See Ref. [6] for an overview of hints for deviations from the $\mathrm{\Lambda }\mathrm{CDM}$ model.

Established observational facts that cannot be described in this framework include the microphysical composition of the Dark Matter [9] and the origin of the matter-antimatter asymmetry of the Universe [10]; it is not clear whether the explanations for other tensions or anomalies [6, 11] require the existence of New Physics.

By this we mean the ability to impose both an upper and a lower bound on a quantity from data.

Notable exceptions are Starobinsky inflation [12], Higgs inflation [107], and the QCD-WI model (1) used here.

In the regime where $\mathrm{\Gamma }$ strongly affects the dynamics of the background field $\phi$, a better approximation is given by $\mathrm{\Gamma }=N_c^5{\alpha }_s^5{T}^3/[2f^2(1+2N_f{N}_c^4{\alpha }_s^{5}T{M}_{\mathrm{pl}}/(\mathcal{V}/3{)}^{1/2})]$ [55].

This breaking is sufficiently soft that $\mathcal{V}$ is protected from the problems [68] which haunt the original warm inflation proposal [61]. Specifying its origin in the UV is not required for the present analysis; it could, e.g., originate from gravitational effects [69].

In addition to finding the inflaton in the laboratory, observing the cosmological bispectrum [99] would provide another test of QCD-WI.

The condition (6) represents a special case of the more general conditions derived in Ref. [36] which were found to be consistent with lattice simulations in Ref. [73]. They also appear to be consistent with the considerations in Refs. [74, 75].

As a parametrization of the duration of reheating, the use of $y$ does not introduce any further assumptions and can (for fixed $\mathcal{V}$ and $g_{*}$) be used equivalently to $T_{\mathrm{re}}$ or $N_{\mathrm{re}}$, only its interpretation as a microphysical coupling constant is ambiguous if Eq. (6) is strongly violated [36, 77].

For the model (1), the constraint imposed by fixing $A_s$ can only be implemented numerically, for the cold plateau models, the analytic relations are, e.g., given in Ref. [43].

More precisely, the authors of Ref. [46] assumed that the first additional MF SAT starts observations halfway through the second year, and the final two instruments (a second additional MF SAT, and the additional LF SAT) start observations at the start of the third year.

We note that the interpretation of ${\sigma }_r$ as a variance is only indicative; especially for $\overline{r}=0.0036$, it is restricted to the $1\sigma$ level due to the non-Gaussian shape of the full likelihood for small values of $r$, while for larger values of $\overline{r}$, its range of validity increases. In any case, the ability to remove foregrounds will have a stronger impact on the precision at which $r$ can be measured than the dependence of ${\sigma }_r$ on $\overline{r}$ studied in Ref. [82], hence our estimates are sufficient for the present purpose.

Practically, this amounts to $N_{\mathrm{re}}=0$. In warm inflation scenarios, there is no reheating in the literal sense because the thermal plasma of particles is already present during inflation. However, in principle, a number of $e$-folds $N_{\mathrm{re}}$ could pass between the moment when $\ddot{a}<0$ and the moment when the radiation’s energy density exceeds that of $\phi$. In Ref. [55], it was estimated that this period is very brief; a more detailed investigation would require computing $\mathrm{\Gamma }$ throughout the entire cosmic history, which goes beyond the scope of the present work.

The precise relations between $M$, $\alpha$ and observables (as well as the quantities ${\phi }_k$, $N_k$, $N_{\mathrm{re}}$, etc.) used below are well known for the models (2)–(4); for a summary we refer the reader to Refs. [43, 77].

The perspectives to constrain the microphysics of reheating from CMB observations [30, 36] in $\alpha$-attractor models have previously been studied in Refs. [42, 43, 74, 75, 77, 92, 93, 94, 95, 96]; a recent discussion relating to the Yukawa coupling used here can be found in Ref. [77].

A possible way to systematically address those uncertainties could be the Schwinger-Keldysh formalism (previously applied to the background evolution [104]) in combination with the open effective field method [105].

The relations in Appendix hold at leading order in the slow-roll parameters; see Appendix B in Ref. [43] for a discussion of higher-order corrections.