Testing for spectral index variations in polarized CMB foregrounds

We present a Bayesian parametric component separation method for polarised microwave sky maps. solve jointly the primary cosmic background (CMB) signal and main Galactic foreground components. For latter, we consider electron-synchrotron radiation thermal dust emission, modelled in frequency as power law modified blackbody respectively. account inter-pixel correlations noise covariance matrices of input maps introduce spatial correlation length prior spectral indices beta. apply our to low-resolution Planck 2018 Low High Frequency Instrument (LFI/HFI) data, including SRoll2 re-processing HFI data. find evidence variation synchrotron index, no depolarisation dust. Using maps, applying wide priors on indices, mean index over unmasked beta-sync = -2.833 +- 0.620. obtain beta-dust 1.429 0.236. Our returns correlated uncertainties all components model. recovered CMB associated uncertainties, constrain optical depth reionization, tau, using cross-spectrum-based likelihood-approximation scheme (momento) be tau 0.0598 0.0059. confirm findings pixel-based likelihood (pixlike). In both cases, result that is consistent with, albeit fraction sigma higher than, found by subtracting spatially uniform templates. While latter sufficient current polarisation data from Planck, next-generation space-borne experiments will need more powerful schemes such one presented here.