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Thin-wall vacuum decay in the presence of a compact dimension

Ignatios Antoniadis, Daniele Bielli, Auttakit Chatrabhuti, Hiroshi Isono

TL;DR

This work analyzes false vacuum decay in the presence of gravity with one compact dimension, extending the Coleman–de Luccia thin-wall framework to an $O(d-1)\times U(1)$ symmetric bounce that lives on a bubble with geometry $S^{d-2}\times S^1$. Using the Israel junction conditions, the authors derive a unique, finite-action Euclidean solution and compute decay rates for transitions between de Sitter, Minkowski, and anti-de Sitter vacua, with explicit results for a five-dimensional setup. A central result is the gravity-corrected bounce action $B=45\pi^3 R_0 S_1^4/\epsilon^3\,\mathcal{B}(\hat{\kappa}_5)$, where $R_0$ is the true-vacuum circle radius and $\hat{\kappa}_d=\kappa_d S_1^2/|\epsilon|$ encodes gravitational effects; the flat-space limit is recovered as gravity decouples ($\hat{\kappa}_d\to0$). The analysis clarifies how a compact dimension modifies the instanton geometry (two radii $\rho$ and $\rho\sigma$) and demonstrates the robustness of the Israel approach in more intricate settings, with implications for higher-dimensional cosmologies and string-inspired vacua. The study also outlines regimes of validity ($\hat{\kappa}_d<1$) and directions for future work beyond thin-wall, including stability analyses and potential Hawking–Moss dominance.

Abstract

We study the problem of false vacuum decay in arbitrary dimensions, in the presence of gravity, and compute the transition probability within the thin-wall approximation, generalising the results of Coleman and de Luccia. In the particular case of one compact dimension, we present explicit formulae for the Euclidean Bounce configuration that drives the transition from a de Sitter to Minkowski or from a Minkowski to anti-de Sitter vacua.

Thin-wall vacuum decay in the presence of a compact dimension

TL;DR

This work analyzes false vacuum decay in the presence of gravity with one compact dimension, extending the Coleman–de Luccia thin-wall framework to an symmetric bounce that lives on a bubble with geometry . Using the Israel junction conditions, the authors derive a unique, finite-action Euclidean solution and compute decay rates for transitions between de Sitter, Minkowski, and anti-de Sitter vacua, with explicit results for a five-dimensional setup. A central result is the gravity-corrected bounce action , where is the true-vacuum circle radius and encodes gravitational effects; the flat-space limit is recovered as gravity decouples (). The analysis clarifies how a compact dimension modifies the instanton geometry (two radii and ) and demonstrates the robustness of the Israel approach in more intricate settings, with implications for higher-dimensional cosmologies and string-inspired vacua. The study also outlines regimes of validity () and directions for future work beyond thin-wall, including stability analyses and potential Hawking–Moss dominance.

Abstract

We study the problem of false vacuum decay in arbitrary dimensions, in the presence of gravity, and compute the transition probability within the thin-wall approximation, generalising the results of Coleman and de Luccia. In the particular case of one compact dimension, we present explicit formulae for the Euclidean Bounce configuration that drives the transition from a de Sitter to Minkowski or from a Minkowski to anti-de Sitter vacua.
Paper Structure (12 sections, 109 equations, 2 figures)

This paper contains 12 sections, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Vacuum Decay in Theories with $O(d)$ Symmetry
  3. The approach of Coleman and de Luccia (CdL)
  4. The approach of Israel junction conditions
  5. Vacuum Decay in Theories with $O(d-1)\times U(1)$ Symmetry
  6. Instanton geometry
  7. Bounce and complications of the CdL approach
  8. Junction conditions with compact dimension
  9. Two explicit examples of vacuum transitions
  10. dS $\rightarrow$ Minkowski
  11. Minkowski $\rightarrow$ AdS
  12. Conclusions

Figures (2)

  • Figure 1: Behaviour of the dimensionless function $\mathcal{B}(\hat{\kappa}_{d})$ describing the gravitational corrections in the dS to Minkowski transition, normalised to the result in the absence of gravity, as a function of the dimensionless parameter $\hat{\kappa}_d$ defined in \ref{['dimensionless-kappa']}. The plot compares the results of the $d=4,5$ non-compact case obtained in section \ref{['sec:theories-with-O(d)-symmetry']} with those of $d=5$ with one compact dimension studied here.
  • Figure 2: Behaviour of the dimensionless function $\mathcal{B}(\hat{\kappa}_{d})$ describing the gravitational corrections in the Minkowski to AdS transition. The plot compares the results of the $d=4,5$ non-compact case obtained in section \ref{['sec:theories-with-O(d)-symmetry']} with those of $d=5$ with one compact dimension studied here.