Forecasting graviton-mass constraints from the full covariance of PTA-astrometry ORF estimators

We develop a full-covariance formalism for pulsar timing array(PTA) – astrometry verlap reduction function (ORF) estimators and use it to forecast graviton-mass constraints from a nanohertz stochastic gravitational-wave background (SGWB). Analytic covariance expressions are derived for auto- and cross-channel ORF estimators, including signal-signal, noise-noise, and signal-noise contributions, and are validated against numerical simulations. For an observational configuration with sensitivities comparable to NANOGrav and Gaia, we obtain an expected joint 90% upper limit of $m_g<4.41\times {10}^{-24}\mathrm{e}\mathrm{V}\mathrm{/}{c}^2$, which remains PTA-dominated and lies at the same order of magnitude as the existing NANOGrav 15-year PTA-only bound. For a future-like configuration with sensitivities comparable to the SKA and Theia/Gaia-NIR, the astrometric channels contribute significantly to the constraining power, and the joint limit improves to $m_g<0.48\times {10}^{-24}\mathrm{e}\mathrm{V}\mathrm{/}{c}^2$. These forecasts indicate that PTA – astrometry multichannel inference provides a viable avenue for improving graviton-mass constraints under next-generation observational conditions.