Chirality of the gravitational-wave background and pulsar-timing arrays

We describe the signatures of a circularly polarized gravitational-wave background on the timing residuals obtained with pulsar-timing arrays. Most generally, the circular polarization will depend on the gravitational-wave direction, and we describe this angular dependence in terms of spherical harmonics. While the amplitude of the monopole (the overall chirality of the gravitational-wave background) cannot be detected, measures of the anisotropy are theoretically conceivable. We provide expressions for the minimum-variance estimators for the circular-polarization anisotropy. We evaluate the smallest detectable signal as a function of the signal-to-noise ratio with which the isotropic GW signal is detected and the number of pulsars (assumed to be roughly uniformly spread throughout the sky) in the survey. We find that the overall dipole of the circular polarization and a few higher overall multipoles, are detectable in a survey with $\gtrsim100$ pulsars if their amplitude is close to maximal and once the isotropic signal is established with a signal-to-noise ratio $\gtrsim400$. Even if the anisotropy can be established, though, there will be limited information on its direction. Similar arguments apply to astrometric searches for gravitational waves.