Magnification of Quasars by Cosmologically Distributed Gravitational Lenses

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Abstract

We present an analytic method for computing the magnification of quasars by cosmologically distributed gravitational lenses. Given the inputs of the observed luminosity function of quasars and the magnification probability distribution of gravitational lenses, we obtain corrections for the effects of magnification on the luminosity function of quasars. We use analytic models of the observed luminosity function of quasars derived in our previous paper and lens models consisting of different combinations of compact objects, isolated galaxies, and clusters of galaxies, with their properties compatible with current observational limits. Our calculations indicate that 8%-80% of the bright (M_B ≈ -30) quasars at z = 3 are enhanced in magnitude-limited samples as a result of the magnification, while 0.2%-1.2% of the faint (M_B ≈ - 24) quasars at z = 3 are reduced at the same time. This is not enough magnification bias to affect significantly the number-magnitude relation of observed quasars and the redshift distribution of bright quasars, at least up to z ≈ 4. The main uncertainties in our estimates of the magnification are caused by the weak constraints on the cosmological density of compact objects with masses greater than 0.01 M_sun, which are the efficient lenses to produce the magnification. We conclude that the magnification bias on observed quasars by cosmologically distributed gravitational lenses is unlikely to be large, within a factor of 3 or so at z ≲ 4, unless there is a large number of compact objects with cosmological density much in excess of Ω_compact ≳ 0.1 in the universe. Our results indicate, however, that even a small magnification bias on samples of bright quasars at high redshifts may have a bearing on searches for multiple-imaged quasars.

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