Consider a scalar field \(\phi\) in \(d\) dimensions, with Euclidean action

\[S = \int \mathup{d}^d x \left[ \frac{1}{2}\nabla_\mu \phi \nabla_\mu \phi + V(\phi) \right] .\]

If the potential \(V\) has a metastable vacuum,

Potential with metastable and stable minima

then the decay rate, \(\Gamma\), is given by[1]

\[\Gamma = \underbrace{\left( \frac{S[\phi_\text{b}]}{2\pi} \right)^{d/2} \left(\frac{\vert\det ' (- \Box + V''(\phi_\text{b}))\vert}{\det (- \Box + V''(\phi_\text{F}))} \right)^{-1/2}}_\text{prefactor} e^{-\left(S[\phi_\text{b}] - S[\phi_\text{F}]\right)} .\]

where \(\phi_\text{b}\) is the inhomogeneous bubble, or bounce, profile, and \(\phi_\text{F}\) is the homogeneous metastable phase, or false vacuum. BubbleDet can compute the prefactor term highlighted above. BubbleDet is applicable for arbitrary potentials, and in any dimension up to seven. It can also compute multi-particle functional determinants, which arise for example in symmetry-breaking transitions. There are several packages which focus on the computation of the exponential factor in the nucleation rate, such as CosmoTransitions[2].

For further details of the physics and mathematics behind BubbleDet, see the acommpanying paper, BubbleDet: A Python package to compute functional determinants for bubble nucleation[3].