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Boost operator approach to the relativistic polarized SZ effect

Erik Rosenberg, Jens Chluba

TL;DR

This work addresses polarization in the relativistic Sunyaev–Zeldovich (SZ) effect by extending the boost-operator approach to track spin-2 polarization, using spin-weighted harmonics and Wigner-D rotations to handle general boost directions. It derives exact, analytic expressions for the polarized Compton collision term in the Thomson limit, including kinematic and intrinsic CMB anisotropy sources, and provides explicit calculations for monopole, quadrupole, and octupole cases with isotropic electron distributions and moving clusters. The authors introduce a compact operator formulation based on Doppler and diffusion operators, ${\mathcal{O}}_{\nu}=-\nu\partial_{\nu}$ and ${\mathcal{D}}_{\nu}=\nu^{-2}\partial_{\nu}\nu^{4}\partial_{\nu}$, to generate polarized distortion spectra via recursion, with spectral functions $\mathcal{B}_k$ capturing temperature corrections up to high order in $\theta_e$. This framework yields a versatile analytic toolkit for modeling polarized SZ signals and anisotropic spectral distortions, enabling efficient calculations and extensions to more complex scattering scenarios, including multiple scatterings and recoil.

Abstract

We extend the recent boost operator formalism for relativistic Compton scattering calculations to also account for polarization. This allows us to provide general, exact expressions for the polarized Sunyaev-Zeldovich (SZ) effect sourced both kinematically and from intrinsic anisotropies of the Cosmic Microwave Background (CMB). The results are given in terms of rational operator functions that can be used to generate distortion spectra that describe the general SZ signal, reproducing the classical polarized SZ results in the appropriate limits. Our derivation allows for clear separation of physical effects in the generation of polarized SZ, and beyond the SZ application provides a general description of the Compton collision term in the Doppler-dominated regime. Through direct computation of important example cases we further illustrate the power of this new method.

Boost operator approach to the relativistic polarized SZ effect

TL;DR

This work addresses polarization in the relativistic Sunyaev–Zeldovich (SZ) effect by extending the boost-operator approach to track spin-2 polarization, using spin-weighted harmonics and Wigner-D rotations to handle general boost directions. It derives exact, analytic expressions for the polarized Compton collision term in the Thomson limit, including kinematic and intrinsic CMB anisotropy sources, and provides explicit calculations for monopole, quadrupole, and octupole cases with isotropic electron distributions and moving clusters. The authors introduce a compact operator formulation based on Doppler and diffusion operators, and , to generate polarized distortion spectra via recursion, with spectral functions capturing temperature corrections up to high order in . This framework yields a versatile analytic toolkit for modeling polarized SZ signals and anisotropic spectral distortions, enabling efficient calculations and extensions to more complex scattering scenarios, including multiple scatterings and recoil.

Abstract

We extend the recent boost operator formalism for relativistic Compton scattering calculations to also account for polarization. This allows us to provide general, exact expressions for the polarized Sunyaev-Zeldovich (SZ) effect sourced both kinematically and from intrinsic anisotropies of the Cosmic Microwave Background (CMB). The results are given in terms of rational operator functions that can be used to generate distortion spectra that describe the general SZ signal, reproducing the classical polarized SZ results in the appropriate limits. Our derivation allows for clear separation of physical effects in the generation of polarized SZ, and beyond the SZ application provides a general description of the Compton collision term in the Doppler-dominated regime. Through direct computation of important example cases we further illustrate the power of this new method.

Paper Structure

This paper contains 25 sections, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering of polarized radiation
  3. Single Scattering
  4. General boost direction
  5. Scattering by isotropic electron distributions
  6. Leading order terms ($p=0$)
  7. Aberration-induced temperature corrections
  8. Including leading order Doppler diffusion corrections
  9. Moving cloud of thermal electrons
  10. Applications to CMB
  11. Scattering of the average CMB monopole
  12. Zero temperature limit
  13. Temperature corrections to the polarized SZ effect
  14. Scattering of primary CMB anisotropies
  15. Conclusions
  16. ...and 10 more sections

Figures (2)

  • Figure 1: Spectral functions $\mathcal{B}_k$. We normalized each of them to have a similar peak amplitude. The focus of the functions moves upwards as $k$ increases.
  • Figure 2: Spectral function $\mathcal{Y}=\sum_{k=0}^{k_{\rm max}} \theta_{\rm e}^k \mathcal{Y}_k$ up to varying orders and for $\theta_{\rm e}=0.02$ (i.e, $\simeq 10\,{\rm keV}$).