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3 A lens survey with the Planck Surveyor   

Estimates of the numbers of detectable galaxies expected in a Planck survey are listed in Table1. These were calculated by combining the counts discussed above with the estimates of the sensitivity of Planck (Bersanelli et al. (1996)) quoted in Table2. In order to produce conservative estimates, an Einstein--de Sitter world model and an evolving population of lenses are assumed in the calculations. The results can be scaled to match other scenarios using the counts in Figs2(a)&(b). Note that the predicted number of lenses could be increased by a factor of about 5 if both a non-evolving model of lensing galaxies and a non-zero cosmological constant were assumed.

 
Table 1:   The surface densities of both lensed and unlensed sources expected in a Planck survey at different signal-to-noise ratios. The expected sensitivities of a Planck survey are listed in Table2. The surface densities are calculated assuming galaxy evolution model A, an Einstein--de Sitter world model and an evolving population of lensing galaxies. The units of are .

 
Table 2:   The sensitivities and confusion limits expected in a Planck survey. represents the expected galactic cirrus confusion noise (Helou & Beichman (1990)), assuming that the mean galactic background intensity at a wavelength of is . is the expected sensitivity of a Planck all-sky survey (Bersanelli et al. (1996)). represents the flux density at which the surface density of galaxies is expected to exceed ; the expected point source confusion noise .

Between about 0.1 and 1% of the point sources detected in a Planck survey are expected to be lensed by a foreground galaxy. Despite a relatively large ratio of lensed to unlensed galaxies at wavelengths of and , the absolute surface density of detectable sources is expected to be rather small, and so the most useful wavelengths for detecting lenses will be and . Several tens of lenses could be detected per unit solid angle in this systematic survey. Hence, despite the inevitable exclusion of lenses that are hidden by galactic emission at galactic latitudes less than about , several hundred lenses could still be detected in an all-sky survey, increasing the number of known lenses by an order of magnitude. The bright counts of unlensed galaxies in the submillimetre/far-infrared waveband would also be determined accurately in such a survey, and so tight limits could be imposed to the form of evolution of the global star-formation rate at moderate redshifts.

How efficiently could lensed sources be separated from the distant unlensed galaxies that are detected in this survey? A careful analysis of the submillimetre-wave colours of the point sources detected by Planck could probably be used to reduce the size of the sample by a small factor. Colour--colour and colour--magnitude diagrams that are derived from a simulated - sub-field of a Planck survey are presented in Fig.3. The distribution of lensed and unlensed galaxies are clearly different in each plot: lensed galaxies are found exclusively at large redshifts; and if the dust temperature in star-forming galaxies is correlated with their luminosity, then lenses would be expected to have redder colours as compared with unlensed galaxies that emit the same flux densities. The field of points in Fig.3(a) is bounded at small and large redshifts by lines with a gradient of about 0.6. This slope reflects the relative wavelengths of the three observing bands at , and . If the emissivity of dust grains was independent of wavelength, then both lensed and unlensed galaxies would lie on a single line with this slope. The position of each galaxy on this line would be determined by its redshift and dust temperature. The points in Fig.3(a) are spread within a box-like region because a wavelength-dependent emissivity is assumed.

 
Figure 3:  Colour--colour and colour--magnitude diagrams for both lensed and unlensed galaxies in a - sub-field of a Planck all-sky survey. A sub-field is plotted in order to avoid saturating the figure with a very large density of points. The galaxies are selected to exceed the - sensitivity of a Planck survey at a wavelength of . Lensed galaxies are always found at redshifts greater than unity, and are typically redder as compared with unlensed galaxies.

A more vivid representation of the lensed and unlensed galaxies that would be expected in a Planck survey is shown in Fig.4. The simulated colours of the sources expected in a - field are presented, with the lensed galaxies highlighted by white cross-hairs. As shown more quantitatively in Fig.3, the lensed galaxies tend to be rather red as compared with unlensed galaxies that emit the same submillimetre-wave flux density.

 
Figure 4:   A simulated - field of a Planck all-sky survey with no galactic contamination or instrumental noise. The primary survey wavelength is . The flux densities of the lensed and unlensed galaxies at the three wavelengths surveyed by Planck in the submillimetre waveband -- , and -- are represented by the intensity of the red, green and blue colours of the pixels respectively. The lensed sources are highlighted by white cross-hairs. The intensity scale at wavelengths of , and saturates at , and respectively. The pixels are square and are in area. For acceptable reproduction of this figure on paper a high-quality colour printer is required.

In order to diagnose lensed sources unequivocally high-resolution imaging, using either 8-m telescopes or large millimetre/submillimetre-wave interferometer arrays (MIAs; Downes (1996)), would be required in order to detect lensed arcs, rings or multiple images in the candidate sources. At a wavelength of a large MIA could detect a 10-mJy source at a signal-to-noise ratio of about at 0.1-arcsec resolution in a 2-second integration (Brown (1996)). Hence, in a 1-minute integration a large MIA could determine whether or not a detected source was gravitationally lensed. However, the angular resolution of Planck is about and the width of the primary beam of an MIA is about , and so a more accurate position would be required for a candidate lens unless about 100 different pointings of an MIA were used for each follow-up observations. The FIRST space-borne observatory (Beckwith et al. (1993)) will be equipped with a submillimetre-wave bolometer array receiver with a 6-arcmin field of view and a resolution of about . Hence, in a single pointed observation FIRST could locate a candidate lens to within a single primary beam area of a large MIA for an efficient follow-up observation. The submillimetre-wave flux densities of the faintest sources in a Planck survey are expected to be about . These sources could be detected by FIRST at a signal-to-noise ratio of about 10 in a 1-minute integration. Hence, lens candidates could be observed in about three months of FIRST observations, a period comparable with the time required for a shallow FIRST galaxy survey (Rowan-Robinson (1997)). About 100 pointings of a large MIA could also be used to determine an accurate position for a candidate lens within a - Planck resolution element, however several minutes of integration would be required for each candidate.

The most efficient technique for diagnosing gravitational lenses detected in a Planck survey would be to image the fields of sources using FIRST in order to find more accurate positions, and then to image the localised source using a large ground-based MIA. In this way a sample of several hundred lenses, drawn from about candidates, could be catalogued in a concerted programme of observations lasting several months.