Palatini Formalism Research Articles

Propagation and emission of gravitational waves in the weak-field limit within the Palatini formalism

Propagation and emission of gravitational waves in the weak-field limit within the Palatini formalism

Gravity and [formula omitted] flows in higher dimensions

We study systems in arbitrary space-time dimensions where matter, deformed by TT‾-like irrelevant operators, is coupled to gravity in the Palatini formalism. The dynamically equivalent perspective is investigated, wherein the deformation transitions from the matter action to the gravitational one or vice versa. This alternative viewpoint leads to the emergence of Ricci-based gravity theories, thus providing a high-dimensional generalisation of the well-known equivalence between two-dimensional TT‾ deformations and coupling to Jackiw-Teitelboim gravity. This dynamical equivalence is examined within the framework of the recently introduced Lagrangian flow equation, which notably led to the discovery of a direct link between Nambu-Goto theory and TT‾ in d=2, as well as significant insights into nonlinear electrodynamics models in d=4. The investigation involves explicit examples in d=4 dimensions; it builds upon earlier research concerning the metric interpretation of TT‾-like perturbations, incorporates and extends recent findings in the cosmology-related literature associated to the concept of reframing. We focus on scenarios where the resulting modified gravity theories manifest as Born-Infeld and Starobinsky types.

Post-inflationary leptogenesis and dark matter production: metric versus Palatini formalism

We investigate production of non-thermal dark matter particles and heavy sterile neutrinos from inflaton during the reheating era, which is preceded by a slow-roll inflationary epoch with a quartic potential and non-minimal coupling (ξ) between inflaton and gravity. We compare our analysis between metric and Palatini formalism. For the latter, the tensor-to-scalar ratio, r, decreases with ξ. We find that for ξ = 0.5 and number of e-folds ~ 60, r can be as small as ~ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−3) which may be validated at future reaches of upcoming CMB observation such as CMB-S4 etc. We identify the permissible range of Yukawa coupling yχ between inflaton and fermionic DM χ, to be O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−3.5) ≳ yχ ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−20) for metric formalism and O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−4) ≳ yχ ≳ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document} (10−11) for Palatini formalism which is consistent with current PLANCK data and also within the reach of future CMB experiments. For the scenario of leptogenesis via the decay of sterile neutrinos produced from inflaton decay, we also investigate the parameter space involving heavy neutrino mass MN1 and Yukawa coupling yN1 of sterile neutrino with inflaton, which are consistent with current CMB data and successful generation of the observed baryon asymmetry of the universe via leptogenesis. In contrast to metric formalism, in the case of Palatini formalism, for successful leptogenesis to occur, we find that yN1 has a very narrow allowable range and is severely constrained from the consistency with CMB predictions.

Setting the connection free in the Galilei and Carroll expansions of gravity

We obtain a Palatini-type formulation for the Galilei and Carroll expansions of general relativity, where the connection is promoted to a variable. Known versions of these large and small speed of light expansions are derived from the Einstein-Hilbert action and involve dynamical Newton-Cartan or Carroll geometry, along with additional gauge fields at subleading orders. The corresponding Palatini actions that we obtain in this paper are derived from an appropriate expansion of the Einstein-Palatini action, and the connection variable reduces to the Galilei- or Carroll-adapted connection on shell. In particular, we present the Palatini form for the next-to-leading-order Galilean action and recover the known equations of motion. We also compute the leading-order Palatini-type action for the Carrollian case and show that, while it depends on the connection variable, it reduces on shell to the known action of electric Carroll gravity, which only depends on extrinsic curvature. Published by the American Physical Society 2024

Weak-scale Higgs inflation

The present measurement of the Standard Model (SM) parameters suggests that the Higgs effective potential has a maximum at the Higgs field value of approximately 1011GeV, and the electroweak (EW) vacuum is not absolutely stable. To achieve absolute EW stability, a very large Higgs-Ricci scalar non-minimal coupling can be introduced.I study cosmic inflation driven by the Higgs field in this extension of the SM and refer to it as “weak-scale Higgs inflation” because the resulting inflationary Hubble parameter is around the weak scale. The Palatini formulation of gravity with a dimension 6 term is shown to drive successful inflation. I also argue for the UV (in)sensitivity of the predictions, and phenomenological implications. In the metric formulation case, the scenario for stable EW vacuum may be probed by measuring the Higgs coupling in future colliders.

