In preferred axion models with slow heavy-quark decays, axions produced after decoupling form dark radiation that can exceed Planck's ΔNeff bound, excluding much of the parameter space of models D and E.
New confining force solution of QCD axion domain wall problem
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abstract
The serious cosmological problems created by the axion-string/axion-domain-wall system in standard axion models are alleviated by positing the existence of a new confining force. The instantons of this force can generate an axion potential that erases the axion strings long before QCD effects become important, thus preventing QCD-generated axion walls from ever appearing. Axion walls generated by the new confining force would decay so early as not to contribute significantly to the energy in axion dark matter.
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Using $\Delta N_{\rm eff}$ to constrain preferred axion model dark matter
In preferred axion models with slow heavy-quark decays, axions produced after decoupling form dark radiation that can exceed Planck's ΔNeff bound, excluding much of the parameter space of models D and E.