Schedule

In the first part of the talk I am discussing General Relativity, in particular its basic ingredients and its mathematical structure. This will naturally lead the way to consider various modifications or extensions of General Relativity, many of which have been studied recently. Next I will discuss modified gravity models based on generalised geometries and on actions no longer linear in curvature. The main part of the talk will discuss how these many different theories can be studied using a single unified approach which also shows the equivalence of some of these models. Boundary terms in the action will play a crucial role in establishing the equivalence between different theories. The final part discusses the study of cosmological models using dynamical systems techniques.

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 talk, I 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 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. I explore both theories in a plethora of applications, namely, the weak-field limit, cosmology, and irreversible matter creation processes of a specific curvature-matter coupling theory.

Inspired by recent findings involving Ricci-Based Theories, in this talk I will argue why, for theories reproducing GR at low energies which admit a perturbative expansion for the higher order corrections, and for which derivatives of the metric do not appear in the Lagrangian, the terms containing \(R^{\Gamma}_{(\mu\nu)}\) beyond the Einstein-Hilbert term can be understood as redundant operators (in the Effective Field Theory sense) that can be reabsorbed into the interactions of the matter sector. At the same time, the form of the field equations suggests that the metric which describes gravitational perturbations is not the one in the original frame of the theory, namely \(g^{\mu\nu}\), but rather an object \(q^{\mu\nu}\) defined from the derivatives of the Lagrangian with respect to \(R^{\Gamma}_{(\mu\nu)}\). I will then try to outline the construction of a sort of generalized Einstein frame where \(q^{\mu\nu}\) acts as an effective metric.

We work out the junction conditions for the Palatini \(f(\mathcal{R},T)\) extension of General Relativity, where \(f\) is an arbitrary function of the curvature scalar \(\mathcal{R}\) of an independent connection, and of the trace \(T\) of the stress-energy tensor of the matter fields. We find such conditions on the allowed discontinuities of several geometrical and matter quantities, some of which depart from their metric counterparts, and in turn extend their Palatini \(f(\mathcal{R})\) versions via some new \(T\)-dependent terms. Moreover, we also identify some ``exceptional cases" of \(f(\mathcal{R},T)\) Lagrangians such that some of these conditions can be discarded, thus allowing for further discontinuities in \(\mathcal{R}\) and \(T\) and, in contrast with other theories of gravity, they are shown to not give rise to extra components in the matter sector e.g. momentum fluxes and double gravitational layers. We discuss how these junction conditions, together with the non-conservation of the stress-energy tensor ascribed to these theories, may induce non-trivial changes in the shape of specific applications such as traversable thin-shell wormholes.

In this talk we will revise the stability of the four vector irreducible pieces of the torsion and the nonmetricity tensors in the general quadratic metric-affine Lagrangian in 4 dimensions. This analysis highly constrains the theory and reduces the parameter space of the quadratic curvature part from 16 to 5 parameters. We will also mention the case of Weyl-Cartan gravity, proving that the stability of the vector sector completely fixes the dynamics of the full Lagrangian to just an Einstein-Proca theory or pure General Relativity.

I will review a few last findings related to properties of matter in stellar objects in metric-affine theories of gravity.

We consider the most general Quadratic Metric-Affine Gravity setup in the presence of generic matter sources with non-vanishing hypermomentum. The gravitational action consists of all 17 quadratic invariants (both parity even and odd) in torsion and non-metricity as well as their mixings, along with the terms that are linear in the curvature namely the Ricci scalar and the totally antisymmetric Riemann piece. Adding also a matter sector to the latter we first obtain the field equations for the generalized quadratic Theory. Then, using a recent Theorem, we successfully find the exact form of the affine connection under some quite general non-degeneracy conditions. Finally we shall discuss the consequences and also applications of our result.

Gravitational collapse is still poorly understood in the context of \(f(R)\) theories of gravity. The archetypal Oppenheimer-Snyder model (which is an exact solution of General Relativity) is known to be incompatible with the junction conditions of \(f(R)\) gravity, both in the metric and Palatini formalisms. In this talk, we shall explain how the Oppenheimer-Snyder construction must be generalised so as to fit within \(f(R)\) models of gravity. By means of a systematic analysis of the relevant junction conditions, we will show that some paradigmatic vacuum metrics cannot represent spacetime outside collapsing stars in \(f(R)\) gravity.

In this talk I will explain how the pseudoscalar degree of freedom of the quadratic Poincaré Gauge theory of gravity can act as a dark matter candidate. I will give the parameter space of the theory for which such dark matter candidate can account for all predicted cold dark matter, and constrain such parameters based on current astrophysical observations.

We present novel effects of uniform rapid stellar rotation on the minimum mass of stable hydrogen burning in very low mass stars, using an analytic model and relaxing the assumption of spherical symmetry. We obtain an analytic formula for the minimum mass of hydrogen burning as a function of the angular speed of stellar rotation. Further, we show the existence of a maximum mass of stable hydrogen burning in such stars, which is purely an artifact of rapid rotation. The existence of this extremum in mass results in a minimum admissible value of the stellar rotation period of ∼22 minutes, below which a very low mass object does not reach the main sequence, within the ambit of our model. For a given angular speed, we predict a mass range beyond which such an object will not evolve into a main-sequence star.

In the framework of Metric-Affine Gravity, we focus on the dynamical role of the traceless parts of the nonmetricity tensor and construct new static and spherically symmetric black hole solutions with independent shear charges.

