Gravitational lensing time delays: where the standard formula breaks down

The textbook formula for gravitational lensing time delays splits the signal into a geometric path-length term and a Shapiro (gravitational potential) term, with cosmology folded in via angular diameter distances. Starting from exact null geodesics in the Schwarzschild-de Sitter metric, this work derives the first correction to that split and finds it carries no new cosmological dependence — it is purely a higher-order Schwarzschild contribution. The cosmological constant therefore enters time delays only through the standard unlensed distances and redshift prefactor, tightening the theoretical footing for using lensing time delays as a cosmological probe.