Cross-correlation studies with CMB polarization maps

The free-electron population during the reionized epoch rescatters the cosmic microwave background (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 Wilkinson Microwave Anisotropy Probe data, due to the large cosmic variance associated with anisotropy measurements at angular scales of tens of degrees 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 the 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 the first protogalaxies, 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 the 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 the question whether the universe was reionized once or twice.