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.