Inflationary Attractors in $F(R)$ Gravity

Understand

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 $r\simeq \frac{48 (1-n_s)^2}{(4-x)^2}$, 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 $x\ll 1$, these belong to the $R^2$-type of attractors, with $r\sim 3 (1-n_s)^2$, and these are viable theories.