The impact of modelling errors on interferometer calibration for 21 cm power spectra

We study the impact of sky-based calibration errors from source mismodeling on 21\,cm power spectrum measurements with an interferometer and propose a method for suppressing their effects. While emission from faint sources that are not accounted for in calibration catalogs is believed to be spectrally smooth, deviations of true visibilities from model visibilities are not, due to the inherent chromaticity of the interferometer's sky-response (the "wedge"). Thus, unmodeled foregrounds, below the confusion limit of many instruments, introduce frequency structure into gain solutions on the same line-of-sight scales on which we hope to observe the cosmological signal. We derive analytic expressions describing these errors using linearized approximations of the calibration equations and estimate the impact of this bias on measurements of the 21\,cm power spectrum during the Epoch of Reionization (EoR). Given our current precision in primary beam and foreground modeling, this noise will significantly impact the sensitivity of existing experiments that rely on sky-based calibration. Our formalism describes the scaling of calibration with array and sky-model parameters and can be used to guide future instrument design and calibration strategy. We find that sky-based calibration that down-weights long baselines can eliminate contamination in most of the region outside of the wedge with only a modest increase in instrumental noise.