A Malmquist-like Bias in the Inferred Areas of Diamond Caustics and Consequences for the Inferred Time Delays of Gravitationally Lensed Quasars
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2024-02-01 |
| Journal | The Astrophysical Journal |
| Authors | Derek Baldwin, Paul L. Schechter |
| Institutions | IIT@MIT, Kavli Institute for Particle Astrophysics and Cosmology |
| Citations | 4 |
Abstract
Section titled âAbstractâAbstract 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, <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ overflow=âscrollâ> <mml:mi>ln</mml:mi> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>inf</mml:mi> </mml:mrow> </mml:msub> </mml:math> , is proportional to the square of the error in that area, <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ overflow=âscrollâ> <mml:msubsup> <mml:mrow> <mml:mi>Ď</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> . In effect, Malmquistâs correction compensates posthoc 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 <mml:math xmlns:mml=âhttp://www.w3.org/1998/Math/MathMLâ overflow=âscrollâ> <mml:msub> <mml:mrow> <mml:mi>Ď</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ln</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> for a sample of 13 quadruple systems give a median value of 0.16. We identify a second effect, an âinferred magnification bias,â resulting from the combination of selection by apparent magnitude and errors in model magnification. It is strongly anticorrelated with caustic area bias, and almost always leads to underestimates of the Hubble constant. Malmquistâs 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 a Malmquist-style approximation, but is likely to be challenging in practice.