Inflationary Attractors in $F(R)$ Gravity
  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. Moreover, in the case
that $x$ takes larger values in specific ranges and is constant for a given
$F(R)$ gravity, the resulting $r-n_s$ relation has the form $r\sim 3 \alpha
(1-n_s)^2$, 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.