Global portraits of nonminimal inflation: Metric and Palatini formalism

Global portraits of nonminimal inflation: Metric and Palatini formalism

Quasi-static evolution of axially and reflection symmetric large-scale configuration

In this paper, we review a recently offered notion of quasi-static evolution of the axial self-gravitating structures at large scales and the criterium to characterize the corresponding evolutionary aspects under the influence of strong curvature regimes. In doing so, we examine the axial source’s dynamic and quasi-static behavior within the parameters of various modified gravity theories. We address the formalism of these notions and their possible implications in studying the dissipative and anisotropic configuration. We initiate by considering higher-order curvature gravity. The Palatini formalism of [Formula: see text] gravity is also taken into consideration to analyze the behavior of the kinematical as well as the dynamical variables of the proposed problem. The set of invariant velocities is defined to comprehend the concept of quasi-static approximation that enhances the stability of the system in contrast to the dynamic mode. It is identified that vorticity and distinct versions of the structure scalars [Formula: see text], [Formula: see text] and [Formula: see text] play an important role in revealing the significant effects of a fluid’s anisotropy. As another example of evolution, we check the influence of Palatini-based factors on the shearing motion of the object. A comparison-based study of the physical nature of distinct curvature factors on the propagation of the axial source is exhibited. This provides the intriguing platform to grasp the notion of quasi-static evolution together with the distinct curvature factors at current time scenario. The importance of slowly evolving axially symmetric regimes will be addressed through the distinct modified gravitational context. Finally, we share a list of queries that, we believe, deserve to be addressed in the near future.

Critical points in Palatini Higgs inflation with small non-minimal coupling

We investigate inflation driven by the Higgs boson in the Palatini formulation of General Relativity. Our analysis primarily focuses on a small non-minimal coupling of the Higgs field to gravity in the range 0 < ξ ≲ 1. We incorporate the renormalization group running of the relevant parameters as computed within the Standard Model and allow for small corrections. In addition to ξ, our model features two tunable parameters: the low-energy value of the top Yukawa coupling and an effective jump of the Higgs self-interaction. Our results indicate that critical points leading to a large enhancement of the power spectrum can be produced. However, the observed amplitude of perturbations in the CMB cannot be matched within this setting. On the one hand, this makes it difficult to generate a sizable abundance of primordial black holes. On the other hand, our findings can be viewed as very positive since they provide further evidence that Palatini Higgs inflation has favourable high-energy properties due to robustness against quantum corrections.

Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation

We study models where a scalar field has derivative and non-derivative couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation. We consider both the metric formulation and the Palatini formulation. In the Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the same result regardless of whether the connection is unconstrained or the non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is included, the unconstrained case and the zero torsion case are physically different. We reduce all the actions to the Einstein frame with minimally coupled matter, and find the leading order differences between the metric case and the Palatini cases.

Quantum corrections to Higgs inflation in Einstein-Cartan gravity

This paper studies the quantum corrections to the Higgs inflation model in the context of the Einstein-Cartan (E-C) gravity in the large-N limit with N being the number of real scalar components in Higgs. Recently, it is realized that the Higgs inflation in the E-C formalism smoothly connects those in the metric and the Palatini formalisms in the presence of a non-minimal coupling between the Higgs fields and the Nieh-Yan term. This motivates us to investigate the quantum corrections in large-N limit to the E-C Higgs inflation and to clarify how the Ricci curvature squared R2 induced by the quantum corrections succeeds in Ultraviolet (UV)-extending the Higgs inflation in metric formalism while it fails in the Palatini case. We show that a generalized R2-term required for the renormalization in the E-C formalism induces a new scalar degree of freedom (DoF), the scalaron, which gradually decouples with the system due to its increasing mass as approaching the Palatini limit. The presence of the scalaron extends the UV cutoff at vacuum of the original model except for the parameter space close to the Palatini limit. This UV-extension is expected to solve the strong coupling problem that may exist during (p)reheating in the absence of the scalaron.

