Cosmological perturbation theory, instantaneous gauges, and local inertial frames

Linear perturbations of Friedmann-Robertson-Walker universes with any curvature and cosmological constant are studied in a general gauge without decomposition into harmonics. Desirable gauges are selected as those which embody best Mach’s principle: in these gauges local inertial frames can be determined instantaneously via the perturbed Einstein field equations from the distributions of energy and momentum in the universe. The inertial frames are identified by their “accelerations and rotations” with respect to the cosmological frames associated with the “Machian gauges.” In closed spherical universes, integral gauge conditions are imposed to eliminate motions generated by the conformal Killing vectors. The meaning of Traschen’s integral-constraint vectors is thus elucidated. For all three types of Friedmann-Robertson-Walker universes the Machian gauges admit much less residual freedom than the synchronous or generalized harmonic gauge. Mach’s principle is best exhibited in the Machian gauges in closed spherical universes. Independent of any Machian motivation, the general perturbation equations and discussion of gauges are useful for cosmological perturbation theory.