The simplest non-minimal matter–geometry coupling in the <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) cosmology

f(R, T) gravity is an extended theory of gravity in which the gravitational action contains general terms of both the Ricci scalar R and the trace of the energy-momentum tensor T. In this way, f(R, T) models are capable of describing a non-minimal coupling between geometry (through terms in R) and matter (through terms in T). In this article we construct a cosmological model from the simplest non-minimal matter–geometry coupling within the f(R, T) gravity formalism, by means of an effective energy-momentum tensor, given by the sum of the usual matter energy-momentum tensor with a dark energy contribution, with the latter coming from the matter–geometry coupling terms. We apply the energy conditions to our solutions in order to obtain a range of values for the free parameters of the model which yield a healthy and well-behaved scenario. For some values of the free parameters which are submissive to the energy conditions application, it is possible to predict a transition from a decelerated period of the expansion of the universe to a period of acceleration (dark energy era). We also propose further applications of this particular case of the f(R, T) formalism in order to check its reliability in other fields, rather than cosmology.     

Modified <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity models with curvature–matter coupling

A modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss–Bonnet term. The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra force acting on test particles and non-geodesic motion. Moreover, the energy conditions and the stability criterion at the de Sitter point in modified f(G) gravity models with curvature–matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight in the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corresponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter ω smaller than -\frac13-\frac{1}{3}.     

Non-minimal Maxwell-modified Gauss–Bonnet cosmologies: inflation and dark energy

In this paper we show that power-law inflation can be realized in non-minimal gravitational coupling of electromagnetic field with a general function of the Gauss–Bonnet invariant. Such a non-minimal coupling may appear due to quantum corrections. We also consider modified Maxwell-F(G) gravity in which non-minimal coupling between electromagnetic field and f(G) occurs in the framework of modified Gauss–Bonnet gravity. It is shown that inflationary cosmology and late-time accelerated expansion of the universe are possible in such a theory.     

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.     

<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.     

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.     

Quark and strange quark matter in f(R) gravity for Bianchi type I and V space-times

Behaviors of quark matter and strange quark matter which exist in the first seconds of the early Universe in f(R) gravity are studied for Bianchi I and V universes. In this respect, we obtain exact solutions of the modified Einstein field equations by using anisotropy feature of Bianchi I and V space-times. In particular, we investigate exact f(R) functions for Bianchi I as the contribution of strange quark and quark matter. Also, we have concluded that quark matter may contribute to the early acceleration of the universe since quark matter behaves like phantom-type dark energy. Furthermore, obtained f(R) solutions represents early eras of the Universe since f(R) solutions for quark matter coincide with f(R) equations for inflation. From this point, we can reach the conclusion that quarks may be source of the early dark energy of the universe or source of little inflation due to their repulsive force.     

Black-hole solutions in <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity with conformal anomaly

In this paper, we consider F(R)=R+f(R) theory instead of Einstein gravity with conformal anomaly and look for its analytical solutions. Depending on the free parameters, one may obtain both uncharged and charged solutions for some classes of F(R) models. Calculation of Kretschmann scalar shows that there is a singularity located at r=0. The geometry of uncharged (charged) solution corresponds to the Schwarzschild (Reissner–Nordström) singularity. Further, we discuss the viability of our models in detail. We show that these models can be stable, depending on their parameters and in different epochs of the universe.     

The effect of modified gravity on weak lensing

We study the effect of modified gravity on weak lensing in a class of scalar-tensor theory that includes f(R) gravity as a special case. These models are designed to satisfy local gravity constraints by having a large scalar-field mass in a region of high curvature. Matter density perturbations in these models are enhanced at small redshifts because of the presence of a coupling Q that characterizes the strength between dark energy and non-relativistic matter. We compute a convergence power spectrum of weak lensing numerically and show that the spectral index and the amplitude of the spectrum in the linear regime can be significantly modified compared to the ΛCDM model for large values of |Q| of the order of unity. Thus weak lensing provides a powerful tool to constrain such large coupling scalar-tensor models including f(R) gravity.     

