Simplified treatment of kinematic corrections to the SZ effect using the boost operator approach
The Sunyaev-Zeldovich (SZ) effects provide a potent cosmological probe of the large-scale structure in the universe. Here, we present a simplified derivation of the kinematic corrections to the relativistic thermal SZ effect using the boost operator approach. By first performing the thermal average inside the moving electron cloud frame, the thermal and peculiar motion contributions can be naturally separated, leading to a significant simplification of the scattering calculation. The angular dependence of the problem is resummed into the pre-computed Doppler operators avoiding the otherwise cumbersome many-dimensional angular integrals required using conventional approaches. We provide expressions for the relativistic SZ signal, exact to all orders in the electron temperature theta_e and the peculiar velocity beta_p of a given cluster. These reproduce well-known results for the relativistic SZ effects and extend the description to higher orders in the cluster's speed. We also derive a closed-form expression for the thermally-averaged thermal SZ scattering operator by studying the underlying symmetries of the Doppler operators fundamental to the formalism. Through these results, we also demonstrate how the boost operator approach gives clarity to the physical description of the problem at hand, promising to be useful to a wide range of astrophysical problems.