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≤ M200c/(1014 h−1M⊙)≤20 and 0.2≤ zl≤0.5, for zs=1. For the reference offset model, centering offsets follow a Rayleigh distribution with scale 0.05r200c. For 0.366≤ R/r200c≤1, 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 μα−1 (α=0.3, 1.4). In the paired calculation at zs=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.
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BibTeX
@misc{umetsu2026azimuthal,
title = {{Why Azimuthal Averaging Works in Halo Lensing: Symmetry and Power Counting for Nonlinear Shear and Magnification}},
author = {Keiichi Umetsu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08825},
eprint = {2609.08825},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08825},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Why Azimuthal Averaging Works in Halo Lensing: Symmetry and Power Counting for Nonlinear Shear and Magnification
AU - Keiichi Umetsu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.08825
UR - https://arxiv.org/abs/2609.08825
AB - 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≤ M_200c/(10^14 h^−1M_⊙)≤20 and 0.2≤ z_l≤0.5, for z_s=1. For the reference offset model, centering offsets follow a Rayleigh distribution with scale 0.05r_200c. For 0.366≤ R/r_200c≤1, 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^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.
ER -