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Characterizing the polarized dust emission from our Galaxy will be decisivefor the quest for the Cosmic Microwave Background (CMB) primordial $B$-modes.The incomplete modelling of its potentially complex spectral properties couldlead to biases in the CMB polarization analyses and to a spurious detection ofthe tensor-to-scalar ratio $r$. It is crucial for future surveys like the$LiteBIRD$ satellite, which aims at constraining the primordial signal leftoverby Inflation with an accuracy on $r$ of the order 1e-3. Variations of the dustproperties along and between lines of sight lead to distortions of the spectralenergy distribution (SED) that can not be easily anticipated by standardcomponent separation methods. This issue can be tackled with a moment expansionof the dust SED, an innovative parametrization method imposing minimalassumptions on the sky complexity. In this paper, we apply this formalism tothe $B$-mode cross-angular power spectra computed from simulated $LiteBIRD$polarization data at frequencies between 100 and 402 GHz, containing CMB, dustand instrumental noise. The spatial variation of the dust spectral parameters(spectral index $\beta$ and temperature $T$) in our simulations, lead tosignificant biases on $r$ if not properly taken into account. Performing themoment expansion in $\beta$, reduces the bias but do not lead to reliableenough estimates of $r$. We introduce for the first time the expansion of thecross-angular power spectra SED in $\beta$ and $T$, showing that, at the$LiteBIRD$ sensitivity, it is required to take into account the SED complexitydue to temperature variations to prevent analysis biases on $r$. Thanks to thisexpansion and despite the existing correlations between some of the dustmoments and the CMB signal, responsible for a rise of the error on $r$, we canmeasure an unbiased value of $r$ with an uncertainty of $\sigma(r)$=8.8e-4.

Moment expansion of polarized dust SED: a new path towards capturing the CMB $B$-modes with $\textit{LiteBIRD}$