Why Azimuthal Averaging Works in Halo Lensing: Symmetry and Power Counting for Nonlinear Shear and Magnification

Abstract: Cluster weak-lensing analyses often compress two-dimensional lensing fields into azimuthally averaged radial profiles and evaluate nonlinear observables from the averaged convergence and shear. However, the ring average of a nonlinear observable generally differs from the mean-field prediction formed from the averaged fields. This difference, divided by the mean-field prediction, defines the fractional residual. For a complete ring, the difference begins at second order in angular fluctuations. For centered elliptical halos, rotational symmetry makes the shape contribution even in signed ellipticity, while the leading miscentering dipole is orthogonal to the shape quadrupole. We test these predictions using projected triaxial NFW halos at six mass-redshift grid points spanning $$3\leq M_{200\mathrm c}/(10<sup>{14}\,h<sup>{-1}M_\odot)\leq20$$ and $0.2\leq z_l\leq0.5$, for $z_s=1$. For the reference offset model, centering offsets follow a Rayleigh distribution with scale $0.05r_{200\mathrm c}$. For $0.366\leq R/r_{200\mathrm c}\leq1$, where at least $95\%$ of each population satisfies a conservative subcriticality criterion, we calculate the median of the fractional residuals across the retained halos for each population and radius. The largest population-median magnitudes are $0.90\%$ for reduced shear, $0.0093\%$ for inverse magnification, $0.64\%$ for magnification, and $0.18\%$ across the two magnification-bias cases $μ<sup>{α-1}$ ($α=0.3, 1.4$). In the paired calculation at $z_s=2$, the maximum magnitude increases for every observable, consistent with weak-lensing power counting. Across both source planes and the common radial domain, every population median remains below $2\%$ in magnitude. This accuracy follows from first-order cancellation, rotational symmetry, and harmonic orthogonality, with further weak-lensing suppression of the remaining nonlinear terms.