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We study first and second laws of black hole thermodynamics at the apparent
horizon of FRW spacetime in f(R; T;R��T��) gravity, where R, R�� are the Ricci scalar
and Riemann tensor and T is the trace of the energy-momentum tensor T��. We develop
the Friedmann equations for any spatial curvature in this modified theory and show that these equations can be transformed to the form of Clausius relation T_hS_{eff}� = �Q. Here Th is
the horizon temperature, Se� is the entropy which contains contributions both from horizon
entropy and additional entropy term introduced due to the non-equilibrating description and
�Q is the energy flux across the horizon. The generalized second law of thermodynamics is
also established in a more comprehensive form and one can recover the corresponding results
in Einstein, f(R) and f(R; T) gravities. We discuss GSLT in the locality of assumption that
temperature of matter inside the horizon is similar to that of horizon. Finally, we consider
particular models in this theory and generate constraints on the coupling parameter for the
validity of GSLT.

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... This force also helps to elucidate the galactic rotation curves. Sharif and Zubair [20] assumed two models such as v ...

... In this regard, we choose it as ϖ = ± 4. It is interesting to stress that physical feasibility of different gravity models can be achieved by taking the value of coupling parameter within its observed range. The model (8) has been utilized in several investigations based on the stability and viability of various isotropic and anisotropic configured stars [20,21,25]. In this case, becomes ...

... Finally, we obtain the bag model o for strange fluid by combining equations (20) and (21) as ...

This paper investigates some particular anisotropic star models in $f(\mathcal{R},\mathcal{T},\mathcal{Q})$ gravity, where $\mathcal{Q}=\mathcal{R}_{\omega\alpha}\mathcal{T}^{\omega\alpha}$. We adopt a standard model $f(\mathcal{R},\mathcal{T},\mathcal{Q})=\mathcal{R}+\varpi\mathcal{Q}$, where $\varpi$ indicates a coupling constant. We take spherically symmetric spacetime and develop solutions to the modified field equations corresponding to different choices of the matter Lagrangian by applying `embedding class-one' scheme. For this purpose, we utilize $\mathbb{MIT}$ bag model equation of state and investigate some physical aspects of compact models such as RXJ 1856-37,~4U 1820-30,~Cen X-3,~SAX J 1808.4-3658 and Her X-I. We use masses and radii of these stars and employ the vanishing radial pressure condition at the boundary to calculate the value of their respective bag constant $\mathfrak{B_c}$. Further, we fix $\varpi=\pm4$ to analyze the behavior of resulting state variables, anisotropy, mass, compactness, surface redshift as well as energy bounds through graphical interpretation for each star model. Two different physical tests are performed to check the stability of the developed solutions. We conclude that $\varpi=-4$ is more suitable choice for the considered modified model to obtain stable structures of the compact bodies.

... Modified theories have gained more attention to count with the issue of cosmic acceleration [19][20][21][22][23][24][25][26][27][28]. Likewise in general relativity (GR), the instability problem has also been JCAP02(2015)033 widely discussed in modified gravity theories namely, f (R), f (R, T ), where T is the trace of energy momentum tensor, f (G), Brans-Dicke theory etc. ...

We study the implications of $R^n$ extension of Starobinsky model on
dynamical instability of axially symmetric gravitating body. The matter
distribution is considered to be anisotropic for which modified field equations
are formed in context of $f(R)$ gravity. In order to achieve the collapse
equation, we make use of the dynamical equations, extracted from linearly
perturbed contracted Bianchi identities. The collapse equation carries
adiabatic index $\Gamma$ in terms of usual and dark source components, defining
the range of stability/insatbility in Newtonian (N) and post-Newtonian (pN)
eras. It is found that supersymmetric supergravity $f(R)$ model represents the
more practical substitute of higher order curvature corrections.

... It would be interesting to explore different cosmic features in this theory. Recently, we have studied the thermodynamic properties in f (R, T, R µν T µν ) gravity and found that matter geometry coupling give rise to non-equilibrium description of thermodynamics [39]. Therefore the non-equilibrium picture of thermodynamic laws is presented. ...

We discuss the validity of the energy conditions in a newly modified theory named as f (R, T, R µν T µν) gravity, where R and T represent the scalar curvature and trace of the energy-momentum tensor. The corresponding energy conditions are derived which appear to be more general and can reduce to the familiar forms of these conditions in general relativity, f (R) and f (R, T) theories. The general inequalities are presented in terms of recent values of Hubble, deceleration, jerk and snap parameters. In particular, we use two specific models recently developed in literature to study concrete application of these conditions as well as Dolgov-Kawasaki instability. Finally, we explore f (R, T) gravity as a specific case to this modified theory for exponential and power law models.

... At boundary, these equations guarantee consistency of the solution (33)-(36) (which we have calculated for inner geometry) with the outer region and will be modified undoubtedly after adding the additional source. Equations (17) and (33) provide the radial and temporal metric components that will be used for the construction of anisotropic solution, i.e., for σ = 0 in the inner geometry. The relation between source Θ ρ η and geometric deformation t * has been expressed through Equations (21)- (23). ...

In this paper, we consider isotropic solution and extend it to two different exact well-behaved spherical anisotropic solutions through minimal geometric deformation method in f(R,T,RρηTρη) gravity. We only deform the radial metric component that separates the field equations into two sets corresponding to their original sources. The first set corresponds to perfect matter distribution while the other set exhibits the effects of additional source, i.e., anisotropy. The isotropic system is resolved by assuming the metric potentials proposed by Krori-Barua while the second set needs one constraint to be solved. The physical acceptability and consistency of the obtained solutions are analyzed through graphical analysis of effective matter components and energy bounds. We also examine mass, surface redshift and compactness of the resulting solutions. For particular values of the decoupling parameter, our both solutions turn out to be viable and stable. We conclude that this curvature-matter coupling gravity provides more stable solutions corresponding to a self-gravitating geometry.

... Bamba and Geng [28] found that GSLT holds for the FRW universe with the same temperature inside and outside the apparent horizon in generalized teleparallel theory. Sharif and Zubair [29] checked the validity of first and second laws of thermodynamics at the apparent horizon for both equilibrium as well as nonequilibrium descriptions in ( , ) gravity and found that GSLT holds in both phantom as well as nonphantom phases of the universe. Abdolmaleki and Najafi [30] explored the validity of GSLT for isotropic and homogeneous universe filled with radiation and matter surrounded by apparent horizon with Hawking temperature in (G) gravity. ...

This paper explores the nonequilibrium behavior of thermodynamics at the apparent horizon of isotropic and homogeneous universe model in f(G,T) gravity ( G and T represent the Gauss-Bonnet invariant and trace of the energy-momentum tensor, resp.). We construct the corresponding field equations and analyze the first as well as generalized second law of thermodynamics in this scenario. It is found that an auxiliary term corresponding to entropy production appears due to the nonequilibrium picture of thermodynamics in first law. The universal condition for the validity of generalized second law of thermodynamics is also obtained. Finally, we check the validity of generalized second law of thermodynamics for the reconstructed f(G,T) models (de Sitter and power-law solutions). We conclude that this law holds for suitable choices of free parameters.

... These results are applicable to the situation of minimal geometry-matter couplings. The thermodynamics of nonminimally coupled theories like L = f (R , T (m) ) + 16πGL m [46] (where T (m) = g µν T µν (m) ) and L = f (R , T (m) , R µν T µν (m) ) + 16πGL m [47] have been attempted using the traditional formulation as in [10] for f (R) gravity. However, more profound thermodynamic properties may hide in these theories, as there is direct energy exchange between spacetime geometry and the energy-matter content under nonminimal curvaturematter couplings [21,29,30]. ...

Inspired by the Wald-Kodama entropy $S=A/(4G_{\text{eff}})$ where $A$ is the
horizon area and $G_{\text{eff}}$ is the effective gravitational coupling
strength in modified gravity with field equation $R_{\mu\nu}-Rg_{\mu\nu}/2=$
$8\pi G_{\text{eff}} T_{\mu\nu}^{\text{(eff)}}$, we develop a unified and
compact formulation in which the Friedmann equations can be derived from
thermodynamics of the Universe. The Hawking and Misner-Sharp masses are
generalized by replacing Newton's constant $G$ with $G_{\text{eff}}$, and the
unified first law of equilibrium thermodynamics is supplemented by a
nonequilibrium energy dissipation term $\mathcal{E}$ which arises from the
revised continuity equation of the perfect-fluid effective matter content and
is related to the evolution of $G_{\text{eff}}$. By identifying the mass as the
total internal energy, the unified first law for the interior and its smooth
transit to the apparent horizon yield both Friedmann equations, while the
nonequilibrium Clausius relation with entropy production provides an
alternative derivation on the horizon. We also analyze the equilibrium
situation $G_{\text{eff}}=G=\text{constant}$, provide a viability test of the
generalized geometric masses, and discuss the continuity/conservation equation
along with a second "bi-$G$" unified framework. Finally, the general
formulation is applied to the FRW cosmology of minimally coupled $f(R)$,
generalized Brans-Dicke, scalar-tensor-chameleon, quadratic and
$f(R,\mathcal{G})$ generalized Gauss-Bonnet gravity. In these theories we also
analyze the $f(R)$-Brans-Dicke equivalence, show that the generalized
Misner-Sharp mass by Cai et el. is actually the pure mass of physical matter in
our formulation, and find that the chameleon effect causes extra energy
dissipation and entropy production.

