Gravitational lensing time delay beyond the Shapiro/geometry split

TL;DR

This paper derives the exact structure of gravitational lensing time delays from Schwarzschild-de Sitter geodesics, identifying higher-order corrections to the standard geometrical and Shapiro delay split without adding new cosmological dependencies.

Contribution

It provides a precise derivation of time delay structure from exact geodesics, revealing higher-order corrections and clarifying the role of the cosmological constant.

Findings

Standard formula recovered as leading order in small-angle expansion.

First correction identified as intrinsic to Schwarzschild metric, not adding new cosmological dependence.

Cosmological constant affects only angular diameter distances and redshift factors up to this order.

Abstract

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.