We have reviewed many aspects of f (R) theories studied extensively over the past decade. This burst of activities is strongly motivated by the observational discovery of dark energy. The idea is that the gravitational law may be modified on cosmological scales to give rise to the late-time acceleration, while Newton’s gravity needs to be recovered on solar-system scales. In fact, f (R) theories can be regarded as the simplest extension of General Relativity.

The possibility of the late-time cosmic acceleration in metric f (R) gravity was first suggested by Capozziello in 2002 [113]. Even if f (R) gravity looks like a simple theory, successful f (R) dark energy models need to satisfy a number of conditions for consistency with successful cosmological evolution (a late-time accelerated epoch preceded by a matter era) and with local gravity tests on solar-system scales. We summarize the conditions under which metric f (R) dark energy models are viable:

- 1.
- for , where is the Ricci scalar today. This is required to avoid a ghost state.
- 2.
- for . This is required to avoid the negative mass squared of a scalar-field degree of freedom (tachyon).
- 3.
- for . This is required for the presence of the matter era and for consistency with local gravity constraints.
- 4.
- at . This is required for the stability and the presence of a late-time de Sitter solution. Note that there is another fixed point that can be responsible for the cosmic acceleration (with an effective equation of state ).

We clarified why the above conditions are required by providing detailed explanation about the background cosmological dynamics (Section 4), local gravity constraints (Section 5), and cosmological perturbations (Sections 6 – 8).

After the first suggestion of dark energy scenarios based on metric f (R) gravity, it took almost five years to construct viable models that satisfy all the above conditions [26, 382, 31, 306, 568, 35, 587]. In particular, the models (4.83), (4.84), and (4.89) allow appreciable deviation from the CDM model during the late cosmological evolution, while the early cosmological dynamics is similar to that of the CDM. The modification of gravity manifests itself in the evolution of cosmological perturbations through the change of the effective gravitational coupling. As we discussed in Sections 8 and 13, this leaves a number of interesting observational signatures such as the modification to the galaxy and CMB power spectra and the effect on weak lensing. This is very important to distinguish f (R) dark energy models from the CDM model in future high-precision observations.

As we showed in Section 2, the action in metric f (R) gravity can be transformed to that in the Einstein frame. In the Einstein frame, non-relativistic matter couples to a scalar-field degree of freedom (scalaron) with a coupling of the order of unity (). For the consistency of metric f (R) gravity with local gravity constraints, we require that the chameleon mechanism [344, 343] is at work to suppress such a large coupling. This is a non-linear regime in which the linear expansion of the Ricci scalar into the (cosmological) background value and the perturbation is no longer valid, that is, the condition holds in the region of high density. As long as a spherically symmetric body has a thin-shell, the effective matter coupling is suppressed to avoid the propagation of the fifth force. In Section 5 we provided detailed explanation about the chameleon mechanism in f (R) gravity and showed that the models (4.83) and (4.84) are consistent with present experimental bounds of local gravity tests for .

The construction of successful f (R) dark energy models triggered the study of spherically symmetric solutions in those models. Originally it was claimed that a curvature singularity present in the models (4.83) and (4.84) may be accessed in the strong gravitational background like neutron stars [266, 349]. Meanwhile, for the Schwarzschild interior and exterior background with a constant density star, one can approximately derive analytic thin-shell solutions in metric f (R) and Brans–Dicke theory by taking into account the backreaction of gravitational potentials [594]. In fact, as we discussed in Section 11, a static star configuration in the f (R) model (4.84) was numerically found both for the constant density star and the star with a polytropic equation of state [43, 600, 42]. Since the relativistic pressure is strong around the center of the star, the choice of correct boundary conditions along the line of [594] is important to obtain static solutions numerically.

