Geometry of non-Gaussianity in transient non-attractor inflation

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.

💡 Research Summary

This paper conducts a comprehensive analysis of the geometry of non-Gaussianity in transient non-attractor inflation models, which are prime candidates for generating abundant primordial black holes (PBHs) and scalar-induced gravitational waves (SIGWs). Recognizing that the formation of both PBHs and SIGWs is sensitive to non-linearities, the study focuses on the bispectrum as the leading non-linear effect, moving beyond the extensively studied power spectrum.

The authors introduce the concept of “geometry” to collectively describe the scale- and shape-dependence of the bispectrum. To disentangle these dependencies, they define a novel scale-dependent shape correlator. This tool quantitatively measures the similarity between a given bispectrum and standard templates (local, equilateral, folded) at specific comoving scales, allowing for a systematic analysis of how the bispectrum’s configuration preference changes with scale.

The core findings are derived from analytical and numerical investigations of several motivated models, with a detailed focus on a transient ultra-slow-roll (USR) scenario. The analysis reveals a clear dichotomy:

1. For scales most affected by mechanisms like phase transitions or particle production (including the peak of the power spectrum), the bispectrum is strongest near the equilateral configuration.
2. In contrast, non-attractor phases (like USR) tend to produce correlations strongest near the squeezed configuration.
3. The folded configuration is found to be unlikely to be significant in these models.

Given the complexity of the full bispectrum, the authors propose a simplified estimator, termed a “local-like” bispectrum. This estimator resembles the local-type non-Gaussianity template but incorporates two key modifications: it includes the scale-dependence of the power spectrum, and it allows the amplitude of non-Gaussianity (f_NL) itself to be a function of scale, calculated from the full bispectrum. This estimator effectively captures the main features of the full calculation while being simpler to apply.

The cosmological implications are significant. Incorporating the bispectrum significantly broadens the range of scales with a substantial probability of producing large smoothed density contrasts compared to a linear (power spectrum only) analysis. This suggests that non-linearities can alter not only the abundance of PBHs and the amplitude of SIGWs but also their mass and frequency spectra. A particularly noteworthy result hints that the second-highest peak of the power spectrum may end up producing more PBHs than the highest peak, challenging the usual assumption that the peak scale of the power spectrum directly dictates the dominant PBH mass. This implies that non-Gaussian effects could reshape predictions for PBH and SIGW observables.

In conclusion, the work demonstrates that the bispectrum in transient non-attractor models possesses a rich and scale-dependent geometry that cannot be fully captured by a simple local-type ansatz. Properly accounting for this geometry is crucial for accurate predictions of PBH and SIGW signatures, potentially revising their expected mass and frequency distributions.