Dust moments: towards a new modelling of the galactic dust emission for CMB B-modes analysis

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 blackbody (MBB) emission law but to what extent is it accurate? This paper addresses this question expanding the dust SED, at the power spectrum level, in 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 today no polarized data 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 moment 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 help 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 CMB 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 larger than the targeted sensitivity for the next generation of CMB B-mode experiments.