Boost operator approach to the relativistic SZ effect

Abstract

The Sunyaev-Zeldovich (SZ) effect provides a powerful cosmological probe. We demonstrate that the corresponding relativistic SZ signal can be accurately calculated using the recently developed boost operator approach. We obtain formally exact expressions for the required differential operator that can be used to generate relativistic temperature and velocity correction functions to any order. Many of the otherwise cumbersome intermediate steps can be avoided and the required boost operator elements can be quickly generated using recurrence relations of the underlying aberration kernel. We confirm previous analytic expressions describing the relativistic SZ effect and give new expressions at third order in the cluster's peculiar velocity, demonstrating the feasibility of the boost operator method. Our derivation also highlights general properties of the boost operator and showcases its application to radiative transfer problems of broader interest.