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Maximum Likelihood Foreground Cleaning for Cosmic Microwave Background Polarimeters in the Presence of Systematic Effects

We extend a general maximum likelihood foreground estimation for cosmic microwave background polarization data to include estimation of instrumental systematic effects. We focus on two particular effects: frequency band measurement uncertainty, and instrumentally induced frequency dependent polarization rotation. We assess the bias induced on the estimation of the $B$-mode polarization signal by these two systematic effects in the presence of instrumental noise and uncertainties in the polarization and spectral index of Galactic dust. Degeneracies between uncertainties in the band and polarization angle calibration measurements and in the dust spectral index and polarization increase the uncertainty in the extracted CMB $B$-mode power, and may give rise to a biased estimate. We provide a quantitative assessment of the potential bias and increased uncertainty in an example experimental configuration. For example, we find that with 10\% polarized dust, tensor to scalar ratio of $r=0.05$, and the instrumental configuration of the EBEX balloon payload, the estimated CMB $B$-mode power spectrum is recovered without bias when the frequency band measurement has 5% uncertainty or less, and the polarization angle calibration has an uncertainty of up to 4$^{\circ}$.