Palatini Formalism Research Articles

Einstein gravity as a gauge theory for the conformal group

General relativity in dimension n = p + q can be formulated as a gauge theory for the conformal group , along with an additional field reducing the structure group down to the Poincaré group . In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ‘first order formulation’ of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan–Palatini formulation of general relativity in which the ‘internal’ orthogonal group is extended to an appropriate parabolic subgroup of the conformal group.

Geometrical Diagnostics for Modified Gravitational Theory with the Different Formalisms

Geometrical diagnostic methods were often applied to distinguish the gravitational models. But it is scarce to investigate the differences between the different formalisms of modified gravitational theories (e.g. the metric formalism and the Palatini formalism). In this paper, we discriminate the gravitational theory with the different formalisms by using the geometrical diagnostic methods. For a considered modified theory of gravity (e.g. the f(R) theory or GBD theory), we can see that the difference between the two formalisms is remarkable according to the diagnostic results. And relative to the ΛCDM model, there are more deviations in metric formalism than those in Palatini formalism, according to the {r, s} diagnostic. Given that the GBD (generalized Brans-Dicke theory) is a time-variable Newton gravitational constant (VG) theory, the differences between the VG theory and the constant-G theory are studied. It indicates that the variation of Newton’s gravitational constant could induce notable effects on geometrical quantities (e.g. r, s and q) in both metric formalism and Palatini formalism.

The Einstein–Hilbert–Palatini formalism in pseudo-Finsler geometry

The Einstein–Hilbert–Palatini formalism in pseudo-Finsler geometry

New dynamical system approach to Palatini f(R) theories and its application to exponential gravity

The approach of dynamical systems is a useful tool to investigate the cosmological history that follows from modified theories of gravity. It provides qualitative information on the typical background solutions in a parametrized family of models, through the computation of the fixed points and their characters (attractor, repeller or saddle), allowing, for instance, the knowledge of which regions on the parameter space of the models generate the desired radiation, matter and dark energy dominated eras. However, the traditional proposal for building dynamical systems for an $f(R)$ theory in the Palatini formalism assumes the invertibility of a function that depends on the specific Lagrangian functional form, which is not true, for example, for the particular theory of exponential gravity ($f(R)=R-\alpha R_*(1-e^{-R/R_*})$). In this work, we propose an alternative choice of variables to treat $f(R)$ models in their Palatini formulation, which do include exponential gravity. We derive some general results that can be applied to a given model of interest and present a complete description of the phase space for exponential gravity. We show that Palatini exponential gravity theories have a final attractor critical point with an effective equation of state parameter $w_{\text{eff}} = -1$ (for $\alpha>1$), $w_{\text{eff}} = -2/3$ (for $\alpha=1$) and $w_{\text{eff}} = 0$ (for $\alpha<1$). Finally, our analytical results are compared with numerical solutions of the field equations.

Covariant origin of the U(1)3 model for Euclidean quantum gravity

If one replaces the constraints of the Ashtekar–Barbero SU(2) gauge theory formulation of Euclidean gravity by their U(1)3 version, one arrives at a consistent model which captures significant structures of its SU(2) version. In particular, it displays a non-trivial realisation of the hypersurface deformation algebra which makes it an interesting testing ground for (Euclidean) quantum gravity as has been emphasised in a recent series of papers due to Varadarajan et al. The simplification from SU(2) to U(1)3 can be performed simply by hand within the Hamiltonian formulation by dropping all non-abelian terms from the Gauss, spatial diffeomorphism, and Hamiltonian constraints respectively. However, one may ask from which Lagrangian formulation this theory descends. For the SU(2) theory it is known that one can choose the Palatini action, Holst action, or (anti-)selfdual action (Euclidean signature) as starting point all leading to equivalent Hamiltonian formulations. In this paper, we systematically analyse this question directly for the U(1)3 theory. Surprisingly, it turns out that the abelian analog of the Palatini or Holst formulation is a consistent but topological theory without propagating degrees of freedom. On the other hand, a twisted abelian analog of the (anti-)selfdual formulation does lead to the desired Hamiltonian formulation. A new aspect of our derivation is that we work with (1) half-density valued tetrads which simplifies the analysis, (2) without the simplicity constraint (which admits one undesired solution that is usually neglected by hand) and (3) without imposing the time gauge from the beginning. As a byproduct, we show that also the non-abelian theory admits a twisted (anti-)selfdual formulation. Finally, we also derive a pure connection formulation of Euclidean GR including a cosmological constant by extending previous work due to Capovilla, Dell, Jacobson, and Peldan which may be an interesting starting point for path integral investigations and displays (Euclidean) GR as a Yang–Mills theory with non-polynomial Lagrangian.

