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

Abstract

We show an analogy at high curvature between a f(R) = R + aRn − 1 + bR2 theory and the α-Attractors. We calculate the expressions of the parameters a, b and n as functions of α and the predictions of the model f(R) = R + aRn − 1 + bR2 on the scalar spectral index ns and the tensor-to-scalar ratio r. We find that the power law correction Rn − 1 allows for a production of gravitational waves enhanced with respect to the one in the Starobinsky model, while maintaining a viable prediction on ns. We numerically reconstruct the full α-Attractors class of models testing the goodness of our high-energy approximation f(R) = R + aRn − 1 + bR2. Moreover, we also investigate the case of a single power law f(R) = γ R2 − δ theory, with γ and δ free parameters. We calculate analytically the predictions of this model on the scalar spectral index ns and the tensor-to-scalar ratio r and the values of δ which are allowed from the current observational results. We find that −0.015 < δ < 0.016, confirming once again the excellent agreement between the Starobinsky model and observation.