... However, without the use of the modified gravity theoretical framework, one frequently needs phantom fluids to describe an accelerated expansion. There are various modified gravity models in the literature, such as f (R) gravity [20], f (R, T) gravity [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37], and f (R, T, R µν T µν ) gravity [38][39][40][41], where R, T, R µν , and T µν stand for the Ricci scalar, the trace of the "energy momentum tensor"(EMT), and the Ricci tensor, respectively. ...

We study the cosmic evolution of non-minimally coupled f ( R , T ) gravity in the presence of matter fluids consisting of collisional self-interacting dark matter and radiation. We study the cosmic evolution in the presence of collisional matter, and we compare the results with those corresponding to non-collisional matter and the Λ -cold-dark-matter ( Λ CDM) model. Particularly, for a flat Friedmann–Lema i ^ tre–Robertson–Walker Universe, we study two non-minimally coupled f ( R , T ) gravity models and we focus our study on the late-time dynamical evolution of the model. Our study is focused on the late-time behavior of the effective equation of the state parameter ω e f f and of the deceleration parameter q as functions of the redshift for a Universe containing collisional and non-collisional dark matter fluids, and we compare both models with the Λ CDM model. As we demonstrate, the resulting picture is well accommodated to the latest observational data on the basis of physical parameters.

This paper deals with the new theoretical model of quintessence anisotropic star in f(T) theory of gravity. The equations of motion in f(T) theory with a static spherically symmetric spacetime in the presence of anisotropic fluid and quintessence field have been solved by using Krori-Barua metric. In this case, we have used the diagonal tetrad field that leads to a linear form f(T) function. We have determined that all the obtained solutions are free from central singularity and potentially stable. The values of the unknown constants existing in Krori and Barua metric have been calculated by using the observed values for mass of the different strange stars PSR J 1614-2230, SAXJ1808.4-3658(SS1), 4U1820-30, PSR J 1614-2230. The physical parameters (anisotropy, stability and redshift) of the stars have been investigated in detail. We have determined the constraint under which results of f(T), theory reduces to general relativity.

This study is conducted to examine the validity of thermodynamical laws in a modified f(T) gravity involving a direct coupling of torsion scalar with matter contents. For this purpose, we consider spatially flat FRW geometry with matter contents as perfect fluid and formulate the first thermodynamical law in this gravity at apparent horizon. It is found that equilibrium description of thermodynamics exists in this modified gravity in a similar way to Einstein and other gravities. Further we discuss generalized second law of thermodynamics at apparent horizon of FRW universe for three different f(T) models using Gibbs law as well as the assumption that temperature of matter within apparent horizon is similar to that of horizon. It is found that for some particular cosmologically consistent values of coupling parameters, GSLT remains valid in observationally consistent cosmic eras.

In this paper, we have formulated the new exact model of quintessence
anisotropic star in $f(R)$ theory of gravity. The dynamical equations in $f(R)$
theory with the anisotropic fluid and quintessence field have been solved by
using Krori-Barua solution. In this case, we have used the Starobinsky model of
$f(R)$ gravity. We have determined that all the obtained solutions are free
from central singularity and potentially stable. The observed values of mass
and radius of the different strange stars PSR J 1614-2230,
SAXJ1808.4-3658(SS1), 4U1820- 30, PSR J 1614-2230 have been used to calculate
the values of unknown constants in Krori and Barua metric. The physical
parameters like anisotropy, stability and redshift of the stars have been
investigated in detail.

We reconstruct f (R, T) theory (where R is the scalar curvature and T is the trace of energy-momentum tensor) in the framework of QCD ghost dark energy models. In this study, we concentrate on particular models of f (R, T) gravity which permits the standard continuity equation in this theory. It is found that reconstructed function can represent phantom and quintessence regimes of the universe in the background of flat FRW universe. In addition, we explore the stability of ghost f (R, T) models.

An equilibrium picture of thermodynamics is discussed at the apparent horizon
of FRW universe in $f(T,T_G)$ gravity, where $T$ represents the torsion
invariant and $T_G$ is the teleparallel equivalent of the Gauss-Bonnet term. It
is found that one can translate the Friedmann equations to the standard form of
first law of thermodynamics. We discuss GSLT in the locality of assumption that
temperature of matter inside the horizon is similar to that of horizon.
Finally, we consider particular models in this theory and generate constraints
on the coupling parameter for the validity of GSLT in terms of recent cosmic
parameters and power law solutions.

We regard theory as an efficient tool to explain the current cosmic acceleration and associate its evolution with the known dark energy models. The numerical scheme is applied to reconstruct theory from dark energy model with constant equation of state parameter and holographic dark energy model. We set the model parameters and as describing the different evolution eras and show the distinctive behavior of each case realized in theory. We also present the future evolution of reconstructed and find that it is consistent with the recent observations.

In this manuscript, we develop the counterpart of Tsallis holographic dark energy (THDE) model in F(R,T) theory (where R represents Ricci scalar and T is trace of energy momentum tensor (EMT)) using the two IR-cutoffs namely Hubble horizon (HH) and Granda-Oliveros (GO). The THDE is proposed on the basis of Tsallis entropy (Tsallis and Cirto, 2013) and the holographic hypothesis (Tavayef, 2018). Cosmic evolution is analyzed using the equation of state (EOS) parameter which results in both quintessence and phantom regimes. The stability analysis of reconstructed models is also made with the help of squared speed of sound. Moreover, we study the thermodynamic picture and developed constraints for the validity of generalized second law of thermodynamics (GSLT). It is remarked that these reconstructed models can be useful to further explore the cosmic issues.

We discuss the dynamical analysis in $f(R,T)$ gravity (where $R$ is Ricci
scalar and $T$ is trace of energy momentum tensor) for gravitating sources
carrying axial symmetry. The self gravitating system is taken to be anisotropic
and line element describes axially symmetric geometry avoiding rotation about
symmetry axis and meridional motions (zero vorticity case). The modified field
equations for axial symmetry in $f(R,T)$ theory are formulated, together with
the dynamical equations. Linearly perturbed dynamical equations lead to the
evolution equation carrying adiabatic index $\Gamma$ that defines impact of
non-minimal matter to geometry coupling on range of instability for Newtonian
(N) and post-Newtonian (pN) approximations.

With the usual definitions for the entropy and the temperature associated
with the apparent horizon, we discuss the first law of the thermodynamics on
the apparent in the general scalar-tensor theory of gravity with the kinetic
term of the scalar field non-minimally coupling to Einstein tensor. We show the
equivalence between the first law of thermodynamics on the apparent horizon and
Friedmann equation for the general models, by using a mass-like function which
is equal to the Misner-Sharp mass on the apparent horizon. The results further
support the universal relationship between the first law of thermodynamics and
Fredmann equation.

This study is conducted to examine the validity of generalized second law of thermodynamics (GSLT) in modified teleparallel gravity involving coupling between a scalar field with the torsion scalar and a boundary term. This theory is very useful since it can reproduce other well-known theories in suitable limits. The power law solution is employed to develop the constraints on coupling parameters for different theories of gravity in the background of thermodynamics properties for all potentials. We have also considered the logarithmic entropy corrected relation and discuss the GSLT both on apparent and event horizons. In case of entropy correction, we have constrained the coupling parameters for quartic and inverse potentials.

We study antigravity in F(R)-theory originating scalar-tensor theories and also in Brans-Dicke models without cosmological constant. For the F(R) theory case, we obtain the Jordan frame antigravity scalar-tensor theory by using a variant of the Lagrange multipliers method and we numerically study the time dependent effective gravitational constant. As we shall demonstrate in detail by using some viable F(R) models, although the initial F(R) models have no antigravity, their scalar-tensor counterpart theories might or not have antigravity, a fact mainly depending on the parameter that characterizes antigravity. Similar results hold true in the Brans-Dicke model, which we also studied numerically. In addition, regarding the Brans-Dicke model we also found some analytic cosmological solutions. Since antigravity is an unwanted feature in gravitational theories, our findings suggest that in the case of F(R) theories, antigravity does not occur in the real world described by the F(R) theory, but might occur in the Jordan frame scalar-tensor counterpart of the F(R) theory, and this happens under certain circumstances. The central goal of our study is to present all different cases in which antigravity might occur in modified gravity models.

We study the cosmological reconstruction of f(R,T) gravity (where R and T denote the Ricci scalar and trace of the energy-momentum tensor) corresponding to the evolution background in FRW universe. It is shown that any cosmological evolution including Λ cold dark matter, phantom or non-phantom eras and possible phase transition from decelerating to accelerating can be reproduced in this theory. We propose some specific forms of Lagrangian in the perspective of de Sitter and power law expansion history. Finally, we formulate the perturbed evolution equations and analyze the stability of some important solutions.

