Detectability of the 21-cm CMB cross-correlation from the epoch of reionization

The 21-cm line fluctuations and the cosmic microwave background (CMB) are powerful probes of the epoch of reionization 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 reionization history. We analytically compute the 21-cm cross-correlation with the CMB temperature anisotropy and polarization, and we calculate the signal-to-noise ratio (S/N) for its detection with Planck together with LOw Frequency ARray (LOFAR), Murchison Widefield Array (MWA) and Square Kilometer Array (SKA). We find, on the one hand, that the 21-cm cross-correlation signal with CMB temperature from the instant reionization can be detected with an S/N of similar to 2 for LOFAR and similar to 10 for SKA. On the other hand, we confirm that the detection of the 21-cm cross-correlation with CMB polarization is practically infeasible.