We apply the “systematic” first-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 nonminimally coupled to gravity within both the “Metric” and “Palatini” formulations. Accommodating Planck 2018 data with a length scale [Formula: see text] larger than Planck length [Formula: see text] requires amending the model. First, we assume [Formula: see text] gravity where we show that an [Formula: see text]-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 [Formula: see text].
Natural Inflation with non-minimal coupling (NMC) to gravity, embodied by a Lagrangian term ξϕ
2
R, is investigated in the context of an extended gravity of the form R + αR
2. The treatment is performed in the Palatini formalism. We discuss various limits of the model “α ≫ 1” and “α ≪ 1” in light of two scenarios of inflation: a “Slow roll” and a “Constant roll” scenario. By analyzing the observational consequences of the model, our results show a significant improvement regarding compatibility between the theoretical results of this model and the observational constraints from Planck 2018 and BICEP/Keck 2018, as exemplified by the tensor-to-scalar ratio and spectral index. Furthermore, a broader range for the parameter space of natural inflation is now compatible with the confidence contours of Planck & BICEP/Keck results.
The joint effects of the contributions of both the NMC to gravity and the αR
2 make a significant improvement: αR
2 gravity influences scalar-tensor ratio values, whereas NMC to gravity has a more significant impact on the spectral index values. Contributions from both terms allow more previously excluded intervals to be included being compatible now with observational data. These conclusions about the roles of NMC to gravity and, particularly, the extended gravity remain mainly valid with a periodic NMC similar in form to the natural inflation potential.
We show that upon applying Palatini f(R), characterized by an αR2 term, within a scenario motivated by a temporal variation of strong coupling constant, then one obtains a quadratic kinetic energy. We do not drop this term, but rather study two extreme cases: α<<1 and α>>1. In both cases, one can generate a kinematically-induced inflationary paradigm. In order to fit the Planck 2018 data, the α>>1 case, called k-inflation, requires a fine tuning adjustment with nonvanishing nonminimal coupling to gravity parameter ξ, whereas the α<<1 case, studied in the constant-roll regime, can fit the data for vanishing ξ. The varying strong coupling inflation scenario remains viable when implemented through a warm inflation scenario with or without f(R) gravity.
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