Weak Lensing of the CMB: A Harmonic Approach

TL;DR

This work develops a harmonic-space, all-sky formalism for weak lensing of the CMB, enabling direct calculation of temperature and polarization power spectra and bispectra from the lensing potential power spectrum C_l^{phi phi} and cross-spectra C_l^{X phi}. It derives exact all-sky expressions for lensed TT, EE, BB, and Theta-E spectra, and provides comprehensive bispectrum formulas for temperature and polarization (including cross-correlations with secondary anisotropies like ISW), along with the flat-sky correspondences. A key finding is that all-sky corrections remain at the ~10% level even on small angular scales, due to the second-order mode-coupling nature of lensing, underscoring the need for all-sky calculations in precision analyses. The paper also analyzes the signal-to-noise prospects for detecting these bispectra, showing strong advantages for magnetic-parity polarization bispectra in a cosmic-variance-limited scenario, though real experiments like Planck face detector-noise and foreground challenges; overall, the framework enables more accurate and cross-corroborated interpretations of CMB lensing and its connections to large-scale structure.

Abstract

Weak lensing of CMB anisotropies and polarization for the power spectra and higher order statistics can be handled directly in harmonic-space without recourse to real-space correlation functions. For the power spectra, this approach not only simplifies the calculations but is also readily generalized from the usual flat-sky approximation to the exact all-sky form by replacing Fourier harmonics with spherical harmonics. Counterintuitively, due to the nonlinear nature of the effect, errors in the flat-sky approximation do not improve on smaller scales. They remain at the 10% level through the acoustic regime and are sufficiently large to merit adoption of the all-sky formalism. For the bispectra, a cosmic variance limited detection of the correlation with secondary anisotropies has an order of magnitude greater signal-to-noise for combinations involving magnetic parity polarization than those involving the temperature alone. Detection of these bispectra will however be severely noise and foreground limited even with the Planck satellite, leaving room for improvement with higher sensitivity experiments. We also provide a general study of the correspondence between flat and all sky potentials, deflection angles, convergence and shear for the power spectra and bispectra.

Figures (5)

• Figure 1: Temperature and temperature-polarization cross power spectra. Shown here are the power spectra of the unlensed and lensed fields, their difference in the all-sky and flat-sky calculations and the error induced by using the flat sky expressions. The oscillatory nature of the difference indicates that lensing smooths the power spectrum.
• Figure 2: Polarization power spectra. The same as in Fig. \ref{['fig:temp']} except for the $E$ and $B$ polarization. We have assumed that the unlensed $B$ spectrum vanishes as appropriate for scalar perturbations.
• Figure 3: Lensing power spectra. The power spectrum of the lensing potential is shown in the top panel as calculated by the flat and all sky approaches for linear and non-linear density perturbations. In the lower panel, the cross correlation with the ISW effect is shown. In both cases, a non-negligible fraction of the power comes from scales where the flat-sky approximation is inadequate.
• Figure 4: Cumulative signal-to-noise in the bispectra as a function of maximum $l$ for a cosmic variance limited experiment and for the Planck satellite. Note that for the cosmic variance limited case (a), bispectra involving the $B$-polarization have a substantial signal-to-noise advantage over the other bispectra. For the Planck satellite (b), we assume that the additional variance comes only from detector noise. In practice, residual foreground contamination and sky-cuts to avoid them will lower the signal-to-noise further.
• Figure 5: Wigner-3$j$ function and approximation. An example of the Wigner-3$j$ symbol relevant to the polarization calculation with $l_2=100$, $m_2=0$, $l_3=50$, $m_3=-2$ is shown as calculated from the recursion relations (solid upper) and analytic approximation (dashed). The difference is shown below (solid lower).