Quantum mechanics of exponential-potential inflation

A spatially flat cosmological scalar field $\phi$ model with an exponential scalar field potential $V\left(\phi \right)\propto \exp (-\frac{\phi }{\sqrt{p}})$, with $p>\mathrm{}1\mathrm{}$, has a time-dependent fixed-point solution which describes an inflating universe with the scale factor evolving as $a\left(t\right)\mathrm{}\propto t^p$. In the limit $p\to \infty$ this model reduces to the usual exponential expansion inflation model. In this paper we derive the inflationary epoch spectra of energy density and gravitational-wave irregularities by perturbatively solving the quantum-mechanical Dirac-Wheeler-DeWitt equations for this model. Our results agree with those found using an essentially classical analysis. We also show, by using two different orderings of factors in the quantum-mechanical Hamiltonian, that operator-ordering ambiguities do not affect the spectrum of irregularities to the order to which we work; they do, however, affect the normalization of the wave function.