GAUGE-INVARIANT COSMOLOGICAL PERTURBATION-THEORY FOR COLLISIONLESS MATTER - APPLICATION TO THE EINSTEIN-LIOUVILLE SYSTEM

Beginning from the Einstein-Liouville coupled system of equations for the description of a universe consisting of massive collisionless part icles (dark matter), this paper presents a totally gauge-invariant fra mework for studying the time development of perturbations in homogeneo us and isotropic cosmological models. Since the Einstein-Liouville sys tem involves infinitely many degrees of freedom (a function of the mom entum variable), emphasis is placed on Ending conditions under which t he dark mass behaves in an essentially hydrodynamic way. It is demonst rated that, for collisionless matter in the late universe, the complet e characterization of cosmological perturbations can be obtained if on e defines in a suitable way eighteen ''geometrically'' independent, no t identically vanishing gauge-invariant variables. These basic variabl es are important because they enable one to divide the infinitesimal p erturbations into physically natural equivalence classes: two infinite simal perturbations delta G(0) and delta G(0)' are said to be equivale nt if there is a transformation of the Lie type which carries delta G( 0) into delta G(0)' and vice versa. Another welcome feature of this fo rmulation is that any gauge-invariant quantity can be constructed dire ctly from the basic variables through purely algebraic and differentia l operations. Comparisons with other work on the Einstein-liouville sy stem are also made.