A Malmquist-like bias in the inferred areas of diamond caustics and consequences for inferred time delays of gravitationally lensed quasars

Abstract

Quasars are quadruply lensed only when they lie within the diamond caustic of a lensing galaxy. This precondition produces a Malmquist-like selection effect in observed populations of quadruply lensed quasars, overestimating the true caustic area. The bias toward high values of the inferred logarithmic area, $\ln A_{inf}$, is proportional to the square of the error in that area, $\sigma^2_{\ln{A}}$. In effect, Malmquist's correction compensates post-hoc for a failure to incorporate a prior into parameter optimization. Inferred time delays are proportional to the square root of the inferred caustic area of the lensing galaxy. Model time delays are biased long, leading to overestimates of the Hubble constant. Crude estimates of $\sigma_{\ln A}$ for a sample of 13 quadruple systems give a median value of 0.16. We identify a second effect, "inferred magnification bias,'' resulting from the combination of selection by apparent magnitude and errors in model magnification. It is strongly anti-correlated with caustic area bias, and almost always leads to underestimates of the Hubble constant. Malmquist's scheme can be adapted to priors on multiple parameters, but for quad lenses, the negative covariances between caustic area and absolute magnitude are poorly known. Inferred magnification bias may even cancel out caustic area bias, depending upon (among other things) the slope of the number magnitude relation for the sample. Proper correction for these combined effects can, in principle, be built into Bayesian modeling schemes as priors, eliminating the need for Malmquist-style approximation, but is likely to be challenging in practice.