An improved formula for the relativistic corrections to the kinematical Sunyaev-Zeldovich effect for clusters of galaxies

TL;DR

This paper presents an improved, highly accurate formula for relativistic corrections to the Sunyaev-Zeldovich effect in galaxy clusters, validated through numerical integration, aiding future observational data analysis.

Contribution

The authors develop a fourth-order accurate analytic formula for relativistic corrections to the Sunyaev-Zeldovich effect, improving upon previous models.

Findings

Analytic formula matches numerical integration with high precision.

Enhanced accuracy for relativistic corrections up to fourth order.

Useful for analyzing upcoming observational data of galaxy clusters.

Abstract

We improve the calculation of Nozawa, Itoh, and Kohyama (1998) to provide a formula for relativistic corrections to the thermal and kinematical Sunyaev-Zeldovich effects that is accurate to fourth order in theta_e = kT_e/m_ec^2, T_e and m_e being the electron temperature and electron mass, respectively. We also carry out a direct numerical integration of the Boltzmann collision term and confirm the excellent accuracy of the analytic formula. This formula will be useful for the analysis of the observational data of the forthcoming experiments of the kinematical Sunyaev-Zeldovich effect for clusters of galaxies.