Black Hole Relics and Inflation: Limits on Blue Perturbation Spectra

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 $n=1.3$.