Axion isocurvature and the model-building problem of low-scale inflation

The pre-inflationary QCD axion is often said to require low-scale inflation. We derive a general isocurvature bound for a light axion with a scale-invariant spectrum, treating its inflationary normalization f I independently of the late-time decay constant f a and allowing for an arbitrary axion dark matter fraction, nontrivial perturbation transfer, and the full anharmonic abundance response. In the minimal benchmark, where axions constitute all of the dark matter, f I=f a at most M Pl, and the angular perturbation is conserved, the least restrictive CMB limit considered gives the 95% C.L. upper bound H I<1.25 times10 10, GeV on the inflationary Hubble scale. For an initial misalignment angle theta i=1, the bound strengthens to H I<2.2 times10 7, GeV, with still stronger constraints near the hilltop. Combining the isocurvature and tensor spectra yields axion-tensor and axion-conditioned Lyth bounds. If the observed curvature perturbation is generated by a canonical cold single-field slow-roll inflaton and the tensor amplitude varies slowly across the observable CMB window, these relations limit the inflaton excursion to phi CMB/M Pl 10-7-10-4 and require M PlV'/V about10-8-10-5 while M Pl 2V''/V about-10-2. This hierarchy favors models with independent control of the inflationary scale, potential derivatives, exit, and reheating, including hybrid, running-mass, and inflection-point models. Reheating and the requirement of Peccei-Quinn non-restoration sharpen these conditions, while nonminimal scenarios can relax them by modifying the primordial fluctuation, its transfer, or the relic abundance.