We discuss the consistency of a recently proposed class of theories described
by an arbitrary function of the Ricci scalar, the trace of the energy-momentum
tensor and the contraction of the Ricci tensor with the energy-momentum tensor.
We briefly discuss the limitations of including the energy-momentum tensor in
the action, as it is a non fundamental quantity, but a quantity that should be
derived from the action. The fact that theories containing non-linear
contractions of the Ricci tensor usually leads to the presence of pathologies
associated with higher-order equations of motion will be shown to constrain the
stability of this class of theories. We provide a general framework and show
that the conformal mode for these theories generally has higher-order equations
of motion and that non-minimal couplings to the matter fields usually lead to
higher-order equations of motion. In order to illustrate such limitations we
explicitly study the cases of a canonical scalar field, a K-essence field and a
massive vector field. Whereas for the scalar field cases it is possible to find
healthy theories, for the vector field case the presence of instabilities is
unavoidable.

In this paper we address the well-known cosmic coincidence problem in the
framework of the f(T) gravity. In order to achieve this, an interaction between
dark energy and dark matter is considered. A constraint equation is obtained
which generates the f(T) models that do not suffer from the coincidence
problem. Due to the absence of a universally accepted interaction term
introduced by a fundamental theory, the study is conducted over three different
forms of chosen interaction terms. As an illustration two widely known models
of f(T) gravity are taken into consideration and used in the setup designed to
study the problem. The study reveals that there exists a perfect solution for
the coincidence problem in the background of the second model while the first
model remains utterly plagued by the phenomenon. This not only shows the
cosmological viability but also the superiority of the second model over its
counterpart.

We analyze pilgrim dark energy model in the background of f(R) gravity by taking IR cut-offs as particle and event horizons as well as conformal age of the universe. We regard the f(R) theory as an effective description for the pilgrim dark energy and reconstruct the function f(R) with the parameter μf(R) for the future cosmic evolution.

We study pilgrim dark energy model by taking IR cut-offs as particle and event horizons as well as conformal age of the universe. We derive evolution equations for fractional energy density and equation of state parameters for pilgrim dark energy. The phantom cosmic evolution is established in these scenarios which is well supported by the cosmological parameters such as deceleration parameter, statefinder parameters and phase space of ω
ϑ
and \(\omega'_{\vartheta}\). We conclude that the consistent value of parameter μ is μ<0 in accordance with the current Planck and WMAP9 results.

In this paper we consider the implications that the effect gravitational
memory would have on primordial black holes, within the theoretical context of
$F(R)$ related scalar-tensor theories. As we will demonstrate, under the
assumption that the initial mass of the primordial black hole is such so that
it evaporates today, this can potentially constrain the $F(R)$ related theories
of gravity. We study two scalar-tensor models and discuss the evolution of
primordial black holes created at some initial time $t_f$ in the early
universe. The results between the two models vary significantly which shows us
that, if the effect of gravitational memory is considered valid, some of the
scalar-tensor models and their corresponding $F(R)$ theories must be further
constrained.

We discuss the cosmological reconstruction of $f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)$ (where $R$, $R_{\alpha\beta}R^{\alpha\beta}$ and $\phi$ represents the Ricci scalar, Ricci invariant and scalar field) corresponding to power law and de Sitter evolution in the framework of FRW universe model. We derive the energy conditions for this modified theory which seem to be more general and can be reduced to some known forms of these conditions in general relativity, $f(R)$ and $f(R,\phi)$ theories. We have presented the general constraints in terms of recent values of snap, jerk, deceleration and Hubble parameters. The energy bounds are analyzed for reconstructed as well as known models in this theory. Finally, the free parameters are analyzed comprehensively.

This paper deals with the interior models of compact stars in the framework of modified \(f(R)\) theory of gravity, which is the generalization of the Einstein?s gravity. In order to complete the study, we have involved solution of Krori and Barua to the static spacetime with fluid source in modified \(f(R)\) theory of gravity. Further, we have matched the interior solution with the exterior solution to determine the constants of Krori and Barua solution. Finally, the constants have been formulated by using the observational data of various compact stars like 4U1820-30, Her X-1, SAX J1808-3658. Using the evaluated form of the solutions, we have discussed the regularity of matter components at the center as well as on the boundary, energy conditions, anisotropy, stability analysis and mass-radius relation of the compact stars 4U1820-30, Her X-1, SAX J1808-3658.

We reconstruct the λCDM model for f (T, T) theory, where T is the torsion scalar and T the trace of the energy-momentum tensor. The result shows that the action of λCDM is a combination of a linear term, a constant (-2λ) and a nonlinear term given by the product -T Fg [T1/3/ 16πG) (16πGT + T + 8λ)], with Fg being a generic function. We show that to maintain conservation of the energy-momentum tensor, we should impose that Fg [y] must be linear on the trace T. This reconstruction decays in f (T) theory for Fg ≡ Q, with Q a constant. Our reconstruction describes the cosmological eras to the present time. The model present stability within the geometric and matter perturbations for the choice Fg = y, where y = (T1/3 /16πG)(16πGT + T + 8λ), except for the geometric part in the de Sitter model. We impose the first and second laws of thermodynamics to λCDM and find the condition where they are satisfied, that is, TA, Geff > 0, however where this is not possible in the cases that we choose, this leads to a breakdown of positive entropy and MisnerSharp energy.

In this paper we study the recently proposed f(R,L) theories from a thermodynamic point of view. The uniqueness of these theories lies in the fact that the space-time curvature is coupled to the baryonic matter instead of exotic matter (in the form of scalar field). We investigate the viability of these theories from the point of view of the thermodynamic stability of the models. To be more precise here we are concerned with the thermodynamics of the apparent horizon of Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime in the background of the f(R,L) theory. We consider several models of f(R,L) theories where both minimal and nonminimal coupling has been considered. Various thermodynamic quantities like entropy, enthalpy, internal energy, Gibbs free energy, etc. are computed and using their allowed ranges various model parameters are constrained.

In this paper, we consider static self-gravitating spherical space-time and determine various anisotropic solutions through the extended gravitational decoupling technique in f(R,T,RλξTλξ) gravity to analyze the influence of electromagnetic field on them. We construct two different sets of modified field equations by employing the transformations on both radial as well as temporal metric potentials. The first set symbolizes the isotropic fluid distribution, thus we take Krori-Barua solution to deal with it. The indefinite second sector comprises the influence of anisotropy. In this regard, we apply some constraints to determine unknowns. Further, we observe the impact of charge as well as decoupling parameter ζ on the developed physical variables (such as energy density, radial and tangential pressures) and anisotropy. We also analyze other physical features of the compact geometry like mass, compactness and redshift along with the energy conditions. Eventually, we find that our both solutions show less stable behavior for higher values of charge near the boundary in this gravity.

The main aim of our study is to explore some relativistic configurations of compact object solution in the background of fR gravity, by adopting the Krori-Barua spacetime. In this regard, we establish the field equations for spherically symmetric spacetime along with charged anisotropic matter source by assuming the specific form of the metric potentials, i.e., νr=Br2+C and λr=Ar2. Further, to calculate the constant values, we consider the Bardeen model as an exterior spacetime at the surface boundary. To ensure the viability of the fR gravity model, the physical characteristics including energy density, pressure components, energy bonds, equilibrium condition, Herrera cracking concept, mass-radius relation, and adiabatic index are analyzed in detail. It is observed that all the outcomes by graphical exploration and tabular figures show that the Bardeen black hole model describes the physically realistic stellar structures.

In present work, we use an extended f(T) gravity namely f(T,T) gravity, representing the coupling of torsion scalar T and the trace of energy-momentum tensor T. In this framework, we find the exact interior anisotropic solutions of compact stars considering Krori–Barua space-time under metric potentials, a(r)=Br2+Cr3,b(r)=Ar3 (A, B, C are the unknown parameters), and applying a well-known model f(T,T)=α1Tn(r)+βT(r)+ϕ,n>1. To close the system of equations, we utilize an equation of state for modified Chaplygin gas model. By the well-known matching conditions of exterior and interior space-time, we evaluate the unknown model parameters. For four different strange stars, PSR-J1614-2230, SAX-J1808.4-3658, 4U 1820-30 and Vela-X-12 (last two are added in “Appendix”), we made complete physical analysis by plotting trajectories for energy conditions, square speed of sound, mass function, compactness factor and surface redshift. It is observed that our results satisfy all the necessary and sufficient physical conditions, hence physically viable.

