Black Hole Relics and Inflation: Limits on Blue Perturbation Spectra

TL;DR

This paper derives upper bounds on blue-tilted primordial power spectra ($n>1$) by considering primordial black hole (PBH) formation and, crucially, the possibility that evaporating PBHs leave stable Planck-mass relics. By normalizing the spectrum to the COBE/DMR quadrupole and treating both hard (radiation) and soft (dust) post-inflation equations of state, the authors relate the PBH formation probability $\\beta(M)$ to the density fluctuations $\\delta(M)$ across ~57 decades of mass, making the PBH relic constraints largely insensitive to cosmic variance or gravitational-wave contributions. They find that with Planck-mass relics and high reheating temperature, the spectral index is bounded by $n \le 1.4$, tightening to $n \le 1.3$ if an extended dust phase is present; the exact bound depends on the reheating temperature and the dust duration. These results provide robust, model-insensitive constraints on the inflationary potential in hybrid scenarios and have potential implications for dark matter, gamma-ray backgrounds, and early-Universe cosmology.

Abstract

Blue primordial power spectra have spectral index $n>1$ and arise naturally in the recently proposed hybrid inflationary scenario. An observational upper limit on {\em n} is derived by normalizing the spectrum at the quadrupole scale and considering the possible overproduction of Planck mass relics formed in the final stage of primordial black hole evaporation. In the inflationary Universe with the maximum reheating temperature compatible with the observed quadrupole anisotropy, the upper limit is $n=1.4$, but it is slightly weaker for lower reheat temperatures. This limit applies over 57 decades of mass and is therefore insensitive to cosmic variance and any gravitational wave contribution to the quadrupole anisotropy. It is also independent of the dark matter content of the Universe and therefore the bias parameter. In some circumstances, there may be an extended dust-like phase between the end of inflation and reheating. In this case, primordial black holes form more abundantly and the upper limit is $n=1.3$.