Exact solutions in multi-dimensional cosmology with shear and bulk viscosity

Abstract

Multidimensional cosmological model describing the evolution of a fluid with shear and bulk viscosity in n Ricci-flat spaces is investigated. The barotropic equation of state for the density and the pressure in each space is assumed. The second equation of state is chosen in the form when the bulk and the shear viscosity coefficients are inversely proportional to the volume of the Universe. The integrability of Einstein equations reads as a colinearity constraint between vectors which are related to constant parameters in the first and second equations of state. We give exact solutions in a Kasner-like form. The processes of dynamical compactification and the entropy production are discussed. The non-singular D-dimensional isotropic viscous solution is singled out.PACS numbers: 04.20.J, 04.60.+n, 03.65.Ge (for the physical space-time) can be a sensible scenario only at primordial epochs, since the standard FRW world model is known to be in a sufficiently good agreement with the observational constraints down to a quite primordial epoch such as the nucleosynthesis era. Hence, it is clear that a reduction process (called dynamical compactification of additional dimensions) is required before such an epoch, to make the internal spaces contracting themself down to unobservable sizes. Herein, we assume that the cosmic fluid (the source of the gravitational field at early stages) is viscous, which might simulate high energy physics processes (such as the particles creation). The effects related to viscosity in 4-dimensional Universe were studied through different viewpoints (see e.g., [2][3][4]8,10,26,30,[33][34][35][36][37][38]44]).Before developing the multidimensional model let us briefly discuss (extensive review of the subject was given by Gron [19]) the main trends in 4-dimensional cosmology with viscous fluid as a source.First, Misner [33] considered neutrino viscosity as a mechanism of reducing the anisotropy in the Early Universe. Stewart [40] and Collins and Stewart [10] proved that it is possible only if initial anisotropies are small enough. Another series of papers which concerns the production of entropy in the viscous Universe was started by Weinberg [44]. Both isotropization and production of entropy during lepton era in models of Bianchi types I,V were considered by Klimek [26]. Caderni and Fabbri [8] calculated coefficients of shear and bulk viscosity in plasma and lepton eras within the model of Bianchi type I. The next approach is connected with obtaining singularity free viscous solutions. The first nonsingular solution was obtained by Murphy [35] within the flat FRW model with fluid possessing a bulk viscosity. However, Belinsky and Khalatnikov [2,3] showed that this solution corresponds to the very peculiar choice of parameters and is unstable with respect to the anisotropy perturbations. Other nonsingular solutions with bulk viscosity were obtained by Novello and Araujo [36], Romero [38], Oliveira and Salim [37].The crucial feature of each viscous cosmological model is assuming of the so called...

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