Detection of Gravitational Lensing in the Cosmic Microwave Background

TL;DR

This paper provides the first robust cross-correlation detection of gravitational lensing of the CMB by large-scale structure, using a quadratic estimator to reconstruct the CMB lensing potential from WMAP data and cross-correlating it with the NVSS galaxy catalog. The method emphasizes inverse-variance filtering, a three-point lensing-bispectrum approach, and a curl-null test to control systematics, achieving 3.4σ significance once systematic uncertainties are included. Extensive handling of NVSS and WMAP systematics—especially declination-gradient marginalization, beam effects, and point-source/bispectrum contamination—enables a robust measurement of the cross-power spectrum C_ℓ^{φg}. The result validates the CMB lensing framework and sets the stage for precise cosmological constraints from upcoming, higher-sensitivity CMB surveys. The work also demonstrates the practicality of data-driven, optimal estimators for higher-order CMB statistics in the presence of realistic instrumental and astrophysical systematics.

Abstract

Gravitational lensing of the cosmic microwave background (CMB), a long-standing prediction of the standard cosmolgical model, is ultimately expected to be an important source of cosmological information, but first detection has not been achieved to date. We report a 3.4 sigma detection, by applying quadratic estimator techniques to all sky maps from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite, and correlating the result with radio galaxy counts from the NRAO VLA Sky Survey (NVSS). We present our methodology including a detailed discussion of potential contaminants. Our error estimates include systematic uncertainties from density gradients in NVSS, beam effects in WMAP, Galactic microwave foregrounds, resolved and unresolved CMB point sources, and the thermal Sunyaev-Zeldovich effect.

Figures (21)

• Figure 1: Left panel: CMB signal power spectrum, and three-year WMAP noise power spectrum. Right panel: Fiducial NVSS signal power spectrum, and NVSS shot noise ($\overline G = 159000$ gal/steradian).
• Figure 2: Left panel: Auto power spectrum $C_\ell^{\phi\phi}$ of the CMB lensing potential, and reconstruction noise power spectrum $N_\ell^{\phi\phi}$ (Eq. (\ref{['eq:nlpp']})) at three-year WMAP noise levels. Right panel: Cross power spectrum $C_\ell^{\phi g}$ between the CMB lensing potential and NVSS galaxy counts, and the effective noise power spectrum $[N_\ell^{\phi\phi} N_\ell^{gg} / 2]^{1/2}$ for detecting the cross-correlation. The "boost" in signal-to-noise between the two cases is sufficient to obtain a several-sigma detection of CMB lensing.
• Figure 3: Mean contribution to the squared total detection significance $\sigma^2$ per NVSS galaxy multipole $\ell_3$ (left panel), and per unit increase in maximum CMB multipole $\ell_{\rm max}^{\rm CMB} = \hbox{max}(\ell_1,\ell_2)$ (right panel). Most of the statistical weight comes from galaxy multipoles near $\ell\sim 50$, and CMB multipoles near $\ell\sim 400$.
• Figure 4: Simulation + analysis pipeline used in this paper; the stages (1)-(7) are described in detail in §\ref{['sec:pipeline']}.
• Figure 5: Detection of CMB lensing via the cross power spectrum $C_\ell^{\phi g}$ between the reconstructed potential and galaxy counts. The three 1$\sigma$ error bars on each bandpower represent different Monte Carlo methods: WMAP simulations vs NVSS simulations (left/black), WMAP data vs NVSS simulations (middle/blue), and WMAP simulations vs NVSS data (right/red). These error bars represent statistical errors only; the result with systematic errors included will be shown in Fig. \ref{['fig:final']}.