Pulsar Timing Array Harmonic Analysis and Source Angular Correlations

Gravitational waves (GWs) influence the arrival times of radio signals coming from pulsars. Here, we investigate the harmonic space approach to describing a pulsar's response to GWs. We derive and discuss the "diagonalized form" of the response, which is a sum of spin-2-weighted spherical harmonics of the GW direction multiplied by normal (spin-weight 0) spherical harmonics of the pulsar direction. We show how this allows many useful objects, for example, the Hellings and Downs two-point function, to be easily calculated. The approach also provides a clear description of the gauge dependence. We then employ this harmonic approach to model the effects of angular correlations in the sky locations of GW sources (sometimes called "statistical isotropy"). To do this, we construct rotationally invariant ensembles made up of many Gaussian subensembles, each of which breaks rotational invariance. Using harmonic techniques, we compute the cosmic covariance and the total covariance of the Hellings and Downs correlation in these models. The results may be used to assess the impact of angular source correlations on the Hellings and Downs correlation, and for optimal reconstruction of the Hellings and Downs curve in models where GW sources have correlated sky locations.