Discriminating weak lensing from intrinsic spin correlations using the curl-gradient decomposition

TL;DR

This work develops a curl–gradient ($E$–$B$) decomposition of the 2D galaxy-shape distortion field to distinguish weak gravitational lensing from intrinsic spin correlations. It shows that lensing produces curl-free ($B=0$) distortions while intrinsic angular-momentum alignments generate significant both $E$- and $B$-modes, enabling robust separation via real-space estimators and local aperture measures. The authors derive inversion relations between observable ellipticity correlations and the $E$/$B$ correlations, provide efficient operator forms in Fourier space, and present practical, local correlators (aperture-like) that can detect intrinsic alignments and gauge noise/systematics. The methodology also offers immediate relevance to CMB polarization analyses, where scalar perturbations produce only $E$-modes and gravity waves produce $B$-modes, informing strategies to extract weak signals from noisier data. Overall, the approach promises improved lensing signal isolation, quantification of intrinsic alignments, and broader applicability to polarization studies in cosmology.

Abstract

The distortion field defined by the ellipticities of galaxy shapes projected on the sky can be uniquely decomposed into a gradient and a curl component. If the observed ellipticities are induced by weak gravitational lensing, then the distortion field is curl free. Here we show that, in contrast, the distortion field resulting from intrinsic spin alignments is not curl free. This provides a powerful discriminant between lensing and intrinsic contributions to observed ellipticity correlations. We also show how these contributions can be disentangled statistically from the ellipticity correlations or computed locally from circular integrals of the ellipticity field. This allows for an unambiguous detection of intrinsic galaxy alignments in the data. When the distortions are dominated by lensing, as occurs at high redshifts, the decomposition provides a valuable tool for understanding properties of the noise and systematic errors. These techniques can be applied equally well to the polarization of the microwave background, where it can be used to separate curl-free scalar perturbations from those produced by gravity waves or defects.

Figures (2)

• Figure 1: The $E$ and $B$-mode correlation functions for intrinsic spin correlations in the model of CNPT (2000). The amplitude of the correlations are determined by the parameters $a$ and $\alpha$, which have been taken to be unity for simplicity. The mean redshift of the sources was taken to be $z_m = 0.1$ and the density correlations were taken to fall off as $r^{-1}$ in the left figure and as $r^{-3/2}$ in the right figure. Also plotted are the differences between $\xi_E$ and $\xi_+$, which is the same as the differences between $\xi_B$ and $\xi_\times$. In the left panel, the projected ellipticity correlations fall off as $\theta^{-1}$ and $\xi_E$ and $\xi_+$ are very nearly the same. This is not the case in general, as can be seen in the right panel where the projected correlations fall as $\theta^{-2}$. However, the $\theta^{-2}$ is also special in that $\xi_E$ and $\xi_B$ are nearly identical.
• Figure 2: Local representations of $E$ (gradient) and $B$ (curl) modes. $E$-modes are either tangential or radial, depending on their sign. $B$-modes can be oriented in either a clockwise or counter-clockwise (shown) direction. Lensing generally brings about only E-modes, while noise and angular momentum correlations can generate both. Local estimators of the $E$ and $B$-modes can be found by doing a radial weighting of these circular integrals (Kaiser et al. 1994).