We derive the field equations and the equations of motion for 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-Civita 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 non-conservation 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.
The Starobinsky model, considered in the framework of the Palatini formalism, in contrast to the metric formulation, does not provide us with a model for inflation, due to the absence of a propagating scalar degree of freedom that can play the role of the inflaton. In the present article we study the Palatini formulation of the Starobinsky model coupled, in general nonminimally, to scalar fields and analyze its inflationary behavior. We consider scalars, minimally or nonminimally coupled to the Starobinsky model, such as a quadratic model, the induced gravity model or the standard Higgs-like i…
The Palatini formulation of the Starobinsky model does not yield a propagating scalaron that can assume the role of the inflaton field as in the conventional metric formulation. In the so-called Palatini-${R}^2$ models this role is assumed by a fundamental scalar field nonminimally coupled to gravity. In this article we consider an extension of the interactions of this field by introducing a field dependence to the $R^2$ coefficient in the form of logarithmic corrections. We examine the resulting predictions of the inflationary observables in the framework of slow-roll approximation and invest…
In the present paper, the 3+1 decomposition of the spacetime onto hypersurface(s) is analysed and established for theories within the Palatini formalism by considering a general function of the Ricci scalar in the gravitational action. The corresponding Gauss-Codazzi relations are obtained and the boundary term that has to be subtracted in the gravitational action is easily deduced. Then, these relations are applied to the so-called ADM decomposition to describe the foliation of the spacetime onto hypersurfaces of constant time within these theories. Finally, the junction conditions are also o…
Two exact solutions for $n=0$ and $n=1$ of the Palatini-modified Lane-Emden equation are found. We have employed these solutions to describe a Palatini-Newtonian neutron star and compared the result with the pure Newtonian counterpart. It turned out that for the negative parameter of the Starobinsky model the star is heavier and larger.
We study minimal and nonminimal couplings of fermions to the Palatini action in $n$ dimensions ($n\geq 3$) from the Lagrangian and Hamiltonian viewpoints. The Lagrangian action considered is not, in general, equivalent to the Einstein-Dirac action principle. However, by choosing properly the coupling parameters, it is possible to give a first-order action fully equivalent to the Einstein-Dirac theory in a spacetime of dimension four. By using a suitable parametrization of the vielbein and the connection, the Hamiltonian analysis of the general Lagrangian is given, which involves manifestly Lor…
In this paper, we review modified $f(R)$ theories of gravity in Palatini formalism. In this framework, , we use the Raychaudhuri's equation along with the requirement that the gravity is attractive, which holds for any geometrical theory of gravity to discuss the energy conditions. Then, to derive these conditions, we obtain an expression for effective pressure and energy density by considering FLRW metric. Energy conditions derived in Palatini version of $f(R)$ Gravity differ from those derived in GR. We will see that the WEC (weak energy condition) derived in Palatini formalism has exactly t…
We study a model of quintessential inflation constructed in $R^2$ modified gravity with a non-minimally coupled scalar field, in the Palatini formalism. Our non-minimal inflaton field is characterised by a simple exponential potential. We find that successful quintessential inflation can be achieved with no fine-tuning on the model parameters. Predictions on the characteristics of dark energy will be tested by observations in the near future, while contrast with existing observations provides insights on the modified gravity background, such as the value of the non-minimal coupling and its run…
We show that, within modified gravity, the non-linear nature of the field equations implies that the usual naive averaging procedure (replacing the microscopic energy-momentum by its cosmological average) is invalid. We discuss then how the averaging should be performed correctly and show that, as a consequence, at classical level the physical masses and geodesics of particles, cosmology and astrophysics in Palatini modified gravity theories are all indistinguishable from the results of general relativity plus a cosmological constant. Palatini gravity is however a different theory from general…
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 speedup problem, laboratory and solar systems 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 …
