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Astron. Astrophys. 341, 653-661 (1999)

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3. Predicted numbers

3.1. Optical arcs

In order to calculate the expected number of lensed arcs on the sky, we use the photometric redshift catalog by Sawicki et al. (1997) for HDF galaxies. This catalog contains redshift information for 848 galaxies and is complete down to a magnitude of 27 in I-band. However, HDF allows detection of sources down to a magnitude limit of 28.5 in I-band, and contains 1577 sources down to 28 in I-band, excluding 43 apparent stars (Sawicki et al. 1997). We use this extra information and complemented the photometric redshift catalog by equally distributing the additional number of optical sources between I-band magnitudes of 27 and 28, and between redshifts of 0 and 5. Since these sources are not expected to be at very low redshifts, where lensing probability is small, we do not expect to have created a systematic bias in our study, other than perhaps underestimate the lensing rate, if all these sources were in fact at high redshifts. Also, since these additional sources have very low magnitudes, at the limit of HDF, we do not expect these sources to make a large contribution to the total number of lensed arcs when the limiting magnitude of lensed search programs are at the bright end. However, in order to calculate the true number of arcs at faint magnitudes, it is essential that these sources be accounted for. The HDF galaxies are within an area of 4.48 arcmin2. We extrapolate the predicted number of lenses in the HDF to the whole sky, by assuming that HDF is an accurate description of the distant universe everywhere on the sky. Since HDF was carefully selected to avoid bright sources, it is likely that we have missed a large number of low redshift galaxies, but, such galaxies are not expected to contribute to the lensing rate.

We have calculated the expected number of gravitationally lensed arcs in by using Eq. (1) as a function of [FORMULA] and [FORMULA], and using [FORMULA] of 10. Since we are using the SIS model, the amplification is simply equal to the ratio of length to width in observed lensing arcs (see, e.g., Wu & Mao 1996), allowing us an easy comparison between observed number of arcs with length to width greater than 10 in Le Fèvre et al. (1994) survey. In Table 1, we list the expected number of strongly lensed arcs in the sky for different [FORMULA] and [FORMULA] values, together with the number of lensed sources at radio and sub-mm wavelengths.


Table 1. Predicted number of lensed optical, radio, and sub-mm sources on the sky due to foreground clusters. [FORMULA] is the expected number of lensed sub-mm sources, with flux densities greater than 50 mJy at 850 µm, towards clusters that are expected to be detected with Planck Surveyor (see Sect. 5.3.1). The horizontal lines across the two optical arc columns contain the current range of observed arc statistics (see Sect. 5.1.1).

3.2. Lensed radio sources

In order to describe the background µJy sources, we describe the redshift and number distribution observed towards the HDF by Richards et al. (1998). The main advantage in using the HDF data is the availability of redshift information for µJy sources. Also, HDF is one of the few areas where a deep radio survey down to a flux limit of [FORMULA] 2 µJy at 1.4 GHz has been carried out. The HDF contains 14 sources with flux densities of the order [FORMULA] 6 to 500 µJy at 8.5 GHz, and 11 of these sources have measured spectroscopic redshifts. We converted the 8.5 GHz flux densities to 1.4 GHz using individual spectral indices as presented by Richards et al. (1998). For sources with no measured spectral indices, we assumed an index of 0.4, the mean spectral index observed for µJy sources (Fomalont et al. 1991; Windhorst et al. 1993; Richards et al. 1998). For the 3 sources with no measured spectroscopic redshifts, we used photometric redshifts from the catalog of Fernández-Soto et al. (1998). We binned the redshift-number distribution in redshift steps of 0.25, and calculated the lensing probability using filled-beam formalism. The predicted number are tabulated in Table 1 for minimum amplifications of 2 and 10 respectively, and down to a flux limit of 10 µJy at 1.4 GHz.

3.3. Lensed sub-mm sources

In order to describe the background sub-mm sources, we again use the redshift and number distribution observed towards the HDF sources by Hughes et al. (1998). The HDF contains 5 sources with flux densities of the order [FORMULA] 2 to 7 mJy. Hughes et al. (1998) studied the probable redshifts of the detected sources by considering the optical counterparts and assigning probabilities for likely associations. In Table 1, we list the expected number of lensed sources on the sky for [FORMULA] and 10. We have also tabulated the expected number of lensed sources towards clusters in Planck all sky survey data, as described in Sect. 5.3.1.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998