xPand: An algorithm for perturbing homogeneous cosmologies
  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.
