Bridging the Population Synthesis of Supermassive Binary Black Holes and the Gravitational Wave Background

Pulsar Timing Arrays (PTAs) constrain population properties of supermassive binary black holes (SMBHBs) through the observation of the gravitational wave background (GWB). Unlike other approaches that interpolate population-synthesis libraries or only consider the mean of the strain spectrum, here we capture its full strain probability density directly from semi-analytic population models. We apply our new method to the semi-analytic SMBHB population model, independently reproducing the parameter estimation for this model performed by the NANOGrav Collaboration with their 15-yr data. We also show the extent to which discrete SMBHB contributions to the GWB resolve degeneracies in the population parameter space. Finally, using the source-count intensity as the intermediate product in our calculation, we map PTA observations, as a proof of principle, to the SMBHB model based on the galaxy merger prescriptions from numerical hydrodynamical simulations "Illustris". We find the effect of delay times τ between kiloparsec and subparsec SMBHB separations following galaxy mergers, finding h c spanning (1 - 6) times10-16 and N c spanning (0.3 - 7.1) times10-3 for τ up to 8 Gyr.