Abstract. We show an analogy at high curvature between a f (R) = R + aR n−1 + bR 2 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+aR n−1 +bR 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 α-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) = γR 2−δ theory, with γ and δ 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 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.

We address the issue of the existence of inequivalent definitions of gravitational mass in R 2 -gravity. We present several definitions of gravitational mass, and discuss the formal relations between them. We then consider the concrete case of a static and spherically symmetric neutron star, and solve numerically the equations of motion for several values of the free parameter of the model. We compare the features of the massradius relations obtained for each definition of gravitational mass, and we comment on their dependence on the free parameter. We then argue that R 2 -gravity is a valuable proxy to discuss the existence of inequivalent definitions of gravitational mass in a generic modified gravity theory, and present some comments on the general case. *

In this paper we establish formulae for the inflationary slow-roll parameters , η and ζ as functions of the Ricci scalar R for f (R) theories of gravity. As examples, we present the analytic and numerical solutions of , η and ζ as functions of the number of e-folds N in two important instances: for the Starobinsky model and for a f (R) reconstruction of the α-Attractors. The highlight of our proposal is to rewrite the slow-roll parameters in terms of f (R), which allows to find directly n s , r, α s and f equil NL as functions of R itself. We obtain that both models indicate a small contribution to the non-Gaussianity parameters, which are in good agreement with current observational constraints.