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

Abstract

This is the third in a series of papers [J. Math. Phys. 40, 3978 (1999); 40, 3995 (1999)], the overall objective of which is the demonstration that a set of 26 gauge-invariant variables, denoted collectively by D and referred to as the complete set of basic variables, can be 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 here is the derivation of a full system of propagation and constraint equations for these basic variables. We show that the constraint equations, which involve only the spatial derivatives of D, are preserved in time along the unperturbed fluid flow lines, i.e., that the time derivative of each constraint equation is identically satisfied as a consequence of the other equations that hold. Let us put things another way. What we prove is the statement that if the constraints in our system are satisfied at one time and the evolution equations are satisfied at all times, then the constraints are satisfied at all times. A further important point is simply this. 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 D since there are too many unknowns. Thus this set must be suitably closed by means of accurate “closure” relations. In order to find them, we observe that the definition of basic gauge-invariant variables gives rise to additional geometrical identities from which an exact method of closure can be determined. Our formalism turns out to be especially appropriate for handling the linearized perturbations in a Bianchi type V universe model where the standard approaches conceptually break down.