  Supermassive black hole binary mergers generate a stochastic gravitational
wave background detectable by pulsar timing arrays. While the amplitude of this
background is subject to significant uncertainties, the frequency dependence is
a robust prediction of general relativity. We show that the effects of new
forces beyond the Standard Model can modify this prediction and introduce
unique features into the spectral shape. In particular, we consider the
possibility that black holes in binaries are charged under a new long-range
force, and we find that pulsar timing arrays are capable of robustly detecting
such forces. Supermassive black holes and their environments can acquire charge
due to high-energy particle production or dark sector interactions, making the
measurement of the spectral shape a powerful test of fundamental physics.
