Constraints on Black Hole/Host Galaxy Co-evolution and Binary Stalling
  Using Pulsar Timing Arrays
  Pulsar timing arrays are now setting increasingly tight limits on the
gravitational wave background from binary supermassive black holes. But as
upper limits grow more constraining, what can be implied about galaxy
evolution? We investigate which astrophysical parameters have the largest
impact on strain spectrum predictions and provide a simple framework to
directly translate between measured values for the parameters of galaxy
evolution and PTA limits on the gravitational wave background of binary
supermassive black holes. We find that the most influential observable is the
relation between a host galaxy's central bulge and its central black hole,
$\mbox{$M_{\bullet}$-$M_{\rm bulge}$}$, which has the largest effect on the
mean value of the characteristic strain amplitude. However, the variance of
each prediction is dominated by uncertainties in the galaxy stellar mass
function. Using this framework with the best published PTA limit, we can set
limits on the shape and scatter of the $\mbox{$M_{\bullet}$-$M_{\rm bulge}$}$
relation. We find our limits to be in contention with strain predictions using
two leading measurements of this relation. We investigate several possible
reasons for this disagreement. If we take the $\mbox{$M_{\bullet}$-$M_{\rm
bulge}$}$ relations to be correct within a simple power-law model for the
gravitational wave background, then the inconsistency is reconcilable by
allowing for an additional "stalling" time between a galaxy merger and
evolution of a binary supermassive black hole to sub-parsec scales, with lower
limits on this timescale of $\sim 1-2$ Gyr.
