One of the most favourable 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 $Λ$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.
NMDC-Palatini cosmology in slow-roll regime is of our interests. We present flat FLRW cosmological NMDC-Palatini field equations and acceleration condition. Late time trajectory is approximated. Chaotic inflation potential is considered here as it is viable in a range of positive coupling constant as constrained by CMB results. Phase portraits show that the NMDC-Palatini gives new saddle-point solutions. In $V \propto ϕ^2$ potential, the NMDC-Palatini effect enlarges the acceleration-allowed region from that of GR case.
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.
We discuss recent results on the cosmology of extended theories of gravity formulated in the Palatini approach, i.e., assuming that metric and connection are independent fields. In particular, we focus on the attempts to explain the cosmic speedup with f(R) theories and on models that avoid the big bang singularity. The field equations for gravity Lagrangians of the form f(g_{μν},{R^α}_{βμν}) (including torsion) are explicitly derived and discussed.
We study nonlinear gravity theories in both the metric and the Palatini (metric-affine) formalisms. The nonlinear character of the gravity lagrangian in the metric formalism causes the appearance of a scalar source of matter in Einstein's equations that can be interpreted as a quintessence field. However, in the Palatini case no new energy sources appear, though the equations of motion get modified in such a way that usual matter can lead to repulsive gravity at very low densities. Thus, the Palatini formalism could provide a mechanism to explain the recent acceleration of the universe without…
The Einstein field equations can be derived in $n$ dimensions ($n>2$) by the variations of the Palatini action. The Killing reduction of 5-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of 4-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same fields equations in Kaluza-Klein theory.
We consider the weak field limit of gravity in the vierbein-Einstein-Palatini formalism, find the action and the equations for perturbations around an arbitrary background, and compare them with the usual metric perturbation equations. We also write the Fierz-Pauli equations for massive gravitons on an arbitrary curved background in this formalism.
We analyze the cosmological implementation of Palatini $f(R)$ theories, constructed with a Nieh-Yan term and solved with respect to the torsion. We consider the relevant case of the quadratic correction to the Hilbert-Palatini action in the Ricci scalar, mimicking the Starobinsky model of the metric formulation. We point out the emergence of peculiar cosmological scenarios, depending on the sign of such correction, able to reproduce bouncing settings and to restore the standard Universe dynamics in the late asymptotic limit. Furthermore, we outline the settling of Little-Rip dynamics, which ca…
We consider several physical scenarios where black holes within classical gravity theories including $R^2$ and Ricci-squared corrections and formulated à la Palatini can be analytically studied.
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 $\mathcal{F}(\mathcal{Q})$, with $\mathcal{Q}\equiv F^{αβ}F_{αβ}$ 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.
In loop quantum cosmology, non-perturbative quantum gravity effects lead to the resolution of the big bang singularity by a quantum bounce without introducing any new degrees of freedom. Though fundamentally discrete, the theory admits a continuum description in terms of an effective Hamiltonian. Here we provide an algorithm to obtain the corresponding effective action, establishing in this way the covariance of the theory for the first time. This result provides new insights on the continuum properties of the discrete structure of quantum geometry and opens new avenues to extract physical pre…
The speed of light is a complicated synthesizer quantity with distinctive origins which lead to coincident values in the standard theory. Due to the fact that different aspects of speed of light do not coincide in the local inertial frame for different Palatini modified gravity theories, when deviating from general relativity, one should consider which aspect of speed of light should be taken into account meticulously and unambiguously. The aim of this study is mainly investigating the modification of the SN Ia luminosity distance, for two f(R) and f(RR) extended theories in the local inertial…
We construct a relativistic metric description of MOND using the Palatini formalism following the \( f(χ)=χ^b \) description of \citet{mendozatula}. We show that in order to recover the non-relativistic MOND regime where, for circular orbits the Tully-Fisher law replaces Kepler's third law, the value of the parameter $ b = 3/2 $, which is coincident with the value found using a pure metric formalism Capozziello et al. (2011). Unlike the pure metric formalism, which yields 4th order field equations, the Palatini approach yields second order field equations, which is a desirable requirement from…
We study the generalized version of energy-momentum squared gravity (EMSG) in the Palatini formalism. This theory allows the existence of a scalar constructed with energy-momentum tensor as $T_{αβ}T^{αβ}$ in the generic action of the theory. We study the most general form of this theory in the Palatini framework and present the underlying field equations. The equations of motion of a massive test particle have been derived. The weak field limit of the theory is explored and the generalized version of the Poisson equation is obtained. Moreover, we explore the cosmological behavior of the theory…
A new model for inflation using modified gravity in the Palatini formalism is constructed. Here non-minimal coupling of scalar field h with the curvature R as a general function f(R,h) is considered. Explicit inflation models for some choices of f(R,h) are developed. By writing an equivalent scalar-tensor action for this model and going over to Einstein frame, slow roll parameters are constructed. There exists a large parameter space which satisfies values of n_s and limits on r compatible with Planck 2018 data. Further, we calculate reheating temperature and the number of e-folds at the end o…
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 consider the Palatini formalism of gravity with cosmological constant $Λ$ coupled to a scalar field $ϕ$ in $n$-dimensions. The $n$-dimensional Einstein equations with $Λ$ can be derived by the variation of the coupled Palatini action provided $n>2$. The Hamiltonian analysis of the coupled action is carried out by a $1+(n-1)$ decomposition of the spacetime. It turns out that both Palatini action and Hilbert action lead to the same geometric dynamics in the presence of $Λ$ and $ϕ$. While, the $n$-dimensional Palatini action could not give a connection dynamics formalism directly.
