Reaching diffraction-limited localization with coherent PTAs

TL;DR

This work shows that coherent matched-filter mapping in pulsar timing arrays, fully incorporating pulsar-distance information (the pulsar term), can reach the diffraction-limited angular resolution for single GW sources. The authors derive aperture-based maps, demonstrate fringe patterns from the pulsar term, and establish a scaling where the resolvable sky area improves as the number of well-constrained pulsar distances increases, with a practical target of about 9 such distances at SNR ~ 10. They also quantify how well-constrained distances and PTA geometry affect localization, discuss current and future distance-measurement capabilities, and argue that including pulsar distances offers both a major sensitivity gain and a path to multi-messenger follow-up and discrimination between astrophysical and cosmological GW backgrounds. Overall, the results advocate for immediate incorporation of distance information in PTA analyses to achieve near-diffraction-limited GW localization in the near term.

Abstract

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: $δθ_{\mathrm{diff}} \sim (1/\mathrm{SNR})(λ_{\mathrm{GW}}/r) \approx 2~\mathrm{arcmin}$, 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.

Figures (2)

• Figure 1: The real part of $\bf L$ (left circular polarization) for an Earth term only matched-filter map (left) and a pulsar term only map (right). Both maps are in response to the same left-handed GW located at the black dot. The red points mark regions of the matched-filter map where pixel values are above $90 \%$ of the maximum pixel (located at the GW position). The array consists of two pulsars with identical distances and $\nu=5$. This figure demonstrates the fringe interference effects from the pulsar term that generates isolated islands of intensity that allow for GW localization far beyond the limit of an Earth-term only analysis.
• Figure 2: We consider a single, left-circularly polarized GW at a fixed position $\hat{n} = (1,\pi/2,0)$. $N_{\rm dist}$ equidistant pulsars are distributed randomly across the sky. The resulting matched-filter map, $\operatorname{Re}[L]$, computed for the pulsar term only is then added to an incoherent map (i.e. the matched-filter map, $\operatorname{Re}[L]$, for a dense PTA, $N_{\rm pulsars} = 4800$, using Earth term only). Adding the coherent maps to the incoherent, Earth-term only map simulates gradually adding individual pulsar distances to a PTA with poorly constrained distances. The two maps are normalized by their maximum pixel value before being added together. To generatre HEALPix maps we choose $N_{\rm side} = 20$. Full spherical maps are simulated for $\nu = 2,4,8$. The curve for $\nu = 160$ (purple) is the projected relation from adding $N_{\rm dist}$ plane waves on a flat patch of sky and measuring the fraction of the sky above $90 \%$ of the maximum. The scaling relation prior to reaching the diffraction limit (dashed lines) is found from the flat sky approximation as $(1/\rm SNR)^{N_{\rm dist}/2}$ (black dotted line). Here we have plotted the fraction of the sky divided by SNR (solid lines) to make a fair comparison between the area above an iso-contour (which scales as $\sim 1/\rm SNR$ for a 2D Gaussian profile) and $(\delta \theta_d )^2 = (\frac{\lambda}{d} (\frac{1}{\rm SNR}) )^2$ (which scales as $\sim (1/\rm SNR)^2$).