## 3. Magnification cross-sections of gravitational lensingFirst of all, we consider the lensing cross-section (Turner, Ostriker & Gott 1984) due to a specific galaxy. Following Turner et al. (1984), we model the mass density profile of the total galaxy matter as the singular isothermal sphere (SIS), whose magnification for a point source is given by (Schneider, Ehlers & Falco 1992; Wu 1996) where is the observed angular position of the source (image position), is the angular radius of Einstein ring and is the velocity dispersion of the lensing galaxy. Note that we only include the contribution of the primary image because here we will not deal with the statistics of multiple images. The dimensionless magnification cross-section for a point source located at produced by a single SIS galaxy at is where the relation of Eq. 5has been employed. Now, let's consider the contributions of an ensemble of galaxies having different luminosities and redshifts. The present-day galaxy luminosity function can be described by the Schechter function (Peebles 1993) where The parameter represents the effectiveness
of the where is the velocity bias between the velocity dispersion of stars and of dark matter particles. The above equation can be further written as if the integral is performed from 0 to . In practice, the galaxy luminosities have the minimum and maximum limits, and, therefore, Eq. 13is the maximum estimate of . The dependent factor is For general FLRW cosmologies, an analytic expression is found (see Eq. 15): where can be calculated through Eq. 6. For a flat universe (), it reduces to (Turner 1990) If and , it reads (Turner et al. 1984) We should point out that the expression of Eq. 11is very
useful. Dividing the expression by , one gets
the fraction of the sky within redshift which
is magnified by the factor greater than We employ denoting the total effective
parameter of all galaxies in producing multiple images. Further
omitting the In the above calculations, we have assumed that the comoving number density of galaxies is constant. However, this may not hold true for the realistic situation. The influence of galaxy evolution on the lensing cross-section should also be taken into account. Zhu and Wu (1997) have include this effect by using the galaxy merging model proposed by Broadhurst et al. (1992), since the scenario of galaxy merging can account for both the redshift distribution and the number counts of galaxies at optical and near-infrared wavelengths (Broadhurst et al. 1992). There are two effects arising from the galaxy merging: The first is that there are more galaxies and hence more lenses in the past. The second is that galaxies are typically less massive in the past and hence less efficient as lenses. As a result of two effects, the total magnification cross-section remains roughly unchanged (Zhu & Wu 1997). © European Southern Observatory (ESO) 1998 Online publication: September 17, 1998 |