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

We calculate the flux of $\gamma$ 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 $\gamma$-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 $\gamma$-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 $2\times {10}^{13}\hspace{0.5em}\mathrm{g}-5\times {10}^{14}\hspace{0.5em}\mathrm{g},$ corresponding to a cosmic reheating temperature of $7\times {10}^7\hspace{0.5em}\mathrm{GeV}-4\times {10}^8\hspace{0.5em}\mathrm{GeV}.$ In this range, we find the upper limit to be $n\lesssim 1.23-\mathrm{1.25.}$ 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.