Impact of internal-delensing biases on searches for primordial B-modes of CMB polarisation

Searches for the imprint of primordial gravitational waves in degree-scale CMB $B$-mode polarisation data must account for significant contamination from gravitational lensing. Fortunately, the lensing effects can be partially removed by combining high-resolution $E$-mode measurements with an estimate of the projected matter distribution. In the near future, experimental characteristics will be such that the latter can be reconstructed internally with high fidelity from the observed CMB, with the $EB$ quadratic estimator providing a large fraction of the signal-to-noise. It is a well-known phenomenon in this context that any overlap in modes between the $B$-field to be delensed and the $B$-field from which the reconstruction is derived leads to a suppression of delensed power going beyond that which can be attributed to a mitigation of the lensing effects. More importantly, the variance associated with this spectrum is also reduced, posing the question of whether the additional power suppression could help better constrain the tensor-to-scalar ratio, $r$. In this paper, we show this is not the case, as suggested but not quantified in previous work. We develop an analytic model for the biased delensed $B$-mode angular power spectrum, which suggests a simple renormalisation prescription to avoid bias on the inferred tensor-to-scalar ratio. With this approach, we learn that the bias necessarily leads to a degradation of the signal-to-noise on a primordial component compared to "unbiased delensing". Next, we assess the impact of removing from the lensing reconstruction any overlapping $B$-modes on our ability to constrain $r$, showing that it is in general advantageous to do this rather than modeling or renormalising the bias. Finally, we verify these results within a maximum-likelihood inference framework applied to simulations.