Inflationary attractors in F(R) gravity

摘要

In this letter we shall demonstrate that the viable F(R) gravities can be classified mainly into two classes of inflationary attractors, either the R-2 attractors or the alpha-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 n(s), namely the r - n(s) 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 similar or equal to 48((1-n(s))(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 R-2-type of attractors, with r similar to 3(1 - n(s))(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 - 3 alpha(1 - n(s))(2), where alpha is a constant. Thus we conclude that the viable F(R) gravities may be classified into two limiting types of r - n(s) relations, one identical to the R-2 model at leading order in x, and one similar to the alpha-attractors r - n(s) relation, for the F(R) gravity models that yield x constant. Finally, we also discuss the case that x is not constant. (C) 2020 The Author(s). Published by Elsevier B.V.