Australia Telescope National Facility

Using the 6dF galaxy redshift survey to detect gravitationally-lensed quasars
Daniel J. Mortlock, Michael J. Drinkwater, PASA, 18 (2), in press.

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Lens statistics

Mortlock & Webster (2000) contains a detailed description of how to calculate the number of spectroscopic lenses expected in a GRS. Briefly, one must integrate over the galaxy (i.e., deflector) and quasar (i.e., source) populations to find the number of lenses which satisfy the following three criteria:

The total flux from the composite object (i.e., the sum of the light from the galaxy and quasar) must be brighter than the survey's flux limit.
The quasar must be bright enough to be detectable in the spectrum of the lens.²Defining the summed quasar images' magnitude as $m_{\rm q}$ and the galaxy's magnitude as $m_{\rm g}$ , this criterion is taken to be satisfied if
$m_{\rm q} \leq m_{\rm g} + \Delta m_{\rm qg}$ (Kochanek 1992). The value of

$\Delta m_{\rm qg}$ depends on the quality of the data and the ``distinctiveness'' of the quasars' spectra.
The galaxy must be sufficiently bright that the lens is identified as an extended source; it is this condition which ensures that only lenses with nearby deflectors are selected.

Figure 10 of Mortlock & Webster (2000) shows how the event rate increases with a survey's magnitude limit and

$\Delta m_{\rm qg}$ , but these results apply to a $B_{\rm J}$ -selected GRS, whereas the 6dF sample is subject to a K-band flux limit. The target list, taken from the 2MASS survey, is also K-selected, but

$\Delta m_{\rm qg}$ must be calculated in one of the optical bands (e.g., B, $B_{\rm J}$ V, R, etc.) covered by the 6dF spectra. A further complication is the paucity of available information about the quasar population at infrared wavelengths. These difficulties were circumvented by performing the calculation in the $B_{\rm J}$ -band (in order to facilitate comparison with Mortlock & Webster 2000), parameterising the deflector and source populations in terms of their $B_{\rm J} - K$ colours.

The $B_{\rm J}$ - and K-band galaxy luminosity functions measured, respectively, by Folkes et al. (1999) and Loveday (2000) are consistent provided the local galaxy population has

$\langle B_{\rm J} - K \rangle \simeq 4$ . Hence the 6dF GRS should be approximately equivalent to $B_{\rm J}$ -selected survey with a magnitude limit of $\sim 17$ .

There have been no systematic K-band quasar surveys, but several attempts have been made to infer the luminosity function at these wavelengths from observations in other bands. Most quasar samples are selected using ultraviolet excess (UVX) methods (e.g., Boyle, Shanks & Peterson 1988), but dust obscuration at these wavelengths may result in serious incompleteness. UVX-selected samples typically have

$\langle B_{\rm J} - K \rangle \simeq 2.5$ , but the radio-selected Parkes Half-Jansky Flat-Spectrum sources (Drinkwater et al. 1997) are considerably redder, with

$2 < B_{\rm J} - K < 10$ (Webster et al. 1995). These discrepancies should be resolved by upcoming infrared surveys (e.g., Warren, Hewett & Foltz 2000), but, for the moment, a range of

$\langle B_{\rm J} - K \rangle$ values must be considered.

The uncertainty in the quasars' colours also affects the determination of

$\Delta m_{\rm qg}$ , their spectral prominence in the survey data. The 6dF spectra will be of similar quality to those obtained during the 2dF survey, for which Mortlock & Webster (2001) estimated

$\Delta m_{\rm qg} \simeq 2$ . However this figure is relevant only if UVX-selected quasars are representative of the population as a whole. Whilst

$\Delta m_{\rm qg}$ is calculated in the $B_{\rm J}$ -band, the 6dF spectra extend past V and R, almost to I, and quasars with

$B_{\rm J} - K \simeq 6$ would be much more prominent in the red end (close to the I-band) of the spectra than UVX-selected objects. In terms of the model used here, this leads to an increase in

$\Delta m_{\rm qg}$ by up to a magnitude, although the change depends on the exact shape of the galaxy and quasar spectra (specifically, their $B_{\rm J} - I$ colours).

**Table 1:** Number of lenses expected in the 6dF GRS
	optical spectroscopy		infrared spectroscopy
$\langle B_{\rm J} - K\rangle_{\rm quasar}$	$\Delta m_{\rm qg}$	$N_{\rm lens}$	$\Delta m_{\rm qg}$	$N_{\rm lens}$
2	1.5	0.3	0.0	0.02
4	2.0	1	2.0	1
6	2.5	3	4.0	14
8	3.0	7	6.0	27

The results of integrating over the deflector and source populations are given in Table 1. With optical spectroscopy the 6dF GRS should contain about 5 lenses if quasars are as red as inferred by Webster et al. (1995), but there would also be a large number of lensed quasars in the sample that are too red to be detected in the optical spectra. The only way to find these objects would be to obtain infrared spectroscopy of the entire sample; the likely yield from this procedure is also given in Table 1. In this case the increase of $N_{\rm lens}$ with

$\Delta m_{\rm qg}$ is not as marked, as such a survey would probe magnitudes at which the quasar luminosity function was much flatter (e.g., Boyle et al. 1988)

Next Section: Conclusions
Title/Abstract Page: Using the 6dF galaxy
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Contents Page: Volume 18, Number 2

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