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Strongly broken Peccei-Quinn symmetry in the early Universe

We consider QCD axion models where the Peccei-Quinn symmetry is badly broken by a larger amount in the past than in the present, in order to avoid the axion isocurvature problem. Specifically we study supersymmetric axion models where the Peccei-Quinn symmetry is dynamically broken by either hidden gauge interactions or the $SU(3)_c$ strong interactions whose dynamical scales are temporarily enhanced by the dynamics of flat directions. The former scenario predicts a large amount of self-interacting dark radiation as the hidden gauge symmetry is weakly coupled in the present Universe. We also show that the observed amount of baryon asymmetry can be generated by the QCD axion dynamics via spontaneous baryogenesis. We briefly comment on the case in which the PQ symmetry is broken by a non-minimal coupling to gravity.