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The $R+R^2$, shortly named "$R^2$" ("Starobinsky") inflationary model,represents a fully consistent example of a one-parameter inflationary scenario.This model has a "graceful exit" from inflation and provides a mechanism forsubsequent creation and final thermalization of the standard matter. Moreover,it produces a very good fit of the observed spectrum of primordialperturbations. In the present paper we show explicitly that the $R^2$inflationary spacetime is an exact solution of a range of weakly non-local(quasi-polynomial) gravitational theories, which provide an ultravioletcompletion of the $R^2$ theory. These theories are ghost-free,super-renormalizable or finite at quantum level, and perturbatively unitary.Their spectrum consists of the graviton and the scalaron that is responsiblefor driving the inflation. Notably, any further extension of the spectrum leadsto propagating ghost degrees of freedom. We are aimed at presenting a detailedconstruction of such theories in the so called Weyl basis. Further, we give aspecial account to the cosmological implications of this theory by consideringperturbations during inflation. The highlight of the non-local model is theprediction of a modified, in comparison to a local $R^2$ model, value for theratio of tensor and scalar power spectra $r$, depending on the parameters ofthe theory. The relevant parameters are under control to be successfullyconfronted with existing observational data. Furthermore, the modified $r$ cansurely meet future observational constraints.

Occurrence of exact $R^2$ inflation in non-local UV-complete gravity