Gravitational lensing time delay beyond the Shapiro/geometry split

Time delays are a key observable in strong gravitational lensing systems. Their theoretical expression is usually written as a sum of a geometrical delay and a Shapiro delay, with cosmology entering through angular diameter distances and a redshift prefactor. In this work we derive this structure from the exact null geodesics of the Schwarzschild-de Sitter metric. The standard formula is recovered as the leading term in a small-angle expansion, and we identify the first correction to the usual geometrical-plus-Shapiro split. Such correction does not introduce any new cosmological dependence: it corresponds instead to a higher-order correction intrinsic to the Schwarzschild part of the metric. As a consequence, up to this order, the cosmological constant enters only through the unlensed angular diameter distances and the unlensed lens-redshift prefactor.