Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of Galaxies

TL;DR

This paper develops a covariant formalism for the Sunyaev-Zeldovich effect in galaxy clusters, clarifying mathematical equivalences and extending to the kinematical effect, aiding precise CMB observations.

Contribution

It derives and compares covariant frequency redistribution functions for the SZ effect, establishing their equivalence and extending the formalism to include the kinematical effect.

Findings

Mathematically proves the equivalence of two redistribution functions in the Thomson limit.

Extends the formalism to include the kinematical Sunyaev-Zeldovich effect.

Clarifies higher-order term discrepancies in the kinematical effect.

Abstract

Starting from a covariant formalism of the Sunyaev-Zeldovich effect for the thermal and non-thermal distributions, we derive the frequency redistribution function identical to Wright's method assuming the smallness of the photon energy (in the Thomson limit). We also derive the redistribution function in the covariant formalism in the Thomson limit. We show that two redistribution functions are mathematically equivalent in the Thomson limit which is fully valid for the cosmic microwave background photon energies. We will also extend the formalism to the kinematical Sunyaev-Zeldovich effect. With the present formalism we will clarify the situation for the discrepancy existed in the higher order terms of the kinematical Sunyaev-Zeldovich effect.