Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

The exterior Cauchy problem is discussed for the fourth‐order theories of gravity derived from the Lagrangian densities L=(−g)1/2 (R+ (1/2)aR2+bRμν Rμν) −κLm. When b≠0, the Cauchy problem can be solved by the standard method already used in general relativity. When b=0, the problem cannot be formulated as in the case where b≠0, since the corresponding fourth‐order theory is shown to be equivalent to a second‐order scalar–tensor theory. This scalar–tensor theory is proved to coincide with one of the models of gravity proposed by O’Hanlon in order to present a covariant version of the massive dilaton theory suggested by Fujii. This result is generalized: The models of O’Hanlon are shown to be indistinguishable from the fourth‐order theories derived from the Lagrangian densities L=(−g)1/2 F(R)−κLm, where F is any real function such that F″(R) does not identically vanish.

The Cauchy Problem for the R + R 2 Theories of Gravity Without Torsion

We examine holographic complexity in the doubly holographic model introduced in [1, 2] to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the induced higher-curvature gravity action on the brane. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(ℛ) gravity in the bulk.

/pdf/quantum-extremal-islands-made-easy-part-iii-complexity-on-40d9geiekx.pdf

Quantum extremal islands made easy. Part III. Complexity on the brane

Various Hamiltonian formulations of f(R) gravity can be found in the literature. Some authors follow the Ostrogradsky treatment of higher derivative theories and introduce as extra variables first order time derivatives of the metric (typically the extrinsic curvature). Some others take advantage of the conformal equivalence of f(R) theory with Einstein's gravity coupled to a scalar field and introduce as an extra variable the scalar curvature R itself, which includes second time derivatives of the metric. We show that, contrarily to some claims, these formulations are related by canonical transformations.

Various Hamiltonian formulations of f (R) gravity and their canonical relationships

We examine holographic complexity in the doubly holographic model introduced in [arXiv:2006.04851][arXiv:2010.00018] to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the induced higher-curvature gravity action on the brane. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(R) gravity in the bulk.

/pdf/quantum-extremal-islands-made-easy-partiii-complexity-on-the-2qjfuxsue5.pdf

Quantum Extremal Islands Made Easy, PartIII: Complexity on the Brane

Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${\cal L}$ is given in terms of a function of the salar curvature $R$ as ${\cal L}=\sqrt{-\det g_{\mu\nu}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric.

On the Canonical Formalism for a Higher-Curvature Gravity

Following the method of Buchbinder and Lyakhovich, we carry out a canonical formalism for a higher-curvature gravity theory in which the Lagrangian density is given in terms of a function of the scalar curvature as , where f is a nonlinear differentiable function of . We comment on the physical significance of the metric taking the Robertson-Walker metric as an example.

A canonical formalism for a higher-curvature gravity

The stability of the extra-dimensional spaces in higher-curvature gravity theories is investigated using the semiclassical approximation to the Wheeler-DeWitt equation. It is shown that, if there exists only one internal space, the space is stable under a certain condition.

Masahiko Kiminami

Papers

On the Canonical Formalism for a Higher-Curvature Gravity

A canonical formalism for a higher-curvature gravity

Semiclassical approach to stability of the extra-dimensional spaces in higher-curvature gravity theories