Lensing Reconstruction with CMB Temperature and Polarization

TL;DR

This paper develops a framework to reconstruct the CMB lensing potential from temperature and polarization data by exploiting the non-Gaussian mode coupling induced by weak lensing. It introduces a temperature-based quadratic estimator and derives its variance, including a previously neglected reconstruction-bias term $N^{(1)}$, which couples to other lensing modes and biases the lensing-potential power spectrum if not removed iteratively. It then constructs an unbiased estimator for the lensing-potential power spectrum $C_L^{\phi\phi}$ by debiasing with $N^{(0)}$ and $N^{(1)}$, and analyzes the resulting variances and covariances, finding Planck-like experiments have negligible inter-bin covariance while highlighting the practical importance of iterative bias subtraction. The work further discusses polarization-based estimators and observational considerations (SZ, ISW, galactic foregrounds) and emphasizes that robust lensing reconstruction is crucial for accurate B-mode decontamination and tests of large-scale structure theory.

Abstract

Weak gravitational lensing by intervening large-scale structure induces a distinct signature in the cosmic microwave background (CMB) that can be used to reconstruct the weak-lensing displacement map. Estimators for individual Fourier modes of this map can be combined to produce an estimator for the lensing-potenial power spectrum. The naive estimator for this quantity will be biased upwards by the uncertainty associated with reconstructing individual modes; we present an iterative scheme for removing this bias. The variance and covariance of the lensing-potenial power spectrum estimator are calculated and evaluated numerically in a $Λ$CDM universe for Planck and future polarization-sensitive CMB experiments.

Figures (3)

• Figure 1: The two quadrilaterals consistent with the constraint ${\mathbf{L}} - {\mathbf{L^{\prime}}} = {{\bf l}_1} + {{\bf l}_2} - {{\bf l}_1}' - {{\bf l}_2}' = 0$ for the variance of the estimator ${\mathbf{d}}_{\Theta\Theta}({\mathbf{L}})$. The lensing modes $\phi({\mathbf{L}})$, $\phi({{\bf l}_1}-{{\bf l}_1}')$, and $\phi({{\bf l}_1}-{{\bf l}_2}')$, depicted as diagonals in the above quadrilaterals, induce non-Gaussian couplings between the modes of the observed temperature map represented as sides of these quadrilaterals. They lead to the three groups of linear terms appearing in Eq. (\ref{['E:4pt,CMB']}).
• Figure 2: Variances with which individual modes ${\mathbf{d}}({\mathbf{L}})$ of the deflection field can be reconstructed by the Planck and reference experiments described in the text. The solid curves are the power spectra $C_{L}^{dd}$ anticipated for our $\Lambda$CDM cosmological model. The upper and lower dashed curves are the zeroth and first-order noise power spectra $N^{(0)}_{\Theta\Theta,\Theta\Theta}(L)$ and $N^{(1)}_{\Theta\Theta,\Theta\Theta}(L)$ respectively for the temperature-based estimator ${\mathbf{d}}_{\Theta\Theta}({\mathbf{L}})$, while the dotted curves are the corresponding noise variances for ${\mathbf{d}}_{EB}({\mathbf{L}})$. A mode ${\mathbf{d}}({\mathbf{L}})$ cannot be reconstructed with signal-to-noise greater than unity when $C_{L}^{dd} \leq N^{(0)}_{\Theta\Theta,\Theta\Theta}(L) + N^{(1)}_{\Theta\Theta,\Theta\Theta}(L)$.
• Figure 3: The ratio $R_{LL^{\prime}}$ for Planck as a function of $L$ for fixed values of $L^{\prime}$. The solid curves correspond to $L^{\prime} = 3, 30, 300$ ascending from bottom to top, while the long-dashed and short-dashed curves correspond to $L^{\prime} = 7, 70, 700$ and $L^{\prime} = 10, 100, 1000$ respectively, again with curves in each sequence appearing from bottom to top in the figure.