Homogeneous Modes of Cosmological Instantons

TL;DR

The paper examines spatially homogeneous ($O(4)$) perturbations of cosmological instantons within Euclidean quantum gravity, addressing the conformal factor problem and counting negative modes via a Hamiltonian phase-space approach. It shows that Hawking-Moss and Coleman-De Luccia instantons typically possess a single negative mode, while Hawking-Turok singular instantons behave differently under regularizations: Kirklin–Turok–Wiseman RP$^4$ eliminates negative modes for substantial inflation, whereas Garriga’s five-dimensional construction preserves them, highlighting the sensitivity to boundary conditions. The results stress that an unconstrained Euclidean path integral is generally ill-defined and may only be meaningful as an approximation for tunneling or with appropriate projections to constrain the configuration space. The work clarifies when Euclidean quantum gravity can plausibly describe inflationary initial conditions or tunneling processes, and points to the need for well-defined projection schemes in nonperturbative settings. It also illuminates the spectral flow among instantons and the role of regularization in determining the viability of Euclidean descriptions of the universe.

Abstract

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.

Figures (2)

• Figure 1: The left sketch shows how $m$ at the singularity varies with the value of $\phi$ at the regular pole. The right sketch shows how the appropriate action of the singular instanton varies with the value of $\phi$ at the regular pole.
• Figure 2: A sketch of the action-confguration space structure of gravity with scalar field. The dashed lines represent the instanton solutions for a given $m$, and the third direction represents the infinity of field configurations respecting this $m$ constraint. The curves indicate if the instantons possess negative modes or not.