After predicting many sub- and super-Chandrasekhar limiting mass white dwarfs from the observations of peculiar type Ia supernovae, researchers proposed various models which can separately explain these two classes of white dwarfs. We have shown that these two peculiar classes of white dwarfs, along with the regular Chandrasekhar white dwarfs, can be explained by a single form of modified gravity, whose effect is significant only in the high-density regime, and it almost vanishes in the low-density regime. Thereby it can explain the violation of the Chandrasekhar mass-limit of \(1.4 M_\odot\). However, so far, there is no direct detection of such white dwarfs, and hence it is difficult to single out one specific theory of gravity. In my talk, I’ll show that gravitational wave observation is one of the prominent ways to single out the exact theory of gravity. We estimate the amplitudes of all the relevant polarization modes of gravitational waves for the peculiar and regular white dwarfs and thereby discuss their possible detections in the future through some of the proposed gravitational wave detectors, such as LISA, ALIA, DECIGO, BBO, or Einstein Telescope. This exploration links the theory with possible observations through the gravitational waves in modified gravity.

We discuss some recent developments for the metric-affine formulation of Chern-Simons gravity, where projective invariance is recovered by enlarging the Pontryagin density definition with nonmetricity depending terms. In particular, we show how analytical solutions for the metric-affine structure can be obtained in spherically symmetric spacetimes, by requiring the absence of dynamical instabilities which we demonstrate to be generated by peculiar components of the affine connection. Finally, we present some exact solutions for the homogeneous and isotropic cosmological background and we discuss in details how the propagation of gravitational waves is affected with respect to the metric formulation.

The amplitude propagation of gravitational waves in an Einstein-Cartan-Sciamma-Kibble (ECSK) theory is studied by assuming a dark matter spin tensor sourcing for spacetime torsion at cosmological scales. The analysis focuses on a weak-torsion regime, such that gravitational wave emission, at leading and subleading orders, does not deviate from standard General Relativity. We show that, in principle, the background torsion induced by an eventual dark matter spin component could lead to an anomalous dampening or amplification of the gravitational wave amplitude, after going across a long cosmological distance. The importance of this torsion-induced anomalous propagation of amplitude for binary black hole mergers is assessed. For realistic late-universe astrophysical scenarios, the effect is tiny and falls below detection thresholds, even for near-future interferometers such as LISA. To detect this effect may not be impossible, but it is still beyond our technological capabilities.

The Chern-Simons theory of gravity is obtained by adding the Pontryagin density to the Einstein-Hilbert action of General Relativity. This additional term is a topological, parity violating term, which arises in different contexts such as quantum field theory, string theory and loop quantum gravity. When the coupling to the Chern-Simons term is promoted to a (pseudo)-scalar field, the field equations are modified, offering interesting theoretical insights and new phenomenological predictions. In this talk I will consider the metric-affine formulation of Chern-Simons gravity, presenting a generalization of the theory in which the symmetry under projective transformations of the affine connection is restored via a modified Pontryagin density, which still retains its topological character. The theory comes in two versions, depending on whether a kinetic term for the scalar field is included in the action or not. However, in both cases, the scalar field has dynamical character, in contrast to the purely metric version of Chern-Simons gravity. I will show how the connection field equations can be solved perturbatively for the torsion and nonmetricity tensors, allowing to study the dynamics of the metric and the scalar field perturbations within the resulting effective scalar tensor theory. As a concrete application I will focus on the evolution of tensor and scalar perturbations on a Schwarzschild background. The parity violating character of the Pontryagin density only affects the axial modes of the black hole, whose quasinormal frequencies are modified with respect to General Relativity and the purely metric version of Chern-Simons theory. Finally, I will discuss the properties of the late time section of the gravitational signal, showing how a distinctive signature of metric-affine Chern-Simons gravity is encoded in the exponents of the power law tails.

The planned next generation of detectors like Einstein Telescope, Cosmic Explorer and space based detectors like LISA are likely detect gravitational waves signals more frequently than now. With such an increased frequency of detection we expect some of the signals to be gravitationally lensed. This lensing phenomenon showcases some interesting features like diffraction and formation of beat patterns. Another opportunity that lensing opens up is to test different theories of gravity. In this work we study gravitational lensing in the context of Palatini f(R) gravity using WKB approximation in the geometric optics limit and beyond.

A successful compactification scenario should explain two, in principle, rather different properties of the multidimensional space-time. First, we need to show why the evolution of three big dimension is different from the evolution of other dimensions. Second, we need to explain why the 3-dimension subspace is almost isotropic one. We present a scenario which address both issures. Starting from rather general totally anisotropic initial condition the evolution of a Universe naturally leads to a product of two isotropic subspaces. This presentation is a brief summary of a set of papers made in collaboration with A.Giakomini, S.Pavluchenko and D. Chirkov.

We discuss the importance of multi-ring images in the optical appearance of spherically symmetric compact objects, when illuminated by an optically and geometrically thin accretion disk. On the one hand, we shall consider some spherically symmetric black hole and wormhole geometries characterized by the presence of a second critical curve, via a uni-parametric family of extensions of the Schwarzschild metric. We will show the presence of additional light rings in the intermediate region between the two critical curves. On the other hand, a sub-case of an analytically tractable extension of the Kerr solution, with both a critical curve and an infinite potential barrier at the object’s center for null geodesics. Our results point out to the existence of multi-ring images with a non-negligible luminosity in shadow observations.

