From the Solar System to cosmological distances: a complete formalism for gravitational wave astrometry

TL;DR

This work develops a complete finite-distance formalism for gravitational-wave astrometry, enabling accurate GW-induced angular deflections for sources at arbitrary distances through distance-dependent form factors and a vector spherical harmonic decomposition. By coupling astrometric deflections of stars and asteroids with pulsar timing residuals, the authors derive distance-aware angular power spectra and cross-covariances, extending GW background probes into the mHz regime accessible to LISA. They demonstrate that finite-distance effects are crucial for Solar System objects (e.g., Main Belt asteroids) while distant stars remain effectively in the infinite-distance limit, and they quantify detectability with SNR forecasts and Fisher analyses across multiple GW background models. The results show that joint astrometric and pulsar-timing measurements, incorporating realistic target distributions (Gaian stars, Main Belt asteroids, and galactic pulsars) and cross-correlations, can yield nontrivial constraints on the GW background, with potential percent-level precision for flat spectra and meaningful discrimination among PTA-like, SMBHB, and cosmological backgrounds. This framework paves the way for synergies between Gaia-like and Roman-like astrometric surveys and next-generation pulsar timing arrays to probe GW backgrounds across a broad frequency range.

Abstract

The presence of a gravitational wave background (GWB) can be established not only via exquisitely precise pulsar timing array (PTA) measurements, but also via astrometric observations. Indeed, the very same background responsible for the delay in the arrival time of pulse causes an apparent displacement of galactic objects as stars and asteroids. In this work we provide a framework that allows to derive the displacement of sources overcoming the usually adopted ``infinite distance'' approximation. We also present how this formalism can be used to study the displacements of objects at distances comparable to the GW wavelength, as asteroids, and of objects with a non-trivial three-dimensional distribution, as stars in the Milky Way. Thus, it can be used to probe frequencies beyond PTA experiments, reaching the mHz GWs, also detectable by LISA. We forecast the capability of observing the astrometric deflection induced by a GWB evaluating the harmonic signal-to-noise ratio including correlations between different probes. We find an SNR greater than one for the relevant cases considered and as a consequence a promising Fisher forecast, suggesting a constraining power up to the percent level for a flat background.

Figures (10)

• Figure 1: Top panel: Regions in the $d_L-f$ plane indicating current and future surveys sensitivities to a GW background together with the source's distances probed by different survey. The lines corresponding to $\tilde{x}=10^{-1},1,10$ indicate the short-distance limit, the transition region, and the region where the long-distance approximation starts to apply, respectively. Bottom panel: Mapping between frequencies, sources' distances and observation times. The correspondence is made in a way that given a frequency, the corresponding distance returns $\tilde{x}=1$.
• Figure 2: Top panels: form factors for the E- and B-mode as function of $\tilde{x}=2\pi f r$ for different values of $\ell$. Bottom panels: behavior of the $\ell$-dependent terms in equation \ref{['Eq:ClBB_ClEE']}. Contrary to the top panel, the curves show that the first multipole gives the highest contribution for any value of $\tilde{x}$.
• Figure 3: Absolute value of the form factor for the E- (left panel) and B-mode (right panel) for different values of the frequency and distance. Given a frequency one can easily understand if for a given source the long distance approximation holds and the corresponding value of the form factor.
• Figure 4: Radial number density probability distribution for stars in the GAIA sample (left panel) and Main Belt asteroids in the NASA Small-Body Database (right panel). Markers indicate the value measured from the catalogs, and dashed lines represents the values of our fitting formula.
• Figure 5: Top panels: angular power spectra for the asteroid distance bins considered in this work. In each plot we compare the full approach that consider the finite distance effects (solid lines) with the incorrect infinite-distance approximation (dashed lines). Bottom panels: ratio of the incorrect-to-correct angular power spectra. The incorrect assumption of the infinite distance limit leads an overestimation of the signal by 3 to 5 orders of magnitude for the first multipoles.