## Author(s)

Çalışkan, Mesut, Chen, Yifan, Dai, Liang, Anil Kumar, Neha, Stomberg, Isak, Xue, Xiao## Abstract

Astrometry, the precise measurement of star motions, offers an alternative avenue to investigate low-frequency gravitational waves through the spatial deflection of photons, complementing pulsar timing arrays reliant on timing residuals. Upcoming data from Gaia, Theia, and Roman can not only cross-check pulsar timing array findings but also explore the uncharted frequency range bridging pulsar timing arrays and LISA. We present an analytical framework to evaluate the feasibility of detecting a gravitational wave background, considering measurement noise and the intrinsic variability of the stochastic background. Furthermore, we highlight astrometry's crucial role in uncovering key properties of the gravitational wave background, such as spectral index and chirality, employing information-matrix analysis. Finally, we simulate the emergence of quadrupolar correlations, commonly referred to as the generalized Hellings-Downs curves.

## Figures

SNR$^{XX'}_k$ plotted against $\xi_k \equiv {I_k}/{(4\pi S^{(n)}_{X,k}/N_{X})}$ for $I_k$ estimators, encompassing PTA-only correlation $zz$ (blue), astrometry-only correlation $EE$ or $BB$ (orange), and PTA-astrometry cross-correlation $zE$ (green). Two options for the highest $\ell$-mode constructed, $\ell_{\rm max} = 10$ (solid lines) and $50$ (dashed lines), are considered. In the case of $zE$ correlation, we assume $S^{(n)}_{z,k}/N_{z} = S^{(n)}_{E,k}/N_{E}$.

The SNR distribution as a function of the power-law SGWB model parameters, the reference strain $\log_{10}A$ and spectral index $\alpha$ as defined in Eq.\,(\ref{Eq:hc_def}), for the four considered observations. The left two correspond to PTAs, while the right two represent astrometric observations. The white regions indicate $\text{SNR}<1$, where the information matrix is not applicable. Yellow stars mark the fiducial parameters from Eq.\,(\ref{Eq:Phi_to_PSD1}), with SNR values of $4.0$, $1.5$, $34.1$, and $43.9$ for NANOGrav, Gaia, SKA, and Theia, respectively. The current exclusion region, defined as $\text{SNR}>10$ for NANOGrav, is highlighted by yellow dashed lines.

The resolution of the two power-law SGWB model parameters, $\sigma(\log_{10}A)$ and $\sigma(\alpha)$, as a function of $\log_{10}A$ and $\alpha$, for the four considered observations. The yellow stars correspond to the fiducial parameters observed by NANOGrav, and the yellow dashed lines are excluded in its $\text{SNR}>10$ region. The resolutions at the fiducial value, from top to bottom, are $\sigma(\log_{10}A) = 0.25$, $1.3$, $0.012$, and $0.0056$, and $\sigma(\alpha) = 0.24$, $1.3$, $0.017$, and $0.0080$.

Posteriors of the reference strain amplitude $\log_{10}A$ and the fraction of circular polarization $v\equiv V_k/I_k$, for astrometric observations (left) and PTA-astrometry synergistic analyses (right). The red stars correspond to the true parameters, including the fiducial value for $\log_{10}A$ and the assumed $v=0.1$. The marginalized distribution of each parameter is presented next to the posterior distribution. The dark and light blue regions represent the $1\sigma$ and $2\sigma$ regions, respectively. The $1\sigma$ uncertainties for $v$ are $\sigma(v) = 0.57$, $0.39$, $0.022$, and $0.022$ for Gaia, NANOGrav$+$Gaia, Theia, and SKA$+$Theia, respectively.

Examples of realizations on the celestial sphere for $V/I = -1$ (top), $0$ (middle), and $1$ (bottom), featuring a total of $432$ patches. Each patch contains information on the real part of the redshift $\delta z$ represented on a circle, spanning from red ($>0$) to blue ($<0$). The real (black arrow) and imaginary (white arrow) components, with lengths proportional to their magnitudes, are also displayed. The measurement noise is not included in these examples. In cases with $V/I = \pm 1$, the real and imaginary arrows are always perpendicular, while in the unpolarized case, they are uncorrelated.

Reconstructions of rotationally invariant power spectra in the spherical harmonic space for $\ell$ ranging from $2$ to $6$. Each reconstruction is derived from a random realization of a map of $108$ patches of $\{\delta z_a\}$ and $\{\delta\hbv{x}_a\}$ using Eqs.\,(\ref{eq:zEBr}) and (\ref{Eq:def_cxx}), in the absence of measurement noise. Total intensity $I$ estimators are depicted in orange, while circular polarization $V$ estimators are shown in blue, assuming $V/I=0.3$. Each spectrum is normalized by its theoretical average value. Violins represent the variance of reconstructions from $10^4$ realizations, and the red dots denote the average values of realizations. Both are consistent with the variance calculation in the denominator of Eq.\,(\ref{eq:snr}).

Distribution of sky-averaged two-point functions from $100$ simulations presented in gray violins for various PTA and astrometric observation channels, displayed for the first frequency bin (left) and weight-averaged across all frequency bins (right). The red lines illustrate the generalized Hellings-Downs curves defined in Eqs.\,(\ref{eq:HD}) and (\ref{Eq:correlations}). The variance from the simulations encompasses both SGWB variance and measurement noises. The central values of the violins, shown in blue, align with the red lines.

Sky-averaged two-point functions from a single simulation for various PTA and astrometric observation channels. Each gray line represents a realization in a specific frequency bin, with opacity inversely proportional to the variance associated with that frequency bin. The black dashed lines depict the weighted averages of all frequency bins, aligning with the generalized Hellings-Downs curves (red) defined in Eqs.\,(\ref{eq:HD}) and (\ref{Eq:correlations}) for observations involving SKA or Theia.

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