Reionization on Large Scales II: Detecting Patchy Reionization through
  Cross Correlation of the Cosmic Microwave Background
  We investigate the effect of patchy reionization on the cosmic microwave
background temperature. An anisotropic optical depth tau (theta) alters the TT
power spectrum on small scales l > 2000. We make use of the correlation between
the matter density and the reionization redshift fields to construct full sky
maps of tau(theta). Patchy reionization transfers CMB power from large scales
to small scales, resulting in a non-zero cross correlation between large and
small angular scales. We show that the patchy tau correlator is sensitive to
small root mean square values tau_rms ~ 0.003 seen in our maps. We include
other secondary anisotropies such as CMB lensing, kinetic and thermal
Sunyaev-Zel'dovich terms, as well as the infrared and point source background,
and show that patchy reionization may be detected in the low frequency channels
~ 90 GHz, particularly for extended reionization histories. If frequency
dependent secondaries can be minimized by a multi-frequency analysis, we show
that even small degrees of patchiness can be detected at high significance. In
this case, we show that models with different values of tau_rms are
distinguishable, even for the same mean value <tau>. The patchy tau correlator
thus provides information that is complementary to what may be obtained from
the polarization and the kinetic Sunyaev-Zel'dovich power spectra.
