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CMB Lensing Reconstruction on the Full Sky

Takemi Okamoto, Wayne Hu

TL;DR

We address reconstructing the CMB lensing potential on the full sky by deriving the minimum-variance quadratic estimators for the lensing potential $φ$ from temperature and polarization, including cross-channel estimators and an efficient all-sky angular-space implementation. The work develops the spin-weight formalism to relate lensing-induced mode coupling to optimal weights, derives the noise covariances $N_L^{αβ}$, and shows that full-sky estimators converge to flat-sky results at high multipoles $L$, with the TE estimator nearly MV even when approximated. It provides a practical, scalable reconstruction method using fast transforms (Healpix) and a framework to handle finite-field, inhomogeneous noise, and foregrounds, with relevance for removing lensing contributions in primordial B-mode searches. Overall, the results enable precise lensing reconstructions on wide-sky CMB surveys, unlocking information about dark matter and dark energy on large scales.

Abstract

Gravitational lensing of the microwave background by the intervening dark matter mainly arises from large-angle fluctuations in the projected gravitational potential and hence offers a unique opportunity to study the physics of the dark sector at large scales. Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky. We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields. We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation. This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.

CMB Lensing Reconstruction on the Full Sky

TL;DR

We address reconstructing the CMB lensing potential on the full sky by deriving the minimum-variance quadratic estimators for the lensing potential from temperature and polarization, including cross-channel estimators and an efficient all-sky angular-space implementation. The work develops the spin-weight formalism to relate lensing-induced mode coupling to optimal weights, derives the noise covariances , and shows that full-sky estimators converge to flat-sky results at high multipoles , with the TE estimator nearly MV even when approximated. It provides a practical, scalable reconstruction method using fast transforms (Healpix) and a framework to handle finite-field, inhomogeneous noise, and foregrounds, with relevance for removing lensing contributions in primordial B-mode searches. Overall, the results enable precise lensing reconstructions on wide-sky CMB surveys, unlocking information about dark matter and dark energy on large scales.

Abstract

Gravitational lensing of the microwave background by the intervening dark matter mainly arises from large-angle fluctuations in the projected gravitational potential and hence offers a unique opportunity to study the physics of the dark sector at large scales. Studies with surveys that cover greater than a percent of the sky will require techniques that incorporate the curvature of the sky. We lay the groundwork for these studies by deriving the full sky minimum variance quadratic estimators of the lensing potential from the CMB temperature and polarization fields. We also present a general technique for constructing these estimators, with harmonic space convolutions replaced by real space products, that is appropriate for both the full sky limit and the flat sky approximation. This also extends previous treatments to include estimators involving the temperature-polarization cross-correlation and should be useful for next generation experiments in which most of the additional information from polarization comes from this channel due to sensitivity limitations.

Paper Structure

This paper contains 7 sections, 70 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. CMB Lensing in Multipole Space
  3. Quadratic Estimators
  4. Efficient Estimators
  5. Discussion
  6. Tensor Representation of Spin Weighted Functions
  7. Flat-sky Approximation

Figures (3)

  • Figure 1: Deflection and noise power spectra for the quadratic and minimum variance estimators, assuming the noise properties of the reference experiment ($\Delta_\Theta=1\mu$K-arcmin; $\Delta_P = \sqrt{2}\mu$K-arcmin; $\theta_{\rm FWHM}=4'$) and a fiducial $\Lambda$CDM cosmology with with parameters $\Omega_c = 0.3$, $\Omega_b=0.05$, $\Omega_\Lambda=0.65$, $h=0.65$, $n=1$, $\delta_H=4.2\times 10^{-5}$ and no gravitational waves.
  • Figure 2: Fractional difference between the approximate and the minimum variance $\Theta E$ estimators, defined as $\delta N_L/N_L\equiv (N_L^{\text{approx.}}-N_L^{\text{mv}})/N_L^{\text{mv}}$.
  • Figure 3: Fractional differences $\delta N_L/N_L$ between the noise in the flat sky and full sky estimators, calculated for the reference experiment.