Searching for Gravitational Waves with Gaia and its Cross-Correlation with PTA: Absolute vs Relative Astrometry

TL;DR

The paper evaluates using Gaia-like astrometry to detect a stochastic gravitational-wave background by comparing absolute astrometric deflections with relative, pairwise angular changes, and by cross-correlating with pulsar timing arrays. It develops corrected theoretical expressions for the relative astrometric response and derives corresponding overlap reduction functions, then carries out Fisher-matrix forecasts to compare sensitivities and dataset combinations. The key finding is that relative astrometry offers no practical advantage for Gaia due to small-angle suppression and Gaia's scanning law, though wide-angle relative measurements can approach absolute astrometric performance; cross-correlating Gaia with PTAs yields modest improvements at frequencies above ~10^-7 Hz. The work provides a robust framework for Gaia-PTA GW searches and clarifies the viability of relative astrometry, with implications for future astrometric surveys and cross-field GW analyses.

Abstract

Astrometric missions like Gaia provide exceptionally precise measurements of stellar positions and proper motions. Gravitational waves traveling between the observer and distant stars can induce small, correlated shifts in these apparent positions, a phenomenon known as astrometric deflection. The precision and scale of astrometric datasets make them well-suited for searching for a stochastic gravitational wave background, whose signature appears in the two-point correlation function of the deflection field across the sky. Although Gaia achieves high accuracy in measuring angular separations in its focal plane, systematic uncertainties in the satellite's absolute orientation limit the precision of absolute position measurements. These orientation errors can be mitigated by focusing on relative angles between star pairs, which effectively cancel out common-mode orientation noise. In this work, we compute the astrometric response and the overlap reduction functions for this relative astrometry approach, correcting previous expressions presented in the literature. We use a Fisher matrix analysis to compare the sensitivity of relative astrometry to that of conventional absolute astrometry. Our analysis shows that while the relative method is theoretically sound, its sensitivity is limited for closely spaced star pairs within a single Gaia field of view. Pairs with large angular separations could provide competitive sensitivity, but are practically inaccessible due to Gaia's scanning law. Finally, we demonstrate that combining astrometric data with observations from pulsar timing arrays leads to slight improvements in sensitivity at frequencies greater than approximately 10^-7 Hz.

Figures (9)

• Figure 1: Comparison of the astrometric ORF (blue, dashed), given by the single expression $\mathcal{T}(\Theta)$, the cross-correlation ORF (purple, dot-dashed) $\mathcal{P}(\Theta)$ (also defined in Ref. Mihaylov:2018uqm), with the Hellings–Downs curve (black, solid), which characterizes redshift correlations. Curves are shown as functions of the angular separation on the sky, $\Theta$. The normalization has been chosen so that they each have a maximum value of one (neglecting the pulsar/star term).
• Figure 2: Geometrical configuration of the direction vectors to two pairs of stars, $(\mathbf{n}_{a1},\mathbf{n}_{a2})$ and $(\mathbf{n}_{b1},\mathbf{n}_{b2})$, in the sky parametrized by five angles. We used the freedom in performing a rigid rotation to align the bisector of the directions $(\mathbf{n}_{a_1},\mathbf{n}_{a_2})$ to the stars of the pair $a$, along the $z$ axis, while the bisector of the directions $(\mathbf{n}_{b_1}, \mathbf{n}_{b_2})$ to the stars of the pair $b$, lies in the $x-z$ plane.
• Figure 3: The plot shows the ORF $\mathcal{H}_{DA}(\Theta, \phi_a, \phi_b)$ as a function of $\Theta$ for $\phi_b = 0$ (solid lines) and $\phi_b = \pi/2$ (dashed lines), and for five values of $\phi_a$ in $(0, \pi/2)$, indicated by different colors as specified by the colorbars above. The black solid curve represents the Hellings–Downs correlation, which is also recovered by averaging $\mathcal{H}_{DA}(\Theta, \phi_a, \phi_b)$ over $\phi_a$ and $\phi_b$.
• Figure 4: The plot shows the cross-correlation ORF $\mathcal{H}_\text{cross}(\Theta, \phi_a)$ between GW-induced redshift and astrometric deflection as a function of the pulsar-pair angular separation $\Theta$. Results are shown for five different values of $\phi_a$ in the range $(0, \pi/2)$, distinguished by color as indicated by the colorbars above. The black solid curve corresponds to the standard Hellings–Downs correlation, which is recovered by averaging $\mathcal{H}_\text{cross}(\Theta, \phi_a)$ over the angle $\phi_a$.
• Figure 5: Signal to Noise Ratio as a function of the total time of observation for a PTA and an astrometry (Gaia-like) experiment. The curves are drawn following Eq. \ref{['eq:SNRs_quick_def']} where the noise values for PTA and the astrometric surveys are specified in Tab. \ref{['tab:noise_values']}, while the signal power spectrum is the one in of Eq. \ref{['eq:signal_PSD']}, with $A_{\rm GW}=10^{-14}$ and $\alpha=0$.