Beyond the horizon: Quantifying the full sky foreground wedge in the cylindrical power spectrum

One of the main obstacles preventing the detection of the redshifted 21 cm signal from neutral hydrogen in the early Universe is the astrophysical foreground emission, which is several orders of magnitude brighter than the signal. The foregrounds, due to their smooth spectra, are expected to predominantly occupy a region in the cylindrical power spectrum known as the foreground wedge. However, the conventional equations describing the extent of the foreground wedge are derived under a flat-sky approximation. This assumption breaks down for tracking wide-field instruments, thus rendering these equations inapplicable in these situations. In this paper we derive equations for the full sky foreground wedge, and show that the foregrounds can potentially extend far beyond what the conventional equations suggest. We also derive the equations that describe a specific bright source in the cylindrical power spectrum space. The validity of both sets of equations is tested against numerical simulations. Many current and upcoming interferometers (e.g., LOFAR, NenuFAR, MWA, SKA) are wide-field phase-tracking instruments. These equations give us new insights into the nature of foreground contamination in the cylindrical power spectra estimated using wide-field instruments. Additionally, they allow us to accurately associate features in the power spectrum with foregrounds or instrumental effects. The equations are also important for correctly selecting the epoch of reionization (EoR) window for foreground avoidance analyses, and for planning 21 cm observations. In future analyses, we recommend using these updated horizon lines to indicate the foreground wedge in the cylindrical power spectrum accurately. The new equations for generating the updated wedge lines are made available in the Python library, pslines.