Astrometry meets Pulsar Timing Arrays: Synergies for Gravitational Wave Detection

TL;DR

This work develops a covariant formalism for how astrometric measurements respond to a stochastic gravitational-wave background (SGWB) and how these data can coherently combine with Pulsar Timing Array (PTA) observations to tighten SGWB constraints. By deriving monopole and dipole overlap reduction functions (ORFs) for astrometry and their cross-correlations with PTAs, the authors set up a Fisher-m forecasting framework to estimate SGWB amplitude, spectral tilt, and dipolar anisotropy, including the impact of star positions and observer velocity. The results show that cross-correlating astrometric deflections with PTA timing residuals can modestly improve SGWB constraints, especially when astrometric precision is high and the star sample is large; a kinematic dipole remains challenging to detect at current capabilities. The work highlights the potential of joint PTA-astrometry analyses to help identify the SGWB origin (astrophysical SMBHBs vs cosmological sources) and guides future mission planning (e.g., Theia, SKA) to reach cosmologically motivated anisotropy levels.

Abstract

High-precision astrometry offers a promising approach to detect low-frequency gravitational waves, complementing pulsar timing array (PTA) observations. We explore the response of astrometric measurements to a stochastic gravitational wave background (SGWB) in synergy with PTA data. Analytical, covariant expressions for this response are derived, accounting for the presence of a possible dipolar anisotropy in the SGWB. We identify the optimal estimator for extracting SGWB information from astrometric observations and examine how sensitivity to SGWB properties varies with the sky positions of stars and pulsars. Using representative examples of current PTA capabilities and near-future astrometric sensitivity, we demonstrate that cross-correlating astrometric and PTA data can improve constraints on SGWB properties, compared to PTA data alone. The improvement is quantified through Fisher forecasts for the SGWB amplitude, spectral tilt, and dipolar anisotropy amplitude. In the future, such joint constraints could play a crucial role in identifying the origin of SGWB signals detected by PTAs.

Figures (9)

• Figure 1: The quantities ${\rm Tr}[{\boldsymbol{H}}_{0}{\boldsymbol{H}}_{0}]$ and ${\rm Tr}[{\boldsymbol{H}}_{1}{\boldsymbol{H}}_{1}]$, associated with response to a SGWB depending on stars positions. (See the main text for our notation.) Motivated by CMB, we choose the kinematic dipole direction ${\bf v}$ (red star) at $(l,b)=(264^{\circ},48^{\circ})$ in galactic coordinates. Each panel shows a different choice of ${\bf n}$, while the stars ${\bf q}$ take the position of each pixel of the map. Upper left panel: ${\bf n}$ pointing towards $(l,b)=(0,0)$. Upper right panel: ${\bf n}$ pointing towards $-{\bf v}$. Lower left panel: ${\bf n}$ at $(l,b)=(0,0)$. Lower right panel: ${\bf n}$ pointing towards the direction $(l,b)=(270.21^{\circ},-75.45^{\circ})$.
• Figure 2: Overlap functions response in terms of the angle between the stars. Left: Angular dependence for ${\rm Tr}[{\boldsymbol{H}}_{0}{\boldsymbol{H}}_{0}]$. Right:${\rm Tr}[{\boldsymbol{H}}_{1}{\boldsymbol{H}}_{1}]$ when choosing ${\bf n} = - {\bf v}$.
• Figure 3: The quantities $\mathbf{K}_{0} \mathbf{K}_{0}^{T}$, and $\mathbf{K}_{1}\mathbf{K}_{1}^{T}$ response to stars and pulsars positions. The dipole direction ${\bf v}$ (red star) is chosen in the direction $(l,b)=(264^{\circ},48^{\circ})$ in galactic coordinates. Each panel shows a different choice of ${\bf n}$, while the pulsars positions ${\bf{x}}$ scan over each pixel of the map. Upper left panel: ${\bf n}$ towards $(l,b)=(0,0)$. Upper right panel: ${\bf n}$ towards $-{\bf v}$. Lower left panel: ${\bf n}$ at $(l,b)=(0,0)$. Lower right panel: ${\bf n}$ towards $(l,b)=(270.21^{\circ},-75.45^{\circ})$.
• Figure 4: Overlap functions response in terms of the angle between the stars and pulsars. Left: Angular dependence for $\mathbf{K}_{0} \mathbf{K}_{0}^{T}$. Right: The quantities $\mathbf{K}_{1} \mathbf{K}_{1}^{T}$ when choosing ${\bf n} = - {\bf v}$.
• Figure 5: The components of the Fisher matrix \ref{['eq:astro_fisher']} are evaluated numerically as a function of the number of stars with the stars uniformly distributed across the sky. The dashed lines have slope $N_{\rm star}^2$ and pass through the numerically evaluated result for $N_{\rm star}=500$. This plot numerically confirms eqs \ref{['eq:traceH']} and \ref{['eq:traceHa']}.