# Lensing in matched exterior and interior Kottler solutions

## TL;DR

This paper derives an analytic expression for gravitational lensing in a matched Kottler spacetime with a cosmological constant, showing how interior mass distribution affects light deflection and magnification.

## Contribution

It provides the first analytic formula for lensing in a matched interior-exterior Kottler spacetime, highlighting interior structure effects on lensing observables.

## Key findings

- Interior mass reduces bending angle and tangential magnification.
- Radial magnification is enhanced by interior effects.
- Numerical results show mild amplification for galaxies and demagnification for clusters.

## Abstract

We analyze gravitational lensing in a matched Kottler spacetime with cosmological constant $\Lambda$, where a uniform-density fluid sphere described by the interior Kottler solution is smoothly joined to the exterior Kottler region. We derive an analytic expression for the deflection angle of light rays traversing both regions, isolating the corrections induced by the interior mass distribution relative to the vacuum Kottler case. The interior terms systematically reduce the bending angle and tangential magnification while enhancing the radial one, leading to a scale-dependent effect on the total magnification. Numerical evaluation shows a mild amplification for galaxy lenses and a demagnification for clusters, reflecting the contrast between compact and diffuse configurations. The induced $\Lambda$-contribution from the interior corrections further reduces the bending, but only slightly, consistent with its role in the vacuum Kottler case. These results demonstrate that lensing observables retain a measurable imprint of interior structure in SdS-like lenses, crucial for refining realistic strong-lensing models.