Title: Reconstructing a $f(R)$ theory from the $α$-Attractors,
Abstract: We show an analogy at high curvature between a $f(R) = R + aR^{n - 1} + bR^2$
theory and the $\alpha$-Attractors. We calculate the expressions of the
parameters $a$, $b$ and $n$ as functions of $\alpha$ and the predictions of the
model $f(R) = R + aR^{n - 1} + bR^2$ on the scalar spectral index $n_{\rm s}$
and the tensor-to-scalar ratio $r$. We find that the power law correction $R^{n
- 1}$ allows for a production of gravitational waves enhanced with respect to
the one in the Starobinsky model, while maintaining a viable prediction on
$n_{\rm s}$. We numerically reconstruct the full $\alpha$-Attractors class of
models testing the goodness of our high-energy approximation $f(R) = R + aR^{n
- 1} + bR^2$. Moreover, we also investigate the case of a single power law
$f(R) = \gamma R^{2 - \delta}$ theory, with $\gamma$ and $\delta$ free
parameters. We calculate analytically the predictions of this model on the
scalar spectral index $n_{\rm s}$ and the tensor-to-scalar ratio $r$ and the
values of $\delta$ which are allowed from the current observational results. We
find that $-0.015 < \delta < 0.016$, confirming once again the excellent
agreement between the Starobinsky model and observation.