We consider extensions of the Standard Model in which a spontaneously broken global chiral Peccei-Quinn (PQ) symmetry arises as an accidental symmetry of an exact $Z_N$ symmetry. For $N = 9$ or $10$, this symmetry can protect the accion - the Nambu-Goldstone boson arising from the spontaneous breaking of the accidental PQ symmetry - against semi-classical gravity effects, thus suppressing gravitational corrections to the effective potential, while it can at the same time provide for the small explicit symmetry breaking term needed to make models with domain wall number $N_{rm DW}>1$, such as the popular Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model ($N_{rm DW}=6$), cosmologically viable even in the case where spontaneous PQ symmetry breaking occurred after inflation. We find that $N=10$ DFSZ accions with mass $m_A approx 3.5$-$4.2,mathrm{meV}$ can account for cold dark matter and simultaneously explain the hints for anomalous cooling of white dwarfs. The proposed helioscope International Axion Observatory - being sensitive to solar DFSZ accions with mass above a few meV - will decisively test this scenario.
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