Small-throat boundary condition for the tunneling wave function of a universe

We propose a small-throat prescription for the wave function of a closed universe in the Lorentzian path integral formalism, motivated by the idea that universe creation may be obtained as the decoupling, or pinch-off, limit of a tunneling geometry connected to another universe through a small throat. Instead of retaining the parent-universe side explicitly, we describe the remaining half-geometry by a minisuperspace path integral with boundary conditions imposed at the throat. To model the finite throat, we introduce a small radiation component parametrized by $ε$ in a closed minisuperspace model with a positive cosmological constant. The radiation term produces two turning points, an inner one q₋~ O(ε) and an outer one q₊~ H⁻², where q is the square of the scale factor. Our prescription imposes the Neumann condition q(0)=0 at the initial endpoint and restricts the initial size qᵢ=q(0) to a small-throat domain 0<|qᵢ|<√ε/H that contains q₋. This restriction selects the Riemann sheet containing the small-throat tunneling saddle and its Picard--Lefschetz cycle, while excluding the unsuppressed saddle associated with the outer turning point q₊. Taking the limit ε→0 after this finite-throat saddle problem has been defined, the small-throat domain collapses to qᵢ → 0, and the saddle action reduces to that of the standard tunneling saddle. Other choices of lapse contour can instead select Hartle--Hawking-type growing branches. In this sense, the tunneling wave function can be obtained as the limiting form of tunneling from an arbitrarily small universe in a Lorentzian path integral, rather than by imposing a boundary condition directly at a vanishing geometry.