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We investigate the modified gravity theories in terms of the effective dark energy models. We compare the cosmic expansion history and the linear growth in different models. We also study the evolution of linear cosmological perturbations in modified theories of gravity assuming the Palatini formalism. We find the stability of the superhorizon metric evolution depends on models. We also study the matter density fluctuation in the general gauge and show the differential equations in super and sub-horizon scales.

We discuss the conformal symmetry between Jordan and Einstein frames considering their relations with the metric and Palatini formalisms for modified gravity. Appropriate conformal transformations are taken into account leading to the evident connection between the gravitational actions in the two mentioned frames and the Hilbert–Einstein action with a cosmological constant. We show that the apparent differences between Palatini and metric formalisms strictly depend on the representation while the number of degrees of freedom is preserved. This means that the dynamical content of both formalism is identical.

The equivalence between metric and Palatini formalisms in f(R)-gravity can be achieved in the general context of theories with divergence-free current. This equivalence is a necessary result of a symmetry which is included in a particular conservation equation of the current. In fact the conservation equation, by an appropriate redefinition of the introduced auxiliary field, may be encoded in a massless scalar field equation.

We derive the field equations and the equations of motion for scalar fields and massive test particles in modified theories of gravity with an arbitrary coupling between geometry and matter by using the Palatini formalism. We show that the independent connection can be expressed as the Levi–Cività connection of an auxiliary, matter Lagrangian dependent metric, which is related with the physical metric by means of a conformal transformation. Similarly to the metric case, the field equations impose the nonconservation of the energy–momentum tensor. We derive the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra-force is obtained in terms of the matter-geometry coupling functions and of their derivatives. Generally, the motion is non-geodesic, and the extra force is orthogonal to the four-velocity. It is pointed out here that the force is of a different nature than in the metric formalism. We also consider the implications of a nonlinear dependence of the action upon the matter Lagrangian.

In this work, we provide a possible geometrical interpretation of the spin of elementary particles. In particular, it is investigated how the wave equations of matter are altered by the addition of an antisymmetric contribution to the metric tensor. In this scenario, the explicit form of the matter wave equations is investigated in a general curved spacetime, and then the equations are particularized to the flat case. Unlike traditional approaches of Nonsymmetric Gravitational Theories (NGT), in which the gravitational field is responsible for breaking the symmetry of the flat Minkowski metric, we find more natural to consider that, in general, the metric of the spacetime could be nonsymmetric even in the flat case. The physical consequences of this assumption are explored in detail. Interestingly enough, it is found that the metric tensor splits into a bosonic and a fermionic; the antisymmetric part of the metric is very sensitive to the spin and turns out to be undetectable for spinless scalar particles. However, fermions couple to it in a nontrivial way (only when there are interactions). In addition, the Pauli coupling is derived automatically as a consequence of the nonsymmetric nature of the metric.

Cosmological solutions for covariant canonical gauge theories of gravity are presented. The underlying covariant canonical transformation framework invokes a dynamical spacetime Hamiltonian consisting of the Einstein–Hilbert term plus a quadratic Riemann tensor invariant with a fundamental dimensionless coupling constant $g_1$. A typical time scale related to this constant, $\tau =\sqrt{8\pi G{g}_1}$, is characteristic for the type of cosmological solutions: for $t\ll \tau$, the quadratic term is dominant, the energy–momentum tensor of matter is not covariantly conserved, and we observe modified dynamics of matter and spacetime. On the other hand, for $t\gg \tau$, the Einstein term dominates and the solution converges to classical cosmology. This is analyzed for different types of matter and dark energy with a constant equation of state. While for a radiation-dominated universe solution, the cosmology does not change, we find for a dark energy universe the well-known de-Sitter space. However, we also identify a special bouncing solution (for $k=0$) which for large times approaches the de-Sitter space again. For a dust-dominated universe (with no pressure), deviations are seen only in the early epoch. In late epoch, the solution asymptotically behaves as the standard dust solution.

