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-Einstein action describe the gravitational field. In this review, we perform a detailed theoretical and phenomenological analysis of two largely explored extensions of $f(R)$ gravity, namely: (i) the hybrid metric-Palatini theory; (ii) and modified gravity with curvature-matter couplings. Relative to the former, it has been established that both metric and Palatini versions of $f(R)$ gravity possess interesting features but also manifest severe drawbacks. A hybrid combination, containing elements from both of these formalisms, turns out to be very successful in accounting for the observed phenomenology and avoids some drawbacks of the original approaches. Relative to the curvature-matter coupling theories, these offer interesting extensions of $f(R)$ gravity, where the explicit nonminimal couplings between an arbitrary function of the scalar curvature $R$ and the Lagrangian density of matter, induces a non-vanishing covariant derivative of the energy-momentum tensor. We extensively explore both theories in a plethora of applications, namely, the weak-field limit, galactic and extragalactic dynamics, cosmology, stellar-type compact objects, irreversible matter creation processes and the quantum cosmology of a specific curvature-matter coupling theory.
We derive the expressions for the local, on-shell closed co-dimension 2 forms in the Palatini formulation of general relativity and explicitly show their on-shell equivalence to those of the metric formulation. When compared to other first order formulations, two subtleties have to be addressed during the construction: off-shell non-metricity and the fact that the transformation of the connection under infinitesimal diffeomorphisms involves second order derivatives of the associated vector fields.
We show that the Palatini--Cartan--Holst formulation of General Relativity in tetrad variables must be complemented with additional requirements on the fields when boundaries are taken into account for the associated BV theory to induce a compatible BFV theory on the boundary.
We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.
Pure $R^2$ gravity has been shown to be equivalent to Einstein gravity with non-zero cosmological constant and a massless scalar field. We show that the Palatini formulation of pure $R^2$ gravity is equivalent to Einstein gravity with non-zero cosmological constant as before but with no massless scalar field. This is an important new development because the massless scalar field is not readily identifiable with any known particle in nature or unknown particles like cold dark matter which are expected to be massive. We then include a non-minimally coupled Higgs field as well as fermions to disc…
Projective invariance is a symmetry of the Palatini version of General Relativity which is not present in the metric formulation. The fact that the Riemann tensor changes nontrivially under projective transformations implies that, unlike in the usual metric approach, in the Palatini formulation this tensor is subject to a gauge freedom, which allows some ambiguities even in its scalar contractions. In this sense, we show that for the Schwarzschild solution there exists a projective gauge in which the (affine) Kretschmann scalar, $K\equiv {R^α}_{βμν}{R_α}^{βμν}$, can be set to vanish everywhere…
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 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…
The Palatini formalism is developed for gravitational theories in flat geometries. We focus on two particularly interesting scenarios. First, we fix the connection to be metric compatible, but we follow a completely covariant approach by imposing the constraints with suitable Lagrange multipliers. For a general quadratic theory we show how torsion naturally propagates and we reproduce the Teleparallel Equivalent of General Relativity as a particular quadratic action that features an additional Lorentz symmetry. We then study the much less explored theories formulated in a geometry with neither…
We study preheating in the Palatini formalism with a quadratic inflaton potential and an added $αR^2$ term. In such models, the oscillating inflaton field repeatedly returns to the plateau of the Einstein frame potential, on which the tachyonic instability fragments the inflaton condensate within less than an e-fold. We find that tachyonic preheating takes place when $α\gtrsim 10^{13}$ and that the energy density of the fragmented field grows with the rate $Γ/H \approx 0.011 \times α^{0.31}$. The model extends the family of plateau models with similar preheating behaviour. Although it contains…
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…
We will briefly discuss the necessary conditions for stability of polytropies in $f(\hat R)$ Palatini gravity and the differences with the General Relativity ones.
