Simultaneous constraints on H0 and w from J1721+8842 under lens-model degeneracies

Determine whether the complete observational dataset for the gravitational lens system J1721+8842—including six measured quasar image time delays and double source-plane lensing constraints—can yield simultaneous constraints on the Hubble constant H0 and the dark energy equation-of-state parameter w when adopting maximally degenerate multi-plane lens models that preserve strong-lensing observables.

Background

J1721+8842 is a rare compound strong-lensing system featuring six images of a single background quasar produced by two intervening galaxies at redshifts z1=0.184 and z2=1.885. The configuration enables both time-delay cosmography (through six measurable time delays) and double source-plane lensing (through two distinct Einstein radii), offering a unique opportunity to constrain cosmological parameters.

A central challenge in time-delay cosmography is the mass-sheet degeneracy (MSD), which can be internal (related to the lens galaxy mass profile) or external (due to line-of-sight structures). In multi-plane systems and DSPLs, generalized MSD transformations can maintain many lensing observables while rescaling time delays, potentially biasing H0. The authors discuss how DSPL characteristics might help mitigate these degeneracies, but note the mathematical possibility of transformations in both lens planes that keep observables unchanged, complicating inference.

Given these degeneracies, the authors explicitly leave to future modeling work the question of whether the full dataset for J1721+8842 can simultaneously constrain both H0 and w despite maximally degenerate lens models, highlighting a concrete unresolved issue crucial to leveraging this system for cosmography.

References

Whether all the available data can constrain both $H_0$ and $w$ simultaneously, even when considering maximally degenerate lensed models is left for the future modeling and cosmology papers on this lens.

— J1721+8842: The first Einstein zig-zag lens  (2411.04177 - Dux et al., 2024) in Section 3 (Discussion)

Estimating $\kappa_{\rm ext}$ is typically done in time-delay cosmography via weighted galaxy number counts or weak lensing \citep[e.g.][]{wells_tdcosmo_2023}, but this has not yet been attempted for compound lensing dark energy constraints, so it remains unknown to what extent an external mass-sheet component from over- or under-dense environments could bias $w$ in practice.

— Harnessing stellar kinematics to constrain dark energy with the double-source-plane gravitational lens SDSS J0946+1006  (2609.08573 - Ballard et al., 8 Sep 2026) in Section 5, “Conclusions”