In the framework of the mimetic approach, we study the $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) gravity with the Lagrange multiplier constraint and the scalar potential. We introduce field equations for the discussed theory and overview their properties. By using the general reconstruction scheme we obtain the power law cosmology model for the $$f(R,R_{\mu \nu }R^{\mu \nu })=R+d(R_{\mu \nu }R^{\mu \nu })^p$$ f ( R , R μ ν R μ ν ) = R + d ( R μ ν R μ ν ) p case as well as the model that describes symmetric bounce. Moreover, we reconstruct model, unifying both matter dominated and accelerated phases, where ordinary matter is neglected. Using inverted reconstruction scheme we recover specific $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) function which give rise to the de-Sitter evolution. Finally, by employing the perfect fluid approach, we demonstrate that this model can realize inflation consistent with the bounds coming from the BICEP2/Keck array and the Planck data. We also discuss the holographic dark energy density in terms of the presented $$f(R,R_{\mu \nu }R^{\mu \nu })$$ f ( R , R μ ν R μ ν ) theory. Thus, it is suggested that the introduced extension of the mimetic regime may describe any given cosmological model.

Our study illuminates the effects of gravitational collapse by considering heat flux anisotropic matter sources within the framework of the f(R) theory of gravity. We assume the non-static spherically symmetric configuration to narrate the nature of the interior spacetime and match it with the Vaidya exterior geometry. Further, we employ the well-known Karmarkar condition, as it reduces the solution-generating method of field equations to a single metric potential. To ensure the viability of our collapsing system, a comprehensive graphical analysis is presented in the context of the Logarithmic-corrected R2 gravity model. Our investigation argues that there are no traces of trapped surfaces in the collapsing procedure. Thus, naked singularity and black hole are not the final fate of the gravitational collapse.

This paper focuses on the analysis of static spherically symmetric anisotropic solutions in the presence of electromagnetic field through the gravitational decoupling approach in f(R,T,RγυTγυ) gravity. We use geometric deformation only on radial metric function and obtain two sets of the field equations. The first set deals with isotropic fluid while the second set yields the influence of anisotropic source. We consider the modified Krori-Barua charged isotropic solution for spherical self-gravitating star to deal with the isotropic system. The second set of the field equations is solved by taking two different constraints. We then investigate physical acceptability of the obtained solutions through graphical analysis of the effective physical variables and energy conditions. We also analyze the effects of charge on different parameters, (i.e., mass, compactness and redshift) for the resulting solutions. It is found that our both solutions are viable as well as stable for specific values of the decoupling parameter φ and charge. We conclude that a self-gravitating star shows more stable behavior in this gravity.

This paper investigates the new interior solution of stellar compact spheres in the framework of metric [Formula: see text] gravity. In this connection, we derived the Einstein field equations for static anisotropic spherically symmetric spacetime in the mechanism of Karmakar condition. The obtained results of the field equations have been studied with well-known Starobinskian model [Formula: see text] by using three different compact stars like [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text]. Moreover, the constants involved in the solution of metric potentials have been determined through smooth matching conditions between the interior geometry and exterior spacetime. Thereafter, the physical significance of the obtained results is examined in the form of fluid variables, equation of state (EoS) parameters, energy conditions, anisotropic stress and stability analysis by using the graphical plot. The approximated values of the constants and the mass-radius relation have been calculated through different stellar star objects ([Formula: see text]-[Formula: see text] ([Formula: see text]), [Formula: see text]-[Formula: see text] ([Formula: see text]) and [Formula: see text]-[Formula: see text] ([Formula: see text])) shown in Table 1. Finally, we have concluded that our considered compact stellar objects with particular choice of [Formula: see text] model in the mechanism of Karmakar condition satisfies all the necessary bounds for potentially stable formation of the stars.

This paper is devoted to develop compact stellar configuration and to evaluate exact interior solutions in the background of f(T,T) gravity, where T be the torsion scalar and T be the trace of energy momentum tensor. We develop a new set up for embedding class I model under well-known Karmarkar condition for LMC X-4 and Vela X-1 compact stars using specific linear function f(T,T)=αT(r)+βT(r)+Φ, where α,β are any arbitrary constants and Φ be the cosmological constant. Considering static spherically symmetric space–time with perfect fluid distribution, we construct the background field equations, which describe the interior of a fluid sphere. To examine the physical viability of the obtained solutions, we have discussed all the relevant physical quantities (density, pressure, equation of state parameter, square speed of sound and equilibrium condition) analytically and graphically. It is observed that our calculated interior solutions for compact stars showed consistency with all necessary and sufficient physical conditions and hence are physically acceptable and interesting.

In this paper, we discuss the cosmic evolution in a modified theory involving non-minimal interaction of geometry and matter, labeled as [Formula: see text] gravity, where [Formula: see text] is the non-minimal interaction term. First, we develop the dynamical [Formula: see text] field equations for Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime and then by using divergence of these equations, we explore its interesting outcome of non-conserved energy–momentum tensor (EMT). The presence of geometry matter coupling in such theories results in non-geodesic test particles motion and hence causes an additional force orthogonal to four-velocity of these particles. By taking these interesting features into account along with a particular choice of Lagrangian [Formula: see text], we explore the resulting expression of energy density. Further, the free model parameters are constrained using energy condition bounds where it is concluded that these values of free parameters are compatible with the recent data.

We examine in this paper the stability analysis in $f(R; T; R_{\mu\nu}T^{\mu\nu})$ modified gravity, where $R$ and $T$ are the Ricci scalar and the trace of the energy-momentum tensor, respectively. By considering the flat Friedmann universe, we obtain the corresponding generalized Friedmann equations and we evaluate the geometrical and matter perturbation functions. The stability is developed using the de Sitter and power-law solutions. We search for application the stability of two particular cases of $f(R; T; R_{\mu\nu}T^{\mu\nu})$ model by solving numerically the perturbation functions obtained.

The present study is elaborated to investigate the validity of thermodynamical laws in a modified teleparallel gravity based on higher-order derivative terms of torsion scalar. For this purpose, we consider spatially flat FRW model filled with perfect fluid matter contents. Firstly, we explore the possibility of existence of equilibrium as well as non-equilibrium picture of thermodynamics in this extended version of teleparallel gravity. Here, we present the first law and the generalized second law of thermodynamics (GSLT) using Hubble horizon. It is found that non-equilibrium description of thermodynamics exists in this theory with the presence of an extra term called as entropy production term. We also establish GSLT using the logarithmic corrected entropy. Further, by taking the equilibrium picture, we discuss validity of GSLT at Hubble horizon for two different F models. Using Gibbs law and the assumption that temperature of matter within Hubble horizon is similar to itself, We use different choices of scale factor to discuss the GSLT validity graphically in all scenarios. It is found that the GSLT is satisfied for a specified range of free parameters in all cases.

In this manuscript, we are interested to address the issue of cosmic expansion in the background of matter-geometry coupling. For this purpose we consider \(f(R,T,Q)\) modified theory (where \(R\) is the Ricci Scalar, \(T\) is the trace of energy-momentum tensor (EMT) \(T_{uv}\) and \(Q=R_{uv}T^{uv}\) is interaction of EMT \(T_{\mu \nu }\) and Ricci Tensor \(R_{uv}\)). We formulate modified field equations in the background of flat Friedmann-Lemaître-Robertson-Walker (FLRW) model which is defined as \(ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2}+dz^{2} )\), where \(a(t)\) represents the scale factor. In this formalism energy density is found using covariant divergence of modified field equations. \(\rho \) involves a contribution from non-minimal matter geometry coupling which helps to study different cosmic eras based on equation of state (EOS). Furthermore, we apply the energy bounds to constrain the model parameters establishing a pathway to discuss the cosmic evolution for best suitable parameters in accordance with recent observations.

In this paper, we have investigated the perfect fluid matter distribution with different quadratic equation of states (EoS) parameters in the form (i) \( p = p_{0} + \alpha \rho + \beta \rho^{2} \), (ii) \( p = \alpha \rho + \frac{{\rho^{2} }}{{\rho_{c} }} \) and (iii) \( p = \alpha \rho^{2} - \rho \) for Marder space–time in \( f(R,T) \) theory. The field equations of \( f(R,T) \) gravity with the inclusion of cosmological parameter \( \varLambda \) have been derived for the first case of Harko et al. (Phys Rev D 84(2):024020, 2011) in the form \( f(R,T) = R + 2f(T) \). The physical parameters are investigated and its properties are studied.

This paper analyzes the anisotropic stellar evolution governed by polytropic equation-of-state in the framework of f(R,T,Q) gravity, where Q = RabTab. We construct the field equations, hydrostatic equilibrium equation and trace equation to obtain their solutions numerically under the influence of σR2 + γQ gravity model, where σ and γ are arbitrary constants. We examine the dependence of various physical characteristics such as radial/tangential pressure, energy density, anisotropic factor, total mass and surface redshift for specific values of the model parameters. The physical acceptability of the considered model is discussed by verifying the validity of energy conditions, causality condition and adiabatic index. We also study the effects arising due to strong nonminimal matter-curvature coupling on anisotropic polytropes. It is found that the polytropic stars are stable and their maximum mass point lies within the required observational Chandrasekhar limit.

