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Motivated by recent studies of the one-bubble inflationary universe scenariothat predicts a low density, negative curvature universe, we investigate theEuclidean vacuum mode functions of a scalar field in a spatially open chart ofde Sitter space which is foliated by hyperbolic time slices. When we considerthe possibility of an open inflationary universe, we are faced with the problemof the initial condition for the quantum fluctuations of the inflaton field,because the inflationary era should not last too long to lose every informationof the initial condition. In the one-bubble scenario in which an open universeis created in an exponentially expanding false vacuum universe triggered byquantum decay of false vacuum, it seems natural that the initial state is thede Sitter-invariant Euclidean vacuum. Here we present explicit expressions forthe Euclidean vacuum mode functions in the open chart for a scalar field witharbitrary mass and curvature coupling.Comment: 18 pages, revtex file, 3 uuencoded compressed postscript figures  appended at the en

Euclidean vacuum mode functions for a scalar field on open de Sitter space