  In cosmology, the quest for primordial $B$-modes in cosmic microwave
background (CMB) observations has highlighted the critical need for a refined
model of the Galactic dust foreground. We investigate diffusion-based modeling
of the dust foreground and its interest for component separation. Under the
assumption of a Gaussian CMB with known cosmology (or covariance matrix), we
show that diffusion models can be trained on examples of dust emission maps
such that their sampling process directly coincides with posterior sampling in
the context of component separation. We illustrate this on simulated mixtures
of dust emission and CMB. We show that common summary statistics (power
spectrum, Minkowski functionals) of the components are well recovered by this
process. We also introduce a model conditioned by the CMB cosmology that
outperforms models trained using a single cosmology on component separation.
Such a model will be used in future work for diffusion-based cosmological
inference.
