Integration contours for the no-boundary wave function of the universe

In the no-boundary proposal for the initial conditions of a closed cosmology, the wave function of the universe is the integral of exp (-action) over a contour of four-geometries and matter-field configurations on compact manifolds having only that boundary necessary to specify the arguments of the wave function. There is no satisfactory covariant Hamiltonian quantum mechanics of closed cosmologies from which the contour may be derived, as there would be for defining the ground states of asymptotically flat spacetimes. No compelling prescription, such as the conformal rotation for asymptotically flat spacetimes, has been advanced. In this paper it is argued that the contour of integration can be constrained by simple physical considerations: (1) the integral defining the wave function should converge; (2) the wave function should satisfy the constraints implementing diffeomorphism invariance; (3) classical spacetime when the universe is large should be a prediction; (4) the correct field theory in curved spacetime should be reproduced in this spacetime; (5) to the extent that wormholes make the cosmological constant dependent on initial conditions the wave function should predict its vanishing. We argue that the convergence criterion is readily satisfied by choosing a suitable complex contour. The constraints will be satisfied if the end points of the contour are suitably restricted. For classical spacetime to be a prediction, the contour must be dominated by one or more saddle points at which the four-metric is complex. We discuss the conditions under which such complex solutions to the Einstein equations arise and their interpretation. Because the action is double valued in the space of complex metrics, every solution of the Einstein equations corresponds to two saddle points: one with Re () >0, the other with Re () 0. They differ only in the sign of their action. We find that criteria (4) and (5) imply that the contour should not be dominated by a saddle point with Re () 0. This restriction may be difficult to satisfy in the path-integral forms of the ``tunneling'' boundary condition proposals of Linde, Vilenkin, and others. Although all of these physical considerations constrain the contour and largely determine the semiclassical predictions of the wave function, there is still remaining freedom. Until fixed by more fundamental considerations, the remaining freedom in the contour means that there are many corresponding no-boundary proposals.