Constraints on Patchy Reionization from Planck CMB Temperature
  Trispectrum
  We present constraints on the patchy reionization by measuring the
trispectrum of the Planck 2015 cosmic microwave background (CMB) temperature
anisotropies. The patchy reionization leads to anisotropies in the CMB optical
depth, and the statistics of the observed CMB anisotropies is altered. We
estimate the trispectrum of the CMB temperature anisotropies to constrain
spatial variation of the optical depth. We show that the measured trispectrum
is consistent with that from the standard lensed CMB simulation at $2\sigma$.
While no evidence of the patchy reionization is found in the Planck 2015
temperature trispectrum, the CMB constraint on the patchy reionization is
significantly improved from previous works. Assuming the analytic bubble-halo
model of Wang and Hu (2006), the constraint obtained in this work rules out the
typical bubble size at the ionization fraction of $\sim0.5$ as $R\gtrsim 10$
Mpc. Further, our constraint implies that large-scale $B$-modes from the patchy
reionization are not a significant contamination in detecting the primordial
gravitational waves of $r\gtrsim0.001$ if the $B$ mode induced by the patchy
reionization is described by Dvorkin et al. (2009). The CMB trispectrum data
starts to provide meaningful constraints on the patchy reionization.