This paper is devoted to the study of the existence and stability of the Einstein universe against scalar inhomogeneous perturbations in the background of f (R, T, Q) gravity (R, T represent the scalar curvature, trace of the energy-momentum tensor and \( Q=R_{\sigma\xi}T^{\sigma\xi}\). We construct static as well as perturbed field equations using the linear equation of state in the presence of perfect fluid and examine the stability regions. Considering conserved and non-conserved cases of the energy-momentum tensor for specific models of this gravity, it is found that the stable Einstein static universe exists for both open as well as closed FRW universe models corresponding to the appropriate choice of parameters.

In this paper, we study the non-static spherically symmetric solution in the scenario of f(R, T, Q) gravity (R stands for the scalar curvature, T denotes the trace component of Tab and Q= RabTab). The junction conditions are developed by taking interior and exterior metrics. For a specified model of this gravity, we formulate the field equations using a perfect fluid and discuss the solution for the dust case with constant scalar curvature and trace of the energy-momentum tensor. We evaluate the mass function for this solution and discuss the mass-radius relation as well as the surface redshift of the star. These show compatible results for suitable choice of parameters. © 2018, Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.

The scalar particle production through a scalar field non-minimally coupled with geometry is investigated in the context of a spatially homogeneous and isotropic universe. In this paper, in order to study the evolution of particle production over time in the case of analytical solutions, we focus on a simple Horndeski theory. We first suppose that the universe is dominated by a scalar field and derive the energy conservation condition. Then from the thermodynamic point of view, the macroscopic non-conservation of the scalar field energy-momentum tensor can be explained as an irreversible production of the scalar particles. Based on the explanation, we obtain a scalar particle production rate and the corresponding entropy. Finally, since the universe, in general, could be regarded as a closed system satisfying the laws of thermodynamics, we naturally impose some thermodynamic constraints on it. The thermodynamic properties of the universe can provide additional constraints on the simple Horndeski theory.

First and second laws of black hole thermodynamics are examined at the apparent horizon of FRW spacetime in f(R,RαβRαβ,ϕ) gravity, where R, RαβRαβ and ϕ are the Ricci invariant, Ricci tensor and the scalar field respectively. In this modified theory, Friedmann equations are formulated for any spatial curvature. These equations can be presented into the form of first law of thermodynamics for ThdSˆh+ThdiSˆh+WdV=dE, where diSˆh is an extra entropy term because of the non-equilibrium presentation of the equations and ThdSˆh+WdV=dE for the equilibrium presentation. The generalized second law of thermodynamics (GSLT) is expressed in an inclusive form where these results can be represented in GR, f(R) and f(R,ϕ) gravities. Finally to check the validity of GSLT, we take some particular models and produce constraints of the parameters.

This paper aims to formulate certain scalar factors associated with matter variables for self-gravitating non-static cylindrical geometry by considering a standard model R+ζQ of f(R,T,Q) gravity, where Q=RφϑTφϑ and ζ is the arbitrary coupling parameter. We split the Riemann tensor orthogonally to calculate four scalars and deduce YTF as complexity factor for the fluid configuration. This scalar incorporates the influence of inhomogeneous energy density, heat flux and pressure anisotropy along with correction terms of the modified gravity. We discuss the dynamics of cylinder by considering two simplest modes of structural evolution. We then take YTF=0 with homologous condition to determine the solution for dissipative as well as non-dissipative scenarios. Finally, we discuss the criterion under which the complexity-free condition shows stable behavior throughout the evolution. It is concluded that complex functional of this theory results in a more complex structure.

We discuss the validity of the energy conditions in a newly modified theory
named as $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity, where $R$ and $T$ represent the
scalar curvature and trace of the energy-momentum tensor. The corresponding
energy conditions are derived which appear to be more general and can reduce to
the familiar forms of these conditions in general relativity, $f(R)$ and
$f(R,T)$ theories. The general inequalities are presented in terms of recent
values of Hubble, deceleration, jerk and snap parameters. In particular, we use
two specific models recently developed in literature to study concrete
application of these conditions as well as Dolgov-Kawasaki instability.
Finally, we explore $f(R,T)$ gravity as a specific case to this modified theory
for exponential and power law models.

We investigate the thermodynamics at the apparent horizon of the FRW universe in f(R, T) theory in the nonequilibrium description. The laws of thermodynamics are discussed for two particular models of the f(R, T) theory. The first law of thermodynamics is expressed in the form of the Clausius relation , where δQ is the energy flux across the horizon and is the entropy production term. Furthermore, the conditions for the generalized second law of thermodynamics to be preserved are established with the constraints of positive temperature and attractive gravity. We illustrate our results for some concrete models in this theory.

Within the context of scalar-tensor gravity, we explore the generalized
second law (GSL) of gravitational thermodynamics. We extend the action of
ordinary scalar-tensor gravity theory to the case in which there is a
non-minimal coupling between the scalar field and the matter field (as
chameleon field). Then, we derive the field equations governing the gravity and
the scalar field. For a FRW universe filled only with ordinary matter, we
obtain the modified Friedmann equations as well as the evolution equation of
the scalar field. Furthermore, we assume the boundary of the universe to be
enclosed by the dynamical apparent horizon which is in thermal equilibrium with
the Hawking temperature. We obtain a general expression for the GSL of
thermodynamics which its validity depends on the scalar-tensor gravity model.
For some viable scalar-tensor models, we first obtain the evolutionary
behaviors of the matter density, the scale factor, the Hubble parameter, the
scalar field, the deceleration parameter as well as the effective equation of
state (EoS) parameter. We conclude that in most of the models, the deceleration
parameter approaches a de Sitter regime at late times, as expected. Also the
effective EoS parameter acts like the LCDM model at late times. Finally, we
examine the validity of the GSL for the selected models.

We propose general f(R, T, RμνTμν) theory as generalization of covariant Hořava-like gravity with dynamical Lorentz symmetry breaking. FRLW cosmological dynamics for several versions of such theory is considered. The reconstruction of the above action is explicitly done, including the numerical reconstruction for the occurrence of ΛCDM universe. De Sitter universe solutions in the presence of non-constant fluid are also presented. The problem of matter instability in f(R, T, RμνTμν) gravity is discussed.

We consider the $f(R,T)$ theory, where $R$ is the scalar curvature and $T$ is
the trace of energy-momentum tensor, as an effective description for the
holographic and new agegraphic dark energy and reconstruct the corresponding
$f(R,T)$ functions. In this study, we concentrate on two particular models of
$f(R,T)$ gravity namely, $R+2A(T)$ and $B(R)+\lambda{T}$. We conclude that the
derived $f(R,T)$ models can represent phantom or quintessence regimes of the
universe which are compatible with the current observational data. In addition,
the conditions to preserve the generalized second law of thermodynamics are
established.

We investigate the validity of the generalized second law (GSL) of gravitational thermodynamics in the framework of f(T) modified teleparallel gravity. We consider a spatially flat FRW universe containing only the pressureless matter. The boundary of the universe is assumed to be enclosed by the Hubble horizon. For two viable f(T) models containing f(T) = T+μ1{(−T)}n and f(T) = T−μ2T(1−eβT0/T), we first calculate the effective equation of state and deceleration parameters. Then, {we investigate the null and strong energy conditions and conclude that a sudden future singularity appears in both models. Furthermore, using a cosmographic analysis we check the viability of two models. Finally, we examine the validity of the GSL and find that for both models it} is satisfied from the early times to the present epoch. But in the future, the GSL is violated for the special ranges of the torsion scalar T.

We consider a gravitational model in which matter is non-minimally coupled to
geometry, with the effective Lagrangian of the gravitational field being given
by an arbitrary function of the Ricci scalar, the trace of the matter
energy-momentum tensor, and the contraction of the Ricci tensor with the matter
energy-momentum tensor. The field equations of the model are obtained in the
metric formalism, and the equation of motion of a massive test particle is
derived. In this type of models the matter energy-momentum tensor is generally
not conserved, and this non-conservation determines the appearance of an
extra-force acting on the particles in motion in the gravitational field. The
Newtonian limit of the model is also considered, and an explicit expression for
the extra-acceleration which depends on the matter density is obtained in the
small velocity limit for dust particles. We also analyze in detail the
so-called Dolgov-Kawasaki instability, and obtain the stability conditions of
the model with respect to local perturbations. A particular class of
gravitational field equations can be obtained by imposing the conservation of
the energy-momentum tensor. We derive the corresponding field equations for the
conservative case by using a Lagrange multiplier method, from a gravitational
action that explicitly contains an independent parameter multiplying the
divergence of the energy-momentum tensor. The cosmological implications of the
model are investigated for both the conservative and non-conservative cases,
and several classes of analytical solutions are obtained.

In this work we explore the consequences that a non-minimal coupling between
geometry and matter can have on the dynamics of perfect fluids. It is argued
that the presence of a static, axially symmetric pressureless fluid does not
imply a Minkowski space-time like as is in General Relativity. This feature can
be atributed to a pressure mimicking mechanism related to the non-minimal
coupling. The case of a spherically symmetric black hole surrounded by fluid
matter is analyzed, and it is shown that under equilibrium conditions the total
fluid mass is about twice that of the black hole. Finally, a generalization of
the Newtonian potential for a fluid element is proposed and its implications
are briefly discussed.

