Remarks on cosmological bulk viscosity in different epochs

TL;DR

This paper explores a numerical similarity between early and present-day bulk viscosities in cosmology, proposes a unifying functional form, and examines the compatibility of bulk viscosity with anisotropic Kasner universes.

Contribution

It introduces a simple analytical formula for bulk viscosity that links early and late universe cosmology and analyzes its compatibility with anisotropic models like Kasner universes.

Findings

Bulk viscosity in the early universe matches present-day calculations with a $ ho^{1/2}$ dependence.

Shear viscosity is incompatible with the Kasner model, but bulk viscosity is compatible with isotropic models.

In a stiff fluid, Kasner exponents are determined solely by bulk viscosity.

Abstract

The intention of this paper is mainly two-fold. \textit{First}, we point out a striking numerical agreement between the bulk viscosity in the lepton era calculated by Husdal (2016) and our own calculations of the present-day bulk viscosity when the functional form is $\zeta\,\sim\sqrt{\rho}$. From a phenomenological point of view, we thus seem to have an ansatz for the viscosity which bridges the infancy of the Universe ($\sim 1$ s) with the present. This can also be looked upon as a kind of symmetry between the early-time cosmology and the present-day cosmology: it is quite remarkable that the kinetic theory-based bulk viscosity in the early universe and the experimentally-based bulk viscosity in the present universe can be covered by the same simple analytical formula. \textit{Second}, we consider the Kasner universe as a typical anisotropic model of Bianchi-type I, investigating whether this geometrical model is compatible with constant viscosity coefficients in the fluid. Perhaps surprisingly, the existence of a shear viscosity turns out to be incompatible with the Kasner model. By contrast, a bulk viscosity is non-problematic in the {\it isotropic} version of the model. In the special case of a Zel'dovich (stiff) fluid, the three equal exponents in the Kasner metric are even determined by the bulk viscosity alone, independent of the value of the fluid energy density. We also give a brief comparison with some other recent approaches to viscous cosmology.