  It is known for some time that the cosmological bound on the invisible axion
scale can be avoided by an early phase of strong QCD. While most approaches
rely on theories where the strong coupling constant is determined through the
expectation value of some scalar field, we show that this mechanism can also be
implemented into the benchmark KSVZ and DFSZ models when the early phase of
strong QCD emerges by the modification of the running coupling during
inflation. For both models the physics that are responsible for making QCD
strong do not displace the axions minimum by too much, so that the efficiency
of the relaxation is controlled by parameters of the theory and the number of
inflationary e-folds. In particular, we consider the case of very efficient
relaxation where the axion abundance is dominated by inflationary quantum and
post-inflationary thermal fluctuations. Within this situation we identify the
parameter space compatible with all cosmological constrains and derive
conditions on the reheating temperature and the QCD scale during inflation that
result in the axion making up all the dark matter. Due to duality below the
Peccei-Quinn scale and a minor influence of the KSVZ and DFSZ fields on the
running, our findings also apply to a minimal 2-form implementation of the
axion.
