Gravitational lensing by smooth inhomogeneities in the Universe

We review the formalism developed by Gunn on the gravitational lensing effect of smooth inhomogeneities in the Universe and derive some new simple analytical expressions for the dispersion in amplification. Two models for the distribution and evolution of density fluctuations are considered, namely a universe dominated by cold dark matter (CDM), and a scale-free universe with an initial density fluctuation spectrum corresponding to white noise. We find that the dispersion in amplification due to large-scale | (0. 5 h^-1 Mpc) | structures is negligible in both models even for sources at redshifts |zₛ 4|⁠, but the dispersion increases substantially when matter distribution on smaller scales is taken into consideration and it is very sensitive to the nature of the mass distribution. In the CDM model, the dispersion remains small for |zₛ 4| even for a smoothing scale as small as |10 h^-1| kpc. In the scale-free model, the dispersion is about 33 per cent for |zₛ 2. 5| if the smoothing scale is |30 h^-1| kpc. Such a large dispersion indicates that shear may be appreciable along some lines-of-sight, resulting in appreciable image distortions and enhanced amplification.