Spectral distortion of cosmic background radiation by scattering on hot electrons. Exact calculations

The spectral distortion of the cosmic background radiation produced by the inverse Compton scattering on hot electrons in clusters of galaxies (thermal Sunyaev-Zeldovich effect) is calculated for arbitrary optical depth and electron temperature. The distortion is found by a numerical solution of the exact Boltzmann equation for the photon distribution function. In the limit of small optical depth and low electron temperature our results confirm the previous analyses. In the opposite limits, our method is the only one that permits to make accurate calculations.