Metric-affine Formalism Research Articles

Revisiting [formula omitted] cosmologies

We review the status of f(R,T) cosmological models, where T is the trace of the energy momentum tensor Tμν. We start focusing on the modified Friedmann equations for the minimally coupled gravitational Lagrangian of the type f(R,T)=R+αeβT+γnTn. We show that in such a minimally coupled case there exists a useful constraining relation between the effective fractionary total matter density with an arbitrary equation of state parameter and the modified gravity parameters. With this association the modified gravity sector can be independently constrained using estimations of the gas mass fraction in galaxy clusters. Using cosmological background cosmic chronometers data and demanding the universe is old enough to accommodate the existence of Galactic globular clusters with ages of at least ∼14 Gyrs we find a narrow range of the modified gravity free parameter space in which this class of theories remains viable for the late time cosmological evolution. This preferred parameter space region accommodates the ΛCDM limit of f(R,T) models. We also work out the non-minimally coupled case in the metric-affine formalism and find that there are no viable cosmologies in the latter situation. However, when analyzing the cosmological dynamics including a radiation component, we find that this energy density interacts with the matter field and it does not scale according to the typical behavior. We conclude stating that f(R,T) gravity is not able to provide a full cosmological scenario and should be ruled out as a modified gravity alternative to the dark energy phenomena.

Conformal metric-affine gravities

We revisit the gauge symmetry related to integrable projective transformations in metric-affine formalism, identifying the gauge field of the Weyl (conformal) symmetry as a dynamical component of the affine connection. In particular, we show how to include the local scaling symmetry as a gauge symmetry of a large class of geometric gravity theories, introducing a compensator dilaton field that naturally gives rise to a Stückelberg sector where a spontaneous breaking mechanism of the conformal symmetry is at work to generate a mass scale for the gauge field. For Ricci-based gravities that include, among others, General Relativity, f(R) and f(R, R μν R μν) theories and the EiBI model, we prove that the on-shell gauge vector associated to the scaling symmetry can be identified with the torsion vector, thus recovering and generalizing conformal invariant theories in the Riemann-Cartan formalism, already present in the literature.

Scalar-metric-affine theories: Can we get ghost-free theories from symmetry?

We reveal the existence of a certain hidden symmetry in general ghost-free scalar-tensor theories which can only be seen when generalizing the geometry of the spacetime from Riemannian. For this purpose, we study scalar-tensor theories in the metric-affine (Palatini) formalism of gravity, which we call scalar-metric-affine theories for short, where the metric and the connection are independent. We show that the projective symmetry, a local symmetry under a shift of the connection, can provide a ghost-free structure of scalar-metric-affine theories. The ghostly sector of the second-order derivative of the scalar is absorbed into the projective gauge mode when the unitary gauge can be imposed. Incidentally, the connection does not have the kinetic term in these theories and then it is just an auxiliary field. We can thus (at least in principle) integrate the connection out and obtain a form of scalar-tensor theories in the Riemannian geometry. The projective symmetry then hides in the ghost-free scalar-tensor theories. As an explicit example, we show the relationship between the quadratic order scalar-metric-affine theory and the quadratic U-degenerate theory. The explicit correspondence between the metric-affine (Palatini) formalism and the metric one could be also useful for analyzing phenomenology such as inflation.

Ghosts in metric-affine higher order curvature gravity

We disprove the widespread belief that higher order curvature theories of gravity in the metric-affine formalism are generally ghost-free. This is clarified by considering a sub-class of theories constructed only with the Ricci tensor and showing that the non-projectively invariant sector propagates ghost-like degrees of freedom. We also explain how these pathologies can be avoided either by imposing a projective symmetry or additional constraints in the gravity sector. Our results put forward that higher order curvature gravity theories generally remain pathological in the metric-affine (and hybrid) formalisms and highlight the key importance of the projective symmetry and/or additional constraints for their physical viability and, by extension, of general metric-affine theories.

Global monopole in Palatinif(R)gravity

We consider the space-time metric generated by a global monopole in an extension of General Relativity (GR) of the form $f(\mathcal{R})=\mathcal{R}-\lambda \mathcal{R}^2$. The theory is formulated in the metric-affine (or Palatini) formalism and exact analytical solutions are obtained. For $\lambda<0$, one finds that the solution has the same characteristics as the Schwarzschild black hole with a monopole charge in Einstein's GR. For $\lambda>0$, instead, the metric is more closely related to the Reissner-Nordstr\"{o}m metric with a monopole charge and, in addition, it possesses a wormhole-like structure that allows for the geodesic completeness of the space-time. Our solution recovers the expected limits when $\lambda=0$ and also at the asymptotic far limit. The angular deflection of light in this spacetime in the weak field regime is also calculated.

