  We compute, from first principles, the quantum fluctuations about instanton
saddle points of the Euclidean path integral for Einstein gravity coupled to a
scalar field. The Euclidean two-point correlator is analytically continued into
the Lorentzian region where it describes the quantum mechanical vacuum
fluctuations in the state described by no boundary proposal initial conditions.
We concentrate on the density perturbations in open inflationary universes
produced from cosmological instantons, describing the differences between
non-singular Coleman-De Luccia and singular Hawking-Turok instantons. We show
how the Euclidean path integral uniquely specifies the fluctuations in both
cases.
