Characterizing bias on large scale CMB $B$-modes after galactic
  foregrounds cleaning
  We study the performance of a typical near-future full sky CMB space mission,
aiming at the characterization of the large scale $B$-modes polarization
anisotropies with precision on $r\sim 0.001$, after a map-based parametric
cleaning of galactic dust and synchrotron, and in the case of spatially varying
astrophysical spectral emission laws. Ignoring the spatial variability of the
spectral emission laws may result in a bias on $r$ as high as
$\mathcal{O}(0.01)$ for realistic models of the variability. However, we show
that the component separation formalism can be extended to suppress this bias
efficiently. We demonstrate this within the context of the semianalytic
formalism of~\citep{Stompor2016}, which we generalize to such cases and use it
to propagate the foreground residuals to a cosmological likelihood on
tensor-to-scalar ratio, $r$. In particular, we investigate the effects due to
introducing extra, independent sets of scaling parameters for different sky
areas and including additional scaling parameters per sky area. We then show
that the residuals resulting in such cases can be efficiently described with
help of extra terms introduced in a model of the covariance of the
component-separated CMB maps and which lead to suppression of the bias on $r$
down to the level lower than the expected statistical uncertainty. We discuss
how these additional terms can be constructed self-consistently from the
available data.
