No Dark Matter Axion During Minimal Higgs Inflation

We study minimal versions of Higgs inflation in the presence of a massless QCD axion. While the inflationary energy scale of the metric variant is too high to accommodate isocurvature bounds, it was argued that Palatini Higgs inflation could evade these constraints. We show, however, that an energy-dependent decay constant enhances isocurvature perturbations, implying that axions can at most constitute a tiny fraction $< 10^{-5}$ of dark matter. This conclusion can be avoided in Einstein-Cartan gravity by an additional coupling of the axion to torsion, albeit for a very specific choice of parameters. Analogous constraints as well as the possibility to alleviate them are relevant for all inflationary models with a non-minimal coupling to gravity.