Path-integral quantum cosmology: A class of exactly soluble scalar-field minisuperspace models with exponential potentials

We study a class of minisuperspace models consisting of a homogeneous isotropic universe with a minimally coupled homogeneous scalar field with a potential $\alpha cosh\left(2\mathrm{}\phi \right)+\beta sinh\left(2\mathrm{}\phi \right)$, where $\alpha$ and $\beta$ are arbitrary parameters. This includes the case of a pure exponential potential $\exp \left(2\mathrm{}\phi \right)$, which arises in the dimensional reduction to four dimensions of five-dimensional Kaluza-Klein theory. We study the classical Lorentzian solutions for the model and find that they exhibit exponential or power-law inflation. We show that the Wheeler-DeWitt equation for this model is exactly soluble. Concentrating on the two particular cases of potentials $cosh\left(2\mathrm{}\phi \right)$ and $\exp \left(2\mathrm{}\phi \right)$, we consider the Euclidean minisuperspace path integral for a propagation amplitude between fixed scale factors and scalar-field configurations. In the gauge $\dot{N}=0\mathrm{}$ (where $N$ is the rescaled lapse function), the path integral reduces, after some essentially trivial functional integrations, to a single nontrivial ordinary integral over $N$. Because the Euclidean action is unbounded from below, $N$ must be integrated along a complex contour for convergence. We find all possible complex contours which lead to solutions of the Wheeler-DeWitt equation or Green's functions of the Wheeler-DeWitt operator, and we give an approximate evaluation of the integral along these contours, using the method of steepest descents. The steepest-descent contours may be dominated by saddle points corresponding to exact solutions to the full Einstein-scalar equations which may be real Euclidean, real Lorentzian, or complex. We elucidate the conditions under which each of these different types of solution arise. For the $\exp \left(2\mathrm{}\phi \right)$ potential, we evaluate the path integral exactly. Although we cannot evaluate the path integral in closed form for the $cosh\left(2\mathrm{}\phi \right)$ potential, we show that for particular $N$ contours the amplitude may be written as a given superposition of exact solutions to the Wheeler-DeWitt equation. By choosing certain initial data for the path-integral amplitude we obtain the amplitude specified by the "no-boundary" proposal of Hartle and Hawking. We discuss the nature of the geometries corresponding to the saddle points of the no-boundary amplitude. We identify the set of classical solutions this proposal picks out in the classical limit.