Cosmological Perturbations from the No Boundary Euclidean Path Integral

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.