Deformed boson statistics and primordial black hole production under slow expansion conditions

Abstract

The Planck scale physics remains largely unexplored, and understanding the complete quantum theory of gravity may unveil the Universe mysteries that lie beyond this scale. This paper investigates the production of primordial black holes within a thermodynamic framework in an expanding Friedmann–Robertson–Walker (FRW) universe. We leverage the established principle that particle collisions can result in black hole formation when particles are confined within the Schwarzschild radius corresponding to their center of mass energy, a concept known as a variant of the hoop conjecture. Initially, we extend the hoop conjecture to apply to an expanding FRW universe. Our findings reveal that the critical formula derived is significantly influenced by the Hubble parameter, which describes the rate of cosmic expansion, and it effectively aligns with established spacetime limits. Subsequently, we utilize our generalized formula to compute the number density of black holes $\left(n_{\mathrm{bh}}\right)$ generated from particle collisions in the hot system of an early expanding FRW universe as a function of temperature and deformation parameter $q$. We propose that at certain temperatures, the density of black holes may dominate. Our analysis indicates that conditions beyond the Planck temperature are plausible under deformed bosonic statistics. Furthermore, we examine scenarios in the early universe when temperatures surpassed the Planck temperature, concluding that deformed statistics are consistent with the extremely high temperatures characteristic of that epoch. Although black holes may proliferate under these elevated thermal conditions, they are predicted to overshadow background radiation when temperatures beyond those typical at the Planck scale. This study contributes to our understanding of primordial black holes and their formation mechanisms in the context of an expanding universe, providing insights into conditions that may have existed shortly after the Big Bang. In fact, we examine the beyond Planck temperature in the early universe and deduce that deformed statistics are consistent with the extremely high temperatures present in the early universe.