General parallel cosmology

General (tele)parallel Relativity, G∥R, is the relativistic completion of Einstein's theories of gravity. The focus of this article is the derivation of the homogeneous and isotropic solution in G∥R. The first-principles derivation, based on a non-trivial realisation of the symmetry, supersedes and unifies previous constructions of Riemannian and teleparallel cosmologies, and establishes the uniqueness of the physical solution. The constitutive law and the form of the material and inertial source currents is presented in the tensor (Palatini) formalism and adapted to the cosmological background, which exhibits novel features absent in the previously studied, static solutions to the theory. The results are contrasted with those in incomplete theories, such as sitting at the three corners of the geometrical trinity which correspond to particular reference frames in G∥R.

Constraining Palatini–Horndeski theory with gravitational waves after GW170817

In this paper, we investigate the possible parameter space of Palatini–Horndeski theory with gravitational waves in a spatially flat Universe. We develop a general method for obtaining the speed of gravitational waves in the Palatini formalism in the cosmological background and we find that if the theory satisfies the following condition: in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light c, then only S = int d^4x sqrt{-g} big [K(phi ,X)-G_{3}(phi ,X){{tilde{Box }}}phi +G_{4}(phi ){tilde{R}}big ] is left as the possible action in Palatini–Horndeski theory. We also find that when G_{5}(phi ,X)ne 0, the tensor part of the connection will propagate and there are two different tensor gravitational wave speeds.

Nonminimally-coupled warm Higgs inflation: Metric vs. Palatini formulations

In this work, we study the non-minimally-coupled Higgs model in the context of warm inflation scenario on both metric and Palatini approaches. We particularly consider a dissipation parameter of the form Γ=CTT with CT being a coupling parameter and focus only on the strong regime of the interaction between inflaton and radiation fluid. We compute all relevant cosmological parameters and constrain the models using the observational Planck 2018 data. We discover that the ns and r values are consistent with the observational bounds. Having used the observational data, we constrain a relation between ξ and λ for the non-minimally-coupled warm Higgs inflation in both metric and Palatini cases. To produce ns and r in agreement with observation, we find that their values are two orders of magnitude higher than those of the usual (cold) non-minimally-coupled Higgs inflation.

Axion quality problem and nonminimal gravitational coupling in the Palatini formulation

Axion quality problem and nonminimal gravitational coupling in the Palatini formulation

Palatini variation in generalized geometry and string effective actions

We develop the Palatini formalism within the framework of generalized Riemannian geometry of Courant algebroids. In this context, the Palatini variation of a generalized Einstein–Hilbert–Palatini action - formed using a generalized metric, a Courant algebroid connection (in contrary to the ordinary case, not necessarily a torsionless one) and a volume form - leads naturally to a proper notion of a generalized Levi-Civita connection and low-energy effective actions of string theory.

Chiral gravitational waves in Palatini-Chern-Simons gravity

We study the parity-breaking higher-curvature gravity theory of Chern-Simons (CS), using the Palatini formulation in which the metric and connection are taken to be independent fields. We first show that Palatini-CS gravity leads to first-order derivative equations of motion and thus avoid the typical instabilities of CS gravity in the metric formalism. As an initial application, we analyze the cosmological propagation of gravitational waves (GWs) in Palatini-CS gravity. We show that, due to parity breaking, the polarizations of GWs suffer two effects during propagation: amplitude birefringence (which changes the polarization ellipticity) and velocity birefringence (which rotates the polarization plane). While amplitude birefringence is known to be present in CS gravity in the metric formalism, velocity birefringence is not present in metric CS gravity for high frequency waves, but now appears in Palatini-CS due to the fact that left-handed and right-handed GW polarizations have a different dispersion relation. In the approximation of small deviations from general relativity (GR), we do find however that velocity birefringence appears at least quadratically in the CS coupling parameter $\ensuremath{\alpha}$, while amplitude birefringence appears linearly in $\ensuremath{\alpha}$. This means that amplitude birefringence will be the most relevant effect in Palatini-CS and hence this model will behave similarly to metric CS. We confirm this by applying current constraints on amplitude and velocity birefringence to Palatini-CS, and showing that those from amplitude birefringence give the tightest bounds.

