Purely Quadratic Non-Gaussianity from Tachyonic Instability: Primordial Black Holes and Scalar-Induced Gravitational Waves

TL;DR

This paper explores how quadratic non-Gaussianity from tachyonic instability affects primordial black hole formation and gravitational wave signals, emphasizing the importance of spectral width in avoiding overproduction.

Contribution

It derives the probability distribution of the compaction function in a quadratic non-Gaussian scenario and highlights the role of spectral width in PBH suppression.

Findings

PBH abundance is exponentially sensitive to perturbation amplitude and correlation coefficient.

Broad spectra tend to fail in suppressing PBH formation, leading to overproduction.

Narrow spectra can suppress PBHs and produce detectable gravitational waves.

Abstract

We investigate primordial black hole (PBH) formation in a cosmological scenario where curvature perturbations follow purely quadratic non-Gaussianity, $\zeta = A(\phi^2-\langle\phi^2\rangle)$, arising from tachyonic instability in multi-component inflationary models. Within an extended Press-Schechter framework based on the compaction function, we derive the probability distribution of the linear compaction function and its asymptotic exponential tail, demonstrating that the PBH abundance is exponentially sensitive not only to the amplitude of perturbations but also to the correlation coefficient $\rho$ between the smoothed field and its radial gradient. We further find that, in this tachyonic amplification scenario, the spectral width of the curvature power spectrum plays a decisive role in avoiding PBH overproduction: broad spectra yield mildly negative $\rho$ and fail to suppress PBH formation, while sufficiently narrow spectra drive $\rho \to -1$, resulting in exponential suppression while maintaining a sizable gravitational-wave signal. Thermal inflation provides a useful benchmark scenario with asteroid-mass PBH dark matter and high-frequency scalar-induced gravitational waves potentially detectable by future space-based interferometers, but its typically broad spectra make it challenging to reconcile PTA observations with PBH constraints.