The Embedded Transparent Lens and Fermat's Least-Time Principle

TL;DR

This paper develops a simplified embedded gravitational lens theory based on Fermat's principle, showing how embedding affects lensing potentials and time delays, with implications for weak and strong lensing observations.

Contribution

It introduces a simplified embedded lens theory extending to any transparent lens, highlighting the impact of embedding on lensing corrections and time delays.

Findings

Embedding causes a few percent correction in weak lensing shear at large impacts.

Embedding affects the potential part of the time delay in strong lensing.

Cosmological constant influences photon deflection in lensing.

Abstract

We present a simplified version of the lowest-order embedded point mass gravitational lens theory and then make the extension of this theory to any embedded transparent lens. Embedding a lens effectively reduces the gravitational potential's range, i.e., partially shields the lensing potential because the lens mass is made a contributor to the mean mass density of the universe and not simply superimposed upon it. We give the time-delay function for the embedded point mass lens from which we can derive the simplified lens equation by applying Fermat's least-time principle. Even though rigorous derivations are only made for the point mass in a flat background, the generalization of the lens equation to lowest-order for any distributed lens in any homogeneous background is obvious. We find from this simplified theory that embedding can introduce corrections above the few percent level in weak lensing shears caused by large clusters but only at large impacts. The potential part of the time delay is also affected in strong lensing at the few percent level. Additionally we again confirm that the presence of a cosmological constant alters the gravitational deflection of passing photons.