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Garay Elizondo, Luis Javier and Halliwel, Jonathan J. and Mena Marugán, Guillermo A.
(1991)
*Path-integral quantum cosmology: a class of exactly soluble scalar-field minisuperspace models with exponential potentials.*
Physical review D, 43
(8).
pp. 2572-2589.
ISSN 0556-2821

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Official URL: http://dx.doi.org/10.1103/PhysRevD.43.2572

## Abstract

We study a class of minisuperspace models consisting of a homogenous isotropic universe with a minimally coupled homogenous scalar field with a potential alpha cosh(2-phi) + beta sinh(2-phi), where alpha and beta are arbitrary parameters. This includes the case of a pure exponential potential exp(2-phi), 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-phi) and exp(2-phi), 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-phi) potential, we evaluate the path integral exactly. Although we cannot evaluate the path integral in closed form for the cosh(2-phi) 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.

Item Type: | Article |
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Additional Information: | © 1991 The American Physical Society. |

Uncontrolled Keywords: | Kaluza-klein theory; Steepest-descent contours; Wave-function; Inflationary universe; Anisotropic universe; Gravitational-field; Boundary-condition; Density-matrix; Initial state; Spacetime |

Subjects: | Sciences > Physics > Physics-Mathematical models Sciences > Physics > Mathematical physics |

ID Code: | 30117 |

Deposited On: | 19 May 2015 08:27 |

Last Modified: | 10 Dec 2018 15:10 |

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