On the radiative and thermodynamic properties of the cosmic radiations using COBE FIRAS instrument data: IV. Sunyaev-Zel’dovich ( μ $\mu$ -type) distortion effect

Abstract

The Sunyaev-Zel’dovich (SZ) effect represents a small spectral distortion to the cosmic microwave background (CMB) radiation, caused by the Compton scattering of CMB photons by the hot gas of galaxy clusters. In an early stage of universe, the SZ effect generates $\mu$ -type of distortions for the CMB spectrum. A $\mu$ -type distortion is created between the double Compton scattering decoupling ( $z \sim 10^{6}$ ) and the thermalization decoupling by the Compton scattering ( $z \sim 10^{5}$ ). In this case, to describe the small spectral distortion of the CMB spectrum, we use the Bose-Einstein ( $\mu$ -type) distribution with a non-zero chemical potential. At present, it is interesting to investigate the effect of this spectral distortion on the integral characteristics of the Bose-Einstein ( $\mu$ -type) spectrum. The thermal radiative and thermodynamic functions are such integral characteristics. These functions are as follows: a) the total radiation power per unit area; b) total energy density; c) number density of photons; d) grand potential density; e) Helmholtz free energy density; f) entropy density; g) heat capacity at constant volume; h) enthalpy density; and i) pressure. Precise analytical expressions are obtained for the temperature dependences of these functions. Using the observational data obtained by the COBE FIRAS, PIXIE, PRISM, and Planck missions, the thermal radiative and thermodynamic functions are calculated. A comparative analysis of the results obtained with the results for the same functions of the CMB spectrum at $T = 2.72548~\mbox{K}$ is carried out. Very small distortions are observed for the thermal radiative and thermodynamic functions. In the redshift range $10^{5} < z < 3 \times10^{6}$ , these functions are calculated. The expressions are obtained for new astrophysical parameters, such as the entropy density/Boltzmann constant and number density, created by the Bose-Einstein ( $\mu$ -type) spectrum.