Relativistic Corrections to the Sunyaev-zel’dovich Effect for Clusters of Galaxies. Ii. Inclusion of Peculiar Velocities

We extend the formalism of the relativistic thermal Sunyaev-Zel’dovich effect to the system moving with a velocity ~ β ≡ ~v/c with respect to the cosmic microwave background radiation. In the present formalism, the kinematic Sunyaev-Zel’dovich effect for the cluster of galaxies with a peculiar velocity ~ β is derived in a straightforward manner by the Lorentz boost of the generalized Kompaneets equation. We give an analytic expression for the kinematic Sunyaev-Zel’dovich effect which is valid up to O(β) with the power series expansion approximation in terms of θe ≡ kBTe/mc , where Te and m are the electron temperature and the electron mass, respectively. It is found that the relativistic corrections to the kinematic Sunyaev-Zel’dovich effect are significant. For a typical electron temperature kBTe = 10keV, one obtains −8.2% and +1.3% corrections from the O(βθe) and O(βθ 2 e) contributions, respectively. The O(β ) correction is extremely small, +0.2% for β = 1/300 at kBTe = 10keV. Therefore it can be safely neglected. These relativistic corrections are directly reflected on the determination of the peculiar velocity β of the cluster of galaxies with the observation of the kinematic Sunyaev-Zel’dovich effect. Subject headings: cosmology: theory — Hubble constant — cosmic microwave background radiation — galaxies: clusters: thermal and kinematic Sunyaev-Zel’dovich effects — plasmas: Compton scattering