Search Results (12)

Einstein–Cartan theory is a generalization of general relativity that introduces spacetime torsion. In this paper, we perform phase space analysis to investigate the evolution of the early universe in Einstein–Cartan theory. By studying the stability of critical points in the dynamical system, we find that there exist two stable critical points which represent an Einstein static solution and an expanding solution, respectively. After analyzing the phase diagram of the dynamical system, we find that the early universe may exhibit an Einstein static state, an oscillating state, or a bouncing state. By assuming the equation of state $\omega$ can decrease over time t, the universe can depart from the initial Einstein static state, oscillating state, or bouncing state and then evolve into an inflationary phase. Then, we analyze four different inflationary evolution cases in Einstein–Cartan theory and find that a time-variable equation of state $\omega$ cannot yield values of $n_s$ and r consistent with observations, while a time-invariant equation of state $\omega$ is supported by the Planck 2018 results. Thus, in Einstein–Cartan theory, the universe likely originates from a bouncing state rather than an Einstein static state or an oscillating state. Full article

Alternative scenarios where the Big Bang singularity of the standard cosmological model is replaced by a bounce, or by an early almost static phase (known as emergent universe) have been frequently studied. We investigate the role of the spinor degrees of freedom in overcoming the initial singularity. We introduce a model which generalizes the Einstein–Cartan–Dirac theory, including local phase invariance of the spinor field supported by a gauge scalar field and certain couplings to the torsion. A natural gauge choice reduces the field equations to that of the Einstein–Dirac theory with a Dirac field potential that has polar and axial spinor currents. We identify a new potential term proportional to the square of the ratio of Dirac scalar and axial scalar, which provides a dark energy contribution dominating in the late-time Universe. In addition, the presence of spinor currents in the potential may induce the bounce of a contracting universe. Full article

In the framework of quantum field theory, we analyze the neutrino oscillations in the presence of a torsion background. We consider the Einstein–Cartan theory and we study the cases of constant torsion and of linearly time-dependent torsion. We derive new neutrino oscillation formulae which depend on the spin orientation. Indeed, the energy splitting induced by the torsion influences oscillation amplitudes and frequencies. This effect is maximal for values of torsion of the same order of the neutrino masses and for very low momenta, and disappears for large values of torsion. Moreover, neutrino oscillation is inhibited for intensities of torsion term much larger than neutrino masses and momentum. The modifications induced by torsion on the $CP$-asymmetry are also presented. Future experiments, such as PTOLEMY, which have as a goal the analysis of the cosmological background of neutrino (which have very low momenta), can provide insights into the effect shown here. Full article

Explicit and spontaneous breaking of spacetime symmetry under diffeomorphisms, local translations, and local Lorentz transformations due to the presence of fixed background fields is examined in Einstein–Cartan theory. In particular, the roles of torsion and violation of local translation invariance are highlighted. The nature of the types of background fields that can arise and how they cause spacetime symmetry breaking is discussed. With explicit breaking, potential no-go results are known to exist, which if not evaded lead to inconsistencies between the Bianchi identities, Noether identities, and the equations of motion. These are examined in detail, and the effects of nondynamical backgrounds and explicit breaking on the energy–momentum tensor when torsion is present are discussed as well. Examples illustrating various features of both explicit and spontaneous breaking of local translations are presented and compared to the case of diffeomorphism breaking. Full article

We consider a cosmological model in the framework of Einstein–Cartan theory with a single scalar torsion $\varphi =\varphi \left(t\right)$ and reconstruct the torsion model corresponding to the holographic dark energy (HDE) density. By studying the corresponding relation between the effective energy density of torsion field ${\rho }_{\varphi }$ and holographic dark energy density ${\rho }_{HDE}$, we naturally obtain a kind of torsion field from the interacting holographic dark energy with interaction term $Q=-2\varphi {\rho }_m$ and ${\rho }_m$ is the energy density of matter. We analyze the reconstructed torsion model and find that the torsion field behaves like the quintessence $(w>-1)$ or quintom (exhibiting a transition from $w>-1$ to $w<-1$) dark energy, depending on the value of the model parameter c. We then perform a stability analysis according to the squared sound speed. It is shown that the model is classically stable in the current epoch for the case of $c<1$. We also investigate the model from the viewpoint of statefinder parameters and it turns out that the statefinder trajectories in the $r-s$ plane behave differently for the three cases of c and also quite distinct from those of other cosmological models. From the trajectories of the statefinder pair $\{q,r\}$, we find that, for all the three cases of c, the universe has a phase transition from deceleration to acceleration, consistently with cosmological observations. In addition, we fit the reconstructed torsion model with the recent Type Ia supernovae (SNe Ia) samples, i.e., the Pantheon sample containing 1048 SNe Ia with the redshift in the range $0.01<z<2.3$ and the Pantheon+ sample with 1701 light curves of 1550 distinct SNe Ia in the range $0.001<z<2.26$. The analysis results show that the limits on the present fractional energy density of matter ${\Omega }_{m0}$ are completely compatible with those of the $\Lambda$CDM model obtained from the latest Planck mission observations at 68% confidence level. The mean value of c constrained from the Pantheon sample corresponds to the quintom-like scenario (i.e., $c<1$) and its mean value from the Pantheon+ sample corresponds to the quintessence-like scenario (i.e., $c\ge 1$). However, both of the Pantheon and Pantheon+ samples cannot distinguish the quintom-like and quintessence-like scenarios at 68% confidence level. Full article

