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Astron. Astrophys. 338, 777-780 (1998)

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3. Magnification cross-sections of gravitational lensing

First 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)

[EQUATION]

where [FORMULA] is the observed angular position of the source (image position), [FORMULA] is the angular radius of Einstein ring and [FORMULA] 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 [FORMULA] produced by a single SIS galaxy at [FORMULA] is

[EQUATION]

[EQUATION]

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)

[EQUATION]

where i indicates the morphological type of galaxies: i=(E, S0, S). The above expression can be converted into the velocity dispersion distribution through the empirical formula between the luminosity and the central dispersion of local galaxies [FORMULA]. We keep the same parameters ([FORMULA]) as those adopted by Kochanek (1996) based on the surveys (Loveday et al. 1992, Marzke et al. 1994), which yield [FORMULA] km/s and [FORMULA] for [FORMULA] galaxies, and the morphological composition [FORMULA] for ([FORMULA]). For the spatial distribution of galaxies, we use a general FLRW cosmological model parametrized by [FORMULA] and [FORMULA], which has been outlined in Sect. 2. Finally, the total dimensionless magnification cross-section by galaxies at redshifts ranging from 0 to [FORMULA] for the distant sources like quasars at [FORMULA] is

[EQUATION]

The parameter [FORMULA] represents the effectiveness of the i-th morphological type of galaxies in producing double images (Turner et al. 1984), which reads

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the velocity bias between the velocity dispersion of stars and of dark matter particles. The above equation can be further written as

[EQUATION]

if the integral is performed from 0 to [FORMULA]. In practice, the galaxy luminosities have the minimum and maximum limits, and, therefore, Eq. 13is the maximum estimate of [FORMULA]. The [FORMULA] dependent factor [FORMULA] is

[EQUATION]

For general FLRW cosmologies, an analytic expression is found (see Eq. 15):

[EQUATION]

where [FORMULA] can be calculated through Eq. 6. For a flat universe ([FORMULA]), it reduces to (Turner 1990)

[EQUATION]

If [FORMULA] and [FORMULA], it reads (Turner et al. 1984)

[EQUATION]

We should point out that the expression of Eq. 11is very useful. Dividing the expression by [FORMULA], one gets the fraction of the sky within redshift [FORMULA] which is magnified by the factor greater than µ:

[EQUATION]

We employ [FORMULA] denoting the total effective parameter of all galaxies in producing multiple images. Further omitting the µ-dependent term in Eq. 18, one obtains the conventional optical depth for multiple images.

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).

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© European Southern Observatory (ESO) 1998

Online publication: September 17, 1998
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