Detectability of the 21 cm-CMB cross-correlation from the EoR

The 21-cm line fluctuations and the cosmic microwave background (CMB) are powerful probes of the epoch of reionisation of the universe. We study the potential of the cross-correlation between 21-cm line fluctuations and CMB anisotropy to obtain further constraints on the reionisation history. We compute analytically the 21-cm cross-correlation with the CMB temperature anisotropy and polarisation, and we calculate the signal-to-noise (SN) ratio for its detection with Planck together with LOFAR, MWA and SKA. We find, on the one hand, that the 21-cm cross-correlation signal with CMB polarisation from the instant reionisation can be detected with an SN ratio of $\sim 1$ for LOFAR and $\sim 10$ for SKA. On the other hand, we confirm that the detection of the 21-cm cross-correlation with CMB polarisation is practically infeasible.