Time delay in matched exterior and interior Kottler solutions

We extend the study of gravitational lensing to photons traversing the lens mass, modeled as a uniform-density fluid described by the interior Kottler (Schwarzschild–de Sitter) solution smoothly matched to the exterior Kottler region. An analytic expression for the time delay is derived, allowing the interior contribution to be explicitly isolated relative to the vacuum Kottler case. This correction is found to systematically enhance the total time delay, an effect corroborated by numerical evaluations for astrophysically relevant lenses at both galaxy and cluster scales. The interior contribution increases the total time delay by a substantial fraction, amounting to several tens of days for galaxies and up to a few years for clusters. These results underscore the importance of accounting for the interior lens structure in accurate modeling of strong lensing time delays.