It is well known that the gravitational lensing by the foreground objects (e.g., galaxies) can alter the apparent brightness of background objects (e.g., quasars), which may contaminate our observations. Three decades ago, Barnothy and Barnothy (1968) proposed that all the quasars were nothing but the gravitationally magnified images of Seyfert galactic nuclei. Press and Gunn (1973) showed that the probability of occurrence of gravitational lensing in an universe is nearly unity. For many years there had been a lack of both convincing observational and theoretical supports for these speculations. However, numerous and unprecedented deep galaxy surveys have recently revealed a considerably large population of faint galaxies (Metcalfe et al. 1996; references therein). This motivates one to readdress the question if the observations of background objects are seriously affected by the gravitational lensing effect of foreground galaxies? For this purpose, Zhu & Wu (1997) have calculated the lensing cross-sections of background quasars by the foreground galaxies, and concluded that, despite the fact that there is a considerably high surface number density of faint galaxies the total lensing cross-sections by galaxies towards a distant source are still rather small, when only a special cosmological model of is considered.
Nevertheless, the optical depth (probability) of gravitational lensing depends sensitively on the cosmological models. It is worthy of examining whether the above claim is valid under general cosmological models. In this paper, we extend our previous work to a variety of cosmological models, which are characterized by the mass density parameter and the normalized cosmological constant (cf., Carroll, Press & Turner 1992). This is well motivated because cosmological models with nonzero cosmological constant have become quite popular recently. Many years ago, Gott, Park & Lee (1989) have given the general expressions for the optical depth and mean image separation in general Friedman-Lema^itre-Robertson-Walker (FLRW) cosmological models. Yet, these expressions are complicated and thereby hard to use in practice. One of the purposes of this paper is thus to simplify the formula. Furthermore, we would like to investigate how the cosmological parameters affect the estimate of the lensing cross-section.
© European Southern Observatory (ESO) 1998
Online publication: September 17, 1998