Early and late-time cosmic acceleration in non-minimal Yang–Mills-<Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

In this paper we show that power-law inflation can be realized in non-minimal gravitational coupling of Yang–Mills field with a general function of the Gauss–Bonnet invariant in the framework of Einstein gravity. Such a non-minimal coupling may appear due to quantum corrections. We also discuss the non-minimal Yang–Mills-f(G) gravity in the framework of modified Gauss–Bonnet action which is widely studied recently. It is shown that both inflation and late-time cosmic acceleration are possible in such a theory.     

Cosmological analysis of scalar field models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity

This paper determines the existence of Noether symmetry in non-minimally coupled f(R, T) gravity admitting minimal coupling with scalar field models. We consider a generalized spacetime which corresponds to different anisotropic and homogeneous universe models. We formulate symmetry generators along with conserved quantities through Noether symmetry technique for direct and indirect curvature–matter coupling. For dust and perfect fluids, we evaluate exact solutions and construct their cosmological analysis through some cosmological parameters. We conclude that decelerated expansion is obtained for the quintessence model with a dust distribution, while a perfect fluid with dominating potential energy over kinetic energy leads to the current cosmic expansion for both phantom as well as quintessence models.     

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.     

Non-minimal warm inflation and perturbations on the warped DGP brane with modified induced gravity

We construct a warm inflation model with inflaton field non-minimally coupled to induced gravity on a warped DGP brane. We incorporate possible modification of the induced gravity on the brane in the spirit of f(R)-gravity. We study cosmological perturbations in this setup. In the case of two field inflation such as warm inflation, usually entropy perturbations are generated. While it is expected that in the case of one field inflation these perturbations to be removed, we show that even in the absence of the radiation field, entropy perturbations are generated in our setup due to non-minimal coupling and modification of the induced gravity. We study the effect of dissipation on the inflation parameters of this extended braneworld scenario.     

Late-time cosmological approach in mimetic <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity

In this paper, we investigate the late-time cosmic acceleration in mimetic f(R, T) gravity with the Lagrange multiplier and potential in a Universe containing, besides radiation and dark energy, a self-interacting (collisional) matter. We obtain through the modified Friedmann equations the main equation that can describe the cosmological evolution. Then, with several models from \(\mathcal {Q}(z)\) and the well-known particular model f(R, T), we perform an analysis of the late-time evolution. We examine the behavior of the Hubble parameter, the dark energy equation of state and the total effective equation of state and in each case we compare the resulting picture with the non-collisional matter (assumed as dust) and also with the collisional matter in mimetic f(R, T) gravity. The results obtained are in good agreement with the observational data and show that in the presence of the collisional matter the dark energy oscillations in mimetic f(R, T) gravity can be damped.     

Does entropic force always imply the Newtonian force law?

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.     

Dark-energy cosmological models in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">G</Emphasis>) gravity

We discuss dark-energy cosmological models in f(G) gravity. For this purpose, a locally rotationally symmetric Bianchi type I cosmological model is considered. First, exact solutions with a well-known form of the f(G) model are explored. One general solution is discussed using a power-law f(G) gravity model and physical quantities are calculated. In particular, Kasner’s universe is recovered and the corresponding f(G) gravity models are reported. Second, the energy conditions for the model under consideration are discussed using graphical analysis. It is concluded that solutions with f(G) = G^5/6 support expansion of universe while those with f(G) = G^1/2 do not favor the current expansion.     

Spatially Homogeneous Rotating Solution in f(R) Gravity and Its Energy Contents

In this paper, the metric approach of f(R) theory of gravity is used to investigate the exact vacuum solutions of spatially homogeneous rotating spacetimes. For this purpose, R is replaced by f(R) in the standard Einstein-Hilbert action and the set of modified Einstein field equations reduce to a single equation. We adopt the assumption of constant Ricci scalar which maybe zero or non-zero. Moreover, the energy density of the non-trivial solution has been evaluated by using the generalized Landau-Lifshitz energy-momentum complex in the perspective of f(R) gravity for some appropriate f(R) model, which turns out to be a constant quantity.     

Wormhole Geometries in f(R,T) Gravity

We study wormhole solutions in the framework of f(R,T) gravity where R is the scalar curvature, and T is the trace of the stress-energy tensor of the matter. We have obtained the shape function of the wormhole by specifying an equation of state for the matter field and imposing the flaring out condition at the throat. We show that in this modified gravity scenario, the matter threading the wormhole may satisfy the energy conditions, so it is the effective stress-energy that is responsible for violation of the null energy condition.     

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.