Flux conservation and random gravitational lensing

This paper presents some approximate methods for calculating the amplification distribution function $$\textit f(A)$$ for a point source observed through a population of randomly placed gravitational lenses. In the limit of small lensing optical depth, the single-lens $$\textit f(A)$$ for point lenses automatically satisfies the general constraint of average flux conservation, for lenses placed either in a thin screen or throughout a Friedmann universe of arbitrary $$\Omega_0$$. Various approximations for extending these results to the non-linear regime are discussed: for cases in which many lenses affect the beam, $$\textit f(A)$$ retains the $$1/A^3$$ form appropriate to the single-lens case. When amplification is expressed in terms of the mean ampification $$\langle A \rangle, \textit f(A/\langle A \rangle)$$ saturates: the probability of a given mplification relative to the mean tends to a finite limit with increasing optical depth. These results are applied to quasars. Previous calculations of quasar–galaxy associations due to lensing by stars in galaxy haloes are found to have overestimated the strength of the effect. Lensing may account for the apparent evolution of the bright end of the quasar luminosity function at $$z\gtrsim2$$, but only if $$\Omega_0\lesssim 0.1$$.