Boundary term in metric <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity: field equations in the metric formalism

The main goal of this paper is to get in a straightforward form the field equations in metric f (R) gravity, using elementary variational principles and adding a boundary term in the action, instead of the usual treatment in an equivalent scalar–tensor approach. We start with a brief review of the Einstein–Hilbert action, together with the Gibbons–York–Hawking boundary term, which is mentioned in some literature, but is generally missing. Next we present in detail the field equations in metric f (R) gravity, including the discussion about boundaries, and we compare with the Gibbons–York–Hawking term in General Relativity. We notice that this boundary term is necessary in order to have a well defined extremal action principle under metric variation.     

Energy distribution in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

The well-known energy problem is discussed in f (R) theory of gravity. We use the generalized Landau–Lifshitz energy–momentum complex in the framework of metric f (R) gravity to evaluate the energy density of plane symmetric solutions for some general f (R) models. In particular, this quantity is found for some popular choices of f (R) models. The constant scalar curvature condition and the stability condition for these models are also discussed. Further, we investigate the energy distribution of cosmic string spacetime.     

The Post-Minkowskian Limit of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity

We formally discuss the post-Minkowskian limit of f(R)-gravity without adopting conformal transformations but developing all the calculations in the original Jordan frame. It is shown that such an approach gives rise, in general, together with the standard massless graviton, to massive scalar modes whose masses are directly related to the analytic parameters of the theory. In this sense, the presence of massless gravitons only is a peculiar feature of General Relativity. This fact is never stressed enough and could have dramatic consequences in detection of gravitational waves. Finally the role of curvature stress-energy tensor of f(R)-gravity is discussed showing that it generalizes the so called Landau-Lifshitz tensor of General Relativity. The further degrees of freedom, giving rise to the massive modes, are directly related to the structure of such a tensor.     

Probing the dark matter issue in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity via gravitational lensing

For a general class of analytic f(R)-gravity theories, we discuss the weak field limit in view of gravitational lensing. Though an additional Yukawa term in the gravitational potential modifies dynamics with respect to the standard Newtonian limit of General Relativity, the motion of massless particles results unaffected thanks to suitable cancellations in the post-Newtonian limit. Thus, all the lensing observables are equal to the ones known from General Relativity. Since f(R)-gravity is claimed, among other things, to be a possible solution to overcome for the need of dark matter in virialized systems, we discuss the impact of our results on the dynamical and gravitational lensing analyses. In this framework, dynamics could, in principle, be able to reproduce the astrophysical observations without recurring to dark matter, but in the case of gravitational lensing we find that dark matter is an unavoidable ingredient. Another important implication is that gravitational lensing, in the post-Newtonian limit, is not able to constrain these extended theories, since their predictions do not differ from General Relativity.     

Gravitational lensing and <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) theories in the Palatini approach

We investigate gravitational lensing in the Palatini approach to the f (R) extended theories of gravity. Starting from an exact solution of the f (R) field equations, which corresponds to the Schwarzschild–de Sitter metric and, on the basis of recent studies on this metric, we focus on some lensing observables, in order to evaluate the effects of the nonlinearity of the gravity Lagrangian. We give estimates for some astrophysical events, and show that these effects are tiny for galactic lenses, but become interesting for extragalactic ones.     

Spherical Symmetric Solution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Model Around Charged Black Hole

A static, asymptotically flat, spherically symmetric solutions is investigated in f(R) theories of gravity for a charged black hole. We have studied the weak field limit of f(R) gravity for the some f(R) model such as f(R)=R+ε h(R). In particular, we consider the case lim  _R→0 h(R)/h′(R)→0 and find the space time metric for f(R)=R+[(m⁴)/(R)]f(R)=R+{\mu^{4}\over R} and f(R)=R ^1+ε theories of gravity far away a charged mass point.     

