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Statistics of Cosmic Microwave Background Polarization

Marc Kamionkowski, Arthur Kosowsky, Albert Stebbins

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive full-sky formalism for cosmic microwave background temperature and polarization maps by expanding polarization as a symmetric trace-free tensor in tensor spherical harmonics, yielding coordinate-independent power spectra C_l^T, C_l^G, C_l^C, and cross-spectra. It derives exact expressions for these multipole moments for scalar and tensor metric perturbations, analyzes their statistical properties under isotropy, and provides practical estimators and a map-simulation method that incorporate cosmic and pixel noise. The work shows that scalar perturbations generate only E-type (G) polarization while tensor perturbations also produce B-type (C) polarization, and parity constraints force certain cross-spectra to vanish; these results enable a clean separation of scalar and tensor contributions and furnish a robust framework for analyzing polarization data, including line-of-sight approaches and small-angle limits. The formalism lays the groundwork for exploiting CMB polarization to detect primordial gravitational waves and to monitor foregrounds in upcoming experiments like MAP and COBRAS/SAMBA.

Abstract

We present a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave background. Temperature maps are analyzed by expanding over the set of spherical harmonics to give multipole moments of the two-point correlation function. Polarization, which is described by a second-rank tensor, can be treated analogously by expanding in the appropriate tensor spherical harmonics. We provide expressions for the complete set of temperature and polarization multipole moments for scalar and tensor metric perturbations. Four sets of multipole moments completely describe isotropic temperature and polarization correlations; for scalar metric perturbations one set is identically zero, giving the possibility of a clean determination of the vector and tensor contributions. The variance with which the multipole moments can be measured in idealized experiments is evaluated, including the effects of detector noise, sky coverage, and beam width. Finally, we construct coordinate-independent polarization two-point correlation functions, express them in terms of the multipole moments, and derive small-angle limits.

Statistics of Cosmic Microwave Background Polarization

TL;DR

<3-5 sentence high-level summary> The paper develops a comprehensive full-sky formalism for cosmic microwave background temperature and polarization maps by expanding polarization as a symmetric trace-free tensor in tensor spherical harmonics, yielding coordinate-independent power spectra C_l^T, C_l^G, C_l^C, and cross-spectra. It derives exact expressions for these multipole moments for scalar and tensor metric perturbations, analyzes their statistical properties under isotropy, and provides practical estimators and a map-simulation method that incorporate cosmic and pixel noise. The work shows that scalar perturbations generate only E-type (G) polarization while tensor perturbations also produce B-type (C) polarization, and parity constraints force certain cross-spectra to vanish; these results enable a clean separation of scalar and tensor contributions and furnish a robust framework for analyzing polarization data, including line-of-sight approaches and small-angle limits. The formalism lays the groundwork for exploiting CMB polarization to detect primordial gravitational waves and to monitor foregrounds in upcoming experiments like MAP and COBRAS/SAMBA.

Abstract

We present a formalism for analyzing a full-sky temperature and polarization map of the cosmic microwave background. Temperature maps are analyzed by expanding over the set of spherical harmonics to give multipole moments of the two-point correlation function. Polarization, which is described by a second-rank tensor, can be treated analogously by expanding in the appropriate tensor spherical harmonics. We provide expressions for the complete set of temperature and polarization multipole moments for scalar and tensor metric perturbations. Four sets of multipole moments completely describe isotropic temperature and polarization correlations; for scalar metric perturbations one set is identically zero, giving the possibility of a clean determination of the vector and tensor contributions. The variance with which the multipole moments can be measured in idealized experiments is evaluated, including the effects of detector noise, sky coverage, and beam width. Finally, we construct coordinate-independent polarization two-point correlation functions, express them in terms of the multipole moments, and derive small-angle limits.

Paper Structure

This paper contains 30 sections, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of Polarization
  3. Review of Stokes Parameters
  4. Scalar and Tensor Harmonic Expansions
  5. Explicit Form of the Harmonics
  6. Statistics of the Multipole Coefficients
  7. Statistical Independence of the Coefficients
  8. Map Simulation
  9. Estimators
  10. Cosmic and Pixel-Noise Variance
  11. Pixel Noise for a Polarization Map
  12. Calculation of the Moments
  13. Scalar Metric Perturbations
  14. Temperature moments
  15. Polarization moments
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The basis tensors $M_{(j)}^{ab}(k)$ traditionally used for polarization are discontinuous in the orientation of the polarization in the directions on the sky both parallel and anti-parallel to ${\bf k}$. The discontinuity is either as depicted in the left panel or as depicted in the right panel. Note that we have just switched the sign of the polarization between the two panels.