detection of arbitrarily low amplitude gravitational wave backgrounds in the absence of instrument noise. In practice, imperfections in removing the lensing signal limit the amplitude to tensor-to-scalar ratios[^2] $r \sim 3\times 10^{-5}$ (@kesden02 [@knox02]; see @seljak03 for refinements from more optimal lensing-reconstruction methods). Heuristically-weighted pseudo-$C_l$ methods do not share the property that the cosmic variance of the estimated $C_l^B$ is immune to $E$-mode power, even if we take care to construct unbiased estimates so that e.g.the mean of our $C_l^B$ estimator decouples from $C_l^E$ [@chon04]. We investigate this issue further here, exploring the limits of detectability for $r$ that arise solely from cosmic variance of the $E$-mode polarization leaking into the variance of quadratic estimates of $C_l^B$.

This paper is organised as follows. Section \[sec:properties\] reviews the properties of the polarization pseudo-$C_l$s and establishes our notation. Following a brief discussion of the exact pseudo-$C_l$ covariance matrices in Section \[sec:exact-cov\], we develop analytic approximations to them that can be rapidly computed in Section \[sec:approx-cov\]. We begin with a toy model, based on a Gaussian weight function applied to a small survey area in the flat-sky limit. (Some of the more involved manipulations are relegated to Appendix \[app:flat\].) This instructive example illustrates 