  We discuss the O(4) 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.
