The R + R², shortly named “R²” (“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 for subsequent creation and final thermalization of the standard matter. Moreover, it produces a very good fit of the observed spectrum of primordial perturbations. In the present paper we show explicitly that the R² inflationary spacetime is an exact solution of a range of weakly non-local (quasi-polynomial) gravitational theories, which provide an ultraviolet completion of the R² 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 responsible for driving the inflation. Notably, any further extension of the spectrum leads to propagating ghost degrees of freedom. We are aimed at presenting a detailed construction of such theories in the so called Weyl basis. Further, we give a special account to the cosmological implications of this theory by considering perturbations during inflation. The highlight of the non-local model is the prediction of a modified, in comparison to a local R² model, value for the ratio of tensor and scalar power spectra r, depending on the parameters of the theory. The relevant parameters are under control to be successfully confronted with existing observational data. Furthermore, the modified r can surely meet future observational constraints.

In this paper we show that there is a universal prediction for the Newtonian potential for a specific class of infinite derivative, ghost-free, quadratic curvature gravity. We show that in order to make such a theory ghost free at a perturbative level, the Newtonian potential always falls-off as $1/r$ in the infrared limit, while at short distances the potential becomes nonsingular. We provide examples which can potentially test the scale of gravitational nonlocality up to 0.004 eV.

We explore a possibility for the Universe to cross the $w=-1$ cosmological constant barrier for the dark energy state parameter. We consider the Universe as a slowly decaying D3-brane. The D3-brane dynamics are approximately described by a nonlocal string tachyon interaction and the backreaction of gravity is incorporated in the closed string tachyon dynamics. In a local effective approximation this model contains one phantom component and one usual field with a simple polynomial interaction. To understand cosmological properties of this system we study toy models with the same scalar fields but with modified interactions. These modifications admit polynomial superpotentials. We find restrictions on these interactions under which it is possible to reach $w=-1$ from below at large time. Explicit solutions with the dark energy state parameter crossing or not crossing the barrier $w=-1$ at large time are presented.

In this paper we provide the criteria for any generally covariant, parity preserving, and torsion-free theory of gravity to possess a stable de Sitter (dS) or anti–de Sitter (AdS) background. By stability we mean the absence of tachyonic or ghostlike states in the perturbative spectrum that can lead to classical instabilities and violation of quantum unitarity. While we find that the usual suspects, the $F(R)$ and $F(G)$ theories, can indeed possess consistent (A)dS backgrounds, $G$ being the Gauss-Bonnet term, another interesting class of theories, string-inspired infinite derivative gravitational theories, can also be consistent around such curved vacuum solutions. Our study should not only be relevant for quantum gravity and early universe cosmology involving ultraviolet physics, but also for modifications of gravity in the infrared sector vying to replace dark energy.