Late-times asymptotic equation of state for a class of nonlocal theories of gravity

We investigate the behavior of the asymptotic late-times effective equation of state for a class of nonlocal theories of gravity. These theories modify the Einstein-Hilbert Lagrangian introducing terms containing negative powers of the d'Alembert operator acting on the Ricci scalar. We find that imposing vanishing initial conditions for the nonlocal content during the radiation-dominated epoch implies the same asymptotic late-times behavior for most of these models. In terms of the effective equation of state of the Universe, we find that asymptotically omega(eff) -> -1, approaching the value given by a cosmological constant. On the other hand, unlike in the case of Lambda CDM, the Hubble factor is a monotonic growing function that diverges asymptotically. We argue that this behavior is not a coincidence and discuss under which conditions this is to be expected in these nonlocal models.