Spacetime torsion is known to be highly suppressed at the end of inflation, which is called preheating. This result was recently shown in (EPJ C (2022)) in the frame of Einstein–Cartan–Brans–Dicke inflation. In this paper, it is shown that a torsionful magnetogenesis in QED effective Lagrangean drives a torsion damping in order to be subsequently amplified by the dynamo effect after the generation of these magnetic fields seeds. This damping on amplification would depend upon the so-called torsion chirality. Here, a cosmic factor $gkK$ is present where K is the contortion vector and k is the wave vector which is connected to the inverse of magnetic coherence length. In a second example, we find Higgs inlationary fields in Einstein–Cartan gravity thick domain walls (DWs). Recently, a modified Einstein–Cartan gravity was given by Shaposhnikov et al. [PRL (2020)] to obtain Higgs-like inflatons as a portal to dark energy. In the case of thick DW, we assume that there is a torsion squared influence, since we are in the early universe where torsion is not so weak as in the late universe as shown by Paul and SenGupta [EPJ C (2019)] in a 5D brane-world. A static DW solution is obtained when the inflationary potential vanishes and Higgs potential is a helical function. Recently, in the absence of inflation, domain wall dynamos were obtained in Einstein–Cartan gravity (EC) where the spins of the nucleons were orthogonal to the wall. Full article

We investigate the Cartan formalism in $F\left(R\right)$ gravity. $F\left(R\right)$ gravity has been introduced as a theory to explain cosmologically accelerated expansions by replacing the Ricci scalar R in the Einstein–Hilbert action with a function of R. As is well-known, $F\left(R\right)$ gravity is rewritten as a scalar–tensor theory by using the conformal transformation. Cartan $F\left(R\right)$ gravity is described based on the Riemann–Cartan geometry formulated by the vierbein-associated local Lorenz symmetry. In the Cartan formalism, the Ricci scalar R is divided into two parts: one derived from the Levi–Civita connection and the other from the torsion. Assuming the spin connection-independent matter action, we have successfully rewritten the action of Cartan $F\left(R\right)$ gravity into the Einstein–Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. red Thus, symmetries in Cartan $F\left(R\right)$ gravity are clearly conserved. The resulting scalar–tensor theory is useful in applications of the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in cosmological microwave background radiation. Full article

Gravity can be formulated as a gauge theory by combining symmetry principles and geometrical methods in a consistent mathematical framework. The gauge approach to gravity leads directly to non-Euclidean, post-Riemannian spacetime geometries, providing the adequate formalism for metric-affine theories of gravity with curvature, torsion and non-metricity. In this paper, we analyze the structure of gauge theories of gravity and consider the relation between fundamental geometrical objects and symmetry principles as well as different spacetime paradigms. Special attention is given to Poincaré gauge theories of gravity, their field equations and Noether conserved currents, which are the sources of gravity. We then discuss several topics of the gauge approach to gravitational phenomena, namely, quadratic Poincaré gauge models, the Einstein-Cartan-Sciama-Kibble theory, the teleparallel equivalent of general relativity, quadratic metric-affine Lagrangians, non-Lorentzian connections, and the breaking of Lorentz invariance in the presence of non-metricity. We also highlight the probing of post-Riemannian geometries with test matter. Finally, we briefly discuss some perspectives regarding the role of both geometrical methods and symmetry principles towards unified field theories and a new spacetime paradigm, motivated from the gauge approach to gravity. Full article

In the Einstein–Cartan gravitational theory with the chameleon field, while changing its mass independently of the density of its environment, we analyze the Friedmann–Einstein equations for the Universe’s evolution with the expansion parameter a being dependent on time only. We analyze the problem of an identification of the chameleon field with quintessence, i.e., a canonical scalar field responsible for dark energy dynamics, and for the acceleration of the Universe’s expansion. We show that since the cosmological constant related to the relic dark energy density is induced by torsion (Astrophys. J.2016, 829, 47), the chameleon field may, in principle, possess some properties of quintessence, such as an influence on the dark energy dynamics and the acceleration of the Universe’s expansion, even in the late-time acceleration, but it cannot be identified with quintessence to the full extent in the classical Einstein–Cartan gravitational theory. Full article

Two of the major open questions in particle physics are: (1) Why do the elementary fermionic particles that are so far observed have such low mass-energy compared to the Planck energy scale? (2) What mechanical energy may be counterbalancing the divergent electrostatic and strong force energies of point-like charged fermions in the vicinity of the Planck scale? In this paper, using a hitherto unrecognised mechanism derived from the non-linear amelioration of the Dirac equation known as the Hehl–Datta equation within the Einstein–Cartan–Sciama–Kibble (ECSK) extension of general relativity, we present detailed numerical estimates suggesting that the mechanical energy arising from the gravitationally coupled self-interaction in the ECSK theory can address both of these questions in tandem. Full article

We discuss the Cauchy problem and the junction conditions within the framework of $f\left(R\right)$ -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable $f\left(R\right)$ -models with torsion. Full article

This article is a review of what could be considered the basic mathematics of Einstein–Cartan theory. We discuss the formalism of principal bundles, principal connections, curvature forms, gauge fields, torsion form, and Bianchi identities, and eventually, we will end up with Einstein–Cartan–Sciama–Kibble field equations and conservation laws in their implicit formulation. Full article