Quantum mechanics of exponential-potential inflation.

Abstract

A spatially flat cosmological scalar field $\ensuremath{\varphi}$ model with an exponential scalar field potential $V(\ensuremath{\varphi})\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\frac{\ensuremath{\varphi}}{\sqrt{p}})$, with $pg1$, has a time-dependent fixed-point solution which describes an inflating universe with the scale factor evolving as $a(t)\ensuremath{\propto}{t}^{p}$. In the limit $p\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.