Cosmic Shear of the Microwave Background: The Curl Diagnostic

TL;DR

The paper analyzes how weak gravitational lensing imprints both gradient and curl components on CMB temperature and polarization, with the curl arising from either inflationary gravitational waves or nonlinear second-order density effects. It develops and applies quadratic estimators to reconstruct both deflection components, showing the curl signal is typically much smaller than the gradient signal but remains a sensitive diagnostic for non-Gaussian foregrounds and systematics. The authors quantify the expected amplitudes, transfer functions, and reconstruction noise, demonstrating that curl measurements can validate gradient reconstructions and identify biases from foregrounds or instrumental artifacts. This curl diagnostic can thus enhance robustness and interpretability of future high-precision CMB lensing analyses.

Abstract

Weak-lensing distortions of the cosmic-microwave-background (CMB) temperature and polarization patterns can reveal important clues to the intervening large-scale structure. The effect of lensing is to deflect the primary temperature and polarization signal to slightly different locations on the sky. Deflections due to density fluctuations, gradient-type for the gradient of the projected gravitational potential, give a direct measure of the mass distribution. Curl-type deflections can be induced by, for example, a primordial background of gravitational waves from inflation or by second-order effects related to lensing by density perturbations. Whereas gradient-type deflections are expected to dominate, we show that curl-type deflections can provide a useful test of systematics and serve to indicate the presence of confusing secondary and foreground non-Gaussian signals.

Figures (3)

• Figure 1: Lensing-deflection power spectra. Here, we show the gradient component from density perturbations (top curve), the curl component from inflationary gravitational waves (dashed curve labeled 'IGWs'), and the curl component from second-order density perturbations. (dot-dashed curve). We have taken the maximum IGW amplitude consistent with the current upper limit to tensor-to-scalar ratio MelOdm03.
• Figure 2: The lensing modification to CMB power spectra for density perturbations and for gravitational waves. Left: Temperature fluctuations. The top curve is the primordial power spectrum. The middle curve is the (additive) contribution to the temperature power spectrum from lensing by density perturbations, and the lower curve is for lensing by gravitational waves, assuming the maximum IGW background consistent with current CMB bounds. Note that for the lower two curves, it is $|\tilde{C}_l - C_l|$ which is plotted. The negative contribution of the $R$ term in Eq. (\ref{['eqn:ttflat']}) allows for $\tilde{C}_l-C_l$ to become negative. Since the coherence scale for $C_l^{\Omega\Omega}$ is so small, the lensed power spectrum reflects closely at high $l$ the primordial temperature power spectrum. Right: Polarization. These curves are: (1) the top (solid) curve is the primordial E-mode power spectrum; (2) the next-highest (solid) curve is the lensing correction to the primordial E-mode power spectrum by density perturbations; (3) the top dashed curve is the B-mode power spectrum resulting from cosmic-shear conversion of E-modes by density perturbations; (4) the lower dashed curve is the B-mode power spectrum from lensing by foreground IGWs; and (5) the lowest solid curve is the E-mode power spectrum resulting from lensing by foreground IGWs. The dot-dash curve is the primordial B power spectrum from the maximal IGW background allowed by current constraints.
• Figure 3: Reconstructed unbinned noise spectra using the temperature-temperature quadratic estimator and the EB polarization combination. We show noise for both grad and curl modes. For reference, we also plot the power spectrum of the deflection angle corresponding to the gradient component.