It is natural to expect a consistent inflationary model of the very early Universe to be an effective theory of quantum gravity, at least at energies much less than the Planck one. For the moment,R+R², or shortlyR², inflation is the most successful in accounting for the latest CMB data from the PLANCK satellite and other experiments. Moreover, recently it was shown to be ultra-violet (UV) complete via an embedding into an analytic infinite derivative (AID) non-local gravity. In this paper, we derive a most general theory of gravity that contributes to perturbed linear equations of motion around maximally symmetric space-times. We show that such a theory is quadratic in the Ricci scalar and the Weyl tensor with AID operators along with the Einstein-Hilbert term and possibly a cosmological constant. We explicitly demonstrate that introduction of the Ricci tensor squared term is redundant. Working in this quadratic AID gravity framework without a cosmological term we prove that for a specified class of space homogeneous space-times, a space of solutions to the equations of motion is identical to the space of backgrounds in a localR²model. We further compute the full second order perturbed action around any background belonging to that class. We proceed by extracting the key inflationary parameters of our model such as a spectral index (n_s), a tensor-to-scalar ratio (r) and a tensor tilt (n_t). It appears thatn_sremains the same as in the localR²inflation in the leading slow-roll approximation, whilerandn_tget modified due to modification of the tensor power spectrum. This class of models allows for any value ofr< 0.07 with a modified consistency relation which can be fixed by future observations of primordialB-modes of the CMB polarization. This makes the UV completeR²gravity a natural target for future CMB probes.

We hereby study the properties of a large class of weakly nonlocal gravitational theories around the (anti–) de Sitter spacetime background. In particular, we explicitly prove that the kinetic operator for the graviton field has the same structure as the one in Einstein-Hilbert theory around any maximally symmetric spacetime. Therefore, the perturbative spectrum is the same as standard general relativity, while the propagator on any maximally symmetric spacetime is a mere generalization of the one from Einstein’s gravity derived and extensively studied in several previous papers. At quantum level the range of theories presented here is superrenormalizable or finite when proper (not affecting the propagator) terms cubic or higher in curvatures are added. Finally, it is proven that for a large class of nonlocal theories, which in their actions do involve neither the Weyl nor the Riemann tensor, the theory is classically equivalent to the Einstein-Hilbert one with a cosmological constant by means of a metric field redefinition at any perturbative order.

In this paper we will study R ²-like inflation in a non-local modification of gravity which contains quadratic in Ricci scalar and Weyl tensor terms with analytic infinite derivative form-factors in the action. It is known that the inflationary solution of the local R + R ² gravity remains a particular exact solution in this model. It was shown earlier that the power spectrum of scalar perturbations generated during inflation in the non-local setup remains the same as in the local R + R ² inflation, whereas the power spectrum of tensor perturbations gets modified due to the non-local Weyl tensor squared term. In the present paper we go beyond 2-point correlators and compute the non-Gaussian parameter f _NL related to 3-point correlations generated during inflation, which we found to be different from those in the original local inflationary model and scenarios alike based on a local gravity. We evaluate non-local corrections to the scalar bi-spectrum which give non-zero contributions to squeezed, equilateral and orthogonal configurations. We show that f _NL ∼ O(1) with an arbitrary sign is achievable in this model based on the choice of form-factors and the scale of non-locality. We present the predictions for the tensor-to-scalar ratio, r, and the tensor tilt, n _t . In contrast to standard inflation in a local gravity, here the possibility n _t > 0 is not excluded. Thus, future CMB data can probe non-local behaviour of gravity at high space-time curvatures.