Light Deflection, Lensing, and Time Delays from Gravitational Potentials and Fermat's Principle in the Presence of a Cosmological Constant

TL;DR

This paper derives how the cosmological constant affects gravitational lensing and time delays using gravitational potential integration and Fermat's Principle, confirming recent exact solutions and revealing new small but non-negligible terms.

Contribution

It introduces a novel derivation of the cosmological constant's effects on lensing and time delays, including new terms in geometrical delay expressions.

Findings

Lambda contributes to gravitational time delay.

A new Lambda-term appears in geometrical time delay.

Lambda effects can be significant at galaxy cluster scales.

Abstract

The contribution of the cosmological constant to the deflection angle and the time delays are derived from the integration of the gravitational potential as well as from Fermat's Principle. The findings are in agreement with recent results using exact solutions to Einstein's equations and reproduce precisely the new $\Lambda$-term in the bending angle and the lens equation. The consequences on time delay expressions are explored. While it is known that $\Lambda$ contributes to the gravitational time delay, it is shown here that a new $\Lambda$-term appears in the geometrical time delay as well. Although these newly derived terms are perhaps small for current observations, they do not cancel out as previously claimed. Moreover, as shown before, at galaxy cluster scale, the $\Lambda$ contribution can be larger than the second-order term in the Einstein deflection angle for several cluster lens systems.