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.
