A unified spin-harmonic framework for correlating pulsar timing, astrometric deflection, and shimmering gravitational wave observations

Abstract

We present a unified spin-weighted harmonic framework that delivers analytic, diagonal expressions for the overlap (correlation) functions of three low frequency gravitational wave observables-pulsar timing redshifts, astrometric deflections, and time-dependent image distortions (``shimmering''). Writing each response in spin-$s$ spherical harmonics and rotating to a basis in which the wave tensor has definite helicity, we obtain compact closed-form series for every auto- and cross-correlation, recovering the Hellings-Downs curve as the $s=0$ limit and deriving its astrometric ($s=\pm 1$) and shimmering ($s=\pm 2$) analogues. The formalism naturally extends to non-standard scalar-breathing, longitudinal, and vector polarisation modes, clarifying when higher-spin observables are (and are not) sourced and providing a complete set of harmonic spectra $C_\ell$ ready for parameter estimation pipelines. These results supply the common theoretical language needed to combine upcoming pulsar timing, Gaia-class astrometric, and high resolution imaging data sets, enabling coherent, multi probe searches for stochastic gravitational wave backgrounds, tests of general relativity and its alternatives across the nano- to micro-hertz gravitational wave band.

Figures (3)

• Figure 1: Angular correlation functions for an Einsteinian‑unpolarised GW background, defined in Eq. \ref{['eq:Gamma_unpol_def']}, in the aligned frame as a function of the separation angle $\beta=\arccos(\hat{n}_1\!\cdot\!\hat{n}_2)$. We show the unique functions only. A lookup table of equivalent functions is shown in Table \ref{['tab:Einsteinian-symmetries']}
• Figure 2: Angular correlation functions for an Einsteinian-unpolarized GW background analogous to the one defined in \ref{['eq:Gamma_unpol_def']} for a scalar/breathing-polarised GW background (left) and a longitudinal-polarised one (right). The correlation functions are evaluated in the reference frame where $\hat{n}_1=\hat{z}$ as a function of $\beta=\arccos(\hat{n}_1\cdot\hat{n}_2)$. As explained in the text, these are the only observables whose correlators can be computed consistently.
• Figure 3: Angular correlation functions for a vectorial‑unpolarised GW background, defined in Eq. \ref{['eq:Gamma_unpol_def']}, in the aligned frame. We show only unique functions and list the equivalent ones in Table \ref{['tab:Vectorial-symmetries']}