The free-electron population during the reionized epoch rescatters CMB
temperature quadrupole and generates a now well-known polarization signal at
large angular scales. While this contribution has been detected in the
temperature-polarization cross power spectrum measured with WMAP data, due to
the large cosmic variance associated with anisotropy measurements at tens of
degree angular scales only limited information related to reionization, such as
the optical depth to electron scattering, can be extracted. The inhomogeneities
in the free-electron population lead to an additional secondary polarization
anisotropy contribution at arcminute scales. While the fluctuation amplitude,
relative to dominant primordial fluctuations, is small, we suggest that a
cross-correlation between arcminute scale CMB polarization data and a tracer
field of the high redshift universe, such as through fluctuations captured by
the 21 cm neutral Hydrogen background or those in the infrared background
related to first proto-galaxies, may allow one to study additional details
related to reionization. For this purpose, we discuss an optimized higher order
correlation measurement, in the form of a three-point function, including
information from large angular scale CMB temperature anisotropies in addition
to arcminute scale polarization signal related to inhomogeneous reionization.
We suggest that the proposed bispectrum can be measured with a substantial
signal-to-noise ratio and does not require all-sky maps of CMB polarization or
that of the tracer field. A measurement such as the one proposed may allow one
to establish the epoch when CMB polarization related to reionization is
generated and to address if the universe was reionized once or twice.Comment: 13 pages, 7 figures; Version in press with Phys. Rev.