Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Propagation and constraint equations (Q2738226)

The authors introduce a set of 26 gauge-invariant variables, denoted collectively by \textbf D \textrm and referred to as the complete set of basic variables, used to describe the equivalence classes of perturbations in a Bianchi type I or type V universe filled with a nonbarotropic perfect fluid. The object of this paper is the derivation of a full system of propagation and constraint equations for these basic variables. It is shown that the constraint equations, which involve only the spatial derivatives of \textbf D\textrm , are preserved in time along the unperturbed fluid flow lines. The time derivative of each constraint equation is identically satisfied as a consequence of the other equations that hold. The main statement is that if the dynamical equations in the considered system are satisfied at all times, then the constraints are satisfied at all times. When the linearized field equations of Einstein's gravity theory are re-expressed in a manifestly gauge-invariant form, an open set of equations is obtained for \textbf D \textrm since there are too many unknowns. Thus, this set must be suitably closed under accurate ``closure'' relations. It is observed that the definition of basic gauge-invariant variables gives rise to additional geometrical identities from which an exact method of closure can be determined. It turns out that the formalism introduced here is especially appropriate for handling the linearized perturbations in a Bianchi type V universe model where the standard approaches conceptually break down.