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.

33 pages, 17 figures. The replacement file contains the version accepted for publication in PRD