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An attempt is made to clarify the quantum theory of the ``slow-rollover'' phase transition which characterizes the new inflationary universe model. We discuss the theory of the upside-down harmonic oscillator as a toy model, with particular emphasis on the fact that the system can be described at late times by a classical probability distribution. An approximate but exactly soluble model for the scalar field is then constructed, based on three principal assumptions: (1) exact de Sit- ter expansion for all time; (2) a quadratic potential function which changes from stable to unstable as a function of time; and (3) an initial state which is thermal in the asymptotic past. It is proposed that this model would be the proper starting point for a perturbative calculation in more realistic models. The scalar field can also be described at late times by a classical probability distribution, and numerical calculations are carried out to illustrate how this distribution depends on the parameters of the model. For a suitable choice of these parameters, a sufficient period of inflation can be easily obtained. Density fluctuations can be calculated exactly in this model, and the results agree very well with those previously obtained using approximate methods.
Guth et al. (Tue,) studied this question.