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.
