Anatomy of Cosmic Tidal Reconstruction

21-cm intensity surveys aim to map neutral hydrogen atoms in the universe through hyper-fine emission. Unfortunately, long-wavelength (low-wavenumber) radial modes are highly contaminated by smooth astrophysical foregrounds that are six orders of magnitude brighter than the cosmological signal. This contamination also leaks into higher radial and angular wavenumber modes and forms a foreground wedge. Cosmic tidal reconstruction aims to extract the large-scale signal from anisotropic features in the local small-scale power spectrum through non-linear tidal interactions; losing small-scale modes to foreground wedge will impair its performance. In this paper, we review tidal interaction theory and estimator construction, and derive the theoretical expressions for the reconstructed spectra. We show the reconstruction is robust against peculiar velocities. Removing low line-of-sight $k$ modes, we demonstrate cross-correlation coefficient $r$ is greater than 0.7 on large scales ($k <0.1$ $h/$Mpc) even with a cutoff value $k^c_{\|}=0.1$ $h/$Mpc. Discarding wedge modes yields $0.3< r < 0.5$ and completely removes the dependency on $k^c_{\|}$. Our theoretical predictions agree with these numerical simulations.