We discuss the $O\left(4\right)\mathrm{}$ invariant perturbation modes of cosmological instantons. These modes are spatially homogeneous in Lorentzian spacetime and thus not relevant to density perturbations. But their properties are important in establishing the meaning of the Euclidean path integral. If negative modes are present, the Euclidean path integral is not well defined, but may nevertheless be useful in an approximate description of the decay of an unstable state. When gravitational dynamics is included, counting negative modes requires a careful treatment of the conformal factor problem. We demonstrate that for an appropriate choice of coordinate on phase space, the second order Euclidean action is bounded below for normalized perturbations and has a finite number of negative modes. We prove that there is a negative mode for many gravitational instantons of the Hawking-Moss or Coleman–De Luccia type, and discuss the associated spectral flow. We also investigate Hawking-Turok constrained instantons, which occur in a generic inflationary model. Implementing the regularization and constraint proposed by Kirklin, Turok and Wiseman, we find that those instantons leading to substantial inflation do not possess negative modes. Using an alternate regularization and constraint motivated by reduction from five dimensions, we find a negative mode is present. These investigations shed new light on the suitability of Euclidean quantum gravity as a potential description of our universe.

• Received 21 February 2001

DOI:https://doi.org/10.1103/PhysRevD.63.123514