Coupling matter in modifiedQgravity

We present a novel theory of gravity by considering an extension of symmetric teleparallel gravity. This is done by introducing, in the framework of the metric-affine formalism, a new class of theories where the nonmetricity $Q$ is nonminimally coupled to the matter Lagrangian. More specifically, we consider a Lagrangian of the form $L \sim f_1(Q) + f_2(Q) L_M$, where $f_1$ and $f_2$ are generic functions of $Q$, and $L_M$ is the matter Lagrangian. This nonminimal coupling entails the nonconservation of the energy-momentum tensor, and consequently the appearance of an extra force. The motivation is to verify whether the subtle improvement of the geometrical formulation, when implemented in the matter sector, would allow more universally consistent and viable realisations of the nonminimal curvature-matter coupling theories. Furthermore, we consider several cosmological applications by presenting the evolution equations and imposing specific functional forms of the functions $f_1(Q)$ and $f_2(Q)$, such as power-law and exponential dependencies of the nonminimal couplings. Cosmological solutions are considered in two general classes of models, and found to feature accelerating expansion at late times.

Galileon and generalized Galileon with projective invariance in a metric-affine formalism

We study scalar-tensor theories respecting the projective invariance in the metric-affine formalism. The metric-affine formalism is a formulation of gravitational theories such that the metric and the connection are independent variables in the first place. In this formalism, the Einstein-Hilbert action has an additional invariance, called the projective invariance, under a shift of the connection. Respecting this invariance for the construction of the scalar-tensor theories, we find that the Galileon terms in curved spacetime are uniquely specified at least up to quartic order which does not coincide with either the covariant Galileon or the covariantized Galileon. We also find an action in the metric-affine formalism which is equivalent to class ${}^2$N-I/Ia of the quadratic degenerated higher order scalar-tensor (DHOST) theory. The structure of DHOST would become clear in the metric-affine formalism since the equivalent action is just linear in the generalized Galileon terms and non-minimal couplings to the Ricci scalar and the Einstein tensor with independent coefficients. The fine-tuned structure of DHOST is obtained by integrating out the connection. In these theories, non-minimal couplings between fermionic fields and the scalar field may be predicted. We discuss possible extensions which could involve theories beyond DHOST.

Dynamical system analysis of hybrid metric-Palatini cosmologies

The so called $f(X)$ hybrid metric-Palatini gravity presents a unique viable generalisation of the $f(R)$ theories within the metric-affine formalism. Here the cosmology of the $f(X)$ theories is studied using the dynamical system approach. The method consists of formulating the propagation equation in terms of suitable (expansion-normalised) variables as an autonomous system. The fixed points of the system then represent exact cosmological solutions described by power-law or de Sitter expansion. The formalism is applied to two classes of $f(X)$ models, revealing both standard cosmological fixed points and new accelerating solutions that can be attractors in the phase space. In addition, the fixed point with vanishing expansion rate are considered with special care in order to characterise the stability of Einstein static spaces and bouncing solutions.

Non-Riemannian geometry: towards new avenues for the physics of modified gravity

Less explored than their metric (Riemannian) counterparts, metric-affine (or Palatini) theories bring an unexpected phenomenology for gravitational physics beyond General Relativity. Lessons of crystalline structures, where the presence of defects in their microstructure requires the use of non-Riemannian geometry for the proper description of their properties in the macroscopic continuum level, are discussed. In this analogy, concepts such as wormholes and geons play a fundamental role. Applications of the metric-affine formalism developed by the authors in the last three years are reviewed.

WEAK FORCES AND NEUTRINO OSCILLATIONS UNDER THE STANDARDS OF HYBRID GRAVITY WITH TORSION

We present a unifying approach where weak forces and neutrino oscillations are interpreted under the same standards of torsional hybrid gravity. This gravitational theory mixes metric and metric-affine formalism in presence of torsion and allows to derive an effective scalar field which gives rise to a running coupling for Dirac matter fields. In this picture, two phenomena occurring at different energy scales can be encompassed under the dynamics of such a single scalar field, which represents the further torsional and curvature degrees of freedom.

