BEYONDPLANCK IX. Bandpass and beam leakage corrections

We discuss the treatment of bandpass and beam leakage corrections in the Bayesian BEYONDPLANCK cosmic microwave background (CMB) analysis pipeline as applied to the Planck LFI measurements. As a preparatory step, we first applied three corrections to the nominal LFI bandpass profiles, including the removal of a known systematic effect in the ground measuring equipment at 61 GHz, along with a smoothing of standing wave ripples and edge regularization. The main net impact of these modifications is an overall shift in the 70 GHz bandpass of +0.6 GHz. We argue that any analysis of LFI data products, either from Planck or BEYONDPLANCK, should use these new bandpasses. In addition, we fit a single free bandpass parameter for each radiometer of the form & UDelta;(i)?=?& UDelta;(0) + & delta;(i), where & UDelta;(0) represents an absolute frequency shift per frequency band and & delta;(i) is a relative shift per detector. The absolute correction is only fitted at 30 GHz, with a full & chi;(2)-based likelihood, resulting in a correction of & UDelta;(30)?=?0.24 & PLUSMN; 0.03 ?GHz. The relative corrections were fitted using a spurious map approach that is fundamentally similar to the method pioneered by the WMAP team, but excluding the introduction of many additional degrees of freedom. All the bandpass parameters were sampled using a standard Metropolis sampler within the main BEYONDPLANCK Gibbs chain and the bandpass uncertainties were thus propagated to all other data products in the analysis. In summary, we find that our bandpass model significantly reduces leakage effects. For beam leakage corrections, we adopted the official Planck LFI beam estimates without any additional degrees of freedom and we only marginalized over the underlying sky model. We note that this is the first time that leakage from beam mismatch has been included for Planck LFI maps.