Under conformal transformation, $f(\Re )$ theory of gravity in Palatini formalism leads to a Brans–Dicke type of scalar-tensor equivalent theory with a wrong sign in the effective kinetic energy term. This means that the effective scalar acts as the dark energy and so late-time cosmic acceleration in the matter-dominated era is accountable. However, we unveil some aspects of Palatini formalism, which reveals the fact that the formalism is not suitable to explain the cosmological evolution of the early universe with $f(\Re )$ gravity alone. Additionally, it is noticed that some authors, in an attempt to explore Noether symmetry of the theory changed the sign of the kinetic term and hence obtained the wrong answer. Here, we make the correction and unmask a very interesting aspect of symmetry analysis. Mathematical inequivalence between Jordan’s and Einstein’s frame in Palatini $f(\Re )$ theory has also been revealed.

We discuss the Palatini formulation of modified gravity including a Yukawa-like term. It is shown that in this formulation, the Yukawa term offers an explanation for the current exponential accelerated expansion of the universe and reduces to the standard Friedmann cosmology in the appropriate limit. We then discuss the scalar-tensor formulation of the model as a metric theory and show that the Yukawa term predicts a power-law acceleration at late-times. The Newtonian limit of the theory is also discussed in context of the Palatini formalism.

We review the recent literature on modified theories of gravity in the Palatini approach. After discussing the motivations that lead to consider alternatives to Einstein's theory and to treat the metric and the connection as independent objects, we review several topics that have been recently studied within this framework. In particular, we provide an in-depth analysis of the cosmic speed-up problem, laboratory and solar system tests, the structure of stellar objects, the Cauchy problem, and bouncing cosmologies. We also discuss the importance of going beyond the f(R) models to capture other phenomenological aspects related with dark matter/energy and quantum gravity.

We argue that the quantum nature of matter and gravity should lead to a discretization of the allowed states of the matter confined in the interior of black holes. To support and illustrate this idea, we consider a quadratic extension of general relativity (GR) formulated à la Palatini and show that nonrotating, electrically charged black holes develop a compact core at the Planck density which is nonsingular if the mass spectrum satisfies a certain discreteness condition. We also find that the area of the core is proportional to the number of charges times the Planck area.

We derive a new set of field equations within the framework of the Palatini formalism. These equations are a natural generalization of the Einstein–Maxwell equations which arise by adding a function , with to the Palatini Lagrangian f(R, Q). The result we obtain can be viewed as the coupling of gravity with a nonlinear extension of the electromagnetic field. In addition, a new method is introduced to solve the algebraic equation associated to the Ricci tensor.

The Palatini $f(R,T)$ gravity theory is considered. The standard Einstein–Hilbert action is replaced by an arbitrary function of the Ricci scalar $R$ and of the trace $T$ of the energy-momentum tensor. In the Palatini approach, the Ricci scalar is a function of the metric and the connection. These two quantities, metric and connection, are taken as independent variables. Then, it is examined whether Palatini $f(R,T)$ gravity theory allows solutions in which lead to violation of causality. The Gödel and Gödel-type spacetimes are considered. In addition, a critical radius, which permits to examine limits for violation of causality, is calculated. It is shown that, for different matter contents, noncausal solutions can be avoided in this Palatini gravitational theory.

This paper copes with the issue of analyzing the conditions to check the stability of the pressure isotropy condition by taking into account a spherically symmetric dissipative astrophysical configuration with the Palatini $f(R)$ gravity theory. We work out a differential equation in terms of the Weyl scalar that has a crucial part in the analysis of evolution of considered system. Using this equation, we devise another stellar equation that characterizes the evolution of anisotropic factor. Later, we assume an axially symmetric configuration and extend our analysis for that particular symmetry. It is worth-observing that the physical factors responsible for compelling an initially isotropic object to trigger pressure anisotropy incorporate energy density, dissipative flux and shear in fluid flow.

