广义相对论中的大尺度潮汐
Large-Scale Tides in General Relativity
摘要：
宇宙学中的密度微扰，即弗里德曼-勒梅特-罗伯逊-沃克（FLRW）时空的球对称绝热微扰，只要它们的波长远大于所有流体组分的声视界，就局部精确等价于不同的FLRW解。这一事实被称为“分离宇宙”范式。然而，没有这样的关系是已知的各向异性绝热扰动，这对应于一个FLRW时空与大尺度潮汐场。在这里，我们提供了一个封闭的，完全相对论的演化方程组的非线性演化的模式，共形费米（CFC）框架的基础上。我们明确地表明，潮汐效应是由外尔张量编码，因此是完全不同的各向异性Bianchi I时空，其中各向异性是由里奇张量的来源。为了关闭系统，必须删除某些高阶导数项。我们证明了这种近似与Hui和Bertschinger（1995）的局地潮近似是等价的。我们还表明，这个非常简单的方程组匹配的密度场的精确演化在二阶，但在第三和更高的顺序失败。这提供了一个有用的，易于使用的框架计算完全相对论性增长的结构在第二顺序。
英文摘要：
Density perturbations in cosmology, i.e. spherically symmetric adiabatic perturbations of a Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, are locally exactly equivalent to a different FLRW solution, as long as their wavelength is much larger than the sound horizon of all fluid components. This fact is known as the "separate universe" paradigm. However, no such relation is known for anisotropic adiabatic perturbations, which correspond to an FLRW spacetime with large-scale tidal fields. Here, we provide a closed, fully relativistic set of evolutionary equations for the nonlinear evolution of such modes, based on the conformal Fermi (CFC) frame. We show explicitly that the tidal effects are encoded by the Weyl tensor, and are hence entirely different from an anisotropic Bianchi I spacetime, where the anisotropy is sourced by the Ricci tensor. In order to close the system, certain higher derivative terms have to be dropped. We show that this approximation is equivalent to the local tidal approximation of Hui and Bertschinger (1995). We also show that this very simple set of equations matches the exact evolution of the density field at second order, but fails at third and higher order. This provides a useful, easy-to-use framework for computing the fully relativistic growth of structure at second order.
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