Measuring patchy reionisation with kSZ$^2$-21 cm correlations
  We study cross-correlations of the kinetic Sunyaev-Zel'dovich effect (kSZ)
and 21 cm signals during the epoch of reionisation (EoR) to measure the effects
of patchy reionisation. Since the kSZ effect is proportional to the
line-of-sight velocity, the kSZ-21 cm cross correlation suffers from
cancellation at small angular scales. We thus focus on the correlation between
the kSZ-squared field (kSZ$^2$) and 21 cm signals. When the global ionisation
fraction is low ($x_e\lesssim 0.7$), the kSZ$^2$ fluctuation is dominated by
rare ionised bubbles which leads to an anti-correlation with the 21 cm signal.
When $0.8\lesssim x_e<1$, the correlation is dominated by small pockets of
neutral regions, leading to a positive correlation. However, at very high
redshifts when $x_e<0.15$, the spin temperature fluctuations change the sign of
the correlation from negative to positive, as weakly ionised regions can have
strong 21 cm signals in this case. To extract this correlation, we find that
Wiener filtering is effective in removing large signals from the primary CMB
anisotropy. The expected signal-to-noise ratios for a $\sim$10-hour integration
of upcoming Square Kilometer Array data cross-correlated with maps from the
current generation of CMB observatories with 3.4~$\mu$K arcmin noise and
1.7~arcmin beam over 100~deg$^2$ are 51, 60, and 37 for $x_e=0.2$, 0.5, and
0.9, respectively.
