Analytic marginalization over CMB calibration and beam uncertainty

With the increased accuracy and angular scale coverage of the recent CMB experiments it has become important to include calibration and beam uncertainties when estimating cosmological parameters. This requires an integration over possible values of the calibration and beam size, which can be performed numerically but greatly increases computation times. We present a fast and general method for marginalization over calibration-type errors by analytical integration. This is worked through for the specific example of CMB calibration and beam uncertainties and the resulting formulae are practical to implement. We show how cosmological parameter constraints from the latest CMB data are changed when calibration/beam uncertainties are taken into account: typically the best fit parameters are shifted and the errors bars are increased by up to fifty per cent for e.g. ns and Ωbh; although as expected there is no change for ΩK, because it is constrained by the positions of the peaks.