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

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1. Introduction

It is now well known that gravitational lensing statistics is a useful probe of the geometry of the universe, especially for the determination of the cosmological constant. In a recent paper (Cooray et al. 1998a; hereafter CQM), we calculated the expected number of gravitationally lensed sources in the Hubble Deep Field (HDF; Williams et al. 1996) due to foreground galaxies as a function of the cosmological parameters, and estimated these parameters based on the observed lensing rate in the HDF. The expected lensing rate was calculated based on the redshift distribution of HDF galaxies as determined by the photometric redshift catalogs. Similar to multiple lensing events due to foreground galaxies, clusters of galaxies lens background sources. Such lensed sources with high magnification appear as arcs, and the number statistics of gravitationally lensed arcs can be used to determine the cosmological parameters (e.g., Wu & Mao 1996; Bartelmann et al. 1998) and study the galaxy evolution at high redshifts (e.g., Bézecourt et al. 1998).

The number statistics of lensed optical arcs have been studied by Wu & Mao (1996), where they considered the effect of [FORMULA] on the predicted lensing rate, and by Bartelmann et al. (1998), where simulations of galaxy clusters were used to calculate the number of lensed sources. The former study relied on the spherical singular isothermal potential to describe foreground lensing clusters, while the latter study used the cluster potentials observed with numerical simulations. In between these two studies, Hamana & Futamase (1997) showed that the evolution of background source population can affect the lensing rate, while Hattori et al. (1997) refined the observed lensing rate by including observational effects, such as seeing.

In the present paper, we extend our previous work on the HDF (CQM) to estimate the number of lensed optical, radio and sub-mm lensed sources on the sky due to foreground galaxy clusters. We describe the background galaxies by the redshift and magnitude or flux distribution of sources in the HDF. We also assume that the HDF luminosity function, as determined by Sawicki et al. (1997), is a valid description of the distant universe. Thus, one of the main differences between the present paper and previous studies involving arc statistics is that we use individual redshifts to calculate lensing probabilities, and use magnitude information to account for various systematic effects, especially magnification bias present in magnitude-limited optical search programs to find lensed arcs towards galaxy clusters. A main difference between CQM and the present work is that we now describe the number density of foreground lensing objects, galaxy clusters, and their evolution using the Press-Schechter theory (PS; Press & Schechter 1974), normalized to the local cluster abundance.

Similar to optical arcs, galaxy clusters are also expected to lens background radio sources. Such lensed sources with high magnification should appear as arcs in radio surveys. The number statistics of lensed radio sources can be used to determine the cosmological parameters, to study the radio source evolution at high redshifts, and as discussed later, properties of star forming galaxies at moderate to high redshifts. The number statistics of lensed radio sources due to foreground clusters were first calculated by Wu & Hammer (1993). They predicted [FORMULA] 10 lensed radio sources on the sky down to a flux density limit of 0.1 mJy, and [FORMULA] 100 lensed radio sources down to 10 µJy at 2.7 GHz (Fig. 10 in Wu & Hammer 1993). At the source detection level of the VLA FIRST survey ([FORMULA] 1 mJy; Becker et al. 1995), there are only [FORMULA] 2 to 3 lensed radio sources expected on the whole sky, and when compared to the area of the survey and its resolution, it is likely that there is no lensed source present. This prediction is compatible with observational attempts to find lensed radio sources; Andernach et al. (1997) searched the FIRST survey near Abell cluster cores and found no convincing candidates, and a statistical analysis of the radio positions towards clusters showed no preferential tangential orientation, as expected from gravitational lensing. Recently, a sub-mm selected source, SMM02399-0136, towards cluster A370 was found to be lensed with an amplification of 2.5 (Ivison et al. 1998). The source was detected at 1.4 GHz, with a flux density of [FORMULA] 525 µJy. This detection prompted us to calculate the expected number of lensed µJy sources present on the sky due to foreground clusters, and to refine the previous predictions in Wu & Hammer (1993). Since the predictions in Wu & Hammer (1993) for sources down to mJy level are still expected to be valid, we will only concentrate on the µJy sources here.

We also extend our calculation to estimate the number of expected sub-mm sources on the whole sky due to foreground clusters. Our calculation is prompted by recent observational results from the new Sub-millimeter Common-User Bolometer Array (SCUBA; see, e.g., Cunningham et al. 1994) on the James Clerk Maxwell Telescope, where a sample of gravitationally lensed sub-mm sources has now been observed by Smail et al. (1997, 1998). The gravitational lensing of background sub-mm sources due to foreground galaxy clusters was first studied by Blain (1997), using a model of a lensing cluster with predicted source counts for background sources. Blain (1997) showed that the surface and flux densities of lensed sources exceed those of galaxies within the lensing cluster, and their values. This behavior, primarily due to the slope of the source counts and the fact that the distance sources are intrinsically brighter at sub-mm wavelengths, has now allowed the observation of moderate to high redshift dusty star forming galaxies, which are amplified through the cluster potentials (e.g., Ivison et al. 1998). Even though lensing of sub-mm sources has been studied in literature, no clear prediction has been made for the total number of sources lensed due to foreground clusters. Also, past calculations have relied mostly on models of background source counts that were based on different evolutionary scenarios for star forming galaxies.

In Sect. 2 we discuss our calculation and its inputs. In Sect. 3 we present the expected number of optical, radio and sub-mm lensed sources, and in Sect. 4 we outline possible systematic errors involved in our calculational method. In Sect. 5, we compare our predicted number of optical arcs in the whole sky to the observed number of arcs in the Le Fèvre et al. (1994) cluster sample. In the same section we discuss the possibility of large area optical arc survey, using the Sloan Digitized Sky Survey (SDSS; Loveday & Pier 1998) as an example. In Sect. 5 discuss the possibility of detecting lensed µJy sources and sub-mm sources. A summary is presented in Sect. 6. We follow the conventions that the Hubble constant, [FORMULA], is 100 h km s-1 Mpc-1, the present mean density in the universe in units of the closure density is [FORMULA], and the present normalized cosmological constant is [FORMULA]. In a flat universe, [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998