Constraining astrophysical observables of Galaxy and Supermassive Black Hole Binary Mergers using Pulsar Timing Arrays

Abstract: We present an analytic model to describe the supermassive black hole binary (SMBHB) merger rate in the Universe with astrophysical observables: galaxy stellar mass function, pair fraction, merger timescale and black hole - host galaxy relations. We construct observational priors and compute the allowed range of the characteristic spectrum $h_c$ of the gravitational wave background (GWB) to be $10^{-16}<h_c<10^{-15}$ at a frequency of $f=1/{\rm yr}$. We exploit our parametrization to tackle the problem of astrophysical inference from Pulsar Timing Array (PTA) observations. We simulate a series of upper limits and detections and use a nested sampling algorithm to explore the parameter space. Corroborating previous results, we find that the current PTA non-detection does not place significant constraints on any observables; however, either future upper limits or detections will significantly enhance our knowledge of the SMBHB population. If a GWB is not detected at a level of $h_c(f=1/{\rm yr})=10^{-17}$, our current understanding of galaxy and SMBHB mergers is disfavoured at a $5\sigma$ level, indicating a combination of severe binary stalling, over-estimating of the SMBH -- host galaxy relations, and extreme dynamical properties of merging SMBHBs. Conversely, future detections of a Square Kilometre Array (SKA)-type array will allow to constrain the normalization of the SMBHB merger rate in the Universe, the time between galaxy pairing and SMBHB merging, the normalization of the SMBH -- host galaxy relations and the dynamical binary properties, including their eccentricity and density of stellar environment.