We review the Einstein Cartan Holst gravity status and introduce the Holst current concept as an additional source for cosmological dynamo amplification

In order to study gravitational waves in any realistic astrophysical scenario, one must consider geometry perturbations up to second order. Here, we present a general technique for studying linear and quadratic perturbations on a spacetime with torsion. Besides the standard metric mode, a “torsionon” perturbation mode appears. This torsional mode will be able to propagate only in a certain kind of theories.

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It is well-known since the works of Utiyama and Kibble that the gravitational force can be obtained by gauging the Lorentz group, which puts gravity on the same footing as the Standard Model fields. The resulting theory - Einstein-Cartan gravity - happens to be very interesting. It may incorporate cosmological inflation driven by the Higgs field of the Standard Model of particle physics. In addition, it contains a four-fermion interaction that originates from torsion associated with spin degrees of freedom. This interaction leads to a novel universal mechanism for producing singlet fermions in the Early Universe. These fermions can play the role of dark matter particles.

Considering the EiBI gravity model, I will summarize how one can construct axisymmetric solutions in Ricci-Based Gravity theories (RBGs) using an algebraic correspondence between RBGs and General Relativity. If time permits, physical implications for the counterpart of the Kerr-Newman black hole coupled to Maxwell electrodynamics will be discussed.

The scrutiny of gravitational theories beyond Einstein-Hilbert enjoys a different standard than flat models probed at collider-level energies. Of the proposed extensions, only a subset is explicitly required to be free of ghosts. Even in such cases, only dipole ghosts are usually targeted, thus still allowing, in general, wrong-sign states to propagate. Moreover, even in rare cases where a thorough spectral analysis is performed, the stability of the resulting action under radiative corrections is uncertain. In this talk, I discuss the strong constraining power of requiring, as standard for lower-spin models, a radiatively stable ghost- and tachyon-free action for the paradigmatic scenario of metric-affine gravity. I will describe the role of non-accidental symmetries and their interplay with diffeomorphism invariance. I will also provide, in a more general way, an overview of the computational challenges on the way of the spectrum of higher-spin theories.

The torsional teleparallel gravity is constructed from a manifold with torsion and zero curvature while respecting the metric compatibility condition. In this talk, I will discuss spherical symmetry in this formalism and present new exact black hole solutions for \(f(T,B)\) gravity and also scalarized solutions from scalar-torsion theories of gravity.

In this talk I will present recent results on observables in teleparallel gravity:

• accretion discs in f(T)-gravity, based on a non-perturbative spherically symmetric solution of the theory

• gravitational wave birefringence in new general relativity

We discuss theories of gravity with independent metric (or frame field) and connection, from the point of view of effective field theory. In general, in addition to the metric, gravitational interactions can be carried by torsion field, or nonmetricity, or both. This theory reduces to the General Relativity at low energies in the natural scenario, but enjoys nontrivial dynamics at high energies as long as dimension four operators, such as curvature squared, are considered. We count such operators of even parity and give explicit bases for the independent ones that contribute to the two-point function. We then give the decomposition of the linearized action on a complete basis of spin projectors and consider various subclasses of Metric-Affine Gravity (MAG) theories. We proceed with discussion of ghost and tachyonic instabilities and construct simple MAGs that contain only a massless graviton and a state of spin/parity 2- or 3- and therefore are stable. Finally, we discuss the renormalisation group flow of the theory.

We give a new prescription of parallel transport of a vector tangent to a curve which is invariant under both of a local general coordinate and a Weyl transformation. Thus, since the length of tangent vector does not change during parallel transport along a closed curve in spacetimes with non-metricity, a second clock effect does not appear in general, not only for the inetgrable Weyl spacetime. We have specially motivated the problem from the point of view of symmetric teleparallel (or Minkowski-Weyl) geometry. We also conclude that as long as nature respects Lorentz symmetry and Weyl symmetry, one can only develop alternative gravity models in the symmetric teleparallel spacetime (Q≠0,T=0,R=0) or the most general non-Riemannian spacetime (Q≠0,T≠0,R≠0).

In the Palatini formalism with an R^2 term, inflation potentials will be flattened to form a plateau in the Einstein frame. In such models, the inflaton field can repeatedly return to this plateau after inflation. This results in an active tachyonic instability which effectively fragments the inflaton condensate in less than an e-fold. We will discuss tachyonic preheating in Palatini R^2 gravity and in the family of plateau models in general.

We discuss a teleparallel version of Newton–Cartan gravity. This theory arises as a formal large-speed-of-light limit of the teleparallel equivalent of general relativity (TEGR). Thus, it provides a geometric (‘covariant’) formulation of the Newtonian limit of TEGR, analogous to standard Newton–Cartan gravity being the geometric formulation of the Newtonian limit of general relativity. We show how by a certain gauge-fixing the standard formulation of Newtonian gravity can be recovered.

We present various aspects of modified gravity inflationary scenarios when studied in the Palatini formalism.

Over the last years, there has been an increasing interest and amount of results in alternative theories of gravitation based on (Lorentz-)Finsler geometry. An important achievement has been an equation obtainable by applying the metric variation to a geometric functional [1,2]. In this presentation, based upon [3], we apply the Palatini formalism to the same Finslerian action. We also clarify the structure of the solutions with torsion, extending the classical case [4]. Then, assuming certain regularity at the lightlike directions, we obtain results which provide: A) The uniqueness of the torsion-free (nonlinear) connection, and B) The equivalence of the vacuum equation with the vanishing of the Finslerian Ricci scalar of such connection.

