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

AuthorsGaray, Luis Javier ; Halliwell, J. J.; Mena Marugán, Guillermo A.
Issue Date1991
PublisherAmerican Physical Society
CitationPhysical Review D - Particles, Fields, Gravitation and Cosmology 43: 2572- 2589 (1991)
AbstractWe study a class of minisuperspace models consisting of a homogeneous isotropic universe with a minimally coupled homogeneous scalar field with a potential cosh(2+2sinh(2), where and 2 are arbitrary parameters. This includes the case of a pure exponential potential exp(2), 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(2) and exp(2), we consider the Euclidean minisuperspace path integral for a propagation amplitude between fixed scale factors and scalar-field configurations. In the gauge N=0 (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(2) potential, we evaluate the path integral exactly. Although we cannot evaluate the path integral in closed form for the cosh(2) 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. © 1991 The American Physical Society.
Identifiersdoi: 10.1103/PhysRevD.43.2572
issn: 0556-2821
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