Can one determine cosmological parameters from multi-plane strong lens systems?

TL;DR

This paper reveals an invariance in multi-plane strong lens systems that prevents unique determination of cosmological parameters without breaking the mass-sheet degeneracy, highlighting a fundamental limitation in using such systems for cosmology.

Contribution

It generalizes the mass-sheet transformation to multi-plane lens systems and discusses its implications for cosmological parameter estimation.

Findings

Mass-sheet transformation exists in two-plane lens systems.

Time-delay ratios are invariant under the transformation.

Cosmological parameters are degenerate with mass-sheet transformations.

Abstract

Strong gravitational lensing of sources with different redshifts has been used to determine cosmological distance ratios, which in turn depend on the expansion history. Hence, such systems are viewed as potential tools for constraining cosmological parameters. Here we show that in lens systems with two distinct source redshifts, of which the nearest one contributes to the light deflection towards the more distant one, there exists an invariance transformation which leaves all strong lensing observables unchanged (except the product of time delay and Hubble constant), generalizing the well-known mass-sheet transformation in single plane lens systems. The transformation preserves the relative distribution of mass and light, so that a `mass-follows-light' assumption does not fix the MST. All time delays (from sources on both planes) scale with the same factor -- time-delay ratios are therefore invariant under the MST. Changing cosmological parameters, and thus distance ratios, is essentially equivalent to such a mass-sheet transformation. As an example, we discuss the double source plane system SDSSJ0946+1006, which has been recently studied by Collett and Auger, and show that variations of cosmological parameters within reasonable ranges lead to only a small mass-sheet transformation in both lens planes. Hence, the ability to extract cosmological information from such systems depends heavily on the ability to break the mass-sheet degeneracy.