The theory of stochastic cosmological lensing

TL;DR

This paper introduces a stochastic formalism for small-scale cosmological lensing, modeling light beam propagation as a diffusion process to better understand matter's effects on light from sources like supernovae.

Contribution

It develops a novel Langevin and Fokker-Planck-Kolmogorov formalism for small-scale lensing, validated against simulations and numerical integrations, and extends the Dyer-Roeder approximation.

Findings

The formalism accurately predicts mean and dispersion of angular distances.

Validation shows good agreement with ray-tracing simulations.

The approach provides a new analytical tool for small-scale lensing studies.

Abstract

On the scale of the light beams subtended by small sources, e.g. supernovae, matter cannot be accurately described as a fluid, which questions the applicability of standard cosmic lensing to those cases. In this article, we propose a new formalism to deal with small-scale lensing as a diffusion process: the Sachs and Jacobi equations governing the propagation of narrow light beams are treated as Langevin equations. We derive the associated Fokker-Planck-Kolmogorov equations, and use them to deduce general analytical results on the mean and dispersion of the angular distance. This formalism is applied to random Einstein-Straus Swiss-cheese models, allowing us to: (1) show an explicit example of the involved calculations; (2) check the validity of the method against both ray-tracing simulations and direct numerical integrations of the Langevin equation. As a byproduct, we obtain a post-Kantowski-Dyer-Roeder approximation, accounting for the effect of tidal distortions on the angular distance, in excellent agreement with numerical results. Besides, the dispersion of the angular distance is correctly reproduced in some regimes.