Degeneracies and modelling choices in double-plane time-delay cosmography

TL;DR

This paper tackles the double-plane mass-sheet degeneracy (MSD) in strong gravitational lensing by explicitly incorporating line-of-sight (LOS) corrections into the double-plane formalism. It shows that LOS contributions modify angular-diameter distances but can be absorbed into transformed lens potentials and a flexible scaling factor, enabling a reformulation where the time-delay function reduces to a single effective scale via the unfolding relation. The authors derive transformed Fermat potentials and time-delay distances under MSD and demonstrate that time-delay ratios remain invariant under the degeneracy, provided the unfolding relation holds. Practically, this yields a prescription to reduce model freedom in double-plane cosmography while safely accounting for MSD, enabling robust $H_0$ constraints even when the true cosmological scaling factor is not fixed, as LOS and mass-sheet effects can be constrained separately (e.g., via velocity dispersions and galaxy counts). Overall, the work clarifies how to disentangle LOS and MSD effects in double-plane lenses to improve time-delay cosmography and dark-energy constraints.

Abstract

Double-plane gravitational lensing is a rare but increasingly observed phenomenon in which the light from a distant source is lensed by two foreground objects at different redshifts. Such systems can be used to provide simultaneous constraints on the Hubble constant $H_0$ and the dark-energy equation of state, independent of and complementary to other probes. However, just as for single-plane gravitational lenses, the precision of these constraints is limited by the so-called mass-sheet degeneracy (MSD) -- a fundamental limit to the knowledge of the mass profiles of lens galaxies and the line of sight that can be obtained from imaging constraints alone. In this work, we show explicitly how contributions from the line of sight appear in double-plane systems. Because these contributions modify angular diameter distances, we argue that cosmological priors should not be used to simply fix the ``cosmological scaling factor'', a ratio of angular diameter distances which is key to the modelling of double-plane lenses. Motivated by this fact, we generalise the double-plane MSD to account for this uncertainty in the scaling factor. While this complicates the time-delay function, we show that, using the ``unfolding relation'', a geometric relation between distances which holds even in the presence of line-of-sight corrections, the uncertainty in the Hubble constant reduces to the familiar mass-sheet transformation of the first lens plane, and a line-of-sight contribution between the observer and the second lens plane. Our main message is therefore a prescription for reducing the degrees of freedom within double-plane models, while still safely accounting for the MSD in measurements of $H_0$.