Rotational symmetry of Coleman–De Luccia instantons in de Sitter vacuum decay

Prove that the Coleman–De Luccia instanton describing de Sitter-to-de Sitter vacuum decay possesses the conjectured O(d) symmetry in general dimensions.

Background

The paper adopts an O(d)-symmetric ansatz for the Coleman–De Luccia instanton, reducing the Euclidean geometry and inflaton profile to functions of a single radial coordinate. This symmetry underlies the subsequent thin-wall construction of two de Sitter regions with different radii joined across a brane.

Although holographic methods have confirmed the analogous symmetry in the anti-de Sitter case, the paper states that the corresponding result remains unresolved for general vacuum decay, including de Sitter-to-de Sitter decay. No proof or counterexample is currently available, so the analysis proceeds under the symmetry assumption.

References

The original analysis by Coleman and De Luccia was based on the assumption that the solution minimizing the action would be $O(d)$-symmetric. This conjectured symmetry has been confirmed in the anti de Sitter case in using holographic methods, but remains open in general.

— Testing holographic computation of entanglement pseudo-entropy in dS\textsubscript{3}/ICFT\textsubscript{2}  (2608.26997 - Li, 27 Aug 2026) in Section 2.1, “Description for dS-dS decay”