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Efficient pulsar distance measurement with multiple nanohertz gravitational-wave sources

Si-Ren Xiao, Ji-Yu Song, Yue Shao, Ling-Feng Wang, Jing-Fei Zhang, Xin Zhang

TL;DR

This paper tackles the challenge of extracting precise pulsar distances from nanohertz GW signals, where a single CGW source yields a $2\pi$ phase ambiguity in the pulsar term. It presents a two-dimensional distance-posterior framework that uses pulsar-pair posteriors from multiple CGW sources, with a two-step reduction of phase degeneracy by treating the pulsar-term phase $\Phi_p$ as an independent parameter and then post-processing to recover $L_p$; parallax priors further constrain the distances. Using a simulated SKA-era PTA with 20 pulsars and white noise of $20\,\mathrm{ns}$, the authors show that incorporating four CGW sources can drive many pulsars within $\lesssim 1.4$ kpc to sub-parsec distance precision over a 15–30 year observing span, surpassing what one-dimensional posteriors achieve. They also test a more realistic CRTS-based SMBHB population and find substantial, though geometry-dependent, gains in distance precision, reinforcing the method’s potential to enhance GW sky localization and enable multi-messenger studies with nHz GWs. Overall, the approach provides a computationally efficient means to exploit pulsar-term information for sub-parsec pulsar distances, which is crucial for reducing host-galaxy candidate sets and improving cosmological and fundamental-physics applications of PTAs.

Abstract

In recent years, pulsar timing arrays (PTAs) have reported evidence for a nanohertz gravitational-wave (GW) background. As radio telescope sensitivity improves, PTAs are also expected to detect continuous gravitational waves from individual supermassive black hole binaries. Nanohertz GWs generate both Earth and pulsar terms in the timing data, and the time delay between the two terms encodes the pulsar distance. Precise pulsar distance measurements are critical to fully exploiting pulsar-term information, which can improve the measurement precision of GW sources' sky position parameters and thus enhance the GW sky-localization capability. In this work, we propose a new pulsar distance estimation method by using pulsar-term phase information from GWs. We construct two-dimensional distance posteriors for pulsar pairs based on the simulated GW signals and combine them to constrain individual pulsar distances. Compared with the existing one-dimensional method, our approach reduces the impact of source-parameter uncertainties on pulsar distance measurements. Considering four GW sources and a PTA of 20 pulsars with a white-noise level of 20 ns, we find that a significant fraction of pulsars at distances $\lesssim 1.4$ kpc can achieve sub-parsec distance precision over a 15-year observation.

Efficient pulsar distance measurement with multiple nanohertz gravitational-wave sources

TL;DR

This paper tackles the challenge of extracting precise pulsar distances from nanohertz GW signals, where a single CGW source yields a phase ambiguity in the pulsar term. It presents a two-dimensional distance-posterior framework that uses pulsar-pair posteriors from multiple CGW sources, with a two-step reduction of phase degeneracy by treating the pulsar-term phase as an independent parameter and then post-processing to recover ; parallax priors further constrain the distances. Using a simulated SKA-era PTA with 20 pulsars and white noise of , the authors show that incorporating four CGW sources can drive many pulsars within kpc to sub-parsec distance precision over a 15–30 year observing span, surpassing what one-dimensional posteriors achieve. They also test a more realistic CRTS-based SMBHB population and find substantial, though geometry-dependent, gains in distance precision, reinforcing the method’s potential to enhance GW sky localization and enable multi-messenger studies with nHz GWs. Overall, the approach provides a computationally efficient means to exploit pulsar-term information for sub-parsec pulsar distances, which is crucial for reducing host-galaxy candidate sets and improving cosmological and fundamental-physics applications of PTAs.

Abstract

In recent years, pulsar timing arrays (PTAs) have reported evidence for a nanohertz gravitational-wave (GW) background. As radio telescope sensitivity improves, PTAs are also expected to detect continuous gravitational waves from individual supermassive black hole binaries. Nanohertz GWs generate both Earth and pulsar terms in the timing data, and the time delay between the two terms encodes the pulsar distance. Precise pulsar distance measurements are critical to fully exploiting pulsar-term information, which can improve the measurement precision of GW sources' sky position parameters and thus enhance the GW sky-localization capability. In this work, we propose a new pulsar distance estimation method by using pulsar-term phase information from GWs. We construct two-dimensional distance posteriors for pulsar pairs based on the simulated GW signals and combine them to constrain individual pulsar distances. Compared with the existing one-dimensional method, our approach reduces the impact of source-parameter uncertainties on pulsar distance measurements. Considering four GW sources and a PTA of 20 pulsars with a white-noise level of 20 ns, we find that a significant fraction of pulsars at distances kpc can achieve sub-parsec distance precision over a 15-year observation.

Paper Structure

This paper contains 10 sections, 28 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Signal model
  4. Pulsar timing array
  5. Parameter estimation
  6. Reconstructing pulsar distances
  7. Result and Discussions
  8. Pulsar distance inference with multiple GW sources
  9. Constraints from a more realistic mock population of GW sources
  10. Conclusions

Figures (4)

  • Figure 1: Posterior distributions of $\mathcal{M}$, $f_0$, and $L_p$ for J0030+0451 and J0613--0200 (labeled as $L_1$ and $L_2$ according to their ordering in our simulated PTA). Contours show the 1$\sigma$ and 2$\sigma$ credible regions. The injected true values are indicated by crosses.
  • Figure 2: Posterior distributions of the distances of two pulsars derived from individual and combined GW sources. The left panel shows the posterior distributions of $L_p$ for J0030+0451 and J0613--0200 (labeled as $L_1$ and $L_2$ according to their ordering in our simulated PTA), obtained from three independent GW sources. The dashed lines indicate the injected true values of $L_p$. The right panel presents the combined posterior distribution of $L_p$ from the three sources. For comparison, the one-dimensional joint posterior distributions (following Ref. Yu:2025tnk) are also shown, indicated by dashed lines in the one-dimensional panels and blue bands in the two-dimensional panel.
  • Figure 3: Distributions of the $1\sigma$ posterior uncertainty $\Delta L_p$ of the pulsar distance, obtained from multiple realizations for different numbers $N$ of GW sources. The orange and blue violins correspond to GW sources with $f_0 = 3\,\mathrm{nHz}$ and $f_0 = 10\,\mathrm{nHz}$, respectively. All sources are assumed to be observed over 30 years. The black dashed line marks the timing parallax uncertainty $\sigma_{L_p}$ for comparison. (a) J0030+0451, $d_{\rm L}=5$ Gpc. (b) J0613--0200, $d_{\rm L}=5$ Gpc. (c) J1600--3053, $d_{\rm L}=5$ Gpc. (d) J1911+1347, $d_{\rm L}=10$ Gpc. (e) J0030+0451, $d_{\rm L}=10$ Gpc. (f) J0613--0200, $d_{\rm L}=10$ Gpc. (g) J1600--3053, $d_{\rm L}=10$ Gpc. (h) J1911+1347, $d_{\rm L}=10$ Gpc.
  • Figure 4: Pulsar distance precision derived from 4 GW observations combined with a timing-parallax prior, shown for each pulsar as a function of the PTA observational timespan. Each curve within a sub-panel illustrates the evolution of an individual pulsar's distance precision for PTA observation time span ranging from 5 to 30 years.