Limitations of CMB B-mode template delensing

Efforts to detect a primordial $B$-mode of CMB polarization generated by inflationary gravitational waves ought to mitigate the large variance associated with the $B$-modes produced by gravitational lensing, a process known as delensing. A popular approach to delensing entails building a lensing $B$-mode template by mimicking the lensing operation, either at gradient order or non-perturbatively, using high-resolution $E$-mode observations and some proxy of the lensing potential. By explicitly calculating all contributions to two-loop order in lensing to the power spectrum of $B$-modes delensed with such a template in the noise-free limit, we are able to show that: (i) corrections to the leading-order calculation of the lensing $B$-mode power spectrum only enter at the $O(1)\,\%$ level because of extensive cancellations between large terms at next-to-leading order; (ii) these cancellations would disappear if a gradient-order template were to be built from unlensed or delensed $E$-modes, giving rise to a residual delensing floor of $O(10)\,\%$ of the original power; (iii) new cancellations arise when the lensed $E$-modes are used in the gradient-order template, allowing for the delensing floor to be as low as $O(1)\,\%$ of the original power in practical applications of this method; and (iv) these new cancellations would disappear for a non-perturbative template constructed from the lensed $E$-modes, reintroducing a residual delensing floor of $O(10)\,\%$. We further show that the gradient-order template outperforms the non-perturbative one in realistic scenarios with noisy estimates of the $E$-mode polarization and lensing potential. We therefore recommend that in practical applications of $B$-mode template delensing, where the template is constructed directly from the (filtered) observed $E$-modes, the gradient-order approach should be used rather than a non-perturbative remapping.