Overconstrained gravitational lens models and the Hubble constant

Abstract

It is well known that measurements of H0 from gravitational lens time delays scale as H0 1 - κE, where κE is the mean convergence at the Einstein radius RE but that all available lens data other than the delays provide no direct constraints on κE. The properties of the radial mass distribution constrained by lens data are RE and the dimensionless quantity ζ = REα″(RE)/(1 - κE), where α″(RE) is the second derivative of the deflection profile at RE. Lens models with too few degrees of freedom, like power-law models with densities ρ ∝ r-n, have a one-to-one correspondence between ζ and κE (for a power-law model, ζ = 2(n - 2) and κE = (3 - n)/2 = (2 - ζ)/4). This means that highly constrained lens models with few parameters quickly lead to very precise but inaccurate estimates of κE 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.