Palatini Formalism Research Articles

3+1 decomposition in modified gravities within the Palatini formalism and some applications

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 obtained by using a decomposition in Gaussian normal coordinates, which coincide with the conditions deduced previously through different approaches.

Tachyonic preheating in Palatini R 2 inflation

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 ≳ 1013 and that the energy density of the fragmented field grows with the rate Γ/H ≈ 0.011 × α0.31. The model extends the family of plateau models with similar preheating behaviour. Although it contains non-canonical quartic kinetic terms in the Einstein frame, we show that, in the first approximation, these can be neglected during both preheating and inflation.

Inflation and Reheating in f(R,h) theory formulated in the Palatini formalism

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 for different choices of f(R,h) and potentials which satisfy 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 of reheating for different values of equation of state parameter for all the constructed models.

Possible effects of hybrid gravity on stellar kinematics in elliptical galaxies

We use the Fundamental Plane of Elliptical Galaxies to constrain the so-called Hybrid Gravity, a modified theory of gravity where General Relativity is improved by further degrees of freedom of metric-affine Palatini formalism of $f(\cal R)$ gravity. Because the Fundamental Plane is connected to the global properties of elliptical galaxies, it is possible to obtain observational constraints on the parameters of Hybrid Gravity in the weak field limit. We analyze also the velocity distribution of elliptical galaxies comparing our theoretical results in the case of Hybrid Gravity with astronomical data for elliptical galaxies. In this way, we are able to constrain the Hybrid Gravity parameters $m_\phi$ and $\phi_0$. We show that the Fundamental Plane, i.e. $v_c/\sigma$ relations, can be used as a standard tool to probe different theories of gravity in the weak field limit. We conclude that Hybrid Gravity is able to explain elliptical galaxies with different stellar kinematics without the dark matter hypothesis.

Is asymptotically Weyl-invariant gravity viable?

We explore the cosmological viability of a theory of gravity defined by the Lagrangian $f(\mathcal{R})=\mathcal{R}^{n\left(\mathcal{R}\right)}$ in the Palatini formalism, where $n\left(\mathcal{R}\right)$ is a dimensionless function of the Palatini scalar curvature $\mathcal{R}$ that interpolates between general relativity when $n\left(\mathcal{R}\right)=1$ and a locally scale-invariant and superficially renormalizable theory when $n\left(\mathcal{R}\right)=2$. We refer to this model as asymptotically Weyl-invariant gravity (AWIG). We analyse perhaps the simplest possible implementation of AWIG. A phase space analysis yields three fixed points with effective equation of states corresponding to de Sitter, radiation and matter-dominated phases. An analysis of the deceleration parameter suggests our model is consistent with an early and late period of accelerated cosmic expansion, with an intermediate period of decelerated expansion. We show that the model contains no obvious curvature singularities. Therefore, AWIG appears to be cosmologically viable, at least for the simple implementation explored.

Critical point Higgs inflation in the Palatini formulation

We study Higgs inflation in the Palatini formulation with the renormalisation group improved potential in the case when loop corrections generate a feature similar to an inflection point. Assuming that there is a threshold correction for the Higgs quartic coupling λ and the top Yukawa coupling yt, we scan the three-dimensional parameter space formed by the two jumps and the non-minimal coupling ξ.The spectral index ns can take any value in the observationally allowed range. The lower limit for the running is αs> −3.5 × 10−3, and αs can be as large as the observational upper limit. Running of the running is small. The tensor-to-scalar ratio is 2.2×10−17< r < 2 × 10−5. We find that slow-roll can be violated near the feature, and a possible period of ultra-slow-roll contributes to the widening of the range of CMB predictions. Nevertheless, for the simplest tree-level action, the Palatini formulation remains distinguishable from the metric formulation even when quantum corrections are taken into account, because of the small tensor-to-scalar ratio.

Does Palatini Higgs inflation conserve unitarity?