The appearance of non trivial torsion for some Ricci dependent theories in the Palatini formalism

As is known from studies of gravity models in the Palatini formalism, there exist two inequivalent definitions of the generalized Ricci tensor in terms of the generalized curvature namely, and . A deep formal investigation of theories with Lagrangians of the form was initiated in Alfonso et al (2017 Class. Quantum Grav. 34 235003). In that work, the authors leave the connection free, and find out that the torsion only appears as a projective mode. This agrees with the widely employed condition of vanishing torsion in these theories as a simple gauge choice. In the present work the complementary scenario is studied namely, the one described by a Lagrangian that depends on the other possible Ricci tensor L = L(R (μν)). The torsion is completely characterized in terms of the metric and the connection, and a rather detailed description of the equations of motion is presented. It is shown that these theories are non trivial even for 1 + 1 space time dimensions, and admit non zero torsion even in this apparently simple case. It is suggested that to impose zero torsion by force may result into an incompatible system. In other words, the presence of torsion may be beneficial for insuring that the equations of motion are well posed. The results of the present paper do not contradict the ones of Alfonso et al (2017 Class. Quantum Grav. 34 235003), as the underlying theories are inequivalent.

Scale-invariance, dynamically induced Planck scale and inflation in the Palatini formulation

We present two scale invariant models of inflation in which the addition of quadratic in curvature terms in the usual Einstein-Hilbert action, in the context of Palatini formulation of gravity, manages to reduce the value of the tensor-to-scalar ratio. In both models the Planck scale is dynamically generated via the vacuum expectation value of the scalar fields.

Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation

Using the second law of local thermodynamics and the first-order Palatini formalism, we formulate relativistic spin hydrodynamics for quantum field theories with Dirac fermions, such as QED and QCD, in a torsionful curved background. We work in a regime where spin density, which is assumed to relax much slower than other non-hydrodynamic modes, is treated as an independent degree of freedom in an extended hydrodynamic description. Spin hydrodynamics in our approach contains only three non-hydrodynamic modes corresponding to a spin vector, whose relaxation time is controlled by a new transport coefficient: the rotational viscosity. We study linear response theory and observe an interesting mode mixing phenomenon between the transverse shear and the spin density modes. We propose several field-theoretical ways to compute the spin relaxation time and the rotational viscosity, via the Green-Kubo formula based on retarded correlation functions.

Inflation by variation of the strong coupling constant: Update for Planck 2018

We apply the “systematic” first-order cosmological perturbation theory method to re-derive the formulation of an inflationary model generated by variation of constants, then to study the case where it is nonminimally coupled to gravity within both the “Metric” and “Palatini” formulations. Accommodating Planck 2018 data with a length scale [Formula: see text] larger than Planck length [Formula: see text] requires amending the model. First, we assume [Formula: see text] gravity where we show that an [Formula: see text]-term within Palatini formulation is able to make the model viable. All along the discussions, we elucidate the origin of the difference between the “Metric” and “Palatini” formalisms, and also highlight the terms dropped when applying the shortcut “potential formulae method,” unlike the “systematic method,” for the observable parameters. Second, another variant of the model, represented by a two-exponentials potential, fits also the data with [Formula: see text].

