Relativistic Corrections to the Sunyaev-Zeldovich Effect for Clusters of Galaxies. II. Inclusion of Peculiar Velocities

We extend the formalism of the relativistic thermal Sunyaev-Zeldovich effect to a system moving with a velocity \b.beta ≡ v/c with respect to the cosmic microwave background radiation. In the present formalism, the kinematic Sunyaev-Zeldovich effect for the cluster of galaxies with a peculiar velocity \b.beta is derived in a straightforward manner by the Lorentz boost of the generalized Kompaneets equation. We give an analytic expression for the kinematic Sunyaev-Zeldovich effect, which is valid up to O(\beta²) with the power series expansion approximation in terms of θ_e ≡ k_BT_e/mc², where T_e and m are the electron temperature and the electron mass, respectively. It is found that the relativistic corrections to the kinematic Sunyaev-Zeldovich effect are significant. For a typical electron temperature k_BT_e = 10 keV, one obtains -8.2% and +1.3% corrections from the O(βθ_e) and O(βθ²_e) contributions, respectively. The O(β²) correction is extremely small: +0.2% for β = 1/300 at k_BT_e = 10 keV. 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-Zeldovich effect.