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arXiv:2605.16150

Gaussian fluctuations in the tunneling probability of a closed universe

L. Salasnich

gr-qcquant-phSubmitted May 15, 2026• Announced May 18, 2026
arXivHTMLPDF

Abstract

We consider the quantum creation of a closed universe within the Euclidean path-integral formalism. An analytical expression for the tunneling probability is derived, including both the exponential suppression and the exact Gaussian prefactor due to quadratic fluctuations around the instanton. The calculation is performed in a fixed-interval minisuperspace formulation, where the Hamiltonian constraint is imposed at the level of the classical instanton, while the full lapse integration is not included beyond the leading semiclassical approximation. The result provides a transparent and self-consistent semiclassical estimate of the nucleation rate, refining previous analyses with the inclusion of Gaussian fluctuations.

Journal reference
Photonic Quantum Waves 1, 2 (2026)
Comments
8 pages, 1 figure, invited paper for the first issue of Photonic and Quantum Waves

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We consider the quantum creation of a closed universe within the Euclidean path-integral formalism. An analytical expression for the tunneling probability is derived, including both the exponential suppression and the exact Gaussian prefactor due to quadratic fluctuations around the instanton. The calculation is performed in a fixed-interval minisuperspace formulation, where the Hamiltonian constraint is imposed at the level of the classical instanton, while the full lapse integration is not included beyond the leading semiclassical approximation. The result provides a transparent and self-consistent semiclassical estimate of the nucleation rate, refining previous analyses with the inclusion of Gaussian fluctuations.
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A quantum tunneling event in field theory is usually depicted as the materialization of a spherical bubble of true vacuum. The bubble appears at rest, and its profile is the time-symmetric slice of Coleman's bounce, the Euclidean solution that controls the decay rate. However, the bubble is spherical only in the leading semiclassical approximation: at the next order the configuration is a Gaussian ensemble of fluctuations around the bounce. We derive the wave functional of the field on the nucleation slice from the half-bounce path integral, and use it to compute the root-mean-square quadrupole moment of a bubble at nucleation. The relative deformation of the bubble wall radius RRR is δR/R∼(ℏ/Sb)1/2δR/R\sim(\hbar/S_b)^{1/2}δR/R∼(ℏ/Sb​)1/2, with SbS_bSb​ the bounce action. Thus δR/R∼0.1δR/R\sim0.1δR/R∼0.1 for a typical cosmological phase transition with Sb/ℏ∼100S_b/\hbar\sim100Sb​/ℏ∼100.
The DeWitt boundary condition can be implemented in the Lorentzian path integral of minisuperspace quantum cosmology by summing only over non-singular paths. In this paper, we present a general definition of the resulting DeWitt kernels, along with specific rules for their construction, and evaluate them rigorously. The lapse integral over the positive half-line yields the Dirichlet Green function of the Wheeler--DeWitt operator, whereas the integral over the whole real line yields the Dirichlet group-averaging kernel; both vanish when either endpoint lies at the singularity. Since the minisuperspace actions are at most quadratic, all lapse integrals are performed in closed form without saddle-point approximations. We obtain the kernels for the closed universe without cosmological constant, for the flat de Sitter universe, and, in terms of Airy functions, for the closed and open de Sitter universes. Throughout, the DeWitt condition is imposed as the boundary condition at the singularity, which selects a self-adjoint constraint operator on the half-line and fixes the kernels uniquely, including their normalization.
We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle.The configurations which allow construction of a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions.The quantum amplitudes consist of an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant.The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property.Accordingly, we resort to the zeta-function method to compute the functional determinants in a systematic way.The effect of the multi-instantons configurations is also carefully considered.To illustrate the instanton calculus in a relevant model, we go to the double-well potential.The second popular case is the periodic-potential where the initial levels split into bands.The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution.
The decay of Bose-Einstein condensates from a metastable ground state into collapse due to macroscopic quantum tunneling is investigated using a semiclassical approximation to Feynman’s path integral formalism. Applying a variational ansatz of a single Gaussian to the wave function determined by the Gross-Pitaevskii equation, a special choice of the Gaussian width parameters yields a mean-field energy functional in Hamiltonian form. The temporal spatial extension of the condensate then is described in the picture of a particle moving in an external potential. In this picture the decay of the condensate wave function is investigated using the bounce trajectory method, accounting for the action of the tunneling orbit in imaginary time and quadratic fluctuations around it, the latter described by the Gelfand-Yaglom differential equation. Additionally, the tunneling formalism is used for describing the condensate wave function by a trial function of a superposition of several Gaussians. Using this ansatz a Hamiltonian form of the mean-field energy does not exist and the formalism of describing the tunneling process has to be extended. The bounce trajectory is described in the parameter space of the Gaussian width parameters and the equations of motion for the imaginary time evolution are obtained by applying a time-dependent variational principle. The action of the bounce trajectory is investigated as a function of the number of Gaussians taken into account and the contribution of the fluctuations is evaluated using the monodromy matrix.