The standard Hellings-Downs (HD) approach to searching for gravitational wave signatures in pulsar timing array (PTA) data has successfully produced evidence for the presence of nanohertz-wavelength gravitational waves. However, it does not, on its own, produce any directional information and is insensitive to polarization. An alternative approach is to construct maps of the polarized power on the sky. In this paper, we present a simple quadratic estimator of the gravitational wave power as a function of direction on the sky that is sensitive to the polarization state of the wave. We describe the full, $S_2\times S_2$, state space of a polarized gravitational wave background across the sky and the Poincaré sphere describing polarization. A natural question arises from this perspective: what is the resolution of a polarized sky map, i.e., effectively how many independent pixels can such a map contain? In other words, how many distinct gravitational waves can a PTA distinguish? It turns out the answer is finite, and is approximately $N_{\mathrm{res}}=16\times 2=32$ per frequency, where 16 is the number of resolvable sky positions and 2 is the number of distinct polarization states, corresponding to an angular resolution of 58°, achieved with a PTA with more than $N_{\text{pulsar}}\gtrsim 20$ pulsars. We demonstrate that the variance of the map is equivalent to the HD significance, while for a single point source, a $3\sigma$ HD signal corresponds to a $5.2\sigma$ map significance.

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• Gravitational waves