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A tremendous international effort is currently dedicated to observing theso-called $B$-modes of the Cosmic Microwave Background (CMB) polarisation. Atthe unprecedented sensitivity level that the new generation of CMB experimentsaims to reach, every uncontrolled instrumental systematic effect willpotentially result in an analysis bias that is larger than the muchsought-after CMB $B$-mode signal. The absolute calibration of the polarisationangle is particularly important in this sense, as any associated error will endup in a leakage from the much larger $E$ modes into $B$ modes. The Crab nebula(Tau A), with its bright microwave synchrotron emission, is one of the fewobjects in the sky that can be used as absolute polarisation calibrators. Inthis paper we review the best current constraints on its polarisation anglefrom 23 to 353 GHz, at typical angular scales for CMB observations, from WMAP,XPOL, Planck and NIKA data. These polarisation angle measurements arecompatible with a constant angle of $-88.19\,^\circ\pm0.33\,^\circ$. We studythe uncertainty on this mean angle, making different considerations on how tocombine the individual measurement errors. For each of the cases, we study thepotential impact on the CMB $B$-mode spectrum and on the recovered $r$parameter, through a likelihood analysis. We find that current constraints onthe Crab polarisation angle, assuming it is constant through microwavefrequencies, allow to calibrate experiments with an accuracy enabling themeasurement of $r\sim0.01$. On the other hand, even under the most optimisticassumptions, current constraints will lead to an important limitation for thedetection of $r\sim10^{-3}$. New realistic measurement of the Crab nebula canchange this situation, by strengthening the assumption of the consistencyacross microwave frequencies and reducing the combined error.

Absolute calibration of the polarisation angle for future CMB $B$-mode experiments from current and future measurements of the Crab nebula