Spectral Redundancy for Calibrating Interferometers and Suppressing the Foreground Wedge in 21\,cm Cosmology

Observations of 21\,cm line from neutral hydrogen promise to be an exciting new probe of astrophysics and cosmology during the Cosmic Dawn and through the Epoch of Reionization (EoR) to when dark energy accelerates the expansion of the Universe. At each of these epochs, separating bright foregrounds from the cosmological signal is a primary challenge that requires exquisite calibration. In this paper, we present a new calibration method called \textsc{nucal} that extends redundant-baseline calibration, allowing spectral variation in antenna responses to be solved for by using correlations between visibilities measuring the same angular Fourier modes at different frequencies. By modeling the chromaticity of the beam-weighted sky with a tunable set of discrete prolate spheroidal sequences (DPSS), we develop a calibration loop that optimizes for spectrally smooth calibrated visibilities. Crucially, this technique does not require explicit models of the sky or the primary beam. With simulations that incorporate realistic source and beam chromaticity, we show that this method solves for unsmooth bandpass features, exposes narrowband interference systematics, and suppresses smooth-spectrum foregrounds below the level of 21\,cm reionization models, even within much of the so-called "wedge" region where current foreground mitigation techniques struggle. We show that this foreground subtraction can be performed with minimal cosmological signal loss for certain well-sampled angular Fourier modes, making spectral-redundant calibration a promising technique for current and next-generation 21\,cm intensity mapping experiments.