Friedmann-like universes with torsion

Abstract

We consider spatially homogeneous and isotropic cosmologies with non-zero torsion. Given the high symmetry of these universes, we adopt a specific form for the torsion tensor that preserves the homogeneity and isotropy of the spatial surfaces. Employing both covariant and metric-based techniques, we derive the torsional versions of the continuity, the Friedmann and the Raychaudhuri equations. These formulae demonstrate how, by playing the role of the spatial curvature, or that of the cosmological constant, torsion can drastically change the evolution of the classic homogeneous and isotropic Friedmann universes. In particular, torsion alone can lead to exponential expansion. For instance, in the presence of torsion, the Milne and the Einstein-de Sitter universes evolve like the de Sitter model. We also show that, by changing the expansion rate of the early universe, torsion can affect the primordial nucleosynthesis of helium-4. We use this sensitivity to impose strong cosmological bounds on the relative strength of the associated torsion field, requiring that its ratio to the Hubble expansion rate lies in the narrow interval (−0.005813, +0.019370) around zero. Interestingly, the introduction of torsion can reduce the production of primordial helium-4, unlike other changes to the standard thermal history of an isotropic universe. Finally, turning to static spacetimes, we find that there exist torsional analogues of the classic Einstein static universe, with all three types of spatial geometry. These models can be stable when the torsion field and the universe's spatial curvature have the appropriate profiles.Since then, the EC theory (also known as ECKS theory) has been formally established and has received considerable recognition, as it provides the simplest classical extension of Einstein's general relativity (see [3] for a recent review and references therein).The EC theory postulates an asymmetric affine connection for the spacetime, in contrast to the symmetric Christoffel symbols of Riemannian spaces. In technical terms, torsion is described by the antisymmetric part of the non-Riemannian affine connection [1]. Therefore, in addition to the metric tensor, there is an independent torsional field, which also contributes to the total gravitational "pull". Geometrically speaking, curvature reflects the fact that the parallel transport of a vector along a closed loop in a Riemannian space depends on the path. The presence of torsion adds extra complications, since the aforementioned loop does not necessarily close. In a sense, curvature forces the spacetime to bend and torsion twists it. Dynamically, spacetime torsion is triggered by the intrinsic angular momentum (spin) of the matter, whereas spacetime curvature is caused by the mere presence of matter. This distinction is reflected in two sets of formulae, known as the Einstein-Cartan and the Cartan field equations.The literature contains a number of suggestions for experimentally testing gravitational theories with non-zero torsion (see [4] for a representativ...

Low-redshift constraints on homogeneous and isotropic universes with torsion

“…Spatially homogeneous and isotropic cosmologies including torsion have recently been studied in [6,7]. In theories with torsion there is a further degree of freedom (in addition to the usual metric), which also gravitates.…”

“…where a is the scale factor and K is the 3-curvature. Standard computational techniques then lead to the following Friedmann, Raychaudhuri and continuity equations [6,7]…”

“…In most of what follows we will make the usual simplifying assumption (also adopted in [6,10], where it is dubbed steady-state torsion) that the relative torsion contribution to the expansion remains constant in time,…”

Friedmann-like universes with weak torsion: a dynamical system approach

“…The general form of torsion permitted by the high symmetry of an FRW host has been given in [4]. Here, following [5], we will consider a sub-class of the allowed torsion fields, with the torsion tensor taking the form…”

“…These were then combined with the primordial nucleosynthesis measurements to constrain the gravitational effects of torsion. Here, following on the work of [5], we investigate the general qualitative behaviour of homogeneous and isotropic torsional cosmologies. Utilising the above named torsion scalar, φ = φ(t), we parametrise the contribution of the torsion field to the universal expansion and to the total (effective) energy density of the universe.…”