The search for anisotropy in the gravitational-wave background with pulsar-timing arrays

Pulsar-timing arrays (PTAs) are seeking gravitational waves from supermassive-black-hole binaries, and there are prospects to complement these searches with stellar-astrometry measurements. Theorists still disagree, however, as to whether the local gravitational-wave background will be statistically isotropic, as arises if it is the summed contributions from many SMBH binaries, or whether it exhibits the type of statistical anisotropy that arises if the local background is dominated by a handful (or even one) bright source. Here we derive, using bipolar spherical harmonics, the optimal PTA estimators for statistical anisotropy in the GW background and simple estimates of the detectability of this anisotropy. We provide results on the smallest detectable amplitude of a dipole anisotropy (and several other low-order multipole moments) and also the smallest detectable amplitude of a "beam" of gravitational waves. Results are presented as a function of the signal-to-noise with which the GW signal is detected and as a function of the number of pulsars (assuming uniform distribution on the sky and equal sensitivity per pulsar). We provide results first for measurements with a single time-domain window function and then show how the results are augmented with the inclusion of time-domain information. The approach here is intended to be conceptually straightforward and to complement the results of more detailed (but correspondingly less intuitive) modeling of the actual measurements.