  The cosmic microwave background (CMB) polarisation and the 21 cm line
fluctuations are powerful probes of cosmological reionisation. We study how the
cross-correlation between the CMB polarisation (E-modes) and the 21 cm line
fluctuations can be used to gain further understanding of the reionisation
history, within the framework of inhomogeneous reionisation. Since the E-mode
polarisation reflects the amplitude of the quadrupole component of the CMB
temperature fluctuations, the angular power spectrum of the cross-correlation
exhibits oscillations at all multipoles. The first peak of the power spectrum
appears at the scale corresponding to the quadrupole at the redshift that is
probed by the 21 cm line fluctuations. The peak reaches its maximum value in
redshift when the average ionisation fraction of the universe is about half. On
the other hand, on small scales, there is a damping that depends on the
duration of reionisation. Thus, the cross-correlation between the CMB
polarisation and the 21 cm line fluctuations has the potential to constrain
accurately the epoch and the duration of reionisation.
