Observation of Gravitational Lensing in the Clumpy Universe
  We discuss how inhomogeneities of the universe affect observations of the gravitational lensing; (1) the bending angle, (2) the lensing statistics and (3) the time delay. In order to take account of the inhomogeneities, the Dyer-Roeder distance is used, which includes a parameter representing the clumpiness of the matter along the line of sight. It is shown analytically that all three combinations of distances appearing in the above observations (1)-(3) are monotonic with respect to the clumpiness in general, for any given set of the density parameter, cosmological constant and redshifts of the lens and the source. Some implications of this result for the observation are presented; the clumpiness decreases both the bending angle and the lensing event rate, while it increases the time delay. We also discuss cosmological tests using the gravitational lensing in the clumpy universe. 
Hideki Asada