This paper shows that if the strong force was unusually strong during cosmic inflation, the axion—a hypothetical particle proposed to solve the “strong‑CP” problem—would have relaxed early and need not obey the usual cosmological upper limit on its energy scale. The axion comes with a scale f_a (the “decay constant”) that normally cannot be arbitrarily large because a large f_a would overproduce axion dark matter under standard cosmology. The authors implement an explicit scenario inside an SU(5) grand unified theory and argue that a period of high‑scale confinement during inflation removes that upper bound.

The authors study two concrete ways to make the strong force strong during inflation. One is a direct coupling between the inflaton (the field that drives inflation, labeled S) and the gauge fields, so that the effective gauge coupling depends on S through a term like α_eff(S/M) tr GµνGµν. The other is a non‑thermal restoration of the SU(5) symmetry during inflation, which changes how the gauge coupling runs with energy (renormalization group running) and can push the theory into a confining phase. In their setup the inflaton controls vacuum values of other fields and thereby the strength of the gauge coupling.

Why this matters is how the axion responds to early strong coupling. Confinement and instanton effects during inflation give the axion an early mass and an effective potential. If that mass exceeds the Hubble expansion rate during inflation, the axion does damped oscillations and relaxes toward the minimum of its potential. Those early oscillations reduce the axion’s misalignment and its eventual energy density today. In short, an early heavy axion can be driven close to its vacuum so it does not overproduce dark matter even if f_a is very large.

The mechanism is shown to work for the main axion constructions. That includes Peccei‑Quinn (PQ) realizations with extra scalars (DFSZ) or heavy fermions (KSVZ), and also the two‑form gauge axion formulation that relies on QCD gauge redundancy rather than a global symmetry. The authors point out that even if the PQ scalar’s vacuum expectation value (VEV) vanishes during inflation, the axion still exists there as the phase of a fermion condensate known as the ’t Hooft determinant. In the DFSZ case that condensate can be made of ordinary quarks and acts like an early‑universe version of the η′ (eta‑prime) meson.