In the conventional metric formulation of gravity, the Higgs Inflation model violates unitarity in the electroweak vacuum in Higgs scattering at the energy scale Λ ∼ MPl /ξ, where ξ ∼ 104 is the non-minimal coupling of the Higgs to the Ricci scalar. In the Palatini formulation it is commonly believed that Λ ∼ MPl /√(ξ), where ξ ∼ 109. Here we reconsider unitarity violation in the electroweak vacuum in the Palatini formulation. We argue that there is no unitarity violation in Higgs scattering in the Palatini non-minimally coupled Standard Model in the electroweak vacuum at energies below the Planck scale. In this case Palatini Higgs Inflation completely conserves unitarity and is consistent at all energies up to those at which quantum gravity becomes important. If true, this would imply that Palatini Higgs Inflation has a significant advantage over metric Higgs Inflation.

Palatini double-well and Coleman-Weinberg potentials with non-minimal coupling

We present the impact of non-minimal coupling ξϕ2 R on the inflationary parameters by taking into account the models of single-field inflation with the inflaton that has a non-zero vacuum expectation value (v) after the period of inflation in Palatini gravity. We discuss the well-known symmetry-breaking type potentials, namely the double-well potential and Coleman-Weinberg potential. We show that the inflationary predictions, n s and r, of these potentials for both ϕ > v and ϕ < v inflation are compatible with the recent measurements within the regions of the v-ξ plane. Finally, we take into account the inflationary predictions of Coleman-Weinberg potential for selected ξ values as a function of v in the Palatini formalism.

Testing F(R) Gravity with the Simulated Data of Gravitational Waves from the Einstein Telescope

Abstract In this paper we analyze the implications of gravitational waves (GWs) as standard sirens on the modified gravity models by using the third-generation GW detector, i.e., the Einstein Telescope. Two viable models in f(R) theories within the Palatini formalism are considered in our analysis ( and ), with the combination of simulated GW data and the latest electromagnetic (EM) observational data (including the recently released Pantheon type Ia supernovae sample, the cosmic chronometer data, and baryon acoustic oscillation distance measurements). Our analysis reveals that the standard siren GWs, which provide an independent and complementary alternative to current experiments, could effectively eliminate the degeneracies among parameters in the two modified gravity models. In addition, we thoroughly investigate the nature of geometrical dark energy in the modified gravity theories with the assistance of Om(z) and statefinder diagnostic analysis. The present analysis makes it clear-cut that the simplest cosmological constant model is still the most preferred by the current data. However, the combination of future naturally improved GW data and the most recent EM observations will reveal the consistency or acknowledge the tension between the ΛCDM model and modified gravity theories.

Cosmological perturbations in Palatini formalism

We investigate cosmological perturbations of scalar-tensor theories in Palatini formalism. First we introduce an action where the Ricci scalar is conformally coupled to a function of a scalar field and its kinetic term and there is also a k-essence term consisting of the scalar and its kinetic term. This action has three frames that are equivalent to one another: the original Jordan frame, the Einstein frame where the metric is redefined, and the Riemann frame where the connection is redefined. For the first time in the literature, we calculate the quadratic action and the sound speed of scalar and tensor perturbations in three different frames and show explicitly that they coincide. Furthermore, we show that for such action the sound speed of gravitational waves is unity. Thus, this model serves as dark energy as well as an inflaton even though the presence of the dependence of the kinetic term of a scalar field in the non-minimal coupling, different from the case in metric formalism. We then proceed to construct the L3 action called Galileon terms in Palatini formalism and compute its perturbations. We found that there are essentially 10 different (inequivalent) definitions in Palatini formalism for a given Galileon term in metric formalism. We also see that, in general, the L3 terms have a ghost due to Ostrogradsky instability and the sound speed of gravitational waves could potentially deviate from unity, in sharp contrast with the case of metric formalism. Interestingly, once we eliminate such a ghost, the sound speed of gravitational waves also becomes unity. Thus, the ghost-free L3 terms in Palatini formalism can still serve as dark energy as well as an inflaton, like the case in metric formalism.

