See, e.g., Refs. [18, 20, 25], and references therein for derivations within quantum field theory.

The inflaton-gluon interaction is mediated by a dimension-five operator and, therefore, must be viewed as an effective field theory (EFT) with a cutoff ${\mathrm{\Lambda }}_{\text{cutoff}}\equiv ({\alpha }_s/8\pi f{)}^{-1}$. This restricts the validity of our computations to the regime $T<{\mathrm{\Lambda }}_{\text{cutoff}}$. We verify a posteriori that all temperatures in Table 1 lie at least one order of magnitude below this EFT cutoff.

More generally, successful warm inflation in the strong regime requires a potential that breaks the symmetry $\varphi \to \varphi +2\pi f$ of the axionlike coupling in Eq. (6) [35].

In terms of this chemical potential, we see that ${\partial }_t{j}_A^0$ is independent of $N_c$ but scales with $N_f^2{\mu }_A$. A heuristic explanation of this scaling is that the production rate of fermions, which is caused by sphaleron transitions and violates chiral symmetry, is proportional to the free energy $N_f{\mu }_A$ liberated by a sphaleron event and, in addition, each event changes $j_A^0$ by $N_f$ [41].

Moreover, SU(2) sphalerons provide an additional source of chirality violation [23], but this effect turns out to be negligible since $(N_c{\alpha }_s{)}^5\gg (2{\alpha }_w{)}^5$, where ${\alpha }_w$ is the weak fine structure constant.

For the bottom quark ${\mathrm{\Gamma }}_{\mathrm{ch}f}\simeq {\mathrm{\Upsilon }}_{\mathrm{sph}}$, provided ${\alpha }_s$ runs to sufficiently low values.

Thermal corrections to the vacuum $\beta$ functions are expected to be negligible at temperatures greatly exceeding the QCD confinement scale [49, 50].

In contrast, the connection between cold inflation and particle physics can be probed only indirectly [58, 59], with the notable exception of Higgs inflation [60] (see, e.g., Ref. [61]).