We discuss a nonlocal modification of gravity obtained adding a term m 2 R□ −2 R to the Einstein-Hilbert action. We find that the mass parameter m only affects the nonradiative sector of the theory, while the graviton remains massless, there is no propagating ghostlike degree of freedom, no vDVZ discontinuity, and no Vainshtein radius below which the theory becomes strongly coupled. For m ¼ OðH 0 Þ the theory therefore recovers all successes of GR at solar system and lab scales, and only deviates from it at cosmological scales. We examine the cosmological consequences of the model and we find that it automatically generates a dynamical dark energy and a self-accelerating evolution. After fixing our only free parameter m so to reproduce the observed value of the dark energy density today, we get a pure prediction for the dark energy equation of state, w DE ≃ −1.14. This value is consistent with the existing data, and could also resolve the possible tension between the Planck data and local measurements of the Hubble parameter.
Most existing theories of dark energy and/or modified gravity, involving a scalar degree of freedom, can be conveniently described within the framework of the Effective Theory of Dark Energy, based on the unitary gauge where the scalar field is uniform. We extend this effective approach by allowing the Lagrangian in unitary gauge to depend on the time derivative of the lapse function. Although this dependence generically signals the presence of an extra scalar degree of freedom, theories that contain only one propagating scalar degree of freedom, in addition to the usual tensor modes, can be constructed by requiring the initial Lagrangian to be degenerate. Starting from a general quadratic action, we derive the dispersion relations for the linear perturbations around Minkowski and a cosmological background. Our analysis directly applies to the recently introduced Degenerate Higher-Order Scalar-Tensor (DHOST) theories. For these theories, we find that one cannot recover a Poisson-like equation in the static linear regime except for the subclass that includes the Horndeski and so-called "beyond Horndeski" theories. We also discuss Lorentz-breaking models inspired by Horava gravity.
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