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Modifications of Einstein's theory of gravitation have been extensively considered in the past years, in connection to both cosmology and quantum gravity. Higher-curvature and higher-derivative gravity theories constitute the main examples of such modifications. These theories exhibit, in general, more degrees of freedom than those found in standard General Relativity; counting, identifying, and retrieving the description/representation of such dynamical variables is currently an open problem, and a decidedly nontrivial one. In this work we review, via both formal arguments and custom-made examples, the most relevant methods to unveil the gravitational degrees of freedom of a given model, discussing the merits, subtleties and pitfalls of the various approaches., 85 pages; typos corrected, references added; conclusions improved

Between the microscopic domain ruled by quantum gravity, and the macroscopic scales described by general relativity, there might be an intermediate, "mesoscopic" regime, where spacetime can still be approximately treated as a differentiable pseudo-Riemannian manifold, with small corrections of quantum gravitational origin. We argue that, unless one accepts to give up the relativity principle, either such a regime does not exist at all (hence, the quantum-to-classical transition is sharp), or the only mesoscopic, tiny corrections conceivable are on the behaviour of physical fields, rather than on the geometric structures., Comment: 7 pages, no figures

We propose the almost-geodesic motion of self-gravitating test bodies as a possible selection rule among metric theories of gravity. Starting from a heuristic statement, the "gravitational weak equivalence principle", we build a formal, operative test able to probe the validity of the principle for any metric theory of gravity, in an arbitrary number of spacetime dimensions. We show that, if the theory admits a well-posed variational formulation, this test singles out only the purely metric theories of gravity. This conclusion reproduces known results in the cases of general relativity (also with a cosmological constant term), and scalar-tensor theories, but extends also to debated or unknown scenarios, such as f(R) and Lanczos-Lovelock theories. We thus provide new tools going beyond the standard methods, where the latter turn out to be inconclusive or inapplicable., 17 pages, no figures. Tiny modifications to match the published version

Equivalence principles played a central role in the development of general relativity. Furthermore, they have provided operative procedures for testing the validity of general relativity, or constraining competing theories of gravitation. This has led to a flourishing of different, and inequivalent, formulations of these principles, with the undesired consequence that often the same name, "equivalence principle", is associated with statements having a quite different physical meaning. In this paper we provide a precise formulation of the several incarnations of the equivalence principle, clarifying their uses and reciprocal relations. We also discuss their possible role as selecting principles in the design and classification of viable theories of gravitation., Comment: 11 pages, no figures