We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions wh…
In the paper we show that the general relativity action (and Lagrangian) in recent Einstein-Palatini formulation is equivalent to the action (and Langrangian) of a gauge field. We begin with a bit of information of the Einstein-Palatini (EP) action, then we present how Einstein fields equations can be derived from it. In the next section, we consider Einstein-Palatini action integral for general relativity with a positive cosmological constant $Λ$ in terms of $\hat{F}$, the curvature of the affine connection's pulled back from de Sitter bundle to Lorentz bundle. We will see that in terms of $\…
We study the renormalization flow of Hilbert-Palatini gravity to lowest non-trivial order. We find evidence for an asymptotically safe high-energy completion based on the existence of an ultraviolet fixed point similar to the Reuter fixed point of quantum Einstein gravity. In order to manage the quantization of the large number of independent degrees of freedom in terms of the metric as well as the connection, we use an on-shell reduction scheme: for this, we quantize all degrees of freedom beyond Einstein gravity at a given order that remain after using the equations of motion at the precedin…
We study preheating in plateau inflation in the Palatini formulation of general relativity, in a special case that resembles Higgs inflation. It was previously shown that the oscillating inflaton field returns to the plateau repeatedly in this model, and this leads to tachyonic production of inflaton particles. We show that a minimally coupled spectator scalar field can be produced even more efficiently by a similar mechanism. The mechanism is purely gravitational, and the scalar field mass can be of order $10^{13}$ GeV, larger than the Hubble scale by many orders of magnitude, making this a c…
We reformulate the Palatini action for general relativity (GR) in terms of moving frames that exhibit local Galilean covariance in a large speed of light expansion. For this, we express the action in terms of variables that are adapted to a Galilean subgroup of the $GL(n,\mathbb{R})$ structure group of a general frame bundle. This leads to a novel Palatini-type formulation of GR that provides a natural starting point for a first-order non-relativistic expansion. In doing so, we show how a comparison of Lorentzian and Newton-Cartan metric-compatibility explains the appearance of torsion in the …
We study models of quartic inflation where the inflaton field $ϕ$ is coupled non-minimally to gravity, $ξϕ^2 R$, and perform a study of quantum corrections in curved space-time at one-loop level. We specifically focus on comparing results between the metric and Palatini theories of gravity. Transformation from the Jordan to the Einstein frame gives different results for the two formulations and by using an effective field theory expansion we derive the appropriate $β$-functions and the renormalisation group improved effective potentials in curved space for both cases in the Einstein frame. In …
We investigate static spherically symmetric solutions in the Palatini kinetically coupled scalar-tensor theory, which reduces to gravity minimally coupled to a scalar field in Einstein frame. Using the fact that the Jordan and Einstein frame are related by a reversible disformal transformation, which can be solved in closed form, we derive the general solution in the Jordan frame and show that it does not contain black holes. There is a wormhole branch and a naked singularity branch between which lies a non-singular asymptotically flat solution with kernel $M_{1,1}\times S^2$. This theory stro…
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 …
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.
We disclose remarkable features of the scalar-tensor theory with the derivative coupling of the scalar field to the curvature in the Palatini formalism. Using the disformal transformations, we show that this theory is free from Otrogradski ghosts. For a special relation between two coupling constants, it is disformally dual to the Einstein gravity minimally coupled to the scalar, which opens the way to constructing several exact solutions. The disformal transformation degenerates at the boundary of the physical region, near which the properties of desingularization are revealed, illustrated by…
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 give a detailed account of the cyclic $L_\infty$-algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two $L_\infty$-algebras in the case of parallelizable spacetimes. By duality, we show that our $L_\infty$-algebras describe the complete BV-BRST formulation of Einste…
In this paper we generalize the off-shell Abbott-Deser-Tekin (ADT) conserved charge formalism to Palatini theory of gravity with torsion and non-metricity. Our construction is based on the coordinate formalism and the independent dynamic fields are the metric and the affine connection. For a general Palatini theory of gravity, which is diffeomorphism invariant up to a boundary term, we obtain the most general expression for off-shell ADT potential. As explicit examples, we derive the off-shell ADT potentials for Einstein-Hilbert action, the most general $L(g_{μν}, R^λ{}_{ναμ}, T^λ{}_{αβ}, Q_{α…
The theory of General Relativity was established on a spacetime manifold equipped with a metric tensor, $(\mathcal{M}_4,\text{g})$, and the connection on $\mathcal{M}_4$ identified with the Levi-Civita one. Even though there are valid reasons to assume a torsionless manifold that preserves the metric, it was shown that dealing away with these assumptions the Levi-Civita condition can be reproduced at the level of equations of motion of GR. It was not long before the equivalence of General Relativity between the two descriptions, known as the Palatini or first-order formalism in which the conne…
