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Searches for the imprint of primordial gravitational waves in degree-scaleCMB $B$-mode polarisation data must account for significant contamination fromgravitational lensing. Fortunately, the lensing effects can be partiallyremoved by combining high-resolution $E$-mode measurements with an estimate ofthe projected matter distribution. In the near future, experimentalcharacteristics will be such that the latter can be reconstructed internallywith high fidelity from the observed CMB, with the $EB$ quadratic estimatorproviding a large fraction of the signal-to-noise. It is a well-knownphenomenon in this context that any overlap in modes between the $B$-field tobe delensed and the $B$-field from which the reconstruction is derived leads toa suppression of delensed power going beyond that which can be attributed to amitigation of the lensing effects. More importantly, the variance associatedwith this spectrum is also reduced, posing the question of whether theadditional power suppression could help better constrain the tensor-to-scalarratio, $r$. In this paper, we show this is not the case, as suggested but notquantified in previous work. We develop an analytic model for the biaseddelensed $B$-mode angular power spectrum, which suggests a simplerenormalisation prescription to avoid bias on the inferred tensor-to-scalarratio. With this approach, we learn that the bias necessarily leads to adegradation of the signal-to-noise on a primordial component compared to"unbiased delensing". Next, we assess the impact of removing from the lensingreconstruction any overlapping $B$-modes on our ability to constrain $r$,showing that it is in general advantageous to do this rather than modeling orrenormalising the bias. Finally, we verify these results within amaximum-likelihood inference framework applied to simulations.

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