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Characterizing accurately the polarized dust emission from our Galaxy will bedecisive for the quest for the Cosmic Microwave Background (CMB) primordial$B$-modes. The incomplete modeling of its potentially complex spectralproperties could lead to biases in the CMB polarization analyses and to aspurious detection of the tensor-to-scalar ratio $r$. Variations of the dustproperties along and between lines of sight lead to unavoidable distortions ofthe spectral energy distribution (SED) that can not be easily anticipated bystandard component separation methods. This issue can be tackled using a momentexpansion of the dust SED, an innovative parametrization method imposingminimal assumptions on the sky complexity. In the recent work [Vacher \emph{etal.} (2022)]\cite{Vacher_2022}, we apply this formalism to the $B$-modecross-angular power spectra computed from simulated \lb{} polarization data atfrequencies between 100 and 402\,GHz, containing CMB, dust and instrumentalnoise. Thanks to the moment expansion, we can measure an unbiased value of thetensor-to-scalar ratio with a dispersion compatible with the target valuesaimed by the instrument.

Contribution to the 2022 Cosmology session of the 56th Rencontres de  Moriond: Moment expansion of polarized dust SED: A new path towards capturing  the CMB $B$-modes with LiteBIRD