Spectral-index decorrelation is a rank problem.
Component separation often assumes polarized dust occupies one column of the mixing matrix: one complex map times one real frequency law. If that factorization dies, 353 GHz is no longer a template for 150 GHz, in amplitude or in angle.
Why a single spectral index fails
Two dust layers with different temperatures and a misaligned field. Drag frequency: the net polarization angle moves. A one-template cleanup cannot follow that rotation.
Two conditions, not one
Tassis & Pavlidou (2015): a frequency-dependent polarization angle needs both a change in the layers’ relative polarized intensities and non-parallel projected fields. Different SEDs with parallel fields only curve the spectrum. Misaligned fields with identical SEDs still factorize.
McBride, Bull & Hensley (2023) found large biases in recovered CMB Q,U in illustrative models when both SED and angle vary along the line of sight. A single modified blackbody can remain approximately unbiased if only temperatures mix.
Planck Collaboration XI (2020) did not find significant broad-sky BB decorrelation between 217 and 353 GHz at a level that would wreck an r ~ 0.01 search. Pelgrims et al. (2021) did see extra 217–353 GHz angle differences toward HI sightlines with several misaligned clouds. The next generation (LiteBIRD-class δr ~ 10⁻³) is asking a harder question than Planck settled.