rates and shifting mass growth from mergers to accretion. According to our mock analysis, the models with delay times longer than 0.5 Gyr (0.8 Gyr), accretion becomes the primary driver of SMBH mass growth beyond $z \sim 6$ (4). In contrast, the SMBH occupation fraction at $z>3$ remains poorly constrained due to its degeneracies with delay time and the galaxy merger rate. These findings highlight both the promise and limitations of using GW observations to probe the coevolution of galaxies and SMBHs.