Stochastic Bubble Completion in de Sitter Space: Power Spectrum, Bispectrum, and Infrared Moment Hierarchies

Abstract

Stochastic bubble nucleation makes the local completion time of a first-order phase transition a random spatial field. We study its power spectrum and equilateral bispectrum in Minkowski and de Sitter backgrounds, with particular emphasis on the infrared consequences of a finite transition duration. In an eternally extended de Sitter history, rare bubbles nucleated in the remote past generate an ancient-bubble tail that produces a hierarchy of infrared-sensitive spatial moments. For the two-point function, the first finite-$k$ correction becomes tail sensitive at $H/\beta =1/5$, while the leading moment becomes singular at $H/\beta =1/3$. The corresponding three-point thresholds are $H/\beta =1/8$ and $1/6$, showing that the bispectrum is more infrared sensitive than the power spectrum. For any finite starting time, however, the ancient-bubble tail is cut off, all spatial moments remain finite, and the strict infrared spectra recover the analytic white-noise limits ${\mathcal{P}}_{\tau }\propto k^3$ and ${\mathcal{B}}_{\tau }^{\mathrm{eq}}\propto k^6$. The eternal-background thresholds instead control the duration dependence of the moment coefficients and the crossover into an intermediate tail-sensitive regime, in agreement with our numerical calculations. We further show that temporal localization of the nucleation history provides a smooth regulator of the ancient-bubble tail and restores the same analytic infrared limits. These results establish a general first-arrival framework for completion-time fluctuations and clarify how de Sitter expansion can generate distinct power-spectrum and bispectrum infrared hierarchies relevant to curvature perturbations and non-Gaussianity.