Overconstrained gravitational lens models and the Hubble constant

It is well known that measurements of H₀ from gravitational lens time delays scale as H₀ ∝ 1 - κ_E, where κ_E is the mean convergence at the Einstein radius R_E but that all available lens data other than the delays provide no direct constraints on κ_E. The properties of the radial mass distribution constrained by lens data are R_E and the dimensionless quantity ξ = R_Eα″(R_E)/(1 - κ_E), where α″(R_E) is the second derivative of the deflection profile at R_E. Lens models with too few degrees of freedom, like power-law models with densities ρ ∝ r^-n, have a one-to-one correspondence between ξ and κ_E (for a power-law model, ξ = 2(n - 2) and κ_E = (3 - n)/2 = (2 - ξ)/4). This means that highly constrained lens models with few parameters quickly lead to very precise but inaccurate estimates of κ_E and hence H₀. Based on experiments with a broad range of plausible dark matter halo models, it is unlikely that any current estimates of H₀ from gravitational lens time delays are more accurate than ${\sim} 10{{\ \rm per\ cent}}$ , regardless of the reported precision.