The cosmic microwave background spectral modification due to thermal Comptonization in galaxy clusters. Analytical consideration

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• We calculate analytically an effect of the thermal Comptonization on the cosmic microwave background (CMB) photons when they propagate through galaxy clusters.
• We estimate a deformation of the CMB spectrum due to multiple Compton scattering of the photons off electrons of an optically-thin hot spherical plasma cloud.
• This new approach allows us to disentangle and determine the optical depth and the electron temperature of a galaxy cluster using the CMB observations.