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Pulsar timing arrays are now setting increasingly tight limits on thegravitational wave background from binary supermassive black holes. But asupper limits grow more constraining, what can be implied about galaxyevolution? We investigate which astrophysical parameters have the largestimpact on strain spectrum predictions and provide a simple framework todirectly translate between measured values for the parameters of galaxyevolution and PTA limits on the gravitational wave background of binarysupermassive black holes. We find that the most influential observable is therelation between a host galaxy's central bulge and its central black hole,$\mbox{$M_{\bullet}$-$M_{\rm bulge}$}$, which has the largest effect on themean value of the characteristic strain amplitude. However, the variance ofeach prediction is dominated by uncertainties in the galaxy stellar massfunction. Using this framework with the best published PTA limit, we can setlimits 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 usingtwo leading measurements of this relation. We investigate several possiblereasons for this disagreement. If we take the $\mbox{$M_{\bullet}$-$M_{\rmbulge}$}$ relations to be correct within a simple power-law model for thegravitational wave background, then the inconsistency is reconcilable byallowing for an additional "stalling" time between a galaxy merger andevolution of a binary supermassive black hole to sub-parsec scales, with lowerlimits on this timescale of $\sim 1-2$ Gyr.

Constraints on Black Hole/Host Galaxy Co-evolution and Binary Stalling Using Pulsar Timing Arrays