Reconstructing a $f(R)$ theory from the $\alpha$-Attractors

Abstract

We show an analogy at high curvature between a $f(R)=R+a{R}^{n-1}+b{R}^2$ 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 $f(R)=R+a{R}^{n-1}+b{R}^2$ on the scalar spectral index $n_{\mathrm{s}}$ 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 $n_{\mathrm{s}}$. We numerically reconstruct the full $\alpha$-Attractors class of models testing the goodness of our high-energy approximation $f(R)=R+a{R}^{n-1}+b{R}^2$. 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 $n_{\mathrm{s}}$ 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.

17 pages, 8 figures. V2: full numerical reconstruction in the appendix. Accepted for publication in JCAP

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