Strong energy condition and the repulsive character of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

The Raychaudhuri equation enables to examine the whole spacetime structure without specific solutions of Einstein’s equations, playing a central role for the understanding of the gravitational interaction in cosmology. In General Relativity, without considering a cosmological constant, a non-positive contribution in the Raychaudhuri equation is usually interpreted as the manifestation of the attractive character of gravity. In this case, particular energy conditions—indeed the strong energy condition—must be assumed in order to guarantee the attractive character. In the context of f(R) gravity, however, even assuming the standard energy conditions one may have a positive contribution to the Raychaudhuri equation. Besides providing a simple way to explain the observed cosmic acceleration, this fact opens the possibility of a repulsive character of this kind of gravity. In order to discuss physical bounds on f(R) models, we address the attractive/non-attractive character of f(R) gravity considering the Raychaudhuri equation and assuming the strong energy condition along with recent estimates of the cosmographic parameters.     

New types of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) gravity

Recently f(T) theories based on modifications of teleparallel gravity, where torsion is the geometric object describing gravity instead of curvature, have been proposed to explain the present cosmic accelerating expansion. The field equations are always second order, remarkably simpler than f(R) theories. In analogy to the f(R) theory, we consider here three types of f(T) gravity, and find that all of them can give rise to cosmic acceleration with interesting features, respectively.     

Exploring plane-symmetric solutions in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

The modified theories of gravity, especially the f(R) gravity, have attracted much attention in the last decade. This paper is devoted to exploring plane-symmetric solutions in the context of metric f(R) gravity. We extend the work on static plane-symmetric vacuum solutions in f(R) gravity already available in the literature [1, 2]. The modified field equations are solved using the assumptions of both constant and nonconstant scalar curvature. Some well-known solutions are recovered with power-law and logarithmic forms of f(R) models.     

Stability analysis of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-AdS black holes

We study the stability of the f(R)-AdS (Schwarzschild–AdS) black hole obtained from f(R) gravity. In order to resolve the difficulty of solving fourth-order linearized equations, we transform f(R) gravity into scalar–tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a dynamical scalaron, showing that all linearized equations are second order. Using the positivity of gravitational potentials and S-deformed technique allows us to guarantee the stability of f(R)-AdS black hole if the scalaron mass squared satisfies the Breitenlohner–Freedman bound. This is confirmed by computing quasinormal frequencies of the scalaron for the f(R)-AdS black hole.     

Non-vacuum Bianchi types <Emphasis Type="Italic">I</Emphasis> and <Emphasis Type="Italic">V</Emphasis> in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

In a recent paper (Sharif and Shamir in Class. Quantum Grav. 26:235020, 2009), we have studied the vacuum solutions of Bianchi types I and V spacetimes in the framework of metric f (R) gravity. Here we extend this work to perfect fluid solutions. For this purpose, we take stiff matter to find energy density and pressure of the universe. In particular, we find two exact solutions in each case which correspond to two models of the universe. The first solution gives a singular model while the second solution provides a non-singular model. The physical behavior of these models has been discussed using some physical quantities. Also, the function of the Ricci scalar is evaluated.     

Dynamics of anisotropic power-law <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) cosmology

Modified theories of gravity have attracted much attention of the researchers in the recent years. In particular, the f(R) theory has been investigated extensively due to important f(R) gravity models in cosmological contexts. This paper is devoted to exploring an anisotropic universe in metric f(R) gravity. A locally rotationally symmetric Bianchi type I cosmological model is considered for this purpose. Exact solutions of modified field equations are obtained for a well-known f(R) gravity model. The energy conditions are also discussed for the model under consideration. The viability of the model is investigated via graphical analysis using the present-day values of cosmological parameters. The model satisfies null energy, weak energy, and dominant energy conditions for a particular range of the anisotropy parameter while the strong energy condition is violated, which shows that the anisotropic universe in f(R) gravity supports the crucial issue of accelerated expansion of the universe.     

Reconstruction of some cosmological models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>,<Emphasis Type="Italic">T</Emphasis>) cosmology

In this paper, we reconstruct cosmological models in the framework of f(R,T) gravity, where R is the Ricci scalar and T is the trace of the stress-energy tensor. We show that the dust fluid reproduces ΛCDM, phantom–non-phantom era and phantom cosmology. Further, we reconstruct different cosmological models, including the Chaplygin gas, and scalar field with some specific forms of f(R,T). Our numerical simulation for the Hubble parameter shows good agreement with the BAO observational data for low redshifts, z<2.     