Recently, a modified theory of gravity was presented, which consists of the superposition of the metric Einstein-Hilbert Lagrangian with an $f(\cal R)$ term constructed à la Palatini. The theory possesses extremely interesting features such as predicting the existence of a long-range scalar field, that explains the late-time cosmic acceleration and passes the local tests, even in the presence of a light scalar field. In this brief report, we consider the possibility that wormholes are supported by this hybrid metric-Palatini gravitational theory. We present here the general conditions for worm…
In the framework of Higgs inflation, we consider the electromagnetic field nonminimally coupled to gravity via the parity-preserving $\propto RF^{2}$ and parity-violating $\propto RF\tilde{F}$ terms. Using the perturbation theory to the leading order in these couplings, we study the generation of the electromagnetic field during the inflation stage. We derive the effective kinetic and axial coupling functions arising in the Einstein frame in the usual metric as well as Palatini formulations of gravity. For both formulations, we determine the power spectrum, energy density, and helicality of th…
General Relativity has so far passed almost all the ground-based and solar-system experiments. Any reasonable extended gravity models should consistently reduce to it at least in the weak field approximation. In this work we derive the gravitational potential for the Palatini formulation of the modified gravity of the L(R) type which admits a de Sitter vacuum solution. We conclude that the Newtonian limit is always obtained in those class of models and the deviations from General Relativity is very small for a slowly moving source.
We perform a phase space analysis of a generalized modified gravity theory with nonminimally coupling between geometry and matter. We apply the dynamical system approach to this generalized model and find that in the cosmological context, different choices of Lagrangian density will apparently result in different phases of the Universe. By carefully choosing the variables, we prove that there is an attractor solution to describe the late time accelerating universe when the modified gravity is chosen in a simple power-law form of the curvature scalar. We further examine the temperature evolutio…
In this work we study how nonminimally coupled theories of gravity modify the usual Friedmann equation, and develop two methods to treat these. The ambiguity in the form of the Lagrangian density of a perfect fluid is emphasized, and the impact of different dominant matter species is assessed. The Cosmological Constant problem is also discussed.
We investigate in detail a family of quintessential inflation models in the context of $R+R^2$ Palatini modified gravity. We find that successful inflation and quintessence are obtained with an inflaton scalar potential that is approximately quadratic in inflation and inverse quartic in quintessence. We show that corrections for the kination period due to Palatini modified gravity are subdominant, while the setup does not challenge constraints on modified gravity from solar system observations and microscopic experiments, in contrast to the metric case. We obtain concrete predictions regarding…
The Ostrogradsky instability of higher derivative Lagrangians is derived from first principles using Control theory and Lyapunov Stability Analysis. This result is then used to argue that Born-Infeld Lagrangians are viable modifications of the Einstein-Hilbert action provided the action is varied in accordance with the Palatini formalism, in contrast to the metric formalism. Finally, the Born-Infeld version of the FRW equations are derived and the cosmological dynamics is studied for matter dominated closed and open universes, and the results are compared with the usual cosmology.
In this paper by deriving the Modified Friedmann equation in the Palatini formulation of $R^2$ gravity, first we discuss the problem of whether in Palatini formulation an additional $R^2$ term in Einstein's General Relativity action can drive an inflation. We show that the Palatini formulation of $R^2$ gravity cannot lead to the gravity-driven inflation as in the metric formalism. If considering no zero radiation and matter energy densities, we obtain that only under rather restrictive assumption about the radiation and matter energy densities there will be a mild power-law inflation $a(t)\sim…
In this paper, the hybrid metric-Palatini gravity is an approach to modified gravity in which is added to the usual Einstein-Hilbert action a supplementary term containing a Palatini-type correction of the form $f({\cal R},T)$. Here, ${\cal R}$ is the Palatini curvature scalar, which is constructed from an independent connection and $T$ is the trace of the energy-momentum tensor. This theory describes a non-minimal coupling between matter and geometry. The modified Einstein field equations in this hybrid metric-Palatini approach are obtained. Then, it is investigated whether this modified theo…
We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor. A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove …
By using the equivalence between metric and Palatini f(R) (or "modified") gravities with omega=0, -3/2 Brans-Dicke theories, it is shown that the Ehlers-Geren-Sachs theorem of general relativity is extended to modified gravity. In the case of metric f(R) gravity previously studied, this agrees with previous literature.
