Discrete self-similarity imprints on primordial-black-hole mass functions

Abstract

We study how discrete self-similarity (DSS) in the critical behavior of scalar-field collapse is imprinted on primordial-black-hole (PBH) mass functions. Using the DSS-modulated critical scaling law found in cosmological simulations, we propagate the near-threshold mass map into normalized PBH mass functions during a {kination} era. We compare Gaussian window function, $k$-space top-hat window function, and real-space top-hat window function, together with a real-space top-hat window function multiplied by a kination transfer function. We find that critical behavior in gravitational collapse produces an irreducible minimum width even for an infinitesimally narrow primordial spectrum. DSS then modulates this critical-scaling profile, generating approximately log-periodic features in mass. At fixed horizon mass, successive equal-phase points of the DSS-modulated critical mass map satisfy $\Delta\ln M_{\rm PBH}=\gamma P_{\rm ln}$. Consequently, in the narrow-spectrum limit, the corresponding structures in the final mass function are expected to satisfy $\Delta\ln m\simeq\gamma P_{\rm ln}$, or $m_{n+1}/m_n\simeq5.6$, for the fiducial Choptuik DSS parameters $\gamma$ and $P_{\rm ln}$. For broad primordial spectra, the convolution over horizon masses dephases the DSS pattern and progressively washes out the critical substructure. We normalize the mass functions so that we may primarily study the profile shape and the survival of DSS substructure, rather than the absolute PBH abundance. These results provide a bridge between cosmological DSS collapse simulations and PBH phenomenology including population observables.