Higher-order theories of gravity: diagnosis, extraction and reformulation via non-metric extra degrees of freedom—a review

Abstract

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.

Equivalence between Scalar-Tensor theories and f(R)-gravity: from the action to cosmological perturbations

“…We explore directly the equivalence between ST and f (R) theories at the GYH boundary term, and it is clear that the boundary term makes the theory well defined mathematical problem. It is important to notice that in the literature the equivalence problem has been widely studied [10][11][12][13][14][15], but in this paper it was shown how the field equations were obtained for ST theories with the GYH boundary term, in complete agreement with previous work [9,14,15,25], but conecting a previous work [22] through the equivalence in the important issue of the boundary for both theories. Also, the condition to get the equation to f, it had to be imposed on the boundary that the variation δf be equal to zero.…”

“…The equivalence between ST and f (R) theories has been broadly studied at the classical level, e.g., in [10][11][12][13][14][15], but also a quantum level [26,27]. In this paper shows the equivalence between the actions and the field equations, but as we will see in the next section, in addition we will show them in the cosmological perturbations.…”

“…The equivalence between theories ST and f (R) has been studied e.g., in [10][11][12][13][14][15]. It is, starting from the ST action, without the kinetic term of the scalar field, we arrive at the action of the gravity theories f (R).…”

“…Brans y R.H. Dicke in 1961, is a particular case of theories ST, where the parameter ω(f) is independent of the scalar field.Another type of generalization to GR are the theories of gravity f (R) [7][8][9], where the lagrangian of Einstein-Hilbert is generalized, replacing the scalar curvature R by a more general function of it, f (R). The gravitational field in this theory is represented by the metric like GR does.The equivalence between theories ST and f (R) has been studied e.g., in [10][11][12][13][14][15]. It is, starting from the ST action, without the kinetic term of the scalar field, we arrive at the action of the gravity theories f (R).…”

Asymptotically Weyl-invariant gravity

“…Gravity is said to be perturbatively non-renormalizable, a fact demonstrated by explicit calculation at the one-loop level including matter content [2], and at the two-loop level without matter [3]. This behaviour stems from the fact that the gravitational coupling introduces a length scale into the theory so that higher-order corrections in a perturbative expansion come with ever-increasing powers of the cut-off scale.Higher-order actions have been explored as a possible solution to this problem [4]. Explicit calculations show that Lagrangians quadratic in the curvature tensor are renormalizable [5].…”