$R^2$ inflation to probe non-perturbative quantum gravity

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, $R+R^2$, or shortly $R^2$, 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 local $R^2$ model. 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 that $n_s$ remains the same as in the local $R^2$ inflation in the leading slow-roll approximation, while $r$ and $n_t$ get modified due to modification of the tensor power spectrum. This class of models allows for any value of $r<0.07$ with a modified consistency relation which can be fixed by future observations of primordial $B$-modes of the CMB polarization. This makes the UV complete $R^2$ gravity a natural target for future CMB probes.