# Boost operator approach to the relativistic SZ effect

## TL;DR

This paper introduces a boost operator method to accurately compute the relativistic Sunyaev-Zeldovich effect, simplifying calculations and enabling higher-order corrections for cosmological studies.

## Contribution

It develops a formalism using boost operators to efficiently derive relativistic corrections to the SZ effect, including third-order velocity effects, improving upon previous methods.

## Key findings

- Confirmed existing analytic expressions for the relativistic SZ effect.
- Derived new third-order velocity correction expressions.
- Demonstrated the efficiency of the boost operator approach.

## Abstract

The Sunyaev-Zeldovich (SZ) effect provides a powerful cosmological probe. We demonstrate that the corresponding relativistic SZ signal can be accurately calculated using the recently developed boost operator approach. We obtain formally exact expressions for the required differential operator that can be used to generate relativistic temperature and velocity correction functions to any order. Many of the otherwise cumbersome intermediate steps can be avoided and the required boost operator elements can be quickly generated using recurrence relations of the underlying aberration kernel. We confirm previous analytic expressions describing the relativistic SZ effect and give new expressions at third order in the cluster's peculiar velocity, demonstrating the feasibility of the boost operator method. Our derivation also highlights general properties of the boost operator and showcases its application to radiative transfer problems of broader interest.