Cross-correlating Astrometric and Timing Residuals to Constrain Stochastic Gravitational-Wave Backgrounds

TL;DR

This work develops a theoretical framework to constrain a stochastic gravitational-wave background by cross-correlating pulsar timing redshifts with astrometric and shimmering distortions of solar-system bodies, using a unified spin-weighted formalism to derive generalized, purely quadrupolar Hellings–Downs–type angular correlations for mixed observables. It provides explicit short-distance and long-distance response functions, casts the cross-correlations in spin-0 angular harmonics, and derives analytic expressions for the angular patterns and their spin-2 angular power spectra. Forecasts of SNR and sensitivity show that, with current astrometric capabilities, cross-correlation is not yet competitive with PTA data alone, but future wide-field, sub-milliarcsecond astrometry (e.g., LSST-era surveys) could access higher GW frequencies and offer an independent cross-check while bridging PTA and LISA frequency bands. The framework highlights the potential for a complementary SGWB probe that is less susceptible to correlated systematics and could validate SGWB detections across independent measurement domains.

Abstract

We investigate the cross-correlation between astrometric and timing-residual observables for distant sources, such as pulsars and galaxies, and equivalent observables for nearby solar system bodies. Using the unified spin-weighted formalism introduced in [1], we derive the angular correlation functions-generalised Hellings-Downs curves-that describe the response of these mixed observables to a stochastic, unpolarised gravitational-wave background (SGWB). We compute the expected signal-to-noise ratio (SNR) and sensitivity for such measurements, focusing on cross-correlations between pulsar timing array (PTA) redshift signals and astrometric or distortion (shimmering) effects induced in solar system objects such as asteroids. Although the current astrometric precision of asteroid tracking does not yet provide competitive constraints relative to PTA-only surveys, the method offers a complementary probe with enhanced sensitivity at higher frequencies. Future wide-field surveys capable of sub-milliarcsecond precision could make this approach a viable tool for detecting or constraining the SGWB. A key advantage of the technique is its reduced susceptibility to correlated systematics across different measurement domains, providing an independent cross-check of PTA detections and a potential observational bridge between PTA and LISA frequency bands.

Figures (4)

• Figure 1: Definition of angles on celestial sphere in \ref{['spin-weight-addition']}.
• Figure 2: Plot of correlation patterns \ref{['final-patterns']} in the aligned frame where $\alpha = 0$ for $\Gamma^{z, \kappa}$, $\Gamma^{z, \omega}$, $\Gamma^{z, \pm\gamma}$ and $\alpha = \frac{\pi}{2}$ for $\Gamma^{z, \pm\delta}$.
• Figure 3: Analytic sensitivity for the cross-correlation between PTA and Vera C. Rubin (asteroids) according to \ref{['SNR']}, \ref{['SNRhc']} compared to the sensitivity curve of only PTA measurements. Three cross-correlation curves are given for the state-of-the-art LSST survey, tracking asteroids with angular resolution $\sigma_{\rm ast} = \sigma_{\rm LSST}$ and cadence $\Delta T = 14$ days, and two potential future missions with $\sigma_{\rm ast} = 10^{-2}\sigma_{\rm LSST}$ and $\Delta T = 1$ day and $\sigma_{\rm ast} = 10^{-4}\sigma_{\rm LSST}$ and $\Delta T = 14$ days respectively. The vertical lines indicate the minimum frequency $\frac{1}{T} = \frac{1}{14\;\text{days}}$ (red) and maximum frequency $\frac{1}{\Delta T} = \frac{1}{1\;\text{day}}$ (green) that LSST is sensitive to.
• Figure 4: Variation of SNR with cadence and angular resolution of asteroid measurement. Different possible survey parameters are shown again, similarly to fig. \ref{['fig:contaldi-golat']}.