Importance of torsion and invariant volumes in Palatini theories of gravity

We study the field equations of extensions of General Relativity formulated within a metric-affine formalism setting torsion to zero (Palatini approach). We find that different (second-order) dynamical equations arise depending on whether torsion is set to zero i) a priori or ii) a posteriori, i.e., before or after considering variations of the action. Considering a generic family of Ricci-squared theories, we show that in both cases the connection can be decomposed as the sum of a Levi-Civita connection and terms depending on a vector field. However, while in case i) this vector field is related to the symmetric part of the connection, in ii) it comes from the torsion part and, therefore, it vanishes once torsion is completely removed. Moreover, the vanishing of this torsion-related vector field immediately implies the vanishing of the antisymmetric part of the Ricci tensor, which therefore plays no role in the dynamics. Related to this, we find that the Levi-Civita part of the connection is due to the existence of an invariant volume associated to an auxiliary metric $h_{\mu\nu}$, which is algebraically related with the physical metric $g_{\mu\nu}$.

Metric-affine formalism of higher derivative scalar fields in cosmology

Higher derivative scalar field theories have received considerable attention for the potentially explanations of the initial stateof the universe or the current cosmic acceleration which they might offer. They have also attracted many interests in the phenomenologicalstudies of infrared modifications of gravity. These theories are mostly studied by the metric variational approach in which only the metricis the fundamental field to account for the gravitation. In this paper we study the higher derivative scalar fields with the metric-affineformalism where the affine connection is treated arbitrarily at the beginning. Because the higher derivative scalar fields couple to theconnection directly in a covariant theory these two formalisms will lead to different results. These differences are suppressed by the powersof the Planck mass and are usually expected to have small effects. But in some cases they may cause non-negligible deviations. We show by ahigher derivative dark energy model that the two formalisms lead to significantly different pictures of the future universe.

Torsion of space-time in f (R) gravity

In this paper, we first review some aspects of the f (R) gravity, and then the concept of the torsion of space-time due to metric-affine formalism in f (R) gravity is studied. Within this formalism, in which the matter action is supposed to be dependent on the connection, we achieve interesting cases including nonzero torsion tensor. Then with the physical interpretation of the torsion of space-time in high energy limit, the modified expression of Mach’s principle in a very strong gravitational region is obtained.

MATTER LAGRANGIANS COUPLED WITH CONNECTIONS

We shall consider here extended theories of gravitation in the metric-affine formalism with matter coupled directly to the connection. A sufficiently general procedure will be exhibited to solve the resulting field equation associated to the connection. As special cases one has the no-coupling case (which is standard in f(R) literature) as well as the cases already analyzed in [1].

ON THE WELL-FORMULATION OF THE INITIAL VALUE PROBLEM OF METRIC-AFFINE f(R)-GRAVITY

We study the well-formulation of the initial value problem of f(R)-gravity in the metric-affine formalism. The problem is discussed in vacuo and in presence of perfect-fluid matter, Klein–Gordon and Yang–Mills fields. Adopting Gaussian normal coordinates, it can be shown that the problem is always well-formulated. Our results refute some criticisms to the viability of f(R)-gravity recently appeared in literature.

Maximal symmetry and metric-affine f(R) gravity

The affine connection in a spacetime with a homogenous and isotropic subspace is derived using the properties of maximally symmetric tensors. The number of degrees of freedom in metric-affine gravity is thereby considerably reduced while the theory allows spatio-temporal torsion and remains non-metric. The Ricci tensor and scalar are calculated in terms of the connection and the field equations derived for the Einstein–Hilbert as well as for f(R) Lagrangians. By considering specific forms of f(R), we demonstrate that the resulting Friedmann equations in the so-called Palatini formalism without torsion and metric-affine formalism with maximal symmetry are in general different in the presence of matter.

F(R) gravity with torsion: the metric-affine approach

The role of torsion in f(R) gravity is considered in the framework of metric-affine formalism. We discuss the field equations in empty space and in the presence of perfect fluid matter, taking into account the analogy with the Palatini formalism. As a result, the extra curvature and torsion degrees of freedom can be dealt as an effective scalar field of a fully geometric origin. From a cosmological point of view, such a geometric description could account for the whole dark side of the universe.

The metric-affine formalism of f(R) gravity

Recently a class of alternative theories of gravity which goes under the name f(R) gravity, has received considerable attention, mainly due to its interesting applications in cosmology. However, the phenomenology of such theories is not only relevant to cosmological scales, especially when it is treated within the framework of the so called Palatini variation, an independent variation with respect to the metric and the connection, which is not considered a priori to be the Levi–Civita connection of the metric. If this connection has its standard geometrical meaning the resulting theory will be a metric-affine theory of gravity, as will be discussed in this talk. The general formalism will be presented and several aspects of the theory will be covered, mainly focusing on the enriched phenomenology that such theories exhibit with respect to General Relativity, relevant not only to large scales (cosmology) but also to small scales (e.g. torsion).