Our approximations are accurate in the regime where the mixing is perturbative, i.e. for $l \gg l_{\rmn{max}}$, where $l_{\rmn{max}}$ is the effective band-limit of the function used to weight the Stokes parameters. It is difficult to construct good approximations for the covariance on large scales, $l \ll l_{\rmn{max}}$. This is partly because of the increased importance of $E$-$B$ mixing there, but also because the polarization power spectra do not go smoothly to zero on large scales in the presence of reionization. (The behaviour of the temperature-anisotropy power spectrum on large scales produces a similar effect for the temperature pseudo-$C_l$ covariance.) Following @efstathiou04, the low-$l$ sector of the covariance matrices can be constructed directly with exact methods, and used to overwrite the inaccurate approximation. Being able to evaluate accurate covariances efficiently is important in any application where multiple evaluations are required. Examples include parameter estimation, where the covariance generally depends on position in parameter space, and optimising the design and analysis of surveys, where one might wish to assess a large number of weighting schemes.

Throughout, we ignore the effects of instrument noise so we deal only with the sample covariance of the $\tilde{C}_l$. This is to highlight the geometric effects 