Gravitational dark matter production in Palatini preheating

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 1013 GeV, larger than the Hubble scale by many orders of magnitude, making this a candidate for superheavy dark matter.

Natural inflation with a nonminimal coupling to gravity

Although natural inflation is a theoretically well-motivated model for cosmic inflation, it is in tension with recent Planck cosmic microwave background anisotropy measurements. We present a way to alleviate this tension by considering a very weak nonminimal coupling of the inflaton field to gravity in both contexts of metric and Palatini formulations of general relativity. We start our discussions with a generic form of the inflaton coupling to the Ricci scalar, then focus on a simple form to do phenomenological study. Our results show that such an extension can bring natural inflation's predictions to a good agreement with the Planck data. Depending on values of the coupling constant ξ and the symmetry breaking scale f, we find that with |ξ|∼ 10-3 and f≳ 2.0 Mpl predictions of the model stay inside 68% CL allowed region until f increases up to 7.7 Mpl, then only inside 95% CL region after f exceeds the latter value. The predictions from the metric and the Palatini theories are very similar due to the simple form of the coupling function we use and the small magnitude of the coupling ξ. Successful reheating can also be realized in this model.

Standard model meets gravity: Electroweak symmetry breaking and inflation

We propose a model for combining the Standard Model (SM) with gravity. It relies on a non-minimal coupling of the Higgs field to the Ricci scalar and on the Palatini formulation of gravity. Without introducing any new degrees of freedom in addition to those of the SM and the graviton, this scenario achieves two goals. First, it generates the electroweak symmetry breaking by a non-perturbative gravitational effect. In this way, it does not only address the hierarchy problem but opens up the possibility to calculate the Higgs mass. Second, the model incorporates inflation at energies below the onset of strong-coupling of the theory. Provided that corrections due to new physics above the scale of inflation are not unnaturally large, we can relate inflationary parameters to data from collider experiments.

Higgs inflation in Einstein-Cartan gravity

We study inflation driven by the Higgs field in the Einstein-Cartan formulation of gravity. In this theory, the presence of the Holst and Nieh-Yan terms with the Higgs field non-minimally coupled to them leads to three additional coupling constants. For a broad range of parameters, we find that inflation is both possible and consistent with observations. In most cases, the spectral index is given by ns=1−2/N⋆ (with N⋆ the number of e-foldings) whereas the tensor-to-scalar ratio r can vary between about 10−10 and 1. Thus, there are scenarios of Higgs inflation in the Einstein-Cartan framework for which the detection of gravitational waves from inflation is possible in the near future. In certain limits, the known models of Higgs inflation in the metric and Palatini formulations of gravity are reproduced. Finally, we discuss the robustness of inflationary dynamics against quantum corrections due to the scalar and fermion fields.

Variational principle and boundary terms in gravity à la Palatini

A general f(R) gravitational theory is considered within the Palatini formalism. By applying the variational principle and the usual conditions on the boundary, we show explicitly that a surface term remains such that as in their metric-compatible counterparts, an additional surface term has to be added in the gravitational action, which plays a fundamental role when calculating the entropy of the black hole.

Minimalistic Musings about the Standard Model

\hspace{2,0cm}{{\bf A tribute to Martinus Veltman 1931-2021}} Martinus ("Tini") Veltman 's early contributions to the Standard Model were essential for its succes. After some nostalgic reminiscences, I turn to the Standard Model with a minimalistic attitude, the point of view that beyond the SM there is only the Planck scale. Known since long, the gravitational force can be obtained as the gauge theory of local Poincar\'e symmetry, called gauge gravity. This gauge theory of gravity embodies {\it per se} a Palatini formulation. This causes the potential of non-minimally coupled Higgs inflation to have an intriguing improved large field behaviour. Some of its effects are experimentally accessible or refutable. The question of quantum corrections is discussed.

