Nonlocal gravity models are constructed to explain the current acceleration of the universe. These models are inspired by the infrared correction appearing in Einstein–Hilbert action. Here, we develop the Hamiltonian formalism of a nonlocal model by considering only terms to quadratic order in Riemann tensor, Ricci tensor and Ricci scalar. We show how to count degrees of freedom using Hamiltonian formalism including Ricci tensor and Ricci scalar terms. In this model, we have also worked out with a choice of a nonlocal action which has only two degrees of freedom equivalent to GR. Finally, we find the existence of additional constraints in Hamiltonian required to remove the ghosts in our full action. We also compare our results with that of obtained using Lagrangian formalism.

Ostrogradsky instability generally appears in nondegenerate higher-order derivative theories and this issue can be resolved by removing any existing degeneracy present in such theories. We consider an action involving terms that are at most quadratic in second derivatives of the scalar field and non-minimally coupled with the curvature tensors. We perform a 3+1 decomposition of the Lagrangian to separate second-order time derivative terms from rest. This decomposition is useful for checking the degeneracy hidden in the Lagrangian and helps us find conditions under which Ostrogradsky instability does not appear. We show that our construction of Lagrangian resembles that of a GR-like theory for a particular case in the unitary gauge. As an example, we calculate the equation of motion for the flat FRW. We also write the action for open and closed cases, free from higher derivatives for a particular choice derived from imposing degeneracy conditions.

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We construct a Hamiltonian for the nonlocal F(R) theory in the present work. By this construction, we demonstrate the nature of the ghost degrees of freedom. Finally, we find conditions that give rise to ghost-free theories.