Inflation by Variation of the Strong Coupling Constant: update for Planck 2018

We apply the "systematic" $1^{st}$ order cosmological perturbation theory method to re-derive the formulation of an inflationary model generated by variation of constants, then to study the case where it is non-minimally coupled to gravity within both the "Metric" and "Palatini" formulations. Accommodating Planck 2018 data with a length scale $\ell$ larger than Planck Length $L_{pl}$ requires amending the model. First, we assume $f(R)$ gravity where we show that an $R^2$-term within Palatini formulation is able to make the model viable. All along the discussions, we elucidate the origin of the difference between the "Metric" and "Palatini" formalisms, and also highlight the terms dropped when applying the shortcut "potential formulae method", unlike the "systematic' method", for the observable parameters. Second, another variant of the model, represented by a two-exponentials potential, fits also the data with $\ell> L_{pl}$.