Error analysis of quadratic power spectrum estimates for cosmic microwave background polarization: sampling covariance

Quadratic methods with heuristic weighting (e.g. pseudo-C-l or correlation function methods) represent an efficient way to estimate power spectra of the cosmic microwave background ( CMB) anisotropies and their polarization. We construct the sample covariance properties of such estimators for CMB polarization, and develop semi-analytic techniques to approximate the pseudo-Cl sample covariance matrices at high Legendre multipoles, taking account of the geometric effects of mode coupling and the mixing between the electric ( E) and magnetic ( B) polarization that arise for observations covering only part of the sky. The E-B mixing ultimately limits the applicability of heuristically weighted quadratic methods to searches for the gravitational-wave signal in the large-angle B-mode polarization, even for methods that can recover ( exactly) unbiased estimates of the B-mode power. We show that for surveys covering 1 or 2 per cent of the sky, the contribution of E-mode power to the covariance of the recovered B-mode power spectrum typically limits the tensor-to-scalar ratio that can be probed with such methods to similar to 0.05.