Importance of intracluster scattering and relativistic corrections from tSZ effect with Cosmic Infrared Background

The Sunyaev-Zeldovich effect towards clusters of galaxies has become a standard probe of cosmology. It is caused by the scattering of photons from the cosmic microwave background (CMB) by the hot cluster electron gas. In a similar manner, other photon backgrounds can be scattered when passing through the cluster medium. This problem has been recently considered for the radio and the cosmic infrared background. Here we revisit the discussion of the cosmic infrared background (CIB) including several additional effects that were omitted before. We discuss the {\it intracluster} scattering of the CIB and the role of {\it relativistic} temperature corrections to the individual cluster and all-sky averaged signals. We show that the all-sky CIB distortion introduced by the scattering of the photon field was underestimated by a factor of $\simeq 1.5$ due to neglecting the intracluster scattering contribution. Energy is essentially transferred twice from the thermal electrons to the CIB. We carefully clarify the origin of various effects in the calculation of the average CIB and also scattered signals. The single-cluster CIB scattering signal also exhibits a clear redshift dependence, which can be used in cosmological analyses, as we describe both analytically and numerically. This may open a new way for cosmological studies with future CMB experiments.