Characterizing gravitational wave stochastic background anisotropy with pulsar timing arrays

Detecting a stochastic gravitational wave background, particularly radiation from individually unresolvable supermassive black hole binary systems, is one of the primary targets for pulsar timing arrays. Increasingly more stringent upper limits are being set on these signals under the assumption that the background radiation is isotropic. However, some level of anisotropy may be present and the characterization of the gravitational wave energy density at different angular scales carries important information. We show that the standard analysis for isotropic backgrounds can be generalized in a conceptually straightforward way to the case of generic anisotropic background radiation by decomposing the angular distribution of the gravitational wave energy density on the sky into multipole moments. We introduce the concept of generalized overlap reduction functions which characterize the effect of the anisotropy multipoles on the correlation of the timing residuals from the pulsars timed by a pulsar timing array. In a search for a signal characterized by a generic anisotropy, the generalized overlap reduction functions play the role of the so-called Hellings and Downs curve used for isotropic radiation. We compute the generalized overlap reduction functions for a generic level of anisotropy and pulsar timing array configuration. We also provide an order of magnitude estimate of the level of anisotropy that can be expected in the background generated by supermassive black hole binary systems.