the primordial tensor-to-scalar ratio $r \la 0.01$, then the dominant power in $B$ modes is expected to come from gravitational-lensing conversion of Gaussian $E$ modes into non-Gaussian $B$ modes. In this limit, our assumption of Gaussianity will be violated and the expressions derived in this paper will require some modification to account for the non-zero connected four-point function. While a full treatment of the pseudo-$C_l$ covariance in the presence of lens-induced $B$ modes is beyond the scope of the current paper, we note that the lensing contribution to $\rmn{cov}(\tilde{C}_l^B,
\tilde{C}_{l'}^B)$ has been calculated recently for uniform weighting in the flat-sky limit [@smith04].

In this section we calculate the exact sample covariance of $\tilde{C}_l^E$ and $\tilde{C}_l^B$ (see also @hansen02b for a similar calculation) and present the results of numerical computations for two types of survey: a full-sky survey with a simple Galactic cut, and a $15\degr$-radius circular survey of the type that is optimal for gravitational wave searches in the presence of instrument noise at the level achievable with current technology [@jaffe00; @lewis02].

For Gaussian CMB fields, the pseudo-multipoles are also Gaussian, so the covariance of their power spectra are given by e.g.$$\rmn{cov}(\tilde{C}_l^E,\tilde{C}_{l'}^E) = \frac{2}{(2l+1)(2l'+1)}
\sum_{mm'} | \langle \tilde{E}_{lm} \tilde{E}_{(lm)'}^* \rangle 