Modified gravity theories have received increased attention lately due to combined motivation coming from high-energy physics, cosmology, and astrophysics. Among numerous alternatives to Einstein's theory of gravity, theories that include higher-order curvature invariants, and specifically the particular class of $f(R)$ theories, have a long history. In the last five years there has been a new stimulus for their study, leading to a number of interesting results. Here $f(R)$ theories of gravity are reviewed in an attempt to comprehensively present their most important aspects and cover the largest possible portion of the relevant literature. All known formalisms are presented---metric, Palatini, and metric affine---and the following topics are discussed: motivation; actions, field equations, and theoretical aspects; equivalence with other theories; cosmological aspects and constraints; viability criteria; and astrophysical applications.

f ( R ) theories of gravity

Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models

Classical generalization of general relativity is considered as gravitational alternative for unified description of the early-time inflation with late-time cosmic acceleration. The structure and cosmological properties of number of modified theories, including traditional $F(R)$ and Hořava-Lifshitz $F(R)$ gravity, scalar-tensor theory, string-inspired and Gauss-Bonnet theory, non-local gravity, non-minimally coupled models, and power-counting renormalizable covariant gravity are discussed. Different representations and relations between such theories are investigated. It is shown that some versions of above theories may be consistent with local tests and may provide qualitatively reasonable unified description of inflation with dark energy epoch. The cosmological reconstruction of different modified gravities is made in great detail. It is demonstrated that eventually any given universe evolution may be reconstructed for the theories under consideration: the explicit reconstruction is applied to accelerating spatially-flat FRW universe. Special attention is paid to Lagrange multiplier constrained and conventional $F(R)$ gravities, for last theory the effective $\Lambda$CDM era and phantom-divide crossing acceleration are obtained. The occurrence of Big Rip and other finite-time future singularities in modified gravity is reviewed as well as its curing via the addition of higher-derivative gravitational invariants.

Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution

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 $f(R,{T}^{\ensuremath{\phi}})$ models, where ${T}^{\ensuremath{\phi}}$ 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 nongeodesic, 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.

$f(R,T)$ gravity

In order to shed some light on the current discussion about f(R)-gravity theories we derive and discuss the bounds imposed by the energy conditions on a general f(R) functional form The null and strong energy conditions in this framework are derived from Raychaudhuri's equation along with the requirement that gravity is attractive, whereas the weak and dominant energy conditions are stated from a comparison with the energy conditions that can be obtained in a direct approach via an effective energy-momentum tensor for f(R) gravity As a concrete application of the energy conditions to locally homogeneous and isotropic f(R) cosmology, the recent estimated values of the deceleration and jerk parameters are used to examine the bounds from the weak energy condition on the parameters of two families of f(R)-gravity theories

Energy conditions in f(R) gravity

Cosmography is an ideal tool to investigate the cosmic expansion history of the Universe in a model-independent way. The equations of motion in modified theories of gravity are usually very complicated; cosmography may select practical models without imposing arbitrary choices a priori. We use the model-independent way to derive $f(z)$ and its derivatives up to fourth order in terms of measurable cosmographic parameters. We then fit those functions into the luminosity distance directly. We perform the MCMC analysis by considering three different sets of cosmographic functions. Using the largest supernovae Ia Pantheon sample, we derive the constraints on the Hubble constant ${H}_{0}$ and the cosmographic functions, and find that the former two terms in Taylor expansion of luminosity distance work dominantly in $f(Q)$ gravity.

Cosmography in f ( Q ) gravity

In this paper, we investigate the dark energy phenomenon by studying the Tsallis holographic dark energy within the framework of Brans–Dicke (BD) scalar–tensor theory of gravity (Brans and Dicke in Phys. Rev. 124:925, 1961). In this context, we choose the BD scalar field $$\phi $$ as a logarithmic function of the average scale factor a(t) and Hubble horizon as the IR cutoff ($$L=H^{-1}$$). We reconstruct two cases of non-interacting and interacting fluid (dark sectors of cosmos) scenario. The physical behavior of the models are discussed with the help of graphical representation to explore the accelerated expansion of the universe. Moreover, the stability of the models are checked through squared sound speed $$v_s^2$$. The well-known cosmological plane i.e., $$\omega _{de}-\omega ^{\prime }_{de}$$ is constructed for our models. We also include comparison of our findings of these dynamical parameters with observational constraints. It is also quite interesting to mention here that the results of deceleration, equation of state parameters and $$\omega _{de}-\omega ^{\prime }_{de}$$ plane coincide with the modern observational data.

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Observational constraint on interacting Tsallis holographic dark energy in logarithmic Brans–Dicke theory

The current interests in the universe motivate us to go beyond Einstein's General theory of relativity One of the interesting proposals comes from a new class of teleparallel gravity named symmetric teleparallel gravity, ie, $f(Q)$ gravity, where the non-metricity term $Q$ is accountable for fundamental interaction These alternative modified theories of gravity's vital role are to deal with the recent interests and to present a realistic cosmological model This manuscript's main objective is to study the traversable wormhole geometries in $f(Q)$ gravity We construct the wormhole geometries for three cases: (i) by assuming a relation between the radial and lateral pressure, (ii) considering phantom energy equation of state (EoS), and (iii) for a specific shape function in the fundamental interaction of gravity (ie for linear form of $f(Q)$) Besides, we discuss two wormhole geometries for a general case of $f(Q)$ with two specific shape functions Then, we discuss the viability of shape functions and the stability analysis of the wormhole solutions for each case We have found that the null energy condition (NEC) violates each wormhole model which concluded that our outcomes are realistic and stable Finally, we discuss the embedding diagrams and volume integral quantifier to have a complete view of wormhole geometries

Traversable wormhole geometries in $f(Q)$ gravity

Abstract Despite having the somewhat successful description of accelerated cosmology, the early evolution of the universe always challenges mankind. Our promising approach lies in a new class of symmetric teleparallel theory of gravity named f ( Q ), where the non-metricity scalar Q is responsible for the gravitational interaction, which may resolve some of the issues. To study the early evolution of the universe, we presume an anisotropic locally rotationally symmetric (LRS) Bianchi-I spacetime and derive the motion equations. We discuss the profiles of energy density, equation of state and skewness parameter and observe that our models archive anisotropic spatial geometry in the early phase of the universe with a possible presence of anisotropic fluid and as time goes on, even in the presence of an anisotropic fluid, the universe could approach isotropy due to inflation and the anisotropy of the fluid fades away at the same time. 

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Sanjay Mandal

Papers

Energy conditions in f(R) gravity

Cosmography in f ( Q ) gravity

Observational constraint on interacting Tsallis holographic dark energy in logarithmic Brans–Dicke theory

Traversable wormhole geometries in $f(Q)$ gravity

Isotropization of locally rotationally symmetric Bianchi-I universe in f(Q)-gravity