Quantum tunneling in the curved spacetime

False vacuum decay is theorized to have occurred frequently throughout the history of the universe, particularly during first-order phase transitions associated with spontaneous symmetry breaking. The decay rate of such a vacuum is governed by Euclidean bounce solutions, which can exhibit a wide range of configurations, even under fixed boundary conditions. In the absence of gravitational effects, it was established over four decades ago—under reasonable assumptions—that the most symmetric bounce solution, namely the O(4)-symmetric one, minimizes the Euclidean action. This renders it the dominant tunneling path in flat spacetime. However, when gravitational effects are taken into account—as is essential in cosmological settings—all prior studies have assumed, without rigorous proof, that the O(4)-symmetric bounce continues to minimize the action. This has remained a longstanding unresolved problem for more than forty years. In this work, we address this issue by employing the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to determine the configuration with the lowest Euclidean action in a metastable AdS false vacuum. Within the Euclidean formalism of Callan and Coleman, we identify the most probable decay channel of the AdS vacuum. The AdS/CFT duality enables us to sidestep the technical challenges intrinsic to metastable gravitational systems. We demonstrate that the Fubini bounce in conformal field theory—which is dual to the Coleman–de Luccia (CdL) bounce in AdS—indeed minimizes the Euclidean action among all finite bounce solutions in a conformal scalar field theory. Consequently, under certain conditions, we establish that the CdL bounce yields the lowest action among all relevant configurations, including both large and thin-wall limits. Time permitting, we also discuss the prefactor of the decay rate, as obtained from one-loop quantum corrections.