Constant-roll inflation in modified f(R,phi ) gravity model using Palatini formalism

In this work, we study a constant-roll inflationary model in the Palatini formalism using modified gravity. Here our action consists a non-minimal coupling of a scalar field phi with Ricci scalar R in a general form of f(R,phi ). Using Palatini approach, we write its equivalent scalar-tensor form in the Einstein frame and then apply the constant-roll condition in the equation of motion for the inflaton field. Later the tensor-to-scalar ratio and the spectral index are calculated using the slow-roll parameters and the results obtained are matched with the Planck 2018 data. We found that the results agree nicely with the observations within the parameter regime under consideration.

Gravitational corrections to electroweak vacuum decay: metric vs. Palatini

We consider the standard Einstein-Hilbert-Higgs action where the Higgs field couples nonminimally with gravity via the term ξh2R, and investigate the stability of the electroweak vacuum in the presence of gravitational corrections in both the metric and Palatini formulations of gravity. In order to identify the differences between the two formalisms analytically, we follow a perturbative approach in which the gravitational corrections are taken into consideration via a leading order expansion in the gravitational coupling constant. Our analysis shows that in the Palatini formalism, the well-known effect of gravity suppressing the vacuum decay probability, becomes milder in comparison with the metric case for any value of the nonminimal coupling ξ. Furthermore, we have found that in the Palatini formalism, the positivity of the gravitational corrections, which is a necessary requirement for the unitarity of the theory, entails the lower bound ξ>−1/12.

Born–Infeld-f(R) gravity with de Sitter solutions

In this study, we consider Born–Infeld-[Formula: see text] gravity in which the [Formula: see text] term enters directly into the square root in the Palatini formulation. We briefly analyzed this model for an explicit [Formula: see text] function which includes positive and negative powers of the curvature scalar. We also show that both the ordinary Born–Infeld-[Formula: see text] and this modification reduce to the same gravitational action form under the conformal approach. Then, we consider the existence of a maximally symmetric vacuum solution for the gravitational field equations and find the de Sitter solution for this modified model.

Dynamical emergence of SUq(2) from the regularization of (2+1)D gravity with a cosmological constant

The quantization of the reduced phase-space of the Einstein-Hilbert action for gravity in $2+1D$ has been shown to bring about the emergence, at the quantum level, of a topological quantum field theory endowed with an $SU_q(2)$ quantum group symmetry structure. We hereby tackle the same problem, but start from the kinematical $SU(2)$ (quantum) Hilbert space of the theory of $2+1D$ gravity with non-zero cosmological constant in the Palatini formalism, and subsequently impose the constraints. We hence show the dynamical emergence of the $SU_q(2)$ quantum group at the quantum level within the spin-foam framework. The regularized curvature constraint is responsible for the effective representations of $SU_q(2)$ that are recovered for any Wilson loop evaluated at the $SU(2)$ group element that encodes the discretization of the space-time curvature induced by the cosmological constant. The extension to the spin-network basis, and consequently to any transition amplitude between its generic states, enables us to derive in full generality the recoupling theory of $SU_q(2)$. We provide constructive examples for the scalar product of two loop states and spin-networks encoding trivalent vertices. We further comment on the diffeomorphism symmetry generated by the implementation of the curvature constraint, and finally derive explicitly the partition function amplitude of the Turaev-Viro model.