Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Formulation

Abstract

In this paper we deal for the first time with gauge-invariant perturbations of anisotropic cosmological models of Bianchi types I and V from a unified point of view. Motivated by Ehlers’ pioneering concepts, the key idea is to identify the gauge-invariant perturbations with the equivalence classes of tangents to one-parameter families of exact solutions to Einstein’s field equations. For cases where these models are filled with a nonbarotropic perfect fluid, we show that a set of 26 “geometrically” independent, not identically vanishing gauge-invariant variables, denoted collectively by D and referred to as the complete set of basic variables, can be used to extract the equivalence classes of tangents from D in a unique way. The set D is complete because it has the following property: any gauge-invariant quantity is obtainable linearly from the basic variables through purely algebraic and differential operations. Mathematically, this approach to the gauge problem is a nontrivial example of the general scheme that we have described in our two previous papers [Int. J. Theor. Phys. 36, 1787, 1817 (1997)], and the new concepts developed were also applied to the construction of a complete set of basic gauge-invariant variables for the cases of a fixed background de Sitter space–time and an almost-Robertson–Walker universe model. Arguments are given that there are a number of advantages to be gained by replacing the coordinate-based method of Bardeen or the covariant formalism of Ellis and Bruni by the present one.