Second order cross-correlation between kSZ and 21 cm fluctuations from the EoR

The measurement of the brightness temperature fluctuations of neutral hydrogen 21 cm lines from the Epoch of Reionisation (EoR) is expected to be a powerful tool for revealing the reionisation process. We study the 21 cm cross-correlation with Cosmic Microwave Background (CMB) temperature anisotropies, focusing on the effect of the patchy reionisation. 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 reionisation model. We show that the kSZ and the 21 cm fluctuations are anti-correlated on the scale corresponding to the typical size of an ionised 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 ionised fraction. Especially, in a highly inhomogeneous reionisation model, the amplitude reaches the order of $100 \mu K^2$ at $\ell \sim 3000$. We also show that second order terms may help in distinguishing between reionisation histories.