Reaching diffraction-limited localization with coherent PTAs

Current pulsar timing array (PTA) analyses do not take full advantage of pulsar distance information, thereby missing out on improved angular resolution and on a potential factor-of-two gain in detection sensitivity for individual gravitational-wave (GW) sources. In this work, we investigate the impact of precise pulsar distance measurements on angular resolution as an extension to previous work measuring the angular resolution of a dense, isotropic PTA [Jow et al., 2025]. We present a coherent map-making technique that utilizes precise pulsar distance measurements to reach the diffraction-limited resolution of an individual source: $\delta \theta_{\mathrm{diff}} \sim (1/\mathrm{SNR})(\lambda_{\mathrm{GW}}/r) \approx 2~\mathrm{arcmin}$, where the SNR refers to the detection strength of the source. With this level of angular resolution, identifying an EM counterpart may become feasible, enabling multi-messenger follow-up. We show that for $\rm SNR=10$, which may be the current sensitivity level using a coherent analysis, the diffraction limit is reached with roughly 9 pulsars. Moreover, angular resolution scales sharply with the number of known pulsar distances as $\sim (1/\mathrm{SNR})^{N_{\mathrm{dist}}/2}$. Thus, each additional pulsar with high signal-to-noise timing and precise distance measurement can improve PTA resolution by an order of magnitude. The distance to the best-timed millisecond pulsar (PSR J0437$-$4715) is already constrained to sub-parsec levels. We argue, therefore, that a coherent analysis of PTA data, fully incorporating pulsar distance information, is timely.