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…
In this work, we present the f(R)-Einstein-Palatini formalism in arbitrary dimensions and the study of consistency applied to brane models, the so-called braneworld sum rules. We show that it is possible a scenario of thick branes in five dimensions with compact extra dimension in the framework of the f(R)-Einstein-Palatini theory by the accomplishment of an assertive criteria.
We analyse a scalar field non-minimally coupled to gravity in the context of a Universe described by the flat Friedmann-Robertson-Walker (F-R-W) metric. The adopted model comprises a Universe filled by the scalar field and standard matter (dark and baryonic). The corresponding field equations are obtained through the Palatini formalism. From the action of the model in the flat F-R-W space-time, a point-like Lagrangian of first order is obtained and the Noether symmetry approach is applied to restrict the forms of the a priori undefined coupling and potential of the scalar field. We show that t…
We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the dynamical system methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth dynamical system. This system is is reduced to a 2D dynamical system of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of dynamical system of Newtonian type with non-smooth right-hand sides in the con…
We consider the early time cosmology of f(R) theories in Palatini formalism and study the conditions that guarantee the existence of homogeneous and isotropic models that avoid the Big Bang singularity. We show that for such models the Big Bang singularity can be replaced by a cosmic bounce without violating any energy condition. In fact, the bounce is possible even for pressureless dust. We give a characterization of such models and discuss their dynamics in the region near the bounce. We also find that power-law lagrangians with a finite number of terms may lead to non-singular universes, wh…
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 …
The evolution of linear cosmological perturbations in modified theories of gravity is investigated assuming the Palatini formalism. It has been discussed about the stability problem in this model based on the equivalence between f(R) gravity and the scalar tensor theory. However, we study this problem in the physical frame where the matter is minimally coupled. In general, the stability of the superhorizon metric evolution depends on models. We show that the deviation from the superhorizon metric evolution is null for a specific choice for the nonlinear Einstein-Hilbert action, $f(\hat{R}) \si…
The compact stars are promising candidates associated with the generation of gravitational waves (GWs). In this work, we study a special type of compact stars known as strange stars in the $f(\mathcal{R})$ gravity Palatini formalism. Here we consider three promising $f(\mathcal{R})$ gravity models viz., Starobinsky, Hu-Sawicki and Gogoi-Goswami models in the domain of MIT Bag model and linear equations of state (EoSs). We compute the stellar structures numerically and constrained the $f(\mathcal{R})$ model parameters with a set of probable strange star candidates. The study shows that the cons…
We consider the cosmological models based on Palatini f(R)-theory for the function f(R)=aR-2bR^2-3c/R, which, when only dust visible matter is considered, is called dune cosmology in view of the shape of the function f(R(a)) (being a the scale factor). We discuss about the meaning of solving the model, and interpret it according to Ehlers-Pirani-Schild framework as defining a Weyl geometry on spacetime. Accordingly, we extend the definitions of luminosity distance, proper distance, and redshift to Weyl geometries and fit the values of parameters to SNIa data. Since the theoretical prediction i…
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…
We present a simple generalisation of the $Λ$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-H…
The inflationary dynamics and preheating in a model with a non-minimally coupled inflaton field in the metric and Palatini formalisms are studied in this paper. We find that in both formalisms, irrespective of the initial conditions, our Universe will evolve into a slow-roll inflationary era and then the scalar field rolls into an oscillating phase. The value of the scalar field at the end of the inflation in the Palatini formalism is always larger than that in the metric one, which becomes more and more obvious with the increase of the absolute value of the coupling parameter $|ξ|$. During th…
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 study the classical Euclidean wormholes in the context of extended theories of gravity. With no loss of generality, we use the dynamical equivalence between $f(\tilde{R})$ gravity and scalar-tensor theories to construct a point-like Lagrangian in the flat FRW space time. We first show the dynamical equivalence between Palatini $f(\tilde{R})$ gravity and the Brans-Dicke theory with self-interacting potential, and then show the dynamical equivalence between the Brans-Dicke theory with self-interacting potential and the minimally coupled O'Hanlon theory. We show the existence of new Euclidean …
When considered in the Palatini formalism, the Starobinsky model does not provide us with a mechanism for inflation due to the absence of a propagating scalar degree of freedom. By (non)--minimally coupling scalar fields to the Starobinsky model in the Palatini formalism we can in principle describe the inflationary epoch. In this article, we focus on the minimally coupled quartic and natural inflation models. Both theories are excluded in their simplest realization since they predict values for the inflationary observables that are outside the limits set by the Planck data. However, with the …
