Peeling off foregrounds with the constrained moment ILC method to unveil
  primordial CMB $B$-modes
  Galactic foregrounds are the main obstacle to observations of the cosmic
microwave background (CMB) $B$-mode polarization. In addition to obscuring the
inflationary $B$-mode signal by several orders of magnitude, Galactic
foregrounds have non-trivial spectral signatures that are partially unknown and
distorted by averaging effects along the line-of-sight, within the pixel/beam
window, and by various analysis choices (e.g., spherical harmonic transforms
and filters). Statistical moment expansion methods provide a powerful tool for
modeling the effective Galactic foreground emission resulting from these
averaging effects in CMB observations, while blind component separation
treatments can handle unknown foregrounds. In this work, we combine these two
approaches to develop a new semi-blind component separation method at the
intersection of parametric and blind methods, called constrained moment ILC
(cMILC). This method adds several constraints to the standard ILC method to
de-project the main statistical moments of the Galactic foreground emission.
Applications to maps are performed in needlet space and when compared to the
NILC method, this helps significantly reducing residual foreground
contamination (bias, variance, and skewness) in the reconstructed CMB $B$-mode
map, power spectrum, and tensor-to-scalar ratio. We consider sky-simulations
for experimental settings similar to those of LiteBIRD and PICO, illustrating
which trade-offs between residual foreground biases and degradation of the
constraint on $r$ can be expected within the new cMILC framework. We also
outline several directions that require more work in preparation for the coming
analysis challenges.