Estimating the Parameters of Extended Gravity Theories with the Schwarzschild Precession of S2 Star

After giving a short overview of previous results on constraining of Extended Gravity by stellar orbits, we discuss the Schwarzschild orbital precession of S2 star assuming the congruence with predictions of General Relativity (GR). At the moment, the S2 star trajectory is remarkably fitted with the first post-Newtonian approximation of GR. In particular, both Keck and VLT (GRAVITY) teams declared that the gravitational redshift near its pericenter passage for the S2 star orbit corresponds to theoretical estimates found with the first post-Newtonian (pN) approximation. In 2020, the GRAVITY Collaboration detected the orbital precession of the S2 star around the supermassive black hole (SMBH) at the Galactic Center and showed that it is close to the GR prediction. Based on this observational fact, we evaluated parameters of the Extended Gravity theories with the Schwarzschild precession of the S2 star. Using the mentioned method, we estimate the orbital precession angles for some Extended Gravity models including power-law f(R), general Yukawa-like corrections, scalar–tensor gravity, and non-local gravity theories formulated in both metric and Palatini formalism. In this consideration, we assume that a gravitational field is spherically symmetric, therefore, alternative theories of gravity could be described only with a few parameters. Specifically, considering the orbital precession, we estimate the range of parameters of these Extended Gravity models for which the orbital precession is like in GR. Then we compare these results with our previous results, which were obtained by fitting the simulated orbits of S2 star to its observed astrometric positions. In case of power-law f(R), generic Yukawa-like correction, scalar–tensor gravity and non-local gravity theories, we were able to obtain a prograde orbital precession, like in GR. According to these results, the method is a useful tool to evaluate parameters of the gravitational potential at the Galactic Center.

Neutron stars in Palatini R+alpha R^2 and R+alpha R^2+beta Q theories

We study solutions of the stellar structure equations for spherically symmetric objects in modified theories of gravity, where the Einstein-Hilbert Lagrangian is replaced by f(R)=R+alpha R^2 and f(R,Q)=R+alpha R^2+beta Q, with R being the Ricci scalar curvature, Q=R_{mu nu }R^{mu nu } and R_{mu nu } the Ricci tensor. We work in the Palatini formalism, where the metric and the connection are assumed to be independent dynamical variables. We focus on stellar solutions in the mass-radius region associated to neutron stars. We illustrate the potential impact of the R^2 and Q terms by studying a range of viable values of alpha and beta . Similarly, we use different equations of state (SLy, FPS, HS(DD2) and HS(TMA)) as a simple way to account for the equation of state uncertainty. Our results show that for certain combinations of the alpha and beta parameters and equation of state, the effect of modifications of general relativity on the properties of stars is sizeable. Therefore, with increasing accuracy in the determination of the equation of state for neutron stars, astrophysical observations may serve as discriminators of modifications of General Relativity.

β-function reconstruction of Palatini inflationary attractors

Attractor inflation is a particularly robust framework for developing inflationary models that are insensitive to the details of the potential. Such models are most often considered in the metric formulation of gravity. However, non-minimal models may not necessarily maintain their attractor nature in the Palatini formalism where the connection is independent of the metric. In this work, we employ the β-function formalism to classify the strong coupling limit of inflationary models in both the metric and the Palatini approaches. Furthermore, we determine the range of values for the non-minimal coupling that lead to theories being observationally indistinguishable in metric and Palatini within current accuracy. Finally, we reconstruct the Jordan frame potential for ξ-attractors by imposing an explicit form for the β-function, demonstrating the effect that the choice of metric or Palatini has on the inflationary observables of the theory.

Preheating in radiatively corrected φ4 inflation with non-minimal coupling in Palatini formulation

We discuss the impact of the preheating stage due to the interaction of the inflaton to fermions in Palatini formulation. In Palatini inflation with large non-minimal coupling, the field is allowed to return to the plateau region during the reheating stage, therefore the average equation of state per oscillations is closer to -1 rather than 1\/3. The incursion in the plateau, however, leads to a highly efficient tachyonic instability, which is able to reheat the Universe in less than one e-fold. By taking prescription II into account, which is discussed in the literature, we calculate the spectral index n_s and the tensor-to-scalar ratio r in the wide range of κ- ξ. We will show the results which are compatible with the data given by the Keck Array/BICEP2 and Planck collaborations.

Boson stars in Palatini gravity

We explore equilibrium solutions of spherically symmetric boson stars in the Palatini formulation of 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 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 without self-interaction terms is considered. Our results show that the existence curves of boson stars in Palatini gravity are fairly similar to those found in general relativity. Major differences are observed for negative values of the coupling parameter which results in a repulsive gravitational component for high enough scalar field density distributions. Adding self-interactions makes the degeneracy between and general relativity even more pronounced, leaving very little room for observational discrimination between the two theories.

