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Detecting a stochastic gravitational wave background, particularly radiationfrom individually unresolvable super-massive black hole binary systems, is oneof the primary targets for Pulsar Timing Arrays. Increasingly more stringentupper limits are being set on these signals under the assumption that thebackground radiation is isotropic. However, some level of anisotropy may bepresent and the characterisation of the power at different angular scalescarries important information. We show that the standard analysis for isotropicbackgrounds can be generalised in a conceptually straightforward way to thecase of generic anisotropic background radiation by decomposing the angulardistribution of the gravitational wave power on the sky into multipole moments.We introduce the concept of generalised overlap reduction functions whichcharacterise the effect of the anisotropy multipoles on the correlation of thetiming residuals from the pulsars timed by a Pulsar Timing Array. In a searchfor a signal characterised by a generic anisotropy, the generalised overlapreduction functions play the role of the so-called Hellings and Downs curveused for isotropic radiation. We compute the generalised overlap reductionfunctions for a generic level of anisotropy and Pulsar Timing Arrayconfiguration. We also provide an order of magnitude estimate of the level ofanisotropy that can be expected in the background generated by super-massiveblack hole binary systems.

Characterising gravitational wave stochastic background anisotropy with Pulsar Timing Arrays