Second order cross-correlation between kinetic Sunyaev-Zel'dovich effect and 21-cm fluctuations from the epoch of reionization

The measurement of the brightness temperature fluctuations of neutral hydrogen 21-cm lines from the epoch of reionization (EoR) is expected to be a powerful tool for revealing the reionization process. We study the 21-cm cross-correlation with cosmic microwave background (CMB) temperature anisotropies, focusing on the effect of the patchy reionization. We calculate, up to second order, the angular power spectrum of the cross-correlation between 21-cm fluctuations and the CMB kinetic Sunyaev-Zel'dovich effect (kSZ) from the EoR, using an analytical reionization model. We show that the kSZ and the 21-cm fluctuations are anti-correlated on the scale corresponding to the typical size of an ionized bubble at the observed redshift of the 21-cm fluctuations. The amplitude of the angular power spectrum of the cross-correlation depends on the fluctuations of the ionized fraction. Especially, in a highly inhomogeneous reionization model, the amplitude reaches the order of 100 mu K(2) at l similar to 3000. We also show that second-order terms may help in distinguishing between reionization histories.