Abstract
We perform a detailed study of the existence and the properties of $O(4)$-invariant instanton solutions in Einstein-scalar theory in the presence of flat potential barriers, i.e. barriers where the second derivative of the potential is small at the top of the barrier. We find a whole zoo of solutions: Hawking-Moss, Coleman--de Luccia (CdL), oscillating instantons, and asymmetric CdL as well as other nonstandard CdL-like solutions with additional negative modes in their spectrum of fluctuations. Our work shows how these different branches of solutions are connected to each other via ``critical'' instantons possessing an extra zero-mode fluctuation. Overall, the space of finite-action Euclidean solutions to these theories with flat barriers is surprisingly rich and intricate. We find that critical instantons provide the key to understanding both the existence and the properties of instanton solutions.
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