Analytical studies on the Sunyaev Zeldovich effect in the cluster of galaxies for three Lorentz frames

TL;DR

This paper analytically examines the Sunyaev-Zeldovich effect in galaxy clusters across three Lorentz frames, deriving formulas applicable to both thermal and nonthermal electron distributions.

Contribution

It provides new analytic expressions for the photon redistribution functions in the SZ effect, clarifies frame relations, and extends the Fokker-Planck expansion approach.

Findings

Derived analytic formulas for photon redistribution functions.

Clarified relations of SZ effects among Lorentz frames.

Extended Fokker-Planck expansion to the diffusion operator.

Abstract

We study the Sunyaev-Zeldovich effect for clusters of galaxies. The Boltzmann equations for the CMB photon distribution function are studied in three Lorentz frames. We clarify the relations of the SZ effects among the different Lorentz frames. We derive analytic expressions for the photon redistribution functions. These formulas are applicable to the nonthermal electron distributions as well as the standard thermal distribution. We show that the Fokker-Planck expansion of the Boltzmann equation can be expanded by the power series of the diffusion operator of the original Kompaneets equation.