CANONICAL QUANTIZATION OF COSMOLOGICAL PERTURBATIONS IN THE ONE-BUBBLE OPEN UNIVERSE

Faddeev and Jackiw's method for constrained systems is used to derive a gauge invariant formulation of cosmological perturbations in the one -bubble inflationary universe, For scalar perturbations in a flat univ erse, reduction of the action to the one with a single physical degree of freedom has been derived in the literature. A straightforward gene ralization of it to the case of an open universe is possible but it is not adequate for quantizing perturbations in the one-bubble universe, because of the lack of Cauchy surfaces inside the bubble. Therefore w e perform the reduction of the action outside the light cone emanating from the center of the bubble or nucleation event, where the natural time constant hypersurfaces are no longer homogeneous and isotropic an d as a result the conventional classification of perturbations in term s of scalar and tensor modes is not possible, Nevertheless, after redu ction of the action we find three decoupled actions for three independ ent degrees of freedom, one of which corresponds to the scalar mode an d the other two to the tensor modes, Implications for the one-bubble o pen inflationary models are briefly discussed. As an application of ou r formalism, the spectrum of long-wavelength gravity waves is simply o btained in terms of the real part of the reflection amplitude for a on e-dimensional scattering problem, where the potential barrier is given in terms of the bubble profile. (C) 1998 Elsevier Science B.V.