CMB E B and T B cross-spectrum estimation via pseudospectrum techniques

We discuss methods for estimating $EB$ and $TB$ spectra of the cosmic microwave background anisotropy maps covering limited sky area. Such odd-parity correlations are expected to vanish whenever parity is not broken. As this is indeed the case in the standard cosmologies, any evidence to the contrary would have a profound impact on our theories of the early Universe. Such correlations could also become a sensitive diagnostic of some particularly insidious instrumental systematics. In this work we introduce three different unbiased estimators based on the so-called standard and pure pseudo-spectrum techniques and later assess their performance by means of extensive Monte Carlo simulations performed for different experimental configurations. We find that a hybrid approach combining a pure estimate of $B$-mode multipoles with a standard one for $E$-mode (or $T$) multipoles, leads to the smallest error bars for both $EB$ (or $TB$ respectively) spectra as well as for the three other polarization-related angular power spectra (i.e., $EE$, $BB$, and $TE$). However, if both $E$ and $B$ multipoles are estimated using the pure technique, the loss of precision for the $EB$ spectrum is not larger than $\ensuremath{\sim}30%$. Moreover, for the experimental configurations considered here, the statistical uncertainties-due to sampling variance and instrumental noise-of the pseudo-spectrum estimates is at most a factor $\ensuremath{\sim}1.4$ for $TT$, $EE$, and $TE$ spectra and a factor $\ensuremath{\sim}2$ for $BB$, $TB$, and $EB$ spectra, higher than the most optimistic Fisher estimate of the variance.