Impact of the cosmological expansion on spectral energy density of radiation in the intergalactic medium and once more about Olbers' paradox

We analyze the impact of the expansion of the Universe on the formation of the total spectral energy density of radiation in the intergalactic medium.Assuming the same proper thermal spectrum for sources, we show how the expansion of the Universe alters the nature of the energy distribution in the thermal spectrum: a decrease in energy density in the Wien range and an increase in the Rayleigh-Jeans range with increasing redshift of the bulk filled with sources.This is due to the cosmological redshift and the growing contribution of large number of distant sources.The numerical estimations also illustrate the main factors that resolve the Olbers' paradox in the expanding Universe: i) the particle horizon, ii) the finiteness of the volume filled with luminous objects, and iii) the cosmological redshift.Applying the obtained expressions to the epoch of reionization made it possible to estimate the concentration of objects of various classes (stars, globular clusters, dwarf galaxies) necessary for complete reionization of hydrogen at z = 6.It is shown that even a small part of globular clusters or dwarf galaxies with a thermal spectrum of moderate temperature, from those predicted by the Press-Schechter formalism and its improvements, is able to completely reionize hydrogen in the intergalactic medium at z = 6.