Summary statistic for pulsar timing arrays
The timing residuals produced by gravitational-wave signals can be described as an incoherent (pulsar term) contribution and a coherent (Earth term) map on the sky, which PTAs measure at the locations of the timed pulsars. The observed Earth term map and the variance induced by the pulsar term contain all of the information about any GW signal available to a PTA (assuming pulsar distances are unknown). Furthermore, any type of signal produces on average the same angular correlation function, the Hellings and Downs curve, which decays steeply with multipole as $C_\ell \propto 1/[(\ell+2)(\ell+1)\ell(\ell-1)]$. This suggests that the signal is inherently low-dimensional and therefore only a small number of parameters are needed to fully characterize it. We present an expression for the PTA likelihood that makes the dependence on the Earth term map and pulsar term variance explicit, and show that only a few spherical harmonic coefficients are needed to capture most of the information about the signal. To quantify this in a realistic setting, we compute the Fisher matrix of the amplitude of a stochastic background or a deterministic point source assuming the noise properties and sky locations of the pulsars in the NANOGrav 15yr dataset. We find that $\ell_{\rm max}=2$ of the Earth term map and the monopole of the pulsar term variance retain $\sim 95\%$ of the information about the signal. For a point source, including the dipole of the pulsar term variance is important to achieve a similar fraction.