Exact general relativistic lensing versus thin lens approximation: the crucial role of the void

TL;DR

This study compares exact general relativistic lensing calculations with the thin lens approximation, revealing significant deviations unless a void is included, where the approximations align closely.

Contribution

It demonstrates the importance of including voids in models to accurately match exact relativistic lensing calculations with the thin lens approximation.

Findings

Thin lens approximation can differ by up to 30% from exact calculations.

Including a void in the density profile improves approximation accuracy.

The Rosenbrock method is effective for integrating geodesic equations.

Abstract

We have used an exact general relativistic model structure within a FRW cosmological background based on a LTB metric to study the gravitational lensing of a cosmological structure. The integration of the geodesic equations turned out to be a delicate task. We realized that the use of the rank 8(7) and 10(11) Runge-Kutta numerical method leads to a numerical effect and is therefore unreliable. The so-called semi-implicit Rosenbrock method, however, turned out to be a viable integration method for our problem. The deviation angle calculated by the integration of the geodesic equations for different density profiles of the model structure was then compared to those of the corresponding thin lens approximation. Using the familiar NFW density profile, it is shown that independent of the truncation details the thin lens approximation differ substantially from the exact relativistic calculation. The difference in the deflection angle for different impact parameters may be up to about 30 percent. However, using the modified NFW density profile with a void before going over to the FRW background, as required by an exact general relativistic model, the thin lens approximation coincides almost exactly with the general relativistic calculation.