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 beta (= 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 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(beta^{2}) with the power series expansion approximation in terms of theta_{e} = k_{B} T_{e}/ mc^{2}, 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-Zel'dovich effect are significant. For a typical electron temperature 10keV, one obtains -8.2% and +1.3% corrections from the O(beta theta_{e}) and O(beta theta_{e}^{2}) contributions, respectively. The O(beta^{2}) correction is extremely small, +0.2% for beta=1/300 at 10keV. Therefore it can be safely neglected. These relativistic corrections are directly reflected on the determination of the peculiar velocity of the cluster of galaxies with the observation of the kinematic Sunyaev-Zel'dovich effect. 
Satoshi Nozawa, Naoki Itoh and Yasuharu Kohyama