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.

中文翻译：

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不同时期宇宙体积粘度的评述

本文的意图主要有两个方面。\textit{First}，我们指出 Husdal (2016) 计算的轻子时代体积粘度与我们自己计算的当前体积粘度之间存在惊人的数值一致性，当函数形式为 $\zeta\,\sim \sqrt{\rho}$。因此，从现象学的角度来看，我们似乎对将宇宙的初期（$\sim 1$ s）与现在联系起来的粘度有一个反驳。这也可以看作是早期宇宙学和现代宇宙学之间的一种对称性：早期宇宙基于动力学理论的体积粘度和现在基于实验的体积粘度是相当了不起的。宇宙可以用同样的简单分析公式覆盖。\textit{第二}，我们将 Kasner 宇宙视为 Bianchi 型 I 的典型各向异性模型，研究该几何模型是否与流体中的恒定粘度系数兼容。也许令人惊讶的是，剪切粘度的存在与 Kasner 模型不相容。相比之下，体积粘度在模型的 {\it isotropic} 版本中没有问题。在 Zel'dovich（刚性）流体的特殊情况下，Kasner 度量中的三个相等指数甚至仅由体积粘度决定，与流体能量密度值无关。我们还与其他一些最近的粘性宇宙学方法进行了简要比较。剪切粘度的存在与 Kasner 模型不相容。相比之下，体积粘度在模型的 {\it isotropic} 版本中没有问题。在 Zel'dovich（刚性）流体的特殊情况下，Kasner 度量中的三个相等指数甚至仅由体积粘度决定，与流体能量密度值无关。我们还与其他一些最近的粘性宇宙学方法进行了简要比较。剪切粘度的存在与 Kasner 模型不相容。相比之下，体积粘度在模型的 {\it isotropic} 版本中没有问题。在 Zel'dovich（刚性）流体的特殊情况下，Kasner 度量中的三个相等指数甚至仅由体积粘度决定，与流体能量密度值无关。我们还与其他一些最近的粘性宇宙学方法进行了简要比较。