xPand：扰动均匀宇宙学的算法
xPand: An algorithm for perturbing homogeneous cosmologies
摘要：
在本文中，我们详细地发展了一种完全几何方法来推导空间均匀背景的扰动方程。这种方法依赖于背景时空的3+1分裂，并且不使用任何特定的坐标集：它仅根据几何量来实现，使用xAct分布中的张量代数包xTensor以及微扰xPert的扩展。我们的算法允许一个获得的扰动方程的所有类型的均匀宇宙学，任何顺序和在所有可能的规范。作为应用，我们恢复了著名的扰动爱因斯坦方程的Friedmann-Lemaitre-Robertson-Walker宇宙学在二阶和Bianchi I宇宙学在一阶。这项工作铺平了道路，这些模型的研究在更高的订单和任何其他扰动比安基宇宙学，通过规避通常过于繁琐的推导扰动方程。
英文摘要：
In this paper, we develop in detail a fully geometrical method for deriving perturbation equations about a spatially homogeneous background. This method relies on the 3+1 splitting of the background space-time and does not use any particular set of coordinates: it is implemented in terms of geometrical quantities only, using the tensor algebra package xTensor in the xAct distribution along with the extension for perturbations xPert. Our algorithm allows one to obtain the perturbation equations for all types of homogeneous cosmologies, up to any order and in all possible gauges. As applications, we recover the well-known perturbed Einstein equations for Friedmann-Lemaitre-Robertson-Walker cosmologies up to second order and for Bianchi I cosmologies at first order. This work paves the way to the study of these models at higher order and to that of any other perturbed Bianchi cosmologies, by circumventing the usually too cumbersome derivation of the perturbed equations.
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