Constraints on the spectral index from primordial black holes

We construct the mass spectrum of primordial black holes (PBH's) using the Press-Schechter formalism and study the contribution of PBH's to the density of the universe, ${\Omega }_{\mathrm{BH}}$, when the power spectrum of the density fluctuation is a simple power-law spectrum. From the condition ${\Omega }_{\mathrm{BH}}<1\mathrm{}$, the constraints on the spectral index $n$ are found for both cases of possible end states of black hole evaporation, one in which black holes evaporate completely without relics and the other in which black hole evaporation ends with relics with mass of the order of $M_{\mathrm{Pl}}$. With the normalization of the fluctuation amplitude to COBE's quadrupole anisotropy measurement, we find that the upper limit of $n$ is somewhat largely deviated from the Harrison-Zel'dovich spectrum.