<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) Cosmology from <Emphasis Type="Italic">q</Emphasis>-theory

From a macroscopic theory of the quantum vacuum in terms of conserved relativistic charges (generically denoted by q ^(a) with label a), we have obtained, in the low-energy limit, a particular type of f(R) model relevant to cosmology. The macroscopic quantum-vacuum theory allows us to distinguish between different phenomenological f(R) models on physical grounds. The text was submitted by the authors in English.     

Future cosmological evolution in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity using two equations of state parameters

We investigate the issues of future oscillations around the phantom divide (FOPD) for f(R) gravity. For this purpose, we introduce two types of energy density and pressure arisen from the f(R)-higher order curvature terms. One has the conventional energy density and pressure even in the beginning of the Jordan frame, whose continuity equation defines the native equation of state w _DE. On the other hand, the other has the different energy density and pressure which do not obviously satisfy the continuity equation. This needs to introduce the effective equation of state w _eff to describe the f(R)-fluid, in addition to the native equation of state [(w)\tilde]_DE\tilde{w}_{\mathrm{DE}}. We show that the FOPD occur in f(R) gravities by introducing two types of equation of state. Finally, we point out that the singularity appears ar x=x _c because the stability condition of f(R) gravity violates.     

Running coupling in electroweak interactions of leptons from <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>)-gravity with torsion

The f(R)-gravitational theory with torsion is considered for one family of leptons; it is found that the torsion tensor gives rise to interactions having the structure of the weak forces, while the intrinsic non-linearity of the f(R) function provides an energy-dependent coupling: in this way, torsional f(R) gravity naturally generates both structure and strength of the electroweak interactions among leptons. This implies that the weak interactions among the lepton fields could be addressed as a geometric effect due to the interactions among spinors induced by the presence of torsion in the most general f(R) gravity. Phenomenological considerations are given.     

<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity solutions for evolving wormholes

The scalar–tensor f(R) theory of gravity is considered in the framework of a simple inhomogeneous space-time model. In this research we use the reconstruction technique to look for possible evolving wormhole solutions within viable f(R) gravity formalism. These f(R) models are then constrained so that they are consistent with existing experimental data. Energy conditions related to the matter threading the wormhole are analyzed graphically and are in general found to obey the null energy conditions (NEC) in regions around the throat, while in the limit \(f(R)=R,\) NEC can be violated at large in regions around the throat.     

Strange stars in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) theories of gravity in the Palatini formalism

In the present work we study strange stars in f(R) theories of gravity in the Palatini formalism. We consider two concrete well-known cases, namely the \(R+R^2/(6 M^2)\) model as well as the \(R-\mu ^4/R\) model for two different values of the mass parameter M or \(\mu \). We integrate the modified Tolman–Oppenheimer–Volkoff equations numerically, and we show the mass-radius diagram for each model separately. The standard case corresponding to the General Relativity is also shown in the same figure for comparison. Our numerical results show that the interior solution can be vastly different depending on the model and/or the value of the parameter of each model. In addition, our findings imply that (i) for the cosmologically interesting values of the mass scales \(M,\mu \) the effect of modified gravity on strange stars is negligible, while (ii) for the values predicting an observable effect, the modified gravity models discussed here would be ruled out by their cosmological effects.     

Graviton and scalar propagations on AdS<Subscript>4</Subscript> space in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravities

We investigate propagations of graviton and additional scalar on four-dimensional anti-de Sitter (AdS₄) space using f(R) gravity models with external sources. It is shown that there is the van Dam–Veltman–Zakharov (vDVZ) discontinuity in f(R) gravity models because f(R) gravity implies GR with additional scalar. This clearly indicates a difference between general relativity and f(R) gravity.     

Energy conditions in modified <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

In this paper, we have considered flat Friedmann–Lemaître–Robertson–Walker metric in the framework of perfect fluid models and modified f(G) gravity (where G is the Gauss Bonnet invariant). Particularly, we have considered particular realistic f(G) configurations that could be used to cure finite-time future singularities arising in the late-time cosmic accelerating epochs. We have then developed the viability bounds of these models induced by weak and null energy conditions, by using the recent estimated numerical figures of the deceleration, Hubble, snap and jerk parameters.