Bounds on the cosmological abundance of primordial black holes from diffuse sky brightness: Single mass spectra

We constrain the mass abundance of unclustered primordial black holes (PBHs), formed with a simple mass distribution and subject to the Hawking evaporation and particle absorption from the environment. Since the radiative flux is proportional to the numerical density, an upper bound is obtained by comparing the calculated and observed diffuse background values (similarly to the Olbers paradox in which point sources are considered) for finite bandwidths. For a significative range of formation redshifts the bounds are better than several values obtained by other arguments ${\Omega }_{\mathrm{pbh}}<~{10}^{-10};$ and they apply to PBHs which are evaporating today.