Statistical Field Theory for Weak Gravitational Lensing
Abstract
Standard weak-lensing calculations treat lensing as a linear remapping of the matter field along the line of sight. We instead formulate lensing as a stochastic field theory for the Sachs optical scalars, driven by random Ricci-focusing and Weyl-shearing fields. With convergence and shear expressed linearly in the Sachs-scalar fluctuations, the resulting path integral generates a diagrammatic expansion for their n-point correlation functions, organised into linear response, nonlinear propagation, and driving-field cumulants. The conventional calculation emerges as the lowest-order, linear-propagation limit. Beyond it, nonlinear Sachs evolution couples to driving-field non-Gaussianity, mixing the matter cumulant hierarchy into the lensing hierarchy. A selection rule governs the couplings: an n-point observable receives a direct contribution from the n-point driving-field cumulant, and its leading hierarchy-mixing correction from the (n+1)-point cumulant via one nonlinear Sachs interaction, with higher cumulants entering only at higher order. The two-point function, for instance, receives corrections from three-point cumulants of Ricci focusing and Weyl shearing through nonlinear Sachs evolution. Rather than a restrictive approximation scheme, the formalism is a paradigm shift: a unified framework naturally accommodating path corrections, higher-order matter statistics, stochasticity, and small-scale effects.
Versions
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BibTeX
@misc{zhang2026statistical,
title = {{Statistical Field Theory for Weak Gravitational Lensing}},
author = {Zheng Zhang and Philip Bull and Chris Clarkson and Andrina Nicola},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.19472},
eprint = {2606.19472},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19472},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Statistical Field Theory for Weak Gravitational Lensing
AU - Zheng Zhang
AU - Philip Bull
AU - Chris Clarkson
AU - Andrina Nicola
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2606.19472
UR - https://arxiv.org/abs/2606.19472
AB - Standard weak-lensing calculations treat lensing as a linear remapping of the matter field along the line of sight. We instead formulate lensing as a stochastic field theory for the Sachs optical scalars, driven by random Ricci-focusing and Weyl-shearing fields. With convergence and shear expressed linearly in the Sachs-scalar fluctuations, the resulting path integral generates a diagrammatic expansion for their n-point correlation functions, organised into linear response, nonlinear propagation, and driving-field cumulants. The conventional calculation emerges as the lowest-order, linear-propagation limit. Beyond it, nonlinear Sachs evolution couples to driving-field non-Gaussianity, mixing the matter cumulant hierarchy into the lensing hierarchy. A selection rule governs the couplings: an n-point observable receives a direct contribution from the n-point driving-field cumulant, and its leading hierarchy-mixing correction from the (n+1)-point cumulant via one nonlinear Sachs interaction, with higher cumulants entering only at higher order. The two-point function, for instance, receives corrections from three-point cumulants of Ricci focusing and Weyl shearing through nonlinear Sachs evolution. Rather than a restrictive approximation scheme, the formalism is a paradigm shift: a unified framework naturally accommodating path corrections, higher-order matter statistics, stochasticity, and small-scale effects.
ER -