References:

[1] C. Pfeifer and M. R. Wohlfart: Finsler geometric extension of Einstein gravity. Phys. Rev. D (Vol.85, No.6), 2012.

[2] M. Hohmann, C. Pfeifer and N. Voicu: Finsler gravity action from variational completion. Phys. Rev. D 100, 064035 (2019).

[3] M. Á. Javaloyes, M. Sánchez and F. F. Villaseñor: The Einstein-Hilbert-Palatini formalism in pseudo-Finsler geometry. ArXiv e-prints, arXiv:2108.03197v2 [math.DG] (2021).

[4] A. N. Bernal et al.: On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism. Phys.Lett. B768 (2017) 280-287.

The spectral index of scalar perturbation n and the ratio of tensor-to-scalar perturbation r are often used to estimate the consistency of inflation models with the observed results. Extending the theory of gravity to the scalar-tensor type theory, we show in the presentation how very different models can give the same values for n and r. This suggests that these quantities may not be sufficient to determine whether or not the model is consistent with the observations. We analyze this problem and offer solutions for what and how the model should be further evaluated.

In this paper, the relationship between two basic definitions of the energy-momentum tensor in field theory is discussed. Hilbert's definition is based on the variation derivative of the Lagrangian with respect to the metric tensor. By definition, the resulting tensor is symmetric. The conservation of this tensor, on the other hand, requires the use of field equations. Noether's definition is based on the symmetry of the Lagrangian with respect to the diffeomorphism invariance. This tensor is not symmetric, but it can be symmetrized with the use of the Belinfante-Rosenfeld procedure. The conservation of this tensor is self-evident from its definition. Noether's tensor vanishes on shell modulo a divergence of an arbitrary superpotential. For most essential applications, such as Yang-Mills models, two tensors become equal despite having extremely different features. Moreover, they provide meaningful (and measurable) conserved quantities. In this study, we show that in some natural field models the tensors are actually equivalent. Our analysis is based on a teleparallel space-time and differential-form representation. We examine the examples of the free p-form gauge field theory, the GR in the coframe representation, and the metric-free electrodynamics.

We study single field slow-roll inflation in the presence of \(F(R)\) gravity in the Palatini formulation. In contrast to metric \(F(R)\), when rewritten in terms of an auxiliary field and moved to the Einstein frame, Palatini \(F(R)\) does not develop a new dynamical degree of freedom. However, it is not possible to solve analytically the constraint equation of the auxiliary field for a general \(F(R)\). We propose a method that allows us to circumvent this issue and compute the inflationary observables. We apply this method to test scenarios of the form \(F(R)=R+αR^n\) and find that, as in the previously known \(n=2\) case, a large \(α\) suppresses the tensor-to-scalar ratio \(r\). We also find that models with \(F(R)\) increasing faster than \(R^2\) for large \(R\) suffer from numerous problems, with possible implications on the theoretically allowed UV behaviour of such Palatini models.

We derive conservation laws in STEGR with direct application of Noether's theorem. This approach allows to construct covariant conserved currents, corresponding superpotentials and invariant charges. By calculating currents, one can obtain local characteristics of gravitational field like energy density. Surface integration of superpotentials gives charges which correspond to global quantities of the system like mass, momentum, etc. To test our results for the obtained currents and superpotentials, we calculate the energy density measured by freely falling observer in the simple solutions (Schwartzchild black hole, FRLW) and total mass of the Schwartzchild black hole. We find ambiguities in obtaining the connection, which explicitly affect the values of conserved quantities, and discuss a constructive solution to this problem.

We consider an inflationary scenario in the framework of the Hybrid Metric Palatini Gravity in which a supplementary term, containing a Palatini type correction is added to the standard GR action. This configuration can be reformulated into the equivalent form of a scalar-tensor theory, in which a low mass scalar field can explain the observed gravitational phenomenology, by generating long-range forces that pass all the local tests without invoking any kind of screening mechanism, chameleon scenario for instance. We look over the possibility that this effective scalar field may generate the matter content of the early Universe.

All energy is gravitational energy. That is the consequence of the equivalence principle, according to which gravity is the universal interaction. The physical charges of this interaction have remained undisclosed, but the Advent of the Geometrical Trinity opened a new approach to this foundational problem. Here it is shown to provide a background-independent unification of the previous, noncovariant approaches of Bergmann-Thomson, Cooperstock, Einstein, von Freud, Landau-Lifshitz, Papapetrou and Weinberg. First, the Noether currents are derived for a generic Palatini theory of gravity coupled with generic matter fields, and then the canonical i.e. the unique charges are robustly derived and analysed, particularly in the metric teleparallel and the symmetric teleparallel versions of General Relativity. These results, and their application to black holes and gravitational waves, are new.

Discussion leader: Adrià Delhom I Latorre

Prof. Davi Rodrigues will speak at the historic Tartu Old Observatory on the topic "Why do galaxies rotate?"

The canonical generator \(G\) of local symmetries in Poincare gauge theory is constructed as an integral over a spatial section \(\Sigma\) of spacetime. Its regularity (differentiability) on the phase space is ensured by adding a suitable surface term, an integral over the boundary of at infinity, which represents the asymptotic canonical charge. For black hole solutions, \(\Sigma\) has two boundaries, one at infinity and the other at horizon. It is shown that the canonical charge at horizon defines entropy, whereas the regularity of G implies the first law of black hole thermodynamics. The approach is tested for several black hole types.

