  The detection of a stochastic gravitational-wave signal from the
superposition of many inspiraling supermassive black holes with pulsar timing
arrays (PTAs) is likely to occur within the next decade. With this detection
will come the opportunity to learn about the processes that drive
black-hole-binary systems toward merger through their effects on the
gravitational-wave spectrum. We use Bayesian methods to investigate the extent
to which effects other than gravitational-wave emission can be distinguished
using PTA observations. We show that, even in the absence of a detection, it is
possible to place interesting constraints on these dynamical effects for
conservative predictions of the population of tightly bound supermassive
black-hole binaries. For instance, if we assume a relatively weak signal
consistent with a low number of bound binaries and a low black-hole-mass to
galaxy-mass correlation, we still find that a non-detection by a simulated
array, with a sensitivity that should be reached in practice within a few
years, disfavors gravitational-wave-dominated evolution with an odds ratio of
$\sim$30:1. Such a finding would suggest either that all existing astrophysical
models for the population of tightly bound binaries are overly optimistic, or
else that some dynamical effect other than gravitational-wave emission is
actually dominating binary evolution even at the relatively high
frequencies/small orbital separations probed by PTAs.
