Pulsar Timing Constraints on the Fermi Massive Black-Hole Binary Blazar Population

Blazars are a sub-population of quasars 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 has been detected in a number of bright sources, and has been connected to the orbital motion of a putative massive black hole binary. If this were indeed the case, those 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. Using mock populations informed by the luminosity functions for BL Lacertae objects and flat-spectrum radio quasars with redshifts $z \le 2$, we calculate the expected gravitational wave background and compare it to recent pulsar timing array upper limits. The two are consistent only if a fraction $\lesssim 10^{-3}$ of blazars hosts a binary with orbital periods $<5$ years. We therefore conclude that binarity cannot significantly explain year-like quasi-periodicity in blazars.