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

We show an analogy at high curvature between a f (R) = R + aR(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 Rn-1 + b R-2 on the scalar spectral index n 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 s. We numerically reconstruct the full alpha -Attractorsclass of models testing the goodness of our high-energy approximation f (R) = R + a Rn-1 + b R-2. Moreover, we also investigate the case of a single power law f (R) = gamma R2-delta theory, with gamma and delta free parameters. We calculate analytically the predictions of this model on the scalar spectral index n 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.