Hamiltonian Analysis of Nonlocal F(R) Gravity Models

Abstract

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. I. INTRODUCTIONThe theoretical explanation of the ongoing accelerated expansion of the universe[1-6] is one of the most unsolved cosmological problems. Achieving accelerated expansion by adding a constant to Einstein-Hilbert(E-H) action suffers from fine-tuning problem [7]. There are many ways to explain the accelerated epoch of the current universe. In this context modified gravity theories have been developed by modifying the E-H action, for example, F(R) theories of gravity [8][9][10][11][12][13]. In recent times another class of most popular modified gravity theories is nonlocal gravity models [14,15]. These models are motivated by Einstein action's ultraviolet (UV) and infrared (IR) corrections. They also provide a theoretical explanation of the accelerated expansion of the current universe. Some salient features of nonlocal gravity models are: (1) they can be employed to study cosmology in both the infrared and the ultraviolet, (2) valid cosmological perturbation theory, and (3) the resulting cosmology has good agreement with most observations. In this direction, the nonlocal terms with Ricci scalar R and F (R) are extensively studied along with Einstein Hilbert action. These terms involve analytic transcendental functions of the covariant d'Alembert operator . Initially, as developed by Wetterich, models with correction terms like R −1 R are shown to be effective IR corrected nonlocal gravity model [16]. Further, Deser and Woodard[17] introduce a general form asRf ( 1 R) which can be responsible for the late-time cosmic expansion of the universe. In this line, for a recent review on this topic, refer to [18].The nonlocal theory can have an equivalent scalar-tensor form by introducing auxiliary variables. Higher derivative terms with d'Alembertian operator and curvature can be considered higher derivative field theories. According to Ostrogradsky theorem[19], non-degenerate higher derivative Lagrangians are cursed with instabilities (popularly known Ostrogradsky instabilities). Generally, these instabilities are easy to identify by linear momentum terms within the Hamiltonian, which make it unbounded above and below depending on the structure [20]. However degenerate theories [21][22][23][24][25][26][27][28][29][30](for review, refer to [31,32]) in this regard is free from these instabilities by reducing the phase space non trivially [33] In order to check the appearance of Ostrogradsky instability in any higher derivative theory, a Hamiltonian analysis would be required. The Hamiltonian analysis of these types of theories can be performed by the Dirac method of constraint system [34][35][36][37][38]. The Hamiltonian analysis of the nonlocal model with inverse powers of d'Alembertian operators acting only on Ricci scalar is performed in [39]. A more general analysis of constructing Hami...