Non-Local Gravity Cosmology: an Overview

We discuss some main aspects of theories of gravity containing non-local terms in view of cosmological applications. In particular, we consider various extensions of General Relativity based on geometrical invariants as $f(R, \Box^{-1} R)$, $f({\cal G}, \Box^{-1} {\cal G})$ and $f(T, \Box^{-1} T)$ gravity where $R$ is the Ricci curvature scalar, $\cal G$ is the Gauss-Bonnet topological invariant, $T$ the torsion scalar and the operator $\Box^{-1}$ gives rise to non-locality. After selecting their functional form by using Noether Symmetries, we find out exact solutions in a cosmological background. It is possible to reduce the dynamics of selected models and to find analytic solutions for the equations of motion. As a general feature of the approach, it is possible to address the accelerated expansion of the Hubble flow at various epochs, in particular the dark energy issues, by taking into account non-locality corrections to the gravitational Lagrangian. On the other hand, it is possible to search for gravitational non-local effects also at astrophysical scales. In this perspective, we search for symmetries of $f(R, \Box^{-1} R)$ gravity also in a spherically symmetric background and constrain the free parameters, Specifically, by taking into account the S2 star orbiting around the Galactic Centre SgrA$^*$, it is possible to study how non-locality affects stellar orbits around such a massive self-gravitating object.