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.