The constants Λ and the CDM in the standard model of cosmology arise as integration constants in the derivation of Einstein’s equations from the gauge theory of the Lorentz group.

A recent revival of the link between conformal symmetry and projective invariance in the framework of metric-affine gravity theories motivated a systematic study of a general strategy to build a conformal version of any modified gravity. In this oral presentation will be clarified the underlying gauge symmetry associated to conformal invariance and its theoretical and phenomenological implications deriving from the explicit solutions of field equations in the metric affine formalism.

I will present a transparent and concise method to write generic gauge theory, including gravity gauge theories, and extract the corresponding symmetries from the Lagrangian equations of motion. This method can be applied to theories where the symmetries are broken by the presence of nondynamical fields. As an example, I will discuss unimodular gravity in the first order formalism, where invariance under diffeomorphisms is partially broken. Still, the method can be used to read off the remaining symmetries. Remarkably, the symmetry algebra was calculated and it is possible to verify that it is has the expected properties.

Weyl Transverse-Diffeomorphisms (WTDiff) gravity is a theory so closely related to General Relativity (GR) that one can wonder to what extent they are really different. Unimodular gravity can be thought as a simple gauge fixing of WTDiff and hence completely equivalent to it. WTDiff and GR are based on two different gauge symmetries: WTDiff is based on Transverse Diffeomorphisms and Weyl-rescalings of an auxiliary metric, whereas GR is based on the full group of diffeomorphisms. The formal difference appears due to the existence of a fiduciary background structure, a fixed volume form, in WTDiff theories.

In this talk I will present an overview as complete as possible of all the situations and regimes in which one might suspect that some differences between these two theories might arise. This overview contains analysis in the classical, semiclassical and quantum regimes.

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The equivalence principle naturally provides gravity with a geometrical character, thus making the metric-affine framework be a natural land for gravity theories. However, the precise geometry we employ admits a certain flexibility and, in particular, Einstein's gravity can be equivalently ascribed to the three independent objects that characterise a connection, i.e., curvature, torsion and non-metricity. After reviewing these three alternative descriptions of gravity, I will uncover how pathologies generally arise beyond these GR equivalents and illustrate with some examples how to get around them.

Within the framework of teleparallel gravity, a flat affine connection is used as a dynamical field in addition to the metric tensor. Depending on the class of theories, this connection may further be restricted by imposing either vanishing torsion or vanishing nonmetricity. In the field of cosmology, most commonly a particular, simple, homogeneous and isotropic connection has been chosen, and its background dynamics and perturbations have been studied for different theories. While this does lead to a set of cosmological dynamical equations, it is far from being the most general cosmological dynamics in teleparallel gravity. This presentation gives a complete classification of all homogeneous and isotropic teleparallel geometries (general, metric and symmetric), as well as their perturbations. For the latter, gauge transformations and gauge invariant quantities are presented. It is shown that:

• general teleparallel gravity admits two spatially curved branches with three spatially flat limits,
• symmetric teleparallel gravity admits one spatially curved branch with three spatially flat limits,
• metric teleparallel gravity admits two spatially curved branches with one spatially flat limit,

and that the two latter cases arise as limits of the first.

Teleparallel gravity offers numerous possible classes of theories and arbitrary models in each class. One such example is f(T) gravity where any functional form of the torsion scalar is included. The problem of reducing the space of viable functions has led to several recent interesting avenues of research. Traditionally, models in theories like f(T) gravity would be chosen to meet specific observational or theoretical conditions, which would then be constrained against various observations using Bayesian methods. Recent works in the literature have proposed the use of machine learning techniques to reverse the way models are selected in theories like f(T) gravity so that observational data can be used to define the functional form of the model. In this talk, I will cover some of these new approaches and discuss how they compare with previous methods.

It is a very interesting field of modified gravity research now, constructing modifications starting from alternative geometrical foundations. It gives very exciting new angles to look at general relativity, in particular because no simple modifications of this sort appear to be healthy. I will briefly review the foundational issues of modified teleparallel gravity, then discuss modified symmetric teleparallel models, and compare the two approaches.

Teleparallel gravity is a peculiar limit in the metric affine frameworks of gravity where curvature is taken to zero and still a classically equivalent to general relativity can be found. In its original formulation teleparallel gravity breaks Lorentz invariance at minimum at the boundary and in its most generic case fully. Later there was a realization that a spin connection can be introduced for which Lorentz transformations of the couple tetrad and spin connection leaves the action invariant. This is referred to as the covariant formulation of teleparallel gravity. In this talk I will give deeper insights of Lorentz symmetries in teleparallel theories of gravity referring to both a simple observation as well as to the Hamiltonian analysis and highlight the relation to physical degrees of freedom.

In this paper, we have constructed the cosmological model of the universe that shows the bouncing scenario. We have obtained specific form of the function f(Q) that shows the bouncing behavior at the backdrop of homogeneous and isotropic space-time, Q being the nonmetricity. The dynamics of the model has been studied with the help of critical points and the stability behavior has been discussed.