The actions for bosonic fields typically contain terms quadratic in the derivatives of the fields. This is not the case in the Palatini approach to general relativity. The action does not contain any derivatives of the metric and it only contains terms linear in the derivatives of the connection. In general relativity the covariant derivative of the metric vanishes, so it is not possible to include such terms in the action. However, in more general theories this is not the case. In this paper I consider an action which is quadratic in the derivatives of the metric and connection and show that …
Under conformal transformation, f(R) 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, the effective scalar acts as the dark energy and so late-time cosmic acceleration in matter-dominated era is accountable. However, we unveil some aspects of Palatini formalism, which clearly reveals the fact that the formalism is not suitable to explain the cosmological evolution of the early universe with $f(\mathfrak{R})$ gravity alone. Additionally, it is noticed that some authors, in a…
According to theoretical physics the cosmological constant (CC) is expected to be much larger in magnitude than other energy densities in the universe, which is in stark contrast to the observed Big Bang evolution. We address this old CC problem not by introducing an extremely fine-tuned counterterm, but in the context of modified gravity in the Palatini formalism. In our model the large CC term is filtered out, and it does not prevent a standard cosmological evolution. We discuss the filter effect in the epochs of radiation and matter domination as well as in the asymptotic de Sitter future. …
We consider the Palatini formulation of $f(R,T)$ gravity theory, in which a nonminimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similarly to the metric case, the fiel…
We study new FRW type cosmological models of modified gravity treated on the background of Palatini approach. These models are generalization of Einstein gravity by the presence of a scalar field non-minimally coupled to the curvature. The models employ Starobinsky's term in the Lagrangian and dust matter. Therefore, as a by-product, an exhausted cosmological analysis of general relativity amended by quadratic term is presented. We investigate dynamics of our models, confront them with the currently available astrophysical data as well as against LCDM model. We have used the dynamical system m…
Analytical solutions of nonminimally coupled scalar field cosmology corresponding to critical scalar field values constitute a potential challenge to the recent first-order thermodynamics of scalar-tensor gravity (a formalism picturing general relativity as the zero-temperature equilibrium state for modified gravity). The critical solutions are unstable with respect to homogeneous perturbations, hence unphysical.
In the framework of classical scale invariance, we consider quadratic gravity in the Palatini formalism and investigate the inflationary predictions of the theory. Our model corresponds to a two-field scalar-tensor theory, that involves the Higgs field and an extra scalar field stemming from a gauge $U(1)_X$ extension of the Standard Model, which contains an extra gauge boson and three right-handed neutrinos. Both scalar fields couple nonminimally to gravity and induce the Planck scale dynamically, once they develop vacuum expectation values. By means of the Gildener-Weinberg approach, we desc…
In the context of modified gravity, we point out how the Palatini version of these theories is singled out as a very special case corresponding to the unique fixed point of a transformation involving a special conformal rescaling of the metric. This mathematical peculiarity signals deeply rooted problems which make the theory unphysical.
We explore equilibrium solutions of spherically symmetric boson stars in the Palatini formulation of $f(\mathcal{R})$ gravity. We account for the modifications introduced in the gravitational sector by using a recently established correspondence between modified gravity with scalar matter and general relativity with modified scalar matter. We focus on the quadratic theory $f(\mathcal{R})=R+ξR^2$ and compare its solutions with those found in general relativity, exploring both positive and negative values of the coupling parameter $ξ$. As matter source, a complex, massive scalar field with and w…
It is well known that one cannot apply a conformal transformation to $f(T)$ gravity to obtain a minimally coupled scalar field model, and thus no Einstein frame exists for $f(T)$ gravity. Furthermore nonminimally coupled "teleparallel dark energy models" are not conformally equivalent to $f(T)$ gravity. However, it can be shown that $f(T)$ gravity is conformally equivalent to a teleparallel phantom scalar field model with a nonminimal coupling to a boundary term only. In this work, we extend this analysis by considering a recently studied extended class of models, known as $f(T,B)$ gravity, wh…
We shall investigate the properties of a congruence of geodesics in the framework of Palatini f(R) theories. We shall evaluate the modified geodesic deviation equation and the Raychaudhuri's equation and show that f(R) Palatini theories do not necessarily lead to attractive forces. Also we shall study energy condition for f(R) Palatini gravity via a perturbative analysis of the Raychaudhuri's equation.