A generalized Brans-Dicke (GBD) theory in the framework of Palatini formalism are proposed in this paper. We derive the field equations by using the variational approach and obtain the linearized equations by using the weak-field approximation method. We show various properties of the geometrical scalar field in the Palatini-formalism of GBD theory: it is massless and source-free, which are different from the results given in the metric-formalism of GBD theory. Also, we investigate the polarization modes of gravitational waves (GWs) by using the the geodesic deviation method and the Newman-Pen…
We derive the dynamical equations for a non-local gravity model in the Palatini formalism and we discuss some of the properties of this model. We have shown that, in some specific cases, the vacuum solutions of general relativity are also vacuum solutions of the non-local model, so we conclude that, at least in this case, the singularities of Einstein's gravity are not removed.
We consider, in Palatini formalism, a modified gravity of which the scalar field derivative couples to Einstein tensor. In this scenario, Ricci scalar, Ricci tensor and Einstein tensor are functions of connection field. As a result, the connection field gives rise to relation, $h_{μν} = f g_{μν}$ between effective metric, $h_{μν}$ and the usual metric $g_{μν}$ where $f \,=\,1 - κϕ^{,α}ϕ_{,α}/2 $. In FLRW universe, NMDC coupling constant is limited in a range of $ -2/ \dotϕ^{2} < κ\leq \infty $ preserving Lorentz signature of the effective metric. Slowly-rolling regime provides $κ< 0$ forbiddin…
Cosmological models based on f(R)-gravity may exhibit a natural acceleration mechanism without introducing a dark energy component. In this paper, we investigate cosmological consequences of the so-called Hu-Sawicki f(R)-gravity in the Palatini formalism. We derive theoretical constraints on the model parameters and perform a statistical analysis to determine the parametric space allowed by current observational data. We find that this class of models is indistinguishable from the standard ΛCDM model at the background level. Differently, from previous results in the metric approach, we show th…
We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function f(R^abR_ab) to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the bes…
We compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives. We find that there is a class of theories for which the two formalisms are equivalent. This class contains the Palatini version of Lovelock theory, but also more Lagrangians that are not Lovelock, but respect certain symmetries. For the general case, we find that imposing the Levi-Civita connection as an Ansatz, the Palatini formalism is contained within the metric formalism, in th…
We investigate cosmological consequences of a class of exponential $f(R)$ gravity in the Palatini formalism. By using the current largest type Ia Supernova sample along with determinations of the cosmic expansion at intermediary and high-$z$ we impose tight constraints on the model parameters. Differently from other $f(R)$ models, we find solutions of transient acceleration, in which the large-scale modification of gravity will drive the Universe to a new decelerated era in the future. We also show that a viable cosmological history with the usual matter-dominated era followed by an accelerati…
It is shown that a quadratic gravitational Lagrangian in the Palatini formulation is able to capture different aspects of quantum gravity phenomenology in a single framework. In particular, in this theory field excitations propagating with different energy-densities perceive different background metrics, a fundamental characteristic of the DSR and Rainbow Gravity approaches. This theory, however, avoids the so-called soccer ball problem. Also, the resulting isotropic and anisotropic cosmologies are free from the big bang singularity. This singularity avoidance occurs non-perturbatively and sha…
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.
In a Palatini $f(\mathcal{R})$-model, we define chonodynamical effects due to the choice of atomic clocks as standard reference clocks and we develop a formalism able to quantitatively separate them from the usual effective dark sources one has in extended theories. We apply the formalism to Hubble drift and briefly discuss the issue about the physical frame. In particular, we argue that there is no physical frame in the sense one does different things in different frames and that, in a sense, is the physical characteristic of extended gravity. As an example, we discuss how Jordan frame may be…
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.
The main objective of this thesis is to discuss scalar-tensor theories in the Palatini approach. Both scalar-tensor theories and Palatini formalism are means of alternating classical theory of gravity, general relativity, in order to account for phenomena being seemingly unexplainable on the ground of the Einstein theory or to serve as toy models used to test limitations of the theory in question. In the literature, both Palatini approach and scalar-tensor theories have been widely discussed, but there are very few - if none - authors writing about a merge of these two ideas. The present paper…
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.
