Analytical Study on the Sunyaev-Zeldovich Effect for Clusters of Galaxies. II. comparison of covariant formalisms

TL;DR

This paper compares two covariant formalisms for modeling the Sunyaev-Zeldovich effect in galaxy clusters, demonstrating their mathematical equivalence and providing analytic expressions within the Thomson approximation.

Contribution

It establishes the equivalence of two covariant formalisms for the Sunyaev-Zeldovich effect and derives analytic redistribution functions in the Thomson limit.

Findings

Two formalisms are mathematically equivalent in the Thomson approximation

Analytic expressions for redistribution functions are derived

The work provides a theoretical foundation for SZ effect analysis in galaxy clusters

Abstract

We study a covariant formalism for the Sunyaev-Zeldovich effects developed in the previous papers by the present authors, and derive analytic expressions for the redistribution functions in the Thomson approximation. We also explore another covariant formalism recently developed by Poutanen and Vurm. We show that the two formalisms are mathematically equivalent in the Thomson approximation which is fully valid for the cosmic microwave background photon energies. The present finding will establish a theoretical foundation for the analysis of the Sunyaev-Zeldovich effects for the clusters of galaxies.