Measuring lensing magnification of quasars by large scale structure using the variability-luminosity relation

We introduce a technique to measure gravitational lensing magnification using the variability of type I quasars. Quasars' variability amplitudes and luminosities are tightly correlated, on average. Magnification due to gravitational lensing increases the quasars' apparent luminosity, while leaving the variability amplitude unchanged. Therefore, the mean magnification of an ensemble of quasars can be measured through the mean shift in the variability-luminosity relation. As a proof of principle, we use this technique to measure the magnification of quasars spectroscopically identified in the Sloan Digital Sky Survey (SDSS), due to gravitational lensing by galaxy clusters in the SDSS MaxBCG catalog. The Palomar-QUEST Variability Survey, reduced using the DeepSky pipeline, provides variability data for the sources. We measure the average quasar magnification as a function of scaled distance (r/R ₂₀₀) from the nearest cluster; our measurements are consistent with expectations assuming Navarro-Frenk-White cluster profiles, particularly after accounting for the known uncertainty in the clusters' centers. Variability-based lensing measurements are a valuable complement to shape-based techniques because their systematic errors are very different, and also because the variability measurements are amenable to photometric errors of a few percent and to depths seen in current wide-field surveys. Given the volume data of the expected from current and upcoming surveys, this new technique has the potential to be competitive with weak lensing shear measurements of large-scale structure.