  Characterizing accurately the polarized dust emission from our Galaxy will be
decisive for the quest for the Cosmic Microwave Background (CMB) primordial
$B$-modes. The incomplete modeling of its potentially complex spectral
properties could lead to biases in the CMB polarization analyses and to a
spurious detection of the tensor-to-scalar ratio $r$. Variations of the dust
properties along and between lines of sight lead to unavoidable distortions of
the spectral energy distribution (SED) that can not be easily anticipated by
standard component separation methods. This issue can be tackled using a moment
expansion of the dust SED, an innovative parametrization method imposing
minimal assumptions on the sky complexity. In the recent work [Vacher \emph{et
al.} (2022)]\cite{Vacher_2022}, we apply this formalism to the $B$-mode
cross-angular power spectra computed from simulated \lb{} polarization data at
frequencies between 100 and 402\,GHz, containing CMB, dust and instrumental
noise. Thanks to the moment expansion, we can measure an unbiased value of the
tensor-to-scalar ratio with a dispersion compatible with the target values
aimed by the instrument.
