Pulsar timing constraints on the Fermi massive black hole binary blazar population

Blazars are a subpopulation of active galactic nuclei whose jets are nearly aligned with the line-of-sight, which tend to exhibit multi-wavelength variability on a variety of timescales. Quasi-periodic variability on year-like timescales is sometimes thought to be due to binary orbital motion. If this is indeed the case, then such blazar binaries would contribute to the nanohertz gravitational-wave stochastic background. We test the binary hypothesis for the blazar population observed by the \textit{Fermi} Gamma-Ray Space Telescope, which consists of BL Lacertae objects and flat-spectrum radio quasars. We calculate what the inferred GW background would be from mock populations informed by the luminosity functions for BL Lacertae objects and flat-spectrum radio quasars with redshifts $z \le 2$ and compare the amplitude distributions to recent pulsar timing array upper limits. To make the inferred background amplitudes consistent with pulsar-timing upper limits, we obtain upper limits on the binary fraction for the blazar population, which approximately range between $\approx 0.01\% - 0.1\%$. We discuss implications for the binary fraction for the unaligned AGN population and conclude that binarity alone cannot explain year-like quasi-periodicity in blazars.