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.
Two approaches to the study of cosmological density perturbations in modified theories of Palatini gravity have recently been discussed. These utilise, respectively, a generalisation of Birkhoff's theorem and a direct linearization of the gravitational field equations. In this paper these approaches are compared and contrasted. The general form of the gravitational lagrangian for which the two frameworks yield identical results in the long-wavelength limit is derived. This class of models includes the case where the lagrangian is a power-law of the Ricci curvature scalar. The evolution of dens…
f(R) gravity theories in the Palatini formalism has been recently used as an alternative way to explain the observed late-time cosmic acceleration with no need of invoking either dark energy or extra spatial dimension. However, its applications have shown that some subtleties of these theories need a more profound examination. Here we are interested in the conformal aspects of the Palatini approach in extended theories of gravity. As is well known, extremization of the gravitational action a la Palatini, naturally "selects" a new metric h related to the metric g of the subjacent manifold by a …
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.
A modified model of gravity with additional positive and negative powers of the scalar curvature, $R$, in the gravitational action is studied. This is done using the Palatini variational principle. It is demonstrated that using such a model might prove useful to explain both the early time inflation and the late time cosmic acceleration without the need for any form of dark energy.
In Cuzinatto et al. [Phys. Rev. D 93, 124034 (2016)], it has been demonstrated that theories of gravity in which the Lagrangian includes terms depending on the scalar curvature $R$ and its derivatives up to order $n$, i.e. $f\left(R,\nabla_μR,\nabla_{μ_{1}}\nabla_{μ_{2}}R,\dots,\nabla_{μ_{1}}\dots\nabla_{μ_{n}}R\right)$ theories of gravity, are equivalent to scalar-multitensorial theories in the Jordan frame. In particular, in the metric and Palatini formalisms, this scalar-multitensorial equivalent scenario shows a structure that resembles that of the Brans-Dicke theories with a kinetic term …
This paper aims to analyze the generalization of Lema\^ıtre-Tolman-Bondi (LTB) spacetime for dissipative dust under the influence of Palatini $f(R)$ gravity. We explore the modified field equations, kinematical variables, and mass function in this scenario. We construct Bianchi identities using conservation and differential equations for shear, expansion, and curvature scalar in the background of Palatini $f(R)$ gravity. We calculated the scalar functions coming from the orthogonal decomposition of the Riemann tensor in this framework. These scalar functions known as structure scalars have bee…
By establishing that Palatini formulation of $L(R)$ gravity is equivalent to $ω=-3/2$ Brans-Dicke theory, we show that energy-momentum tensor is covariantly conserved in this type of modified gravity theory.
We introduce a new approach to modified gravity which generalizes the recently proposed hybrid metric-Palatini gravity. The gravitational action is taken to depend on a general function of both the metric and Palatini curvature scalars. The dynamical equivalence with a non-minimally coupled bi-scalar field gravitational theory is proved. The evolution of cosmological solutions is studied using dynamical systems techniques.
A simple cooling model of white dwarf stars is re-analyzed in Palatini $f(R)$ gravity. Modified gravity affects the white dwarf structures and consequently their ages. We find that the resulting super-Chandrasekhar white dwarfs need more time to cool down than sub-Chandrasekhar ones, or when compared to the Newtonian models.
