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