Non-minimal (self-)running inflation: metric vs. Palatini formulation

We consider a model of quartic inflation where the inflaton is coupled non-minimally to gravity and the self-induced radiative corrections to its effective potential are dominant. We perform a comparative analysis considering two different formulations of gravity, metric or Palatini, and two different choices for the renormalization scale, widely known as prescription I and II. Moreover we comment on the eventual compatibility of the results with the final data release of the Planck mission.

Palatini quadratic gravity: spontaneous breaking of gauged scale symmetry and inflation

We study quadratic gravity R^2+R_{[mu nu ]}^2 in the Palatini formalism where the connection and the metric are independent. This action has a gauged scale symmetry (also known as Weyl gauge symmetry) of Weyl gauge field v_mu = (tilde{Gamma }_mu -Gamma _mu )/2, with tilde{Gamma }_mu (Gamma _mu ) the trace of the Palatini (Levi-Civita) connection, respectively. The underlying geometry is non-metric due to the R_{[mu nu ]}^2 term acting as a gauge kinetic term for v_mu . We show that this theory has an elegant spontaneous breaking of gauged scale symmetry and mass generation in the absence of matter, where the necessary scalar field (phi ) is not added ad-hoc to this purpose but is “extracted” from the R^2 term. The gauge field becomes massive by absorbing the derivative term partial _mu ln phi of the Stueckelberg field (“dilaton”). In the broken phase one finds the Einstein–Proca action of v_mu of mass proportional to the Planck scale Msim langle phi rangle , and a positive cosmological constant. Below this scale v_mu decouples, the connection becomes Levi-Civita and metricity and Einstein gravity are recovered. These results remain valid in the presence of non-minimally coupled scalar field (Higgs-like) with Palatini connection and the potential is computed. In this case the theory gives successful inflation and a specific prediction for the tensor-to-scalar ratio 0.007le rle 0.01 for current spectral index n_s (at 95% CL) and N=60 efolds. This value of r is mildly larger than in inflation in Weyl quadratic gravity of similar symmetry, due to different non-metricity. This establishes a connection between non-metricity and inflation predictions and enables us to test such theories by future CMB experiments.

Linearized physics and gravitational-waves polarizations in the Palatini formalism of GBD theory

A generalized Brans-Dicke (GBD) theory in the framework of Palatini formalism is 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 geodesic deviation method and the Newman-Penrose method in the Palatini-GBD theory. It is observed that there are three polarizations modes and four oscillations in the Palatini-GBD theory. Concretely, they are the two transverse tensor (+) and (×) standard polarization modes, and one breathing mode (with two oscillations). The results of GWs polarization in the Palatini-GBD theory are different from that in the metric-GBD theory, where there are four polarizations modes: the two standard tensorial modes (+ and ×), a scalar breathing mode, and a massive scalar mode that is a mix of the longitudinal and the breathing polarization. Comparing with the Palatini-f(R˜) theory and the General Relativity, we can see that the extra breathing mode of GWs polarization can be found in the Palatini-GBD theory. At last, the expression of the parameterized post Newton (PPN) parameter is derived, which could pass through the experimental test.

General Slow-Roll Inflation in f(R) Gravity under the Palatini Approach

Slow-roll inflation is analyzed in the context of modified gravity within the Palatini formalism. As shown in the literature, inflation in this framework requires the presence of non-traceless matter; otherwise, it does not occur just as a consequence of the nonlinear gravitational terms of the action. Nevertheless, by including a single scalar field that plays the role of the inflaton, slow-roll inflation can be performed in these theories, where the equations lead to an effective potential that modifies the dynamics. We obtain the general slow-roll parameters and analyze a simple model to illustrate the differences introduced by the gravitational terms under the Palatini approach, and the modifications on the spectral index and the tensor to scalar ratio predicted by the model.