In this work, we shall study a class of [Formula: see text] gravity models which during the inflationary era, which is the large curvature regime, result to an effective inflationary Lagrangian that contains a rescaled Einstein–Hilbert term [Formula: see text] in the presence of a canonical minimally coupled scalar field. The dimensionless parameter [Formula: see text] is chosen to take values in the range [Formula: see text] and the main motivation for studying these rescaled Einstein–Hilbert [Formula: see text] gravities, is the fact that the rescaled action may render an otherwise incompatible canonical scalar field theory with the Swampland criteria, to be compatible with the Swampland criteria and specifically the distance conjecture and the de Sitter conjecture, without considering the refined de Sitter conjecture however. As we will show, by studying a large number of inflationary potentials appearing in the 2018 Planck collaboration paper for the constraints on inflation, the simultaneous compatibility with both the Planck constraints and the aforementioned two Swampland criteria is achieved for some models, and the main characteristic of the models for which this is possible is the small value that the parameter [Formula: see text] must take.

In this paper, we worked in the framework of an inflationary [Formula: see text] theory in the presence of a canonical scalar field. More specifically, the [Formula: see text] gravity. The values of the dimensionless parameters [Formula: see text] and [Formula: see text] are taken to be [Formula: see text] and [Formula: see text]. The motivation for this study was the striking similarities between the slow-roll parameters of the inflationary model used in this work and the ones obtained by the rescaled Einstein–Hilbert gravity inflation [Formula: see text]. We examined a variety of potentials to determine if they agree with the current Planck Constraints. In addition, we checked whether these models satisfy the Swampland Criteria and we specified the exact region of the parameter space that produces viable results for each model. As we mention in Sec. 4, the inflationary [Formula: see text] theory used in this work cannot produce a positive [Formula: see text] which implies that the stochastic gravitational wave background will not be detectable.

In this work we examine the Swampland criteria for a specific class of rescaled f (R) gravitational models, that are capable of unifying the primordial era of the Universe with the late-time era with the inclusion of string corrections. In particular, we develop separately the theoretical framework of Gauss-Bonnet and Chern-Simons theories considering that, the rescale parameter is constrained to reside in the area 0 < α < 1. As showcased, in the context of the aforementioned theories, a viable inflationary phenomenology consistent with the latest Planck data can be obtained for both cases for a wide variety of values. The Swampland criteria which where examined are satisfied, not necessarily simultaneously, for small values of the rescale parameter, which is in agreement with the case of a canonical scalar field with absent string corrective terms. The Gauss-Bonnet model is also further constrained, in order to obtain a propagation velocity of tensor perturbations which coincides with that of light, according to the recent observations from the GW170817. As a result of this assumption the degrees of freedom of the theory are reduced. An interesting feature which arises from the overall phenomenology is that, due to the inclusion of string corrections the tensor spectral index of primordial perturbations is now capable of obtaining a positive value which is not possible in the case of the canonical scalar field. Last but not least, the power-law model which is known to be incompatible with observations is now rendered viable by including a parity violating term and as showcased, it satisfies the Swampland criteria as well.