Variance of dust temperature and spectral index in

Thermal dust is the major polarized foreground hindering the detection of primordial cosmic microwave background (CMB) B-modes. Its signal exhibits complex behavior in frequency space, arising from the combined variation in our Galaxy of the orientation of magnetic fields and the spectral properties of dust grains aligned with magnetic field lines. In this work, we present a new framework for analyzing the thermal dust signal using polarized microwave data. We introduce residual maps, represented as complex quantities, which capture deviations of the local polarized spectral energy distribution (SED) from the mean complex SED averaged over the sky mask. We present simple predictions that relate the values of the statistical correlation and covariances between the residual maps to the physical properties of the emitting aligned grains. Testing these predictions provides valuable information about the nature of the dust signal. We evaluated our predictions using Planck data over a 97% mask excluding the inner Galactic plane. Despite its simplicity, our model captures a significant part of the statistical properties of the data. For the <ns0:monospace>SRoll2</ns0:monospace> version of the data, the spectral dependence of the covariances between residual maps is compatible with a dust model that includes only temperature variations rather than spectral index variations. In contrast, for the PR4 Planck official release, it is incompatible with both models. Our methodology can be used to analyze future high-precision polarization data and to build more accurate dust models for use by the CMB community.