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