Geometry of non-Gaussianity in transient non-attractor inflation

Abstract

Inflationary models predicting abundant primordial black holes (PBHs) and large amplitude of scalar-induced gravitational waves (SIGWs) often rely on amplified fluctuations over limited scales, typically driven by phase transitions, particle production, or departures from slow-roll evolution. While the power spectrum of these models has been extensively studied, higher-order correlations are much less understood. Motivated by the complex physics involved and the fact that PBH and SIGW formation are both sensitive to non-linearities, we present a detailed study of the bispectrum as the leading non-linear effect in these scenarios. We refer to the scale- and shape-dependence of the bispectrum collectively as its geometry; and define a scale-dependent shape correlator to disentangle the two dependencies. Generally, we find that for the scales most affected by phase transitions and particle production (including the power spectrum peak), the bispectrum is strongest near the equilateral configuration, while non-attractor phases tend to produce correlations near the squeezed configuration. We further propose a simplified bispectrum estimator, resembling local-type non-Gaussianity but with scale-dependent amplitude, that captures the main features of the full bispectrum. As an implication of our results, we show that incorporating the bispectrum significantly broadens the range of scales with a substantial probability of large smoothed density contrasts compared to linear analysis. This suggests that non-linearities can alter not only PBH abundance and SIGW amplitude but also their mass and frequency spectra. In particular, and in contrast with the usual assumption, our results hint that the second-highest peak of the power spectrum may produce more PBHs than the highest peak.