Debiasing cosmological parameters from large-scale foreground contamination in Cosmic Microwave Background data

Current and future Cosmic Microwave Background (CMB) experiments aim to achieve high-precision reconstruction of the CMB polarization signal, with the most ambitious objective being the detection of primordial $B$ modes sourced by cosmic inflation. Given the expected low amplitude of the signal, its estimate-parametrized by the tensor-to-scalar ratio $r$-is highly susceptible to contamination from Galactic foreground residuals that remain after component separation. In this work, we introduce a model-independent procedure to construct a spectral template of residual foreground contamination in the observed angular power spectrum. Specifically, a cleaned multifrequency set of foreground-emission maps is blindly reconstructed from the observed data using the Generalized Needlet Internal Linear Combination (GNILC) technique. These maps are then combined with the weights adopted for CMB reconstruction, yielding an estimate of the spatial distribution of foreground residuals after component separation. The power spectrum of this estimated residual map is incorporated into the spectral model of the cosmological likelihood. We validate the proposed method using realistic simulations of a LiteBIRD-like experiment processed with two Internal Linear Combination (ILC) component-separation techniques, focusing on constraints on the tensor-to-scalar ratio. When the foreground contribution is not included in the model, the resulting $r$ posteriors are biased, irrespective of its input value, the assumed foreground model, or the adopted masking strategy. Conversely, when the residual template is included in the likelihood, the analysis yields unbiased estimates of $r$ for all considered cases, thereby demonstrating the robustness of the proposed procedure. The pipeline has been made publicly available as part of the BROOM Python package (https://github.com/alecarones/broom).