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 formalis…
We investigate the corrections to the inflationary cosmological dynamics due to a $R^2$ term in the Palatini formulation which may arise as quantum corrections to the effective Lagrangian in early universe. We found that the standard Friedmann equation will not be changed when the scalar field is in the potential energy dominated era. However, in the kinetic energy dominated era, the standard Friedmann equation will be modified and in the case of closed and flat universe, the Modified Friedmann equation will automatically require that the initial kinetic energy density of the scalar field must…
We construct gravitational modifications that go beyond Horndeski, namely theories with extended nonminimal derivative couplings, in which the coefficient functions depend not only on the scalar field but also on its kinetic energy. Such theories prove to be ghost-free in a cosmological background. We investigate the early-time cosmology and show that a de Sitter inflationary phase can be realized as a pure result of the novel gravitational couplings. Additionally, we study the late-time evolution, where we obtain an effective dark energy sector which arises from the scalar field and its exten…
Cosmology in extended theories of gravity is considered assuming the Palatini variational principle, for which the metric and connection are independent variables. The field equations are derived to linear order in perturbations about the homogeneous and isotropic but possibly spatially curved background. The results are presented in a unified form applicable to a broad class of gravity theories allowing arbitrary scalar-tensor couplings and nonlinear dependence on the Ricci scalar in the gravitational action. The gauge-ready formalism exploited here makes it possible to obtain the equations i…
In this work we show the existence of traversable wormhole and time-machine solutions in a modified theory of gravity where matter and curvature are nonminimally coupled. Those solutions present a nontrivial redshift function and exist even in the presence of ordinary matter which satisfies the dominant energy condition.
We have derived a modified Lane-Emden equation for the Starobinsky model in Palatini gravity which is numerically solvable. Comparing the results to the ones provided by General Relativity we observe a significant difference depending on the theory parameter for the $M-R$ relations.
Einstein's General Relativity (GR) is possibly one of the greatest intellectual achievements ever conceived by the human mind. In fact, over the last century, GR has proven to be an extremely successful theory, with a well established experimental footing. However, the discovery of the late-time cosmic acceleration, which represents a new imbalance in the governing gravitational field equations, has forced theorists and experimentalists to question whether GR is the correct relativistic theory of gravitation, and has spurred much research in modified gravity, where extensions of the Hilbert-Ei…
Wormhole geometries in curvature-matter coupled modified gravity are explored, by considering an explicit nonminimal coupling between an arbitrary function of the scalar curvature, R, and the Lagrangian density of matter. It is the effective stress-energy tensor containing the coupling between matter and the higher order curvature derivatives that is responsible for the null energy condition violation, and consequently for supporting the respective wormhole geometries. The general restrictions imposed by the null energy condition violation are presented in the presence of a nonminimal R-matter…
We consider special classes of Palatini f(R) theories, featured by additional Loop Quantum Gravity inspired terms, with the aim of identifying a set of modified Ashtekar canonical variables, which still preserve the SU(2) gauge structure of the standard theory. In particular, we allow for affine connection to be endowed with torsion, which turns out to depend on the additional scalar degree affecting Palatini f(R) gravity, and in this respect we successfully construct a novel Gauss constraint. We analyze the role of the additional scalar field, outlining as it acquires a dynamical character by…
In the present work we study strange stars in $f(R)$ theories of gravity in the Palatini formalism. We consider two concrete well-known cases, namely the $R+R^2/(6 M^2)$ model as well as the $R-μ^4/R$ model for two different values of the mass parameter $M$ or $μ$. We integrate the modified Tolman-Oppenheimer-Volkoff equations numerically, and we show the mass-radius diagram for each model separately. The standard case corresponding to the General Relativity is also shown in the same figure for comparison. Our numerical results show that the interior solution can be vastly different depending …
Some parts of the substellar evolution, such as fragmentation of a gaseous cloud and Jupiter-like planet's cooling, are demonstrated to be impacted by Palatini $f(\bar R)$ gravity. Using simple models describing those processes we show that the opacity mass limit as well as cooling time of jovian planets differ in modified gravity.