The model proposed by Starobinsky in 1980 is the first model of inflation in the early universe. Inflation occurs in the regime , which is followed by the reheating phase with an oscillating Ricci scalar. In Section 3 we studied the dynamics of inflation and (p)reheating (with and without nonminimal couplings between a field and ) in detail. As we showed in Section 7, this model is well consistent with the WMAP 5-year bounds of the spectral index of curvature perturbations and of the tensor-to-scalar ratio . It predicts the values of smaller than the order of 0.01, unlike the chaotic inflation model with . It will be of interest to see whether this model continues to be favored in future observations.

Besides metric f (R) gravity, there is another formalism dubbed the Palatini formalism in which the metric and the connection are treated as independent variables when we vary the action (see Section 9). The Palatini f (R) gravity gives rise to the specific trace equation (9.2) that does not have a propagating degree of freedom. Cosmologically we showed that even for the model (, ) it is possible to realize a sequence of radiation, matter, and de Sitter epochs (unlike the same model in metric f (R) gravity). However the Palatini f (R) gravity is plagued by a number of shortcomings such as the inconsistency with observations of large-scale structure, the conflict with Standard Model of particle physics, and the divergent behavior of the Ricci scalar at the surface of a static spherically symmetric star with a polytropic equation of state with . The only way to avoid these problems is that the f (R) models need to be extremely close to the CDM model. This property is different from metric f (R) gravity in which the deviation from the CDM model can be significant for of the order of the Ricci scalar today.

In Brans–Dicke (BD) theories with the action (10.1), expressed in the Einstein frame, non-relativistic matter is coupled to a scalar field with a constant coupling . As we showed in in Section 10.1, this coupling is related to the BD parameter with the relation . These theories include metric and Palatini f (R) gravity theories as special cases where the coupling is given by (i.e., ) and (i.e., ), respectively. In BD theories with the coupling of the order of unity we constructed a scalar-field potential responsible for the late-time cosmic acceleration, while satisfying local gravity constraints through the chameleon mechanism. This corresponds to the generalization of metric f (R) gravity, which covers the models (4.83) and (4.84) as specific cases. We discussed a number of observational signatures in those models such as the effects on the matter power spectrum and weak lensing.

Besides the Ricci scalar , there are other scalar quantities such as and constructed from the Ricci tensor and the Riemann tensor . For the Gauss–Bonnet (GB) curvature invariant one can avoid the appearance of spurious spin-2 ghosts. There are dark energy models in which the Lagrangian density is given by , where is an arbitrary function in terms of . In fact, it is possible to explain the late-time cosmic acceleration for the models such as (12.16) and (12.17), while at the same time local gravity constraints are satisfied. However density perturbations in perfect fluids exhibit violent negative instabilities during both the radiation and the matter domination, irrespective of the form of . The growth of perturbations gets stronger on smaller scales, which is incompatible with the observed galaxy spectrum unless the deviation from GR is very small. Hence these models are effectively ruled out from this Ultra-Violet instability. This implies that metric f (R) gravity may correspond to the marginal theory that can avoid such instability problems.

In Section 13 we discussed other aspects of f (R) gravity and modified gravity theories – such as weak lensing, thermodynamics and horizon entropy, Noether symmetry in f (R) gravity, unified f (R) models of inflation and dark energy, f (R) theories in extra dimensions, Vainshtein mechanism, DGP model, and Galileon field. Up to early 2010 the number of papers that include the word “f (R)” in the title is over 460, and more than 1050 papers including the words “f (R)” or “modified gravity” or “Gauss–Bonnet” have been written so far. This shows how this field is rich and fruitful in application to many aspects to gravity and cosmology.

Although in this review we have focused on f (R) gravity and some extended theories such as BD theory and Gauss–Bonnet gravity, there are other classes of modified gravity theories, e.g., Einstein–Aether theory [325], tensor-vector-scalar theory of gravity [76], ghost condensation [38], Lorentz violating theories [144, 282, 389], and Hořava–Lifshitz gravity [305]. There are also attempts to study f (R) gravity in the context of Hořava–Lifshitz gravity [346, 347]. We hope that future high-precision observations can distinguish between these modified gravity theories, in connection to solving the fundamental problems for the origin of inflation, dark matter, and dark energy.

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