| Sumario: | In this letter we shall demonstrate that the viable F (R) gravities can be classified mainly into two classes of inflationary attractors, either the R2 attractors or the α-attractors. To show this, we shall derive the most general relation between the tensor-to-scalar ratio r and the spectral index of primordial curvature perturbations ns , namely the r −ns relation, by assuming that the slow-roll condition constrains the values of the slow-roll indices. As we show, the relation between the tensor-to-scalar ratio and the spectral index of the primordial curvature perturbations has the form r 48(1−ns )2 (4−x)2 , where the dimensionless parameter x contains higher derivatives of the F (R) gravity function with respect to the Ricci scalar, and it is a function of the e-foldings number N and may also be a function of the free parameters of the various F (R) gravity models. For F (R) gravities which have a spectral index compatible with the observational data and also yield x << 1, these belong to the R2-type of attractors, with r ∼ 3(1 − ns )2, and these are viable theories. Moreover, in the case that x takes larger values in specific ranges and is constant for a given F (R) gravity, the resulting r − ns relation has the form r ∼ 3α(1 −ns )2, where α is a constant. Thus we conclude that the viable F (R) gravities may be classified into two limiting types of r − ns relations, one identical to the R2 model at leading order in x, and one similar to the α-attractors r − ns relation, for the F (R) gravity models that yield x constant. Finally, we also discuss the case that x is not constant.
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