The non-thermal Sunyaev-Zel'dovich effect in clusters of galaxies

In this paper we provide a general derivation of the non-thermal Sunyaev-Zel'dovich (SZ) effect in galaxy clusters which is exact in the Thomson limit to any approximation order in the optical depth tau. The general approach we use also allows us to obtain an exact derivation of the thermal SZ effect in a self-consistent framework. Such a general derivation is obtained using the full relativistic formalism and overcoming the limitations of the Kompaneets and of the single scattering approximations. We compare our exact results with those obtained at different approximation orders in tau and we give estimates of the precision fit. We verified that the third order approximation yields a quite good description of the spectral distortion induced by the Comptonization of CMB photons in the cluster atmosphere. In our general derivation, we show that the spectral shape of the thermal and non-thermal SZ effect depends not only on the frequency but also on the cluster parameters, like the electron pressure and optical depth and from the energy spectrum of the electron population. We also show that the spatial distribution of the thermal and non-thermal SZ effect in clusters depends on a combination of the cluster parameters and on the spectral features of the effect. To have a consistent description of the SZ effect in clusters containing non-thermal phenomena, we also evaluate in a consistent way - for the first time - the total SZ effect produced by a combination of thermal and nonthermal electron population residing in the same environment, like is the case in radio-halo clusters. In this context, we show that the location of the zero of the total SZ effect increases non-linearly with increasing values of the pressure ratio between the non-thermal and thermal electron populations and its determination provides a unique way to determine the pressure of the relativistic particles residing in the cluster atmosphere. We discuss in detail both the spectral and the spatial features of the total (thermal plus non-thermal) SZ effect and we provide specific predictions for a well studied radio-halo cluster like A2163. Our general derivation allows also to discuss the overall SZ effect produced by a combination of different thermal populations residing in the cluster atmosphere. Such a general derivation of the SZ effect allows to consider also the CMB Comptonization induced by several electron populations. In this context, we discuss how the combined observations of the thermal and non-thermal SZ effect and of the other non-thermal emission features occurring in clusters (radio-halo, hard X-ray and EUV excesses) provide relevant constraints of the spectrum of the relativistic electron population and, in turn, on the presence and on the origin of non-thermal phenomena in galaxy clusters. We finally discuss how SZ experiments with high sensitivity and narrow-band spectral coverage, beyond the coming PLANCK satellite, can definitely probe the presence of a non-thermal SZ effect in galaxy clusters and disentangle this source of bias from the cosmologically relevant thermal SZ effect.