Diffuse γ -ray background and primordial black hole constraints on the spectral index of density fluctuations

We calculate the flux of γ rays emitted from primordial black holes (PBHs) which are formed by a ``blue'' power-law spectrum of density fluctuations in the early universe. Gamma-ray emission from such PBHs may contribute significantly to the observed extragalactic diffuse γ-ray background (DGB). Using the observed DGB flux from the imaging Compton Telescope (COMPTEL) and the Energetic Gamma Ray Experiment Telescope (EGRET) as the upper limit of γ-ray flux from PBHs, we derive the upper limit on the spectral index n of the density fluctuations. The range of initial PBH masses which can contribute to the DGB is 2×10¹³ g-5×10¹⁴ g, corresponding to a cosmic reheating temperature of 7×10⁷ GeV-4×10⁸ GeV. In this range, we find the upper limit to be n1.23-1.25. This limit is stronger than those derived from the energy density in PBHs or PBH relics and matches the value of n required to explain the cosmic microwave background anisotropy.