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The approximate gravitational lensing multiple plane mass sheet degeneracy

Luca Teodori

TL;DR

This work extends the Mass Sheet Degeneracy (MSD) to systems with multiple lens planes by incorporating lens-lens coupling and line-of-sight shear. It develops an approximate, first-order MSD-reduced framework in which internal and external mass sheets are absorbed into redefined angular diameter distances, yielding effective distances $D_A^{\rm eff}$ and reduced quantities that preserve the same structural form as the original equations. While time delays from all sources can be measured, a residual $H_0$-mass-sheet degeneracy remains, meaning $H_0$ cannot be uniquely recovered without external priors such as stellar kinematics or cosmological constraints on LOS convergences. The paper demonstrates this with a concrete double-source example and discusses how priors can shrink the allowed degeneracy window, but emphasizes that multiple-plane setups do not inherently break the MSD; they instead offer additional observables in the form of differential convergences that can help constrain the system in practice.

Abstract

Strong gravitational lensing has to deal with many modeling degeneracies, the most notable being the Mass Sheet Degeneracy (MSD). We review the MSD when one needs to model more lens planes, each one with an internal mass sheet. We take into account the non-linear lens-lens coupling and line of sight effects, the latter treated as external mass sheets with associated shear. If second order shear terms on external and internal mass sheets can be neglected, we show that the MSD is always retained, and the mass sheets influence can be reabsorbed in the redefinition of angular diameter distances. In particular, internal and external mass sheets can be placed on the same footing. The version of the MSD discussed here does not require any particular relation between the internal mass sheets in the different planes. Even when including time delays from all sources, a residual degeneracy involving time delays, mass sheets and $ H_0 $ remains. We develop a framework which shows what can actually be constrained in multiple plane lens systems.

The approximate gravitational lensing multiple plane mass sheet degeneracy

TL;DR

This work extends the Mass Sheet Degeneracy (MSD) to systems with multiple lens planes by incorporating lens-lens coupling and line-of-sight shear. It develops an approximate, first-order MSD-reduced framework in which internal and external mass sheets are absorbed into redefined angular diameter distances, yielding effective distances and reduced quantities that preserve the same structural form as the original equations. While time delays from all sources can be measured, a residual -mass-sheet degeneracy remains, meaning cannot be uniquely recovered without external priors such as stellar kinematics or cosmological constraints on LOS convergences. The paper demonstrates this with a concrete double-source example and discusses how priors can shrink the allowed degeneracy window, but emphasizes that multiple-plane setups do not inherently break the MSD; they instead offer additional observables in the form of differential convergences that can help constrain the system in practice.

Abstract

Strong gravitational lensing has to deal with many modeling degeneracies, the most notable being the Mass Sheet Degeneracy (MSD). We review the MSD when one needs to model more lens planes, each one with an internal mass sheet. We take into account the non-linear lens-lens coupling and line of sight effects, the latter treated as external mass sheets with associated shear. If second order shear terms on external and internal mass sheets can be neglected, we show that the MSD is always retained, and the mass sheets influence can be reabsorbed in the redefinition of angular diameter distances. In particular, internal and external mass sheets can be placed on the same footing. The version of the MSD discussed here does not require any particular relation between the internal mass sheets in the different planes. Even when including time delays from all sources, a residual degeneracy involving time delays, mass sheets and remains. We develop a framework which shows what can actually be constrained in multiple plane lens systems.
Paper Structure (23 sections, 95 equations, 1 figure, 2 tables)

This paper contains 23 sections, 95 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The multiple plane lens equation with external and internal sheets
  3. The multiple plane MSD
  4. The rescaling
  5. Internal and external sheets on the same footing
  6. Effective angular diameter distances
  7. Time delays in the MSD-reduced model
  8. An example of a multiple source system non-breaking the MSD
  9. Stellar kinematics.
  10. Time delays.
  11. Induced shear.
  12. Constraining the $H_0$ degeneracy window.
  13. What modelers one can do, in practice
  14. Higher orders in internal mass sheets
  15. Comparison with previous works
  16. ...and 8 more sections

Figures (1)

  • Figure 1: Relation between the convergence factors lying on the time-delay distance $\tilde{D}_{\rm dt}^n/D_{\rm dt}^n$ and $h_0 := H_0/(100km\per s\per M\parsec)$, for the example system described in Sec. \ref{['s:cluster']} and Table \ref{['tab:mock']}. Bands are regions which would be hypothetically excluded from stellar kinematics measurements with precision $\delta_\sigma=0.05$.