In [ T. Jacobson Phys. Rev. Lett. 75 1260 (1995)] it was shown that the Einstein equation can be derived as a local constitutive equation for an equilibrium spacetime thermodynamics. More recently, in the attempt to extend the same approach to the case of f(R) theories of gravity, it was found that a nonequilibrium setting is indeed required in order to fully describe both this theory as well as classical general relativity (GR) [ C. Eling, R. Guedens and T. Jacobson Phys. Rev. Lett. 96 121301 (2006)]. Here, elaborating on this point, we show that the dissipative character leading to nonequilibrium spacetime thermodynamics is actually related—both in GR as well as in f(R) gravity—to nonlocal heat fluxes associated with the purely gravitational/internal degrees of freedom of the theory. In particular, in the case of GR we show that the internal entropy production term is identical to the so-called tidal heating term of Hartle-Hawking. Similarly, for the case of f(R) gravity, we show that dissipative effects can be associated with the generalization of this term plus a scalar contribution whose presence is clearly justified within the scalar-tensor representation of the theory. Finally, we show that the allowed gravitational degrees of freedom can be fixed by the kinematics of the local spacetime causal structure, through the specific equivalence principle formulation. In this sense, the thermodynamical description seems to go beyond Einstein’s theory as an intrinsic property of gravitation.

In the context of f(R,T) theories of gravity, we study the evolution of
scalar cosmological perturbations in the metric formalism. According to
restrictions on the background evolution, a specific model within these
theories is assumed in order to guarantee the standard continuity equation.
Using a completely general procedure, we find the complete set of differential
equations for the matter density perturbations. In the case of sub-Hubble
modes, the density contrast evolution reduces to a second-order equation. We
show that for well-motivated f(R,T) Lagrangians the quasistatic approximation
yields to very different results from the ones derived in the frame of the
concordance {\Lambda}CDM model constraining severely the viability of such
theories.

We review different dark energy cosmologies. In particular, we present the
$\Lambda$CDM cosmology, Little Rip and Pseudo-Rip universes, the phantom and
quintessence cosmologies with Type I, II, III and IV finite-time future
singularities and non-singular dark energy universes. In the first part, we
explain the $\Lambda$CDM model and well-established observational tests which
constrain the current cosmic acceleration. After that, we investigate the dark
fluid universe where a fluid has quite general equation of state (EoS)
[including inhomogeneous or imperfect EoS]. All the above dark energy
cosmologies for different fluids are explicitly realized, and their properties
are also explored. It is shown that all the above dark energy universes may
mimic the $\Lambda$CDM model currently, consistent with the recent
observational data. Furthermore, special attention is paid to the equivalence
of different dark energy models. We consider single and multiple scalar field
theories, tachyon scalar theory and holographic dark energy as models for
current acceleration with the features of quintessence/phantom cosmology, and
demonstrate their equivalence to the corresponding fluid descriptions. In the
second part, we study another equivalent class of dark energy models which
includes $F(R)$ gravity as well as $F(R)$ Ho\v{r}ava-Lifshitz gravity and the
teleparallel $f(T)$ gravity. The cosmology of such models representing the
$\Lambda$CDM-like universe or the accelerating expansion with the
quintessence/phantom nature is described. Finally, we approach the problem of
testing dark energy and alternative gravity models to general relativity by
cosmography. We show that degeneration among parameters can be removed by
accurate data analysis of large data samples and also present the examples.

We explore the thermodynamics of dark energy taking into account the existence of the observer's event horizon in accelerated universes. Except for the initial stage of Chaplygin gas dominated expansion, the generalized second law of gravitational thermodynamics is fulfilled and the temperature of the phantom fluid results positive. This substantially extends the work of Pollock and Singh [M.D. Pollock, T.P. Singh, Class. Quantum Grav. 6 (1989) 901] on the thermodynamics of super-inflationary expansion.

The field equations of a generalized f(R) type gravity model, in which there is an arbitrary coupling between matter and geometry, are obtained. The equations of motion for test particles are derived from a variational principle in the particular case in which the Lagrange density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the model is also considered. The perihelion precession of an elliptical planetary orbit in the presence of an extra force is obtained in a general form, and the magnitude of the extra gravitational effects is constrained in the case of a constant extra force by using Solar System observations.

We study thermodynamics of the apparent horizon in F(R) gravity. In particular, we demonstrate that an F(R) gravity model with realizing a crossing of the phantom divide can satisfy the second law of thermodynamics in the effective phantom phase as well as non-phantom one.

The energy conditions and the Dolgov-Kawasaki criterion in generalized $f(R)$
gravity with arbitrary coupling between matter and geometry are derived in this
paper, which are quite general and can degenerate to the well-known energy
conditions in GR and $f(R)$ gravity with non-minimal coupling and non-coupling
as special cases. In order to get some insight on the meaning of these energy
conditions and the Dolgov- Kawasaki criterion, we apply them to a class of
models in the FRW cosmology and give some corresponding results.

A non-equilibrium picture of thermodynamics is discussed at the apparent
horizon of FRW universe in $f(R,T)$ gravity, where $R$ is the Ricci scalar and
$T$ is the trace of the energy-momentum tensor. We take two forms of the
energy-momentum tensor of dark components and demonstrate that equilibrium
description of thermodynamics is not achievable in both cases. We check the
validity of the first and second law of thermodynamics in this scenario. It is
shown that the Friedmann equations can be expressed in the form of first law of
thermodynamics $T_hdS'_h+T_hd_{\jmath}S'=-dE'+W'dV$, where $d_{\jmath}S'$ is
the entropy production term. Finally, we conclude that the second law of
thermodynamics holds both in phantom and non-phantom phases.

We study the generalized second law (GSL) of thermodynamics in $f(T)$
cosmology. We consider the universe as a closed bounded system filled with $n$
component fluids in the thermal equilibrium with the cosmological boundary. We
use two different cosmic horizons: the future event horizon and the apparent
horizon. We show the conditions under which the GSL will be valid in specific
scenarios of the quintessence and the phantom energy dominated eras. Further we
associate two different entropies with the cosmological horizons: with a
logarithmic correction term and a power-law correction term. We also find the
conditions for the GSL to be satisfied or violated by imposing constraints on
model parameters.

We consider cosmological scenarios based on $f(R,T)$ theories of gravity ($R$
is the Ricci scalar and $T$ is the trace of the energy-momentum tensor) and
numerically reconstruct the function $f(R,T)$ which is able to reproduce the
same expansion history generated, in the standard General Relativity theory, by
dark matter and holographic dark energy. We consider two special $f(R,T)$
models: in the first instance, we investigate the modification $R + 2f(T)$,
i.e. the usual Einstein-Hilbert term plus a $f(T)$ correction. In the second
instance, we consider a $f(R)+\lambda T$ theory, i.e. a $T$ correction to the
renown $f(R)$ theory of gravity.

We consider f(R,T) modified theories of gravity, where the gravitational
Lagrangian is given by an arbitrary function of the Ricci scalar R and of the
trace of the stress-energy tensor T. We obtain the gravitational field
equations in the metric formalism, as well as the equations of motion for test
particles, which follow from the covariant divergence of the stress-energy
tensor. Generally, the gravitational field equations depend on the nature of
the matter source. The field equations of several particular models,
corresponding to some explicit forms of the function f(R,T), are also
presented. An important case, which is analyzed in detail, is represented by
scalar field models. We write down the action and briefly consider the
cosmological implications of the $f(R,T^{\phi})$ models, where $T^{\phi}$ is
the trace of the stress-energy tensor of a self-interacting scalar field. The
equations of motion of the test particles are also obtained from a variational
principle. The motion of massive test particles is non-geodesic, and takes
place in the presence of an extra force orthogonal to the four-velocity. The
Newtonian limit of the equation of motion is further analyzed. Finally, we
provide a constraint on the magnitude of the extra-acceleration by analyzing
the perihelion precession of the planet Mercury in the framework of the present
model.

We generalize the f(R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L
m
. We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert–Einstein Lagrange density are also derived.

We investigate the validity of the generalized second law of thermodynamics
in a universe governed by Horava-Lifshitz gravity. Under the equilibrium
assumption, that is in the late-time cosmological regime, we calculate
separately the entropy time-variation for the matter fluid and, using the
modified entropy relation, that of the apparent horizon itself. We find that
under detailed balance the generalized second law is generally valid for flat
and closed geometry and it is conditionally valid for an open universe, while
beyond detailed balance it is only conditionally valid for all curvatures.
Furthermore, we also follow the effective approach showing that it can lead to
misleading results. The non-complete validity of the generalized second law
could either provide a suggestion for its different application, or act as an
additional problematic feature of Horava-Lifshitz gravity.

