A Gaussian Covariance Matrix for Joint Pre- and Post-Reconstruction Full-Shape Power Spectrum Analysis

We apply the Gaussian covariance formalism to develop a semi-analytical covariance model for the joint analysis of pre-reconstruction, post-reconstruction, and cross full-shape galaxy power spectra. We model the reconstruction-reduced, scale-dependent cross shot noise using displacement-field statistics and introduce a new estimator that directly measures this term. Using the measured power spectra and the modeled shot-noise predictions as inputs, we construct the Gaussian covariance while accounting for correlations between the pre- and post-reconstruction density fields. We validate the resulting semi-analytical Gaussian covariance against mock catalogues. Using emulator-based parameter inference, we demonstrate that the semi-analytical Gaussian covariance adequately captures the dominant contribution to the covariance structure of the full data vector ($P_{\ell}^{\rm pre}, P_{\ell}^{\rm post}, P_{\ell}^{\rm cross}$). For the joint fit to these three power spectra, it yields cosmological constraints consistent with those obtained using the mock-based numerical covariance over the adopted fitting ranges: $k_{\rm max}=0.18\,h\,{\rm Mpc}^{-1}$ for $P_{\rm pre}$ and $P_{\rm post}$, and $k_{\rm max}=0.12\,h\,{\rm Mpc}^{-1}$ for $P_{\rm cross}$.