Abstract
We consider Friedmann-like universes with torsion and take a step towards studying their stability. In so doing, we apply dynamical-system techniques to an autonomous system of differential equations, which monitors the evolution of these models via the associated density parameters. Assuming relatively weak torsion, we identify the system’s equilibrium points. These are found to represent homogeneous and isotropic spacetimes with nonzero torsion that undergo accelerated expansion. We then examine the linear stability of the aforementioned fixed points. Our results indicate that Friedmann-like cosmologies with weak torsion are generally stable attractors, either asymptotically or in the Lyapunov sense. In addition, depending on the equation of state of the matter, the equilibrium states can also act as intermediate saddle points, marking a transition from a torsional to a torsion-free universe.
Highlights
Extensions of general relativity that go beyond the boundaries of the Riemannian geometry, by allowing for an asymmetric affine connection, have a long history in the literature
With one exception that leads to a “saddle point”, Friedmann-like cosmologies equipped with a weak torsion field are generally stable attractors, either asymptotically or in the Lyapunov
Dynamical system techniques have been extensively used to study the qualitative evolution of a wide range of cosmological solutions
Summary
Extensions of general relativity that go beyond the boundaries of the Riemannian geometry, by allowing for an asymmetric affine connection, have a long history in the literature. Assuming relatively weak torsion, we are able to recast the associated Einstein-Cartan equations into an autonomous dynamical system and identify its critical (fixed) equilibrium points. As expected, these include the familiar torsion-free Friedmann models, with varying 3-curvature and a nonzero cosmological constant. In the presence of (weak) torsion, on the other hand, we find that all the critical points correspond to spatially flat cosmologies and that they all undergo accelerated, de Sitter-like, expansion Put another way, the torsional equilibrium states identified in this work are flat Friedmanntype universes, which are either Λ-dominated or filled with non-conventional matter (dark energy or phantom).
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