We study the relation between the thermodynamics and field equations of generalized gravity theories on the dynamical trapping horizon with sphere symmetry. We assume the entropy of dynamical horizon as the Noether charge associated with the Kodama vector and point out that it satisfies the second law when a Gibbs equation holds. We generalize two kinds of Gibbs equations to Gauss-Bonnet gravity on any trapping horizon. Based on the quasi-local gravitational energy found recently for $f(R)$ gravity and scalar-tensor gravity in some special cases, we also build up the Gibbs equations, where the nonequilibrium entropy production, which is usually invoked to balance the energy conservation, is just absorbed into the modified Wald entropy in the FRW spacetime with slowly varying horizon. Moreover, the equilibrium thermodynamic identity remains valid for $f(R)$ gravity in a static spacetime. Our work provides an alternative treatment to reinterpret the nonequilibrium correction and supports the idea that the horizon thermodynamics is universal for generalized gravity theories. Comment: 23 pages, no figure, minor changes, accepted for publication in Phys. Rev. D

It has previously been shown that the Einstein equation can be derived from the requirement that the Clausius relation dS=deltaQ/T hold for all local acceleration horizons through each spacetime point, where is one-quarter the horizon area change in Planck units and deltaQ and T are the energy flux across the horizon and the Unruh temperature seen by an accelerating observer just inside the horizon. Here we show that a curvature correction to the entropy that is polynomial in the Ricci scalar requires a nonequilibrium treatment. The corresponding field equation is derived from the entropy balance relation dS=deltaQ/T+diS, where diS is a bulk viscosity entropy production term that we determine by imposing energy-momentum conservation. Entropy production can also be included in pure Einstein theory by allowing for shear viscosity of the horizon.

With the help of a masslike function which has a dimension of energy and is equal to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon dE=T(A)dS(A) can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, nonlinear, and scalar-tensor theories. This result strongly suggests that the relationship between the first law of thermodynamics of the apparent horizon and the Friedmann equation is not just a simple coincidence, but rather a more profound physical connection.

We investigate the generalized second law of thermodynamics (GSL) in generalized theories of gravity. We examine the total entropy evolution with time including the horizon entropy, the non-equilibrium entropy production, and the entropy of all matter, field and energy components. We derive a universal condition to protect the generalized second law and study its validity in different gravity theories. In Einstein gravity, (even in the phantom-dominated universe with a Schwarzschild black hole), Lovelock gravity, and braneworld gravity, we show that the condition to keep the GSL can always be satisfied. In $f(R)$ gravity and scalar-tensor gravity, the condition to protect the GSL can also hold because the gravity is always attractive and the effective Newton constant should be approximate constant satisfying the experimental bounds.

We investigate the cosmological reconstruction in modified f(R, T) gravity, where R is the Ricci scalar and T the trace of the stress–energy tensor. Special attention is attached to the case in which the function f is given by f(R, T) = f1(R) + f2(T). The use of auxiliary scalar field is considered with two known examples for the scale factor corresponding to an expanding universe. In the first example, where ordinary matter is usually neglected for obtaining the unification of matter dominated and accelerated phases with f(R) gravity, it is shown in this paper that this unification can be obtained without neglecting ordinary matter. In the second example, as in f(R) gravity, model of f(R, T) gravity with transition of matter dominated phase to the acceleration phase is obtained. In both cases, linear function of the trace is assumed for f2(T) and it is obtained that f1(R) is proportional to a power of R with exponents depending on the input parameters.

In this paper, we study the behavior of perfect fluid and massless
scalar field for homogeneous and anisotropic Bianchi type I universe
model in f(R,T) gravity, where R is the Ricci scalar and T is the trace
of the energy-momentum tensor. We assume the variation law of mean
Hubble parameter to obtain exact solutions of the modified field
equations. The physical and kinematical quantities are discussed for
both models in future evolution of the universe. We check the validity
of null energy condition and conclude that our perfect fluid solution
can behave like phantom model. Finally, we find that perfect fluid
solutions correspond to massless scalar field models.

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.

There are a number of similarities between black-hole physics and thermodynamics. Most striking is the similarity in the behaviors of black-hole area and of entropy: Both quantities tend to increase irreversibly. In this paper we make this similarity the basis of a thermodynamic approach to black-hole physics. After a brief review of the elements of the theory of information, we discuss black-hole physics from the point of view of information theory. We show that it is natural to introduce the concept of black-hole entropy as the measure of information about a black-hole interior which is inaccessible to an exterior observer. Considerations of simplicity and consistency, and dimensional arguments indicate that the black-hole entropy is equal to the ratio of the black-hole area to the square of the Planck length times a dimensionless constant of order unity. A different approach making use of the specific properties of Kerr black holes and of concepts from information theory leads to the same conclusion, and suggests a definite value for the constant. The physical content of the concept of black-hole entropy derives from the following generalized version of the second law: When common entropy goes down a black hole, the common entropy in the black-hole exterior plus the black-hole entropy never decreases. The validity of this version of the second law is supported by an argument from information theory as well as by several examples.

The first law and the generalized second law (GSL) of thermodynamics for the generalized f(R)f(R) gravity with curvature–matter coupling are studied in the spatially homogeneous, isotropic FRW universe. The research results show that the field equations of the generalized f(R)f(R) gravity with curvature–matter coupling can be cast to the form of the first law of thermodynamics with the so-called the entropy production terms dS¯ and the GSL can be given by considering the FRW universe filled only with ordinary matter enclosed by the dynamical apparent horizon with the Hawking temperature. Furthermore, as a concrete example, by utilizing the GSL the constraints on the gravitational model with f1(R)=R+αRlf1(R)=R+αRl and f2(R)=Rmf2(R)=Rm are also discussed. It is worth noting these results given by us are quite general and can degenerate to the ones in Einsteinʼs general relativity and pure f(R)f(R) gravity with non-coupling and non-minimal coupling as special cases. Comparing with the case of Einsteinʼs general relativity, the appearance of the entropy production term dS¯ in the first law of thermodynamics demonstrates that the horizon thermodynamics is non-equilibrium one for generalized f(R)f(R) gravity with curvature–matter coupling, which is consistent with the arguments given in Akbar and Cai (2007) [13] and Eling et al. (2006) [18].

We discuss the validity of the energy conditions in a newly modified theory
named as $f(R,T,R_{\mu\nu}T^{\mu\nu})$ gravity, where $R$ and $T$ represent the
scalar curvature and trace of the energy-momentum tensor. The corresponding
energy conditions are derived which appear to be more general and can reduce to
the familiar forms of these conditions in general relativity, $f(R)$ and
$f(R,T)$ theories. The general inequalities are presented in terms of recent
values of Hubble, deceleration, jerk and snap parameters. In particular, we use
two specific models recently developed in literature to study concrete
application of these conditions as well as Dolgov-Kawasaki instability.
Finally, we explore $f(R,T)$ gravity as a specific case to this modified theory
for exponential and power law models.

The energy conditions are derived in the context of $f(R,T)$ gravity, where
$R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor,
which can reduce to the well-known conditions in $f(R)$ gravity and general
relativity. We present the general inequalities set by the energy conditions in
terms of Hubble, deceleration, jerk and snap parameters. In this study, we
concentrate on two particular models of $f(R,T)$ gravity namely,
$f(R)+\lambda{T}$ and $R+2f(T)$. The exact power-law solutions are obtained for
these two cases in homogeneous and isotropic $f(R,T)$ cosmology. Finally, we
find certain constraints which have to be satisfied to ensure that power law
solutions may be stable and match the bounds prescribed by the energy
conditions.

We consider modified theories of gravity with a direct coupling between
matter and geometry, denoted by an arbitrary function in terms of the Ricci
scalar. Due to such a coupling, the matter stress tensor is no longer conserved
and there is an energy transfer between the two components. By solving the
conservation equation, we argue that the matter system should gain energy in
this interaction, as demanded by the second law of thermodynamics. In a
cosmological setting, we show that although this kind of interaction may
account for cosmic acceleration, this latter together with direction of the
energy transfer constrain the coupling function.

It is well known that by applying the first law of thermodynamics to the
apparent horizon of a Friedmann-Robertson-Walker universe, one can derive the
corresponding Friedmann equations in Einstein, Gauss-Bonnet, and more general
Lovelock gravity. Is this a generic feature of any gravitational theory? Is the
prescription applicable to other gravities? In this paper we would like to
address the above questions by examining the same procedure for Horava-Lifshitz
gravity. We find that in Horava-Lifshitz gravity, this approach does not work
and we fail to reproduce a corresponding Friedmann equation in this theory by
applying the first law of thermodynamics on the apparent horizon, together with
the appropriate expression for the entropy in Horava-Lifshitz gravity. The
reason for this failure seems to be due to the fact that Horava-Lifshitz
gravity is not diffeomorphism invariant, and thus, the corresponding field
equation cannot be derived from the first law around horizon in the spacetime.
Without this, it implies that the specific gravitational theory is not
consistent, which shows an additional problematic feature of Horrava-Lifshitz
gravity. Nevertheless, if we still take the area formula of geometric entropy
and regard Horava-Lifshitz sector in the Friedmann equation as an effective
dark radiation, we are able to extract the corresponding Friedmann equation
from the first law of thermodynamics.

