Non-Gaussianity in inflationary scenarios for primordial black holes

TL;DR

This paper calculates the non-Gaussianity parameter $f_{NL}$ in inflationary models with peaks in the primordial power spectrum, providing insights into PBH and SIGW production.

Contribution

It offers an analytic approximation for $f_{NL}$ during inflation with multiple phases, aiding the design and analysis of models producing primordial black holes and gravitational waves.

Findings

$f_{NL}$ is approximately constant near the power spectrum peak.

Conditions for large $f_{NL}$ impacting PBH and SIGW production are identified.

Analytic formulae for the power spectrum's rise and fall are provided.

Abstract

Working in an idealised framework in which a series of phases of evolution defined by the second slow-roll parameter $\eta$ are matched together, we calculate the reduced bispectrum, $f_{\rm NL}$, for models of inflation with a large peak in their primordial power spectra. We find $f_{\rm NL}$ is typically approximately constant over scales at which the peak is located, and provide an analytic approximation for this value. This allows us to identify the conditions under which $f_{\rm NL}$ is large enough to have a significant impact on the resulting production of primordial black holes (PBHs) and scalar induced gravitational waves (SIGWs). Together with analytic formulae for the gradient of the rise and fall in the power spectrum, this provides a toolkit for designing or quickly analysing inflationary models that produce PBHs and SIGWs.