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 } \mbox { $g$ } -5 \times 10 ^ { 14 } \mbox { $g$ } , corresponding to a cosmic reheating temperature of 7 \times 10 ^ { 7 } \mbox { $GeV$ } -4 \times 10 ^ { 8 } \mbox { $GeV$ } .
In this range , we find the upper limit to be n \lesssim 1.23 - 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 .