Inflation with R (αβ) terms in the Palatini formulation

We study inflation with the most general non-degenerate gravitational action that depends on the symmetric part of the Ricci tensor coupled to a scalar field in the Palatini formulation of gravity. We use field redefinitions to shift the effect of the Ricci terms from gravity to the scalar field, and apply the result to slow-roll inflation.As examples, we consider actions quadratic and cubic in the Ricci tensor. In the quadratic case the results are similar to the case R + α R 2 that has been studied earlier: the tensor-to-scalar ratio r can be suppressed by an arbitrary amount, while the scalar spectrum is unaffected. In the cubic case, r can be suppressed by at most a factor of 2/9, and the change in the scalar spectral index n s can be large.

Gravitational origin of dark matter and Majorana neutrino mass with non-minimal quartic inflation

We propose a minimal framework to address successful quartic inflation, dark matter (DM) production in the early universe and non-vanishing tiny Majorana neutrino mass from a common gravitational origin point of view. In this setup, the quartic inflation is revived successfully via non-minimal coupling of inflaton to gravity while the production of DM takes place from purely gravitational effects through a misalignment mechanism. The generation of light Majorana neutrino mass is aided by explicit breaking of global lepton number symmetry through Planck suppressed operators involving non-zero vacuum expectation value (VEV) of the inflaton field. We present a detailed study of the DM yield in presence of non-minimal inflation, considering both the metric and the Palatini formalisms of gravity wherever appropriate. We reach at some interesting and different results in the DM sector compared to the earlier works in the similar direction with minimal inflationary background. Restricting to the light DM regime ( O ( 1 ) keV- O ( 100 ) MeV ) where classical production is expected to dominate over the quantum production, we numerically predict the DM mass by varying the DM quartic and non-minimal coupling, to be consistent with relic density requirements. We also obtain some non-trivial dependence of DM phenomenology on some of the relevant parameters of the inflation sector e.g. non minimal coupling and inflaton VEV. To explore the dependence on inflationary parameters further, we also estimate the DM relic using the same mechanism for two other inflationary models consistent with latest data and observe that one of these models predicts different range of DM mass up to hundreds of TeV.

Generation of an electromagnetic field nonminimally coupled to gravity during Higgs inflation

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 the generated electromagnetic fields for different values of the nonminimal coupling constants. Finally, we estimate the maximal present-day magnitude of the magnetic field as $10^{-14}-10^{-15}\,$G with the correlation length of order 10 pc.

Ultraviolet behaviour of Higgs inflation models

We study the ultraviolet behaviour of Higgs inflation models above the apparent unitarity violation scale arising from the large non minimal coupling to gravity, by computing on-shell 4-point scattering amplitudes in the presence of a large inflaton background, away from the electroweak vacuum. We find that all tree-level amplitudes are well behaved at high energies below the inflaton background that can thus take values up to the Planck scale. This result holds in both the metric and Palatini formulation, and is independent of the frame (Jordan or Einstein) as expected. The same result also holds if an R2 term is added to the action.

Extended interactions in the Palatini-R 2 inflation

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 investigate the possible implications of the reheating process on these predictions. We find that the model predictions lie in the favoured region of the observations and that the assumed scalar field dependence of the R 2 coupling tends to decrease the value of the spectral index n s and potentially enhance the tensor-to-scalar ratio r, circumventing the highly suppressed values commonly cited for models considered in the same framework.

Scale-invariant quadratic gravity and inflation in the Palatini formalism

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, which 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 describe the inflationary dynamics in terms of a single scalar degree of freedom along the flat direction of the tree-level potential. The one-loop effective potential in the Einstein frame exhibits plateaus on both sides of the minimum and thus the model can accommodate both small and large field inflation. The inflationary predictions of the model are found to comply with the latest bounds set by the Planck collaboration for a wide range of parameters and the effect of the quadratic in curvature terms is to reduce the value of the tensor-to-scalar ratio.