Modified gravities based on non-metricity of spacetime have gained an increased attention in recent years. In particular, f(Q) gravity models come as the simplest generalization of the symmetric teleparallel equivalent of general relativity, and are intensively used in cosmological applications. However, the question of the number of physical degrees of freedom and their behavior is not clear. In this talk we will review the literature and recent progress on the Hamiltonian formalism for symmetric teleparallel and f(Q) gravity, compare to the controversial case of f(T) gravity, and derive conclusions that might be of help for future research.

In this work, we have investigated some rip cosmological models in an extended symmetric teleparallel gravity theory. We consider the form f(Q, T) = aQ^m+bT in the Einstein–Hilbert action and expressed the field equations and the dynamical parameters in terms of the non-metricity 'Q'. Three rip models such as Little Rip, Big Rip and Pseudo Rip are presented. The energy conditions and the cosmographic parameters are derived and analysed for all these models.

By constructing two-dimensional, non-local Lorentz invariant gravitational actions based on the torsion tensor, we discuss the physical meaning of the remnant symmetries associated with the near-horizon geometry experienced by a radial observer in Schwarzschild spacetime. These symmetries, which represent special or privileged diads, acquire the form of uniformly accelerated (Rindler) observers whose constant acceleration is proportional to the black hole mass M.

New high-precision observations are now possible to constrain different gravity theories. To examine the accelerated expansion of the Universe, we used the newly proposed f(Q,T) gravity, where Q is the non-metricity, and T is the trace of the energy-momentum tensor. The investigation is carried out using a parameterized effective equation of state with two parameters, m and n. We have also considered the linear form of f(Q,T) = Q + b T, where b is constant. By constraining the model with the recently published 1048 Pantheon sample, we were able to find the best fitting values for the parameters b, m, and n. The model appears to be in good agreement with the observations. Finally, we analyzed the behavior of the deceleration parameter and equation of state parameter. The results support the feasibility of f(Q,T) as a promising theory of gravity, illuminating a new direction towards explaining the Universe’s dark sector.

Initiated with a question on the mathematical nature of the connection in the so-called "translations gauge theory" formulation of Teleparallel Equivalent to General Relativity (TEGR) Theory, our explorations led us to consider a novel approach using a Cartan connection. The mathematically admitted underlying structure of a gauge theory for a symmetry group \(G\) relies on a principal \(G\)-bundle on which is chosen a connection (the gauge field). The bundle formalism with connection allows to distinguish in the structure of TEGR as "translations gauge theory" that the algebra of translations is indeed necessary to build the torsion, which could resemble the field strength of that gravity. However, the corresponding field clearly does not present the structure of a connection. Nevertheless, the translation field can be integrated in a Cartan connection, as shown in our approach, which allows to keep the interpretation of the translation field as related to a connection. The coupling to matter can be performed thanks to a Cartan connection and a theorem by Sharpe to obtain a consistent and complete description of TEGR that confirms its difference with the usual structures of gauge theories. The approach to STGR involves a closer examination of bundle reduction and soldering form that appear to mark the specificity of gravity. Although the frame field remains the best candidate for what could be recognised as a gauge field, the introduction of a distinct soldering form through a bundle reduction giving birth to the metric on the base leads to a specific structure. To bring such a framework closer to a theory with a connection will require more investigations.

In this paper, we have explored the field equations of \(f(T, B)\) gravity and determined the dynamical parameters with the hyperbolic function of the Hubble parameter. The accelerating behavior has been observed and the behavior of the equation of state parameter indicates the \(\Lambda CDM\) model at a late time. The role of model parameters in assessing accelerating behavior has been emphasized. The stability of the model is analyzed by using scalar perturbation.

If General Relativity [GR] offers an extremely successful framework to describe the gravitational interaction, whose achievements range from the solar system to the cosmological scale, the necessity to question the framework of GR is clear at both the experimental and theoretical levels. Since not all of the current puzzles can be purely reduced to quantum correction problems, this motivates the study of alternative theories of gravitation already at the classical level.

Among the many possible directions one can follow, an interesting one consists in the addition of new degrees of freedom in the theory (in addition to the spacetime metric). In this respect, the simplest candidate is a scalar field. Alternative theories of gravity including a scalar field have attracted a lot of attention in the past decade. This is especially true for Horndeski gravity; the most general scalar-tensor theory of gravity in 4D, based on the same geometrical framework as GR, including a real scalar field and presenting second order field equations.

The aim of this talk will be to introduce recent results of our research in the context of teleparallel theories of gravity extended by scalar fields. I will start by a short synopsis of the main features of teleparallel theories of gravity compared to metric based theories (especially GR). I will then discuss recent results regarding hairy black holes for some teleparallel theories of gravity endowed with a non-minimally coupled real scalar field.

If time permits, I will comment on how one could possibly extend results obtained for metric based scalar-tensor theories of gravity using the framework of teleparallel gravity.

In this paper, we explore the possibility to find exact solutions for Teleparallel Gravity (TG) of the type of spherically symmetric Lemaître-Tolman-Bondi (LTB) dust models. We apply to the LTB metric the formalism of TG in its extension to \(f(T,B)\) models, which can be seen as the analogous from the Schwarzschild solution in General Relativity. An exact LTB solution is obtained which is compatible with a specific \(f(T,B)\) model whose observational constraints are cosmologically viable in a standard spatially flat Robertson-Walker geometry.