In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects
cause black holes to create and emit particles as if they were hot bodies with temperature
\frachk2pk » 10 - 6 ( \fracM\odot M )° K\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K
where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black
hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon
of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons
in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so
much entropy per baryon.

Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics at apparent horizon of Friedmann–Robertson–Walker (FRW) universe, which indicates intrinsic thermodynamic properties of apparent horizon of spacetime. In the present Letter we deal with the so-called f(R) gravity, whose action is a function of the curvature scalar R. In the setup of FRW universe, we show that the field equations can also be cast to a similar form, , at the apparent horizon, where W=(ρ−P)/2, E is the energy of perfect fluid with energy density ρ and pressure P inside the apparent horizon. T and S=Af′(R)/4G are temperature and entropy associated with the apparent horizon, respectively. Compared to the case of Einstein's general relativity, an additional term appears here. The appearance of the additional term is consistent with the argument recently given by Eling et al. [C. Eling, R. Guedens, T. Jacobson, Phys. Rev. Lett. 96 (2006) 121301, gr-qc/0602001] that the horizon thermodynamics is non-equilibrium one for the f(R) gravity.

Recently there has been a proposal for modified gravitational f(R) actions which include a direct coupling between the matter action and the Ricci scalar, R. Of particular interest is the specific case where both the action and the coupling are linear in R. It is shown that such an action leads to a theory of gravity which includes higher order derivatives of the matter fields without introducing more dynamics in the gravity sector and, therefore, cannot be a viable theory for gravitation. (c) 2008 Elsevier B.V. All rights reserved.

We show that it is possible to obtain a picture of equilibrium thermodynamics on the apparent horizon in the expanding cosmological background for a wide class of modified gravity theories with the Lagrangian density f(R,ϕ,X), where R is the Ricci scalar and X is the kinetic energy of a scalar field ϕ. This comes from a suitable definition of an energy–momentum tensor of the “dark” component that respects to a local energy conservation in the Jordan frame. In this framework the horizon entropy S corresponding to equilibrium thermodynamics is equal to a quarter of the horizon area A in units of gravitational constant G, as in Einstein gravity. For a flat cosmological background with a decreasing Hubble parameter, S globally increases with time, as it happens for viable f(R) inflation and dark energy models. We also show that the equilibrium description in terms of the horizon entropy S is convenient because it takes into account the contribution of both the horizon entropy in non-equilibrium thermodynamics and an entropy production term.

In this paper we first obtain Friedmann equations for the (n−1)-dimensional brane embedded in the (n+1)-dimensional bulk, with intrinsic curvature term of the brane included in the action (DGP model). Then, we show that one can always rewrite the Friedmann equations in the form of the first law of thermodynamics, , at apparent horizon on the brane, regardless of whether there is the intrinsic curvature term on the brane or a cosmological constant in the bulk. Using the first law, we extract the entropy expression of the apparent horizon on the brane. We also show that in the case without the intrinsic curvature term, the entropy expressions are the same by using the apparent horizon on the brane and by using the bulk geometry. When the intrinsic curvature appears, the entropy of apparent horizon on the brane has two parts, one part follows the n-dimensional area formula on the brane, and the other part is the same as the entropy in the case without the intrinsic curvature term. As an interesting result, in the warped DGP model, the entropy expression in the bulk and on the brane are not the same. This is reasonable, since in this model gravity on the brane has two parts, one induced from the (n+1)-dimensional bulk gravity and the other due to the intrinsic curvature term on the brane.

In this Letter which is an extension of the work [G. Izquierdo, D. Pavón, Phys. Lett. B 639 (2006) 1], we study the conditions required for validity of the generalized second law in phantom-dominated universe in the presence of Schwarzschild black hole. Our study is independent of the origin of the phantom like behavior of the considered universe. We also discuss the generalized second law in the neighborhood of transition (from quintessence to phantom regime) time. We show that even for a constant equation of state parameter, the generalized second law may be satisfied provided that the temperature is not taken as de Sitter temperature. It is shown that in models with (only) a transition from quintessence to phantom regime the generalized second law does not hold in the transition epoch.

In this paper we discuss thermodynamics of apparent horizon of an n-dimensional Friedmann–Robertson–Walker (FRW) universe embedded in an (n+1)-dimensional AdS space–time. By using the method of unified first law, we give the explicit entropy expression of the apparent horizon of the FRW universe. In the large horizon radius limit, this entropy reduces to the n-dimensional area formula, while in the small horizon radius limit, it obeys the (n+1)-dimensional area formula. We also discuss the corresponding bulk geometry and study the apparent horizon extended into the bulk. We calculate the entropy of this apparent horizon by using the area formula of the (n+1)-dimensional bulk. It turns out that both methods give the same result for the apparent horizon entropy. In addition, we show that the Friedmann equation on the brane can be rewritten to a form of the first law, , at the apparent horizon.

We explore thermodynamics of the apparent horizon in $f(T)$ gravity with both
equilibrium and non-equilibrium descriptions. We find the same dual
equilibrium/non-equilibrium formulation for $f(T)$ as for $f(R)$ gravity. In
particular, we show that the second law of thermodynamics can be satisfied for
the universe with the same temperature of the outside and inside the apparent
horizon.

We investigate thermodynamics of the apparent horizon in
f(R) gravity in the Palatini formalism with non-equilibrium and
equilibrium descriptions. We demonstrate that it is more
transparent to understand the horizon entropy in the equilibrium
framework than that in the non-equilibrium one. Furthermore, we
show that the second law of thermodynamics can be explicitly
verified in both phantom and non-phantom phases for the same
temperature of the universe outside and inside the apparent
horizon.

We explicitly show that the equations of motion for modified gravity theories
of $F(R)$-gravity, the scalar-Gauss-Bonnet gravity, $F(\mathcal{G})$-gravity
and the non-local gravity are equivalent to the Clausius relation in
thermodynamics. In addition, we discuss the relation between the expression of
the entropy and the contribution from the modified gravity as well as the
matter to the definition of the energy flux (heat).

Expressions are derived for the mass of a stationary axisymmetric solution of the Einstein equations containing a black hole surrounded by matter and for the difference in mass between two neighboring such solutions. Two of the quantities which appear in these expressions, namely the area A of the event horizon and the ``surface gravity'' kappa of the black hole, have a close analogy with entropy and temperature respectively. This analogy suggests the formulation of four laws of black hole mechanics which correspond to and in some ways transcend the four laws of thermodynamics.

It is well known that, for a wide class of spacetimes with horizons, Einstein equations near the horizon can be written as a thermodynamic identity. It is also known that the Einstein tensor acquires a highly symmetric form near static, as well as stationary, horizons. We show that, for generic static spacetimes, this highly symmetric form of the Einstein tensor leads quite naturally and generically to the interpretation of the near-horizon field equations as a thermodynamic identity. We further extend this result to generic static spacetimes in Lanczos-Lovelock gravity, and show that the near-horizon field equations again represent a thermodynamic identity in all these models. These results confirm the conjecture that this thermodynamic perspective of gravity extends far beyond Einstein's theory. Comment: RevTeX 4; 10 pages; no figures

We consider a general, classical theory of gravity in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian. In any such theory, to each vector field, $\xi^a$, on spacetime one can associate a local symmetry and, hence, a Noether current $(n-1)$-form, ${\bf j}$, and (for solutions to the field equations) a Noether charge $(n-2)$-form, ${\bf Q}$. Assuming only that the theory admits stationary black hole solutions with a bifurcate Killing horizon, and that the canonical mass and angular momentum of solutions are well defined at infinity, we show that the first law of black hole mechanics always holds for perturbations to nearby stationary black hole solutions. The quantity playing the role of black hole entropy in this formula is simply $2 \pi$ times the integral over $\Sigma$ of the Noether charge $(n-2)$-form associated with the horizon Killing field, normalized so as to have unit surface gravity. Furthermore, we show that this black hole entropy always is given by a local geometrical expression on the horizon of the black hole. We thereby obtain a natural candidate for the entropy of a dynamical black hole in a general theory of gravity. Our results show that the validity of the ``second law" of black hole mechanics in dynamical evolution from an initially stationary black hole to a final stationary state is equivalent to the positivity of a total Noether flux, and thus may be intimately related to the positive energy properties of the theory. The relationship between the derivation of our formula for black hole entropy and the derivation via ``Euclidean methods" also is explained.

The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation ffiQ = TdS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with ffiQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air. The four laws of black hole mechanics, which are analogous to those of thermodynamics, were originally derived from the classical Einstein equation[1]. With the discovery of the q...

We consider alternative theories of gravity with a direct coupling between matter and the Ricci scalar We study the relation between these theories and ordinary scalar-tensor gravity, or scalar-tensor theories which include non-standard couplings between the scalar and matter. We then analyze the motion of matter in such theories, its implications for the Equivalence Principle, and the recent claim that they can alleviate the dark matter problem in galaxies. Comment: typos corrected, minor changes, version published in CQG