Absolute calibration of the polarisation angle for future CMB $B$-mode
  experiments from current and future measurements of the Crab nebula
  A tremendous international effort is currently dedicated to observing the
so-called $B$-modes of the Cosmic Microwave Background (CMB) polarisation. At
the unprecedented sensitivity level that the new generation of CMB experiments
aims to reach, every uncontrolled instrumental systematic effect will
potentially result in an analysis bias that is larger than the much
sought-after CMB $B$-mode signal. The absolute calibration of the polarisation
angle is particularly important in this sense, as any associated error will end
up 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 few
objects in the sky that can be used as absolute polarisation calibrators. In
this paper we review the best current constraints on its polarisation angle
from 23 to 353 GHz, at typical angular scales for CMB observations, from WMAP,
XPOL, Planck and NIKA data. These polarisation angle measurements are
compatible with a constant angle of $-88.19\,^\circ\pm0.33\,^\circ$. We study
the uncertainty on this mean angle, making different considerations on how to
combine the individual measurement errors. For each of the cases, we study the
potential impact on the CMB $B$-mode spectrum and on the recovered $r$
parameter, through a likelihood analysis. We find that current constraints on
the Crab polarisation angle, assuming it is constant through microwave
frequencies, allow to calibrate experiments with an accuracy enabling the
measurement of $r\sim0.01$. On the other hand, even under the most optimistic
assumptions, current constraints will lead to an important limitation for the
detection of $r\sim10^{-3}$. New realistic measurement of the Crab nebula can
change this situation, by strengthening the assumption of the consistency
across microwave frequencies and reducing the combined error.
