  Motivated by recent studies of the one-bubble inflationary universe scenario
that predicts a low density, negative curvature universe, we investigate the
Euclidean vacuum mode functions of a scalar field in a spatially open chart of
de Sitter space which is foliated by hyperbolic time slices. When we consider
the possibility of an open inflationary universe, we are faced with the problem
of the initial condition for the quantum fluctuations of the inflaton field,
because the inflationary era should not last too long to lose every information
of the initial condition. In the one-bubble scenario in which an open universe
is created in an exponentially expanding false vacuum universe triggered by
quantum decay of false vacuum, it seems natural that the initial state is the
de Sitter-invariant Euclidean vacuum. Here we present explicit expressions for
the Euclidean vacuum mode functions in the open chart for a scalar field with
arbitrary mass and curvature coupling.
