The approximate gravitational lensing multiple plane mass sheet degeneracy

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.

💡 Research Summary

This paper revisits the mass‑sheet degeneracy (MSD) in the context of strong gravitational lensing when multiple lens planes are involved. The authors start by recalling that in single‑plane lensing the MSD arises because adding a uniform convergence (a “mass sheet”) and simultaneously rescaling the source plane leaves all observable image positions unchanged, while affecting time‑delay distances and thus the inferred Hubble constant H₀. They then extend the formalism to N + 1 planes (N lens planes plus a source plane) and write the full recursive lens equation (Eq. 1) that includes external convergence κ(zₙ,zₘ) and shear Γ(zₙ,zₘ) from line‑of‑sight (LOS) structures, as well as possible internal mass sheets (κ_cₘ, Γ_cₘ) located at the redshifts of the lens planes themselves.

The key step is to show that, under the tidal (first‑order) approximation where all convergence and shear terms are small, the combined effect of internal and external sheets can be absorbed into a set of “total” convergence and shear fields, κ_tot and Γ_tot, defined as the sum of the cosmological contribution and the contributions from discrete sheets (Eqs. 20‑22). With these definitions the original lens equation can be rewritten in a form identical to an MSD‑reduced equation (Eq. 5) by introducing scaling factors λₙ, λₙₘ, and λ_αₙ (Eqs. 23‑27). The λ factors are simply λₙ = 1/(1 – κ_tot(zₙ,0)), and similarly for λₙₘ, while λ_αₙ remains a free normalization of the deflection angles.

Crucially, the authors introduce “effective angular‑diameter distances” D_eff(zₙ,zₘ) = (1 – κ_tot(zₙ,zₘ)) D_A(zₙ,zₘ). Substituting D_eff for the usual FRW distances in the lens equation yields exactly the same algebraic structure as the MSD‑reduced model. This demonstrates that the presence of any number of internal or external mass sheets can be re‑interpreted as a modification of the distance ratios, and that the observable quantities are the ratios D_eff(zₙ,zₘ)/D_eff(zₙ,0) together with the reduced shears Γ_tot/(1 – κ_tot). Because there are N free λ_αₙ parameters but N(N + 1)/2 distance ratios, many of the distance combinations are degenerate with the λ_αₙ factors, leaving a substantial portion of the distance information unobservable.

When time‑delay information is added (Section IV), the same λ_αₙ factors appear in the expression for the Fermat potential, meaning that even with perfect knowledge of all time delays the degeneracy between mass sheets and H₀ persists. In other words, a simultaneous rescaling of the mass sheets and H₀ leaves the predicted time delays unchanged, so H₀ cannot be uniquely determined without external information (e.g., absolute magnifications, stellar kinematics, or priors on the mass distribution).

The paper provides a concrete example (Section V) showing how an internal sheet in one plane can mimic an external convergence, and how the degeneracy manifests in realistic galaxy‑lens configurations. Section VI discusses higher‑order effects: if second‑order shear terms are retained, the MSD is no longer exact but remains an excellent approximation as long as the shear amplitudes are modest.

Finally, the authors compare their framework with earlier works that required fine‑tuned alternating mass sheets across planes to achieve an exact MSD. They argue that such fine‑tuning is unnecessary; the degeneracy is generic once internal and external sheets are treated on equal footing and distances are replaced by their effective counterparts. The conclusion emphasizes that multi‑plane lens systems do not automatically break the MSD, and that careful modeling of both internal and LOS mass sheets is essential for unbiased H₀ measurements, especially in the era of large surveys (LSST, Euclid) that will deliver thousands of multi‑plane lenses.