Derivation from Schwarzschild-de Sitter null geodesics recovers the standard time-delay split as the leading small-angle term, with the first correction intrinsic to the Schwarzschild metric and adding no new cosmological dependence.
The mass-particle in an expanding universe,
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Gravitational lensing time delay beyond the Shapiro/geometry split
Derivation from Schwarzschild-de Sitter null geodesics recovers the standard time-delay split as the leading small-angle term, with the first correction intrinsic to the Schwarzschild metric and adding no new cosmological dependence.