The Multiband Magnification Bias for Gravitational Lenses

We present a generalization of the concept of magnification bias for gravitationally lensed quasars in which the quasars are selected by flux in more than one wavelength band. To illustrate the principle, we consider the case of two-band selection, in which the fluxes in the two bands are uncorrelated, perfectly correlated, or correlated with scatter. For uncorrelated fluxes, we show that the previously held result-that the bias is the product of the single-band biases-is generally false. We demonstrate some important properties of the multiband magnification bias using model luminosity functions inspired by observed correlations among X-ray, optical, infrared, and radio fluxes of quasars. In particular, the bias need not be an increasing function of each flux, and the bias can be extremely large for nonlinear correlations. The latter fact may account for the high lensing rates found in some X-ray/optical- and infrared/radio-selected samples.