CMB-S4: Iterative Internal Delensing and $r$ Constraints

The tightest constraints on the tensor-to-scalar ratio $r$ can only be obtained after removing a substantial fraction of the lensing $B$-mode sample variance. The planned CMB-S4 experiment will remove the lensing $B$-mode signal internally by reconstructing the gravitational lenses from high-resolution observations. We document here a first lensing reconstruction pipeline able to achieve this optimally for arbitrary sky coverage. We make it part of a map-based framework to test CMB-S4 delensing performance and its constraining power on $r$, including inhomogeneous noise and two non-Gaussian Galactic polarized foreground models. The framework performs component-separation of the high-resolution maps, followed by the construction of lensing $B$-mode templates, which are then included in a parametric small-aperture maps cross-spectra-based likelihood for $r$. We find that the lensing reconstruction and framework achieve the expected performance, compatible with the target $\sigma(r) \simeq 5\cdot 10^{-4}$ in the absence of a tensor signal, after an effective removal of $92\%$ to $93\%$ of the lensing $B$-mode variance, depending on the simulation set. The code for the lensing reconstruction can also be used for cross-correlation studies with large-scale structures, lensing spectrum reconstruction, cluster lensing, or other CMB lensing-related purposes. As part of our tests we also demonstrate joint optimal reconstruction of the lensing potential with the lensing curl potential mode, second-order in the density fluctuations.