Generalized multi-plane gravitational lensing: time delays, recursive lens equation, and the mass-sheet transformation

TL;DR

This paper develops a comprehensive framework for multi-plane gravitational lensing, deriving new expressions for time delays and lens equations, and analyzing the invariance under mass-sheet transformations, with implications for cosmological measurements.

Contribution

It introduces a simplified time-delay function, an alternative lens equation form, and characterizes the generalized mass-sheet transformation in multi-plane lensing.

Findings

Derived a simple expression for time delays in multi-plane lensing.

Established an alternative, equivalent form of the lens equation.

Showed that time delays scale uniformly under the mass-sheet transformation.

Abstract

We consider several aspects of the generalized multi-plane gravitational lens theory, in which light rays from a distant source are affected by several main deflectors, and in addition by the tidal gravitational field of the large-scale matter distribution in the Universe when propagating between the main deflectors. Specifically, we derive a simple expression for the time-delay function in this case, making use of the general formalism for treating light propagation in inhomogeneous spacetimes which leads to the characterization of distance matrices between main lens planes. Applying Fermat's principle, an alternative form of the corresponding lens equation is derived, which connects the impact vectors in three consecutive main lens planes, and we show that this form of the lens equation is equivalent to the more standard one. For this, some general relations for cosmological distance matrices are derived. The generalized multi-plane lens situation admits a generalized mass-sheet transformation, which corresponds to uniform isotropic scaling in each lens plane, a corresponding scaling of the deflection angle, and the addition of a tidal matrix (mass sheet plus external shear) to each main lens. We show that the time delay for sources in all lens planes scale with the same factor under this generalized mass-sheet transformation, thus precluding the use of time-delay ratios to break the mass-sheet transformation.