One of the most favorable extensions of General Relativity is the $f(R)$ gravity. $f(R)$ gravity in Palatini formalism can be a realistic alternative to the dark energy problem. In this work, we study a recently introduced dark energy $f(R)$ gravity model along with two other models in cosmological perspectives under the Palatini formalism. First, we study the cosmic expansion history of these models with the help of the important cosmographic parameters, such as the Hubble parameter, luminosity distance, effective equation of state, etc. This study shows that the new model behaves similarly with the other two models as well as with the $\mathrm{\Lambda }$CDM model in some respects in the early or very early phases of the universe. It could predict the present accelerated expansion of the universe somewhat differently from the other models with a peculiar future history of the universe. Within a constrained range of parameters all models show a good agreement with the Union2.1 luminosity distance data. However, the new model shows a quite satisfactory agreement in the whole range of its allowed parameters than that of the other two models. We also obtain cosmological constraints on these models from the observed Hubble data. Further, models have been tested by using $Om(z)$ test and statefinder diagnostics. These diagnostics especially the statefinder diagnostic shows that the evolutionary differences between these models are distinct. The evolutionary trajectories of the new model are completely different from the other two models we have considered.

We study different manifestations of the speed of light in theories of gravity where metric and connection are regarded as independent fields. We find that for a generic gravity theory in a frame with locally vanishing affine connection, the usual degeneracy between different manifestations of the speed of light is broken. In particular, the space-time causal structure constant ($c_{\mathrm{\text{ST}}}$) may become variable in that local frame. For theories of the form $f({ℛ},{{ℛ}}^{\mu \nu }{{ℛ}}_{\mu \nu })$, this variation in $c_{\mathrm{\text{ST}}}$ has an impact on the definition of the luminosity distance (and distance modulus), which can be used to confront the predictions of particular models against Supernovae type Ia (SN Ia) data. We carry out this test for a quadratic gravity model without cosmological constant assuming (i) a constant speed of light and (ii) a varying speed of light (VSL), and find that the latter scenario is favored by the data.

The Born-Infeld gravity is a modified gravity motivated by the quantum gravity, which has the ability to get rid of the big bang and other singularities. We consider the Palatini formalism of the Born-Infeld gravity and investigate the FRW cosmology by using an effective potential. It is shown that the big bang singularity does not always emerge but the scale factor can be bounced. Based on this result, we also study the black hole formation for spherical dust collapse, and find that the dust ball does not always collapse into a singularity.

Here, we present preliminary results from our studies of the variation of different facets of the speed of light in the local inertial frame in Palatini formalism. In these theories, different aspects of the speed of light such as c_ST (the space-time causal structure constant) and c_EM (the electromagnetic wave velocity), do not coincide in the local inertial frame and modify standard ΛCDM model coming from General Relativity. We apply our model to available data such as Union II or Planck 2013. For instance, we can fit the SN Ia distance modulus and CMB shift parameter in the absence of Dark Energy. Also we apply our model to the comoving Hubble radius and find that our favourable model leads to a larger particle horizon at higher redshifts and hence, can solve the horizon problem without the need for inflation.

Using exactly solvable models, it is shown that black hole singularities in different electrically charged configurations can be cured. Our solutions describe black hole space-times with a wormhole giving structure to the otherwise point-like singularity. We show that geodesic completeness is satisfied despite the existence of curvature divergences at the wormhole throat. In some cases, physical observers can go through the wormhole and in other cases the throat lies at an infinite affine distance.

Wormholes may arise as solutions of extensions of General Relativity without violation of the energy conditions. Working in a Palatini approach we consider classical geometries supporting such wormholes. It is shown that the resulting space-times represent explicit realizations of the concept of geon introduced by Wheeler, interpreted as self-consistent bodies generated by an electromagnetic field without sources.

We present an extension of general relativity in which an f(R) term á la Palatini is added to the usual metric Einstein-Hilbert Lagrangian. Expressing the theory in a dynamically equivalent scalar-tensor form, we show that it can pass the Solar System observational tests even if the scalar field is very light or massless. Applications to cosmology and astrophysics, and some exact solutions are discussed.