We study Palatini f(R) cosmology using Noether symmetry approach for the matter dominated universe. In order to construct a point-like Lagrangian in the flat FRW space time, we use the dynamical equivalence between f(R) gravity and scalar-tensor theories. The existence of Noether symmetry of the cosmological f(R) Lagrangian helps us to find out the form of f(R) and the exact solutions for cosmic scale factor. We show that this symmetry always exist for f(R)~R^n and the Noether constant is a function of the Newton's gravitational constant and the current matter content of the universe.
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…
It has recently been shown that f(R) theories formulated in the Palatini variational formalism are able to avoid the big bang singularity yielding instead a bouncing solution. The mechanism responsible for this behavior is similar to that observed in the effective dynamics of loop quantum cosmology and an f(R) theory exactly reproducing that dynamics has been found. I will show here that considering more general actions, with quadratic contributions of the Ricci tensor, results in a much richer phenomenology that yields bouncing solutions even in anisotropic (Bianchi I) scenarios. Some implica…
In this work we derive the covariant and gauge invariant perturbation equations in general theories of $f(R)$ gravity in the Palatini formalism to linear order and calculate the cosmic microwave background (CMB) and matter power spectra for an extensively discussed model, $f(R) = R + α(-R)^β$, which is a possible candidate for the late-time cosmic accelerating expansion found recently. These spectra are discussed and found to be sensitively dependent on the value of $β$. We are thus able to make stringent constraints on $β$ from cosmological data on CMB and matter power spectra: The three-year…
We consider the Palatini formalism of the Born-Infeld gravity. In the Palatini formalism, the propagating mode is only graviton, whose situation is different from that in the metric formalism. We discuss about the FRW cosmology by using an effective potential. Especially we consider the condition that the bouncing could occur. We also give some speculations about the black hole formation
We analyse the most general connection allowed by Einstein-Hilbert theory in Palatini formalism. We also consider a matter lagrangian independent of the affine connection. We show that any solution of the equation of the connection is essentially Levi-Civita up to a term that contains an undetermined 1-form. Finally, it is proved that these connections and Levi-Civita describe a completely equivalent physics.
We consider FRW cosmology in $f(R)= R+ γR^2+δR^3$ modified framework. The Palatini approach reduces its dynamics to the simple generalization of Friedmann equation. Thus we study the dynamics in two-dimensional phase space with some details. After reformulation of the model in the Einstein frame, it reduces to the FRW cosmological model with a homogeneous scalar field and vanishing kinetic energy term. This potential determines the running cosmological constant term as a function of the Ricci scalar. As a result we obtain the emergent dark energy parametrization from the covariant theory. We s…
Recently D. Vollick [Phys. Rev. D68, 063510 (2003)] has shown that the inclusion of the 1/R curvature terms in the gravitational action and the use of the Palatini formalism offer an alternative explanation for cosmological acceleration. In this work we show not only that this model of Vollick does not have a good Newtonian limit, but also that any f(R) theory with a pole of order n in R=0 and its second derivative respect to R evaluated at Ro is not zero, where Ro is the scalar curvature of background, does not have a good Newtonian limit.
We consider compact binary systems in f(R) gravity theories in the Palatini approach and calculate the post-Newtonian parameters to the 1.5PN order using the method of Direct Integration of the Relaxed Einstein equations (DIRE). The Palatini-type modifications of gravity can be formulated as Einsteins gravity with modified response to matter sources, and it is shown in detail how to treat these correctly within the DIRE formalism. Our results explicitly confirm the expectation that for binary black holes the new effects can be absorbed into redefinitions of the binary masses, rendering such sy…
We consider cosmological scalar field evolving under exponential potential of the Non-minimal Derivative Coupling (NMDC) gravity model in Palatini formalism. Slow-roll regime is assumed. GR and metric formalism NMDC cases are compared in this study. Phase portraits show that Palatini NMDC effect restricts acceleration phase into smaller region in the phase space. NMDC effect of the Palatini case suppresses expansion rate than that of the GR while the metric NMDC enhances rate of expansion of the GR case.
We study scalar field inflation in $F(R)$ gravity in the Palatini formulation of general relativity. Unlike in the metric formulation, in the Palatini formulation $F(R)$ gravity does not introduce new degrees of freedom. However, it changes the relations between existing degrees of freedom, including the inflaton and spacetime curvature. Considering the case $F(R)=R+αR^2$, we find that the $R^2$ term decreases the height of the effective inflaton potential. By adjusting the value of $α$, this mechanism can be used to suppress the tensor-to-scalar ratio $r$ without limit in any scalar field mod…
