Reconstructing the distortion function of non-local cosmology: a model-independent approach

We consider non-local modifications of General Relativity given by a distortion function in terms of the inverse of the d'Alembert operator. The inclusion of these terms is motivated by the possibility of reproducing the current accelerated expansion of the Universe starting from non-local gravity. In particular, we propose a model-independent method, based on current observations, to reconstruct the shape of the distortion function, without resorting to a specific cosmological history. We describe a numerical procedure based on Pad\'e polynomials that allows us to study the time evolution of the auxiliary scalar fields introduced to localize non-local gravity action. Thus, adopting suitable boundary conditions, we reconstruct the form of the non-local Lagrangian and infer the best analytical approximation of the numerical outcome. Furthermore, the distortion function turns on during the matter-dominated era, and its effects are delayed to the most recent cosmic times. This provides a natural explanation for the late cosmic acceleration, avoiding any fine-tuning problem of the cosmological constant. Finally, we compare our predictions with previous findings based on enforcing the standard $\Lambda$CDM background in the reconstruction process.