Cosmic Microwave Background Radiation in the Direction of a Moving Cluster of Galaxies with Hot Gas: Relativistic Corrections

It has recently been realized that the relativistic corrections to the spectral distortions of the cosmic microwave background (CMB) measured in the directions of galaxy clusters containing hot gas are significant and should be detectable with forthcoming experiments. In the present paper we calculate the correction terms that are proportional to (V/c)ⁿ, with n = 2-4, and (V_r/c)(kT_e/m_ec²) to the standard formulae describing the spectral distortions caused by the bulk motion (kinematic effect) and thermal motion (thermal effect) of the free electrons for the case of a cluster having a peculiar velocity V (V_r being its radial component). The results of our analytical calculations are confirmed by direct Monte Carlo simulations.