  The characterization of the spectral energy distribution (SED) of dust
emission has become a critical issue in the quest for primordial B-modes. The
dust SED is often approximated by a modified black body (MBB) emission law but
the extent to which this is accurate is unclear. This paper addresses this
question, expanding the dust SED at the power spectrum level. The expansion is
performed by means of moments around the MBB law, related to derivatives with
respect to the dust spectral index. We present the mathematical formalism and
apply it to simulations and Planck total intensity data, from 143 to 857 GHz,
because no polarized data are yet available that provide the required
sensitivity to perform this analysis. With simulations, we demonstrate the
ability of high-order moments to account for spatial variations in MBB
parameters. Neglecting these moments leads to poor fits and a bias in the
recovered dust spectral index. We identify the main moments that are required
to fit the Planck data. The comparison with simulations helps us to disentangle
the respective contributions from dust and the cosmic infrared background to
the high-order moments, but the simulations give an insufficient description of
the actual Planck data. Extending our model to cosmic microwave background
B-mode analyses within a simplified framework, we find that ignoring the dust
SED distortions, or trying to model them with a single decorrelation parameter,
could lead to biases that are larger than the targeted sensitivity for the next
generation of CMB B-mode experiments.
