zbMATH — the first resource for mathematics

Asymptotic regimes of magnetic Bianchi cosmologies. (English) Zbl 1054.83037
Magnetic fields are detected in the universe on a wide range of scales extending beyond galaxy clusters. Magnetic fields should also exist on truly cosmological scales. Such a magnetic field would have to be primordial in origin, that is, one that came into existence at Planck time. The strength of a primordial magnetic field is limited by the anisotropy of the cosmic microwave background [W. H. Kegel, in: Die Erfindung des Universums? Neue Überlegungen zur philosophischen Kosmologie, Insel Verlag, Frankfurt am Main, Leipzig, 126–142 (1997); A. Riotto, Proceedings of the Johns Hopkins Workshop on Current Problems in Particle Theory, Florence, Italy, September 3–5, 2001, World Scientific, River Edge, 189–207 (2003); S. Masi, P. de Bernardis, G. de Troia et al., Imaging the early universe with the boomerang experiment, ibid., 209–222 (2003)].
Obviously, any cosmological model which contains a magnetic field is necessarily anisotropic, since anisotropy is violated by the preferred direction of the magnetic field vector. As a consequence, the Einstein field equations have to be studied in cosmological models that are more general than the homogeneous and isotropic Friedmann-Lemaître cosmological models. The simplest kind of cosmological models that can admit a magnetic field are the Bianchi cosmologies, that is, cosmological models which admit a three-parameter group of isometries acting transitively on space-like hypersurfaces. The cosmological models are thus spatially homogeneous, but not necessarily isotropic. Significant progress has been made in the study of magnetic Bianchi cosmologies, that is, solutions of the Einstein-Maxwell field equations that satisfy the properties of models which contain a barotropic perfect fluid whose four-velocity is orthogonal to the group orbits, and that observers comoving with the fluid measure a pure source-free magnetic field.
In the sense of magnetic Bianchi cosmologies, the paper under review considers the asymptotic dynamics of the Einstein-Maxwell field equations for the class of non-tilted Bianchi cosmologies with a barotropic perfect fluid and a pure homogeneous source-free magnetic field, with special emphasis on models of Bianchi type $$\text{ VII}_0$$. Bianchi cosmologies of type $$\text{ VII}_0$$ are of interest because they represent anisotropic generalizations of the flat Friedmann-Lemaître cosmological models.
The authors demonstrate that the magnetic Bianchi $$\text{ VII}_0$$ cosmologies also exhibit an oscillatory approach to the initial singularity. However, in contrast to the other magnetic Bianchi type models, it establishes rigorously that typical magnetic Bianchi $$\text{ VII}_0$$ cosmologies exhibit the phenomena of asymptotic self-similarity breaking and Weyl curvature dominance in the late-time regime.

MSC:
 83F05 Cosmology 83C75 Space-time singularities, cosmic censorship, etc. 83C50 Electromagnetic fields in general relativity and gravitational theory
Full Text: