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Multiple scattering Sunyaev-Zeldovich signal II: relativistic effects

We study the multiple scattering Sunyaev-Zeldovich (SZ) signature, extending our previous analysis to high-temperature clusters. We consistently treat the anisotropy of the ambient radiation field caused by the first scattering and also consider lowest order kinematic terms. We show that due to temperature corrections monopole through octupole anisotropy of the singly scattered SZ signal attain different spectra in the second scattering. The difference becomes more pronounced at high temperature, and thus could be used to constrain individual line of sight moments of the electron density and temperature profiles. While very challenging from the observational point of view, this further extends the list of possible SZ observables that will be important for 3D cluster-profile reconstruction, possibly helping to break geometric degeneracies caused by projection effects. We also briefly discuss the scattering of primordial CMB anisotropies by SZ clusters.