Moment expansion of polarized dust SED: a new path towards capturing the
  CMB $B$-modes with $\textit{LiteBIRD}$
  Characterizing the polarized dust emission from our Galaxy will be decisive
for the quest for the Cosmic Microwave Background (CMB) primordial $B$-modes.
The incomplete modelling 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 $r$. It is crucial for future surveys like the
$LiteBIRD$ satellite, which aims at constraining the primordial signal leftover
by Inflation with an accuracy on $r$ of the order 1e-3. Variations of the dust
properties along and between lines of sight lead to distortions of the spectral
energy distribution (SED) that can not be easily anticipated by standard
component separation methods. This issue can be tackled with a moment expansion
of the dust SED, an innovative parametrization method imposing minimal
assumptions on the sky complexity. In this paper, we apply this formalism to
the $B$-mode cross-angular power spectra computed from simulated $LiteBIRD$
polarization data at frequencies between 100 and 402 GHz, containing CMB, dust
and instrumental noise. The spatial variation of the dust spectral parameters
(spectral index $\beta$ and temperature $T$) in our simulations, lead to
significant biases on $r$ if not properly taken into account. Performing the
moment expansion in $\beta$, reduces the bias but do not lead to reliable
enough estimates of $r$. We introduce for the first time the expansion of the
cross-angular power spectra SED in $\beta$ and $T$, showing that, at the
$LiteBIRD$ sensitivity, it is required to take into account the SED complexity
due to temperature variations to prevent analysis biases on $r$. Thanks to this
expansion and despite the existing correlations between some of the dust
moments and the CMB signal, responsible for a rise of the error on $r$, we can
measure an unbiased value of $r$ with an uncertainty of $\sigma(r)$=8.8e-4.
