Acceleration of the universe, vacuum metamorphosis, and the large-time asymptotic form of the heat kernel

We investigate the possibility that the late acceleration observed in the rate of expansion of the Universe is due to vacuum quantum effects arising in curved spacetime. The theoretical basis of the vacuum cold dark matter (VCDM), or vacuum metamorphosis, cosmological model of Parker and Raval is reexamined and improved. We show, by means of a manifestly nonperturbative approach, how the infrared behavior of the propagator (related to the large-time asymptotic form of the heat kernel) of a free scalar field in curved spacetime leads to nonperturbative terms in the effective action similar to those appearing in the earlier version of the VCDM model. The asymptotic form that we adopt for the propagator or heat kernel at large proper time s is motivated by, and consistent with, particular cases where the heat kernel has been calculated exactly, namely in de Sitter spacetime, in the Einstein static universe, and in the linearly expanding spatially flat Friedmann-Robertson-Walker (FRW) universe. This large-s asymptotic form generalizes somewhat the one suggested by the Gaussian approximation and the R-summed form of the propagator that earlier served as a theoretical basis for the VCDM model. The vacuum expectation value for the energy-momentum tensor of the free scalar field, obtained through variation of the effective action, exhibits a resonance effect when the scalar curvature R of the spacetime reaches a particular value related to the mass of the field. Modeling our Universe by an FRW spacetime filled with classical matter and radiation, we show that the back reaction caused by this resonance drives the Universe through a transition to an accelerating expansion phase, very much in the same way as originally proposed by Parker and Raval. Our analysis includes higher derivatives that were neglected in the earlier analysis, and takes into account the possible runaway solutions that can follow from these higher-derivative terms. We find that the runaway solutions do not occur if the universe was described by the usual classical FRW solution prior to the growth of vacuum energy density and negative pressure (i.e., vacuum metamorphosis) that causes the transition to an accelerating expansion of the universe in this theory.