Numerical Investigation of Integrable Weyl Geometry in Multidimensional Cosmology

Abstract

The evolution of 4-dimensional and (4+D)-dimensional (D=1, 2) cosmological models based on the integrable Weyl geometry are considered numerically both for empty space-time and for scalar field with nonminimal coupling with gravity. In both cases nonsingular solutions exist only for the open exterior space and flat (with torus topology) interior space. It is shown that in the nonsingular case the scenario of dynamical dimension reduction is realized. Some characteristic features of the considered models and their possible generalizations are discussed.

Vacuum Weyl cosmologies inDdimensions

“…Some properties of the above models have been discovered in numerical calculations for a number of special cases with D = 5 and D = 6 (Konstantinov and Melnikov, 1994). We conclude that some of the multidimensional Weyl cosmologics are nonsingular: there are special flat-space models with eternally increasing or decreasing scale factor (such models are absent in the 4-dimensional approa.ch) and there are more general hyperbolic models with a cosmological bounce generalizing the 4-dimensional ones (Novello et al, 1993).…”

“…However, it should be taken into account that we have considered only one conformal gauge (although, in a certain sense, the most natural one), while, in the others the picture of singularities may change. The choice of a conformal gauge, connected with the choice of a system of measurements, is a separate problem (Staniukovich and Melnikov, 1993), especially when a generalized geometry is used; its solution depends on the specific form of interaction between matter and geometry, which subject is beyond the scope of vacuum cosmologics. Downloaded by [University of Chicago Library] at 14:15 26 December 2014…”

“…Scalar fields play a significant role in modern cosmology (Staniukovich and Melnikov, 1983), in particular, in various inflationary models. However, there is an inherent problem of the origin of this field.…”

“…Since the most reliable results are obtained on the basis of exact solutions, we try to find them in the simplest cases. Recently some results have been obtained in the same scheme using numerical methods (Konstantinov and Melnikov, 1994).…”

“…(3) 1 4 R = fi + ( D -1 ) O w --( D -1)(D -2)LdAW,4 (in and 0, the Riemannian connection Fic is used) and the conformal mapping well-known in STT (Wagoner, 1970), modified for D dimensions (Bronnikov and Melnikov, 1994) :…”