Symmetric teleparallel gravity is constructed with a nonzero nonmetricity tensor while both torsion and curvature are vanishing. In this framework, we find exact scalarised spherically symmetric static solutions in scalar-tensor theories built with a nonminimal coupling between the nonmetricity scalar and a scalar field. It turns out that the Bocharova-Bronnikov-Melnikov-Bekenstein solution has a symmetric teleparallel analogue (in addition to the recently found metric teleparallel analogue), while some other of these solutions describe scalarised black hole configurations that are not known in the Riemannian or metric teleparallel scalar-tensor case. To aid the analysis we also derive no-hair theorems for the theory. Since the symmetric teleparallel scalar-tensor models also include \(f(Q)\) gravity, we shortly discuss this case and further prove a theorem which says that by imposing that the metric functions are the reciprocal of each other (\(g_{rr}=1/g_{tt}\)), the \(f(Q)\) gravity theory reduces to the symmetric teleparallel equivalent of general relativity (plus a cosmological constant), and the metric takes the (Anti)de-Sitter-Schwarzschild form.

In this work, stable traversable wormhole models that satisfy at least the null energy condition (NEC) everywhere are developed in the geometric representation of \(f(R,T)\) gravity. The NEC is satisfied both everywhere in the wormhole spacetime and by using the thin-shell formalism to match the wormhole spacetime with an exterior vacuum spacetime. In the first case, conditions are determined for the particular forms of the shape and redshift functions \(b(r)\) and \(\zeta(r)\) considered that guarantee that the resulting spacetime satisfies at least the NEC everywhere. In the second case, after deriving the junction conditions for the particular form of the theory used, one finds that the radius \(r_\Sigma\) of the thin-shell is not restricted to single values for some values of the throat radius \(r_0\). This allows one to choose \(r_\Sigma\) and \(r_0\) such that the weak energy condition can also be satisfied everywhere.

In the context of symmetric teleparallel gravity the coincident gauge denotes a coordinate system where the affine connection vanishes globally and the covariant derivatives reduce to partial derivatives. Despite the allure for computational ease, explicit examples of this are hard to come by. In the talk we consider general metric and affine connection which both are spherically symmetric and static, and transform this configuration into the coincident gauge. We conclude with a comparison to the Weitzenböck gauge in metric teleparallelism, and remarks about the notion of 'purely inertial connection'.

A review of the Scale Invariant Vacuum (SIV) idea will be presented as related to Weyl Integrable Geometry [1]. The main results related to SIV and inflation [2], the growth of the density fluctuations [3], and the application of the SIV to scale-invariant dynamics of Galaxies, MOND, Dark Matter, and the Dwarf Spheroidals [4] will be highlighted.

[1] Gueorguiev, V. G., Maeder, A., The Scale Invariant Vacuum Paradigm: Main Results and Current Progress. Universe 2022, 8 (4) 213; DOI:10.3390/universe8040213 [gr-qc/2202.08412].

[2] Maeder, A., Gueorguiev, V. G., Scale invariance, horizons, and inflation. MNRAS 504, 4005 (2021) [gr-qc/2104.09314].

[3] Maeder, A., Gueorguiev, V., G., The growth of the density fluctuations in the scale-invariant vacuum theory. Phys. Dark Univ. 25, 100315 (2019) [astro-ph.CO/1811.03495]

[4] Maeder, A.; Gueorguiev, V.G. Scale-invariant dynamics of galaxies, MOND, dark matter, and the dwarf spheroidals. MNRAS 492, 2698 (2019) [gr-qc/2001.04978]

Discussion leader: Christian Pfeifer

See Social Program

I will introduce the Normalized Additional Velocity (NAV) method and show how to apply it, with focus on modified gravity (although it can also be useful for dark matter profiles). The method will be illustrated with several models, including \(f(R)\) Palatini, Eddington-inspired-Born-Infeld (EiBI), MOND and others. It is a complementary and fast approach to study galaxy rotation curves (RCs) directly from the sample distribution, instead of first performing several individual RC fits. It does not cover all the RC properties, but it focuses on the shape of the non-Newtonian contribution for a given sample (we use 122 SPARC galaxies). A relevant advantage, when applying the method to modified gravity models, is that for several models it is possible to use approximations that circumvent the need for solving modified Poisson equations for each one of the galaxies. Among other results, we show that \(f(R)\) Palatini and EiBI gravities cannot be used to replace dark matter in galaxies, while MOND has reasonable results, although with an issue.

Based on https://arxiv.org/abs/2204.03762 and more recent developments.

There is a huge variety of gravitational modifications constructed for theoretical reasons, namely to alleviate the renormalizability issues of General Relativity, as well as for cosmological reasons, namely to successfully describe the Universe evolution and alleviate possible observational tensions. It is time to start using the rich observational datasets of increasing accuracy in order to distinguish between them. We search for signatures and smoking guns of torsional modified gravity in late- and early-time cosmological observations, focusing on gravitational waves and primordial black holes.

We end the last parallel session with a few comments on previous and possible future meetings.

Dynamical systems methods are used to investigate a cosmological model with non-minimally coupled scalar field and asymptotically quadratic potential function. We found that for values of the non-minimal coupling constant parameter \(\frac{3}{16}\lt\xi\lt\frac{1}{4}\) there exists an unstable asymptotic de Sitter state giving rise to non-singular beginning of universe. The energy density associated with this state depends on value of the non-minimal coupling constant and can be much smaller than the Planck energy density. For \(\xi=\frac{1}{4}\) we found that the initial state is in form of the static Einstein universe. Proposed evolutional model, contrary to the seminal Starobinsky's model, do not depend on the specific choice of initial conditions in phase space, moreover, a small change in the model parameters do not change the evolution thus the model is generic and structurally stable. The values of the non-minimal coupling constant can indicate for a new fundamental symmetry in the gravitational theory. We show that Jordan frame and Einstein frame formulation of the theory are physically nonequivalent.

The covariant De Donder-Weyl Hamiltonian theory of the Teleparallel Equivalent of General Relativity is formulated using the methods developed by I. Kanatchikov within his approach of precanonical quantization. Based on the results from the geometric Hamilton-Jacobi theory for variational problems we derive a covariant Hamilton-Jacobi equation for TEGR which generalizes the covariant Hamilton-Jacobi equation for General Relativity which was obtained by Th. De Donder (1930) and P. Hoava (1991). I also comment on the problem of the relation between the canonical HJ equations derived from the canonical ADM formalism and the covariant HJ equations derived from the De Donder-Weyl Hamiltonian theory. The covariant HJ equation can be used for numerical simulations or as a test of the classical limit of quantum TEGR, its quasiclassical study and Bohmian-like formulation. Based on my joint paper with C. Barbachoux: e-Print: 2201.01295 [gr-qc].

In this work, we explore the symmetric teleparallel \(f(Q)\) gravity, which is non-minimally coupled with the fermion field in the Friedmann-Robertson-Walker metric. By using the Lagrange multiplier modified Friedmann equations, Dirac equations for the fermion field are obtained. Using the Noether symmetry method, the form of the coupling between gravity and matter, the self-consistent potential, the symmetry generators, the form of \(f(Q)\) gravity, and the conserved quantity for this model are determined. Cosmological solutions describe the late-time accelerated expansion of the Universe obtained.

We plan to present (i) a reformulation of the classical geometric trinity of gravity which allows discussing quantization of equivalent descriptions of general relativity within a unifying framework, (ii) the covariant Hamiltonian formulation of the trinity based on the polysymplectic formalism and the Poisson-Gerstenhaber brackets of differential forms, and the Dirac-like analysis of constraints, (iii) a quantum formulation of the trinity based on the quantization of the analog of Dirac brackets in the above formalism that leads to three different covariant generalizations of the Schroedinger (or Wheeler-Dewitt) equation for quantum gravity. They treat space-time dimensions on equal footing and describe quantum geometry in terms of Clifford-algebra-valued amplitudes of spin-connection (in GR) or vielbeins (in TEGR), or metric density (in Coincident GR). We briefly discuss a relation to the canonical ADM formulations and the canonical quantization, the emergence of the classical limit, the non-Gaussian statistics of quantum gravitational fluctuations in a cosmological context, a naive estimation of the quantum gravity contribution to the cosmological constant, and the scale when quantum-gravitational fluctuations "foam" classical space-time, which turns out to be significantly sub-Planckian.

In this work, we investigate the dynamical behavior of the Universe in the F(R, G) theory of gravity, where R and G indicate the Ricci scalar and Gauss-Bonnet invariant, respectively. The energy conditions, cosmographic parameters, stability, and the possibility of recreating the mentioned model using a scalar field formalism are all part of our comprehensive research. At late times, the model obtained here exhibits quintessence-like behavior.

When the minimal length approach emerging from noncommutative Heisenberg algebra, generalized uncertainty principle, and thereby integrating gravitational fields to this fundamental theory of quantum mechanics (QM) is thoughtfully extended to general relativity (GR), the possible quantization of the fundamental tensor is suggested. This is a complementary term reconciling principles of QM and GR and comprising noncommutative algebra together with maximal spacelike four-acceleration. That quantization compiles with GR as curvature in relativistic eight-dimensional spacetime tangent bundle, Finsler spacetime, the generalization of the pseudo-Riemannian spacetime, is the recipe applied to derive the quantized metric tensor. This dictates how the affine connection on pseudo-Riemannian manifold is straightforwardly quantized. We have discussed the symmetric property of quantized metric tensor and affine connection.

We examined bouncing cosmological models in an isotropic and homogeneous space-time with the f(R) theory of gravity. The bouncing scale factor was used to study two functional forms of f(R). Along with the cosmographic parameters, the dynamical parameters are calculated and analysed. Both the models’ analyses reveal the presence of bounce. In both models, the violation of strong energy conditions is also shown.

Cosmic inflation is an important paradigm of the early Universe which is so far developed in two equivalent ways, either by geometrical modification of Einstein's general relativity (GR) or by introducing new forms of matter beyond the standard model of particle physics. Starobinsky's R+R^2 inflation based on a geometric modification of GR is one of the most observationally favorable models of cosmic inflation based on a geometric modification of GR. In this talk, I will discuss in detail the fundamental motivations for Starobinsky inflation and present how certain logical steps in the view of its UV completion lead to the emergence of a gravity theory that is non-local in nature. Then I will establish how one can perform studies of the early Universe in the context of non-local gravity and what are the observational consequences in the scope of future CMB and gravitational waves. I will discuss in detail how non-local R^2-like inflation can be observationally distinguishable from the local effective field theories of inflation. Finally, I will comment on the prospects of non-local gravity as a promising candidate for quantum gravity.

In the presentation the cooling process of brown dwarf stars in the framework of DHOST theories will be discussed. The analytical model and its numerical solutions are showcased.