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

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5. Discussion

5.1. Lensed optical arcs

5.1.1. [FORMULA] from observations and predictions

According to Wu & Mao (1996), there are 9 arcs towards 39 clusters with [FORMULA] or roughly 0.2 to 0.3 arc per cluster in the bright EMSS arc surveys (Le Fèvre et al. 1994; Gioia & Luppino 1994). The current predictions for total number of clusters matching the criteria of EMSS arc survey clusters range from [FORMULA] 7500 to 8000 (see, e.g., Bartelmann et al. 1997). Thus we expect a total of [FORMULA] 1500 to 1900 such arcs. This estimate ignores the observational systematic effects in search programs, including observational constraints such as finite seeing (see, e.g., Hattori et al. 1997). Since, the result of such effects is to reduce the observed number, after making an additional correction, we estimate a total number of 1500 to 2500 arcs on the sky, which is slightly higher than the estimate made by Bartelmann et al. (1997). We find that our prediction is roughly in agreement with the observed number when [FORMULA] in a flat universe or [FORMULA] in an open universe. The range of [FORMULA] values when [FORMULA] is in agreement with our previous estimate based on the strong lensing rate in the HDF ([FORMULA] 95% C.I. CQM; also, Kochanek 1996), estimates of cosmological parameters based on the high redshift type Ia supernovae (Riess et al. 1998; [FORMULA]), and galaxy cluster baryonic fraction (Evrard 1997).

However, in order to derive tighter constraints on the cosmological parameters, we need to consider both the statistical and systematic errors in the present calculation, as well as the observed number of lensed arcs. In general, we find that the predicted number in a [FORMULA] dominated universe ([FORMULA]) cannot be used to explain the observed number of lensed arcs, even when we consider the extreme errors in our calculation and the observed statistics.

5.1.2. Future outlook

We have predicted roughly 1500 to 3000 lensed arcs on the sky with I-band magnitudes greater than 22 towards foreground massive clusters. In order to use arc statistics as a probe of the cosmological parameters, it is necessary that reliable results from a large area survey be used. The current observed statistics on lensed arcs come from the optical observations towards X-ray selected clusters in the EMSS sample, which covers an area of [FORMULA] 750 sq. degrees. In the near future, the Sloan Digitized Sky Survey (SDSS) will take both imaging and spectroscopic data over [FORMULA] steradians of the sky. It is likely that the SDSS will image most of the foreground massive clusters, similar to the ones that we have considered here. The optical data from this survey are expected to allow detection of sources down to the I band magnitude of 22. The imaging data will be limited by seeing effects, which is expected to limit the image resolutions to between 1.1 and 1.4 arcsecs. For the purpose of finding lensed luminous arcs, the seeing effects would be of a minor concern; the spatial extent of lensed arcs with length-to-width ratios greater than 10 are not likely to be heavily affected by observational effects. Based on our predictions, it is expected that there will be roughly 375 to 750 arcs in the SDSS imaging data. However, there are various practical limitations which will affect the search for lensed arcs in SDSS data. Especially due to the large volume of data, it is unlikely that one would be able to select lensed arcs by just looking at the images; specific algorithms to find lensed arcs are needed. By testing such algorithms against simulated data, it is likely that selection effects involved in the arc search process can be properly studied. By considering such selection effects and the observed lensing rate of luminous arcs, it may be possible in the future to obtain reliable estimates on the cosmological parameters based on arc statistics.

5.2. Lensed radio sources

We have predicted [FORMULA] 1500 lensed µJy sources, with [FORMULA], for a cosmology with [FORMULA] and [FORMULA]. The number with [FORMULA] for the same cosmology is [FORMULA] 200. When compared with the lensing rate for optical arcs down to I-band magnitude of 22 and amplifications greater than 10, we predict a similar, or slightly lower, rate for the µJy sources, down to a flux density limit of 10 µJy.

In comparison, Wu & Hammer (1993) predicted [FORMULA] 100 sources down to 10 µJy towards clusters. They performed this calculation for a cosmological model of [FORMULA], and using the X-ray luminosity function of Edge et al. (1990). For the same cosmological model, we predict [FORMULA] 0.2 sources with amplifications greater than 10. The difference between two predictions is primarily due to the description of the background sources. We have used redshift information, while Wu & Hammer (1993) used the radio luminosity function with no evolution assumption, an assumption which may have overestimated the number of lensed sources. There are also other differences between the two methods. For example, we have accounted for the galaxy cluster evolution for different cosmological models using PS theory, where the number of available foreground lensing clusters strongly decreases with an increase in the cosmological mass density, [FORMULA]. Such changes have not been accounted in the previous calculation.

5.2.1. Possibility of detection

Unlike optical surveys, radio surveys with interferometers such as the VLA and the MERLIN are subjected to effects arising from instrumental limitations, primarily effects associated with resolution. For example, there is a minimum and a maximum size for sources that can be detected and resolved with an interferometer. The largest angular scale to which the interferometer is sensitive restricts the detection of high amplification sources, which are expected to appear as arcs, with length to width ratios equal to amplification factors. For the VLA A-array at 1.4 GHz, sources larger than [FORMULA] 15" are not likely to be detected. Thus, observations of radio arcs with length to width ratios greater than 10 may not easily be possible. In SIS model for gravitational lensing, most of the lensed sources appear with amplification factors of 2 to 10. However, due to the convolution with synthesized beam, ranging from [FORMULA] 1" to 5", such sources are not likely to appear as arcs. Therefore, detection of lensed sources with small amplifications are likely to be confused with foreground and cluster-member radio sources, requiring a selection process to remove such confusing sources. Most of the confusion is likely to come from cluster member sources, rather than the foreground sources, as there is an overabundance of radio sources in clusters relative to random areas of the sky. As discussed in Cooray et al. (1998b), based on cluster observations at 28.5 GHz, this overabundance is likely to be high as factors of 5 to 7. It is likely that this overabundance exists at low frequencies such as 1.4 GHz. However, certain cluster member sources may easily be identified through source properties and appearances; sources such as wide-angle tail sources are usually found in cluster environments with dense IGM. Such an analysis may be limited to few types of sources, and there is no direct radio property, such as the radio spectral index or luminosity, that can be used to separate cluster member sources from background ones. The identification process of candidate lensed sources needs to consider the optical counterparts of radio sources; a joint analysis between radio and optical data may be required to recover the background radio sources lensed through galaxy cluster potentials. Additional observations, especially redshifts may be required to establish the lensed nature of µJy sources selected towards clusters. This is contrary to optical searches, where lensed galaxies can easily be established due to their arc-like appearances.

By considering the ratio between observed number of optical arcs and arclets and the ratio of surface density of optical to µJy sources, we expect to find [FORMULA] a total of 4 to 6 lensed µJy sources down to 10 µJy at 1.4 GHz towards A2218 and A370. For A370, one such source has already been recovered (Ivison et al. 1998), through the sub-mm observations of Smail et al. (1997). The VLA A-array 1.4 GHz data (Owen & Dwarakanath, in prep.), in which the source was detected allows detection of sources down to a flux limit of 50 µJy beam-1 (5 [FORMULA]). A quick analysis of the same archival data suggests that there is at least one more µJy lensed source towards A370 (Cooray et al., in prep.). It is likely that deep surveys of galaxy clusters with MERLIN and VLA will allow detection of µJy radio sources with amplifications of 2 to 10.

As discussed in Richards et al. (1998; see, also, Cram et al. 1998), µJy sources carry important information on the star formation rate and history. Thus, observational searches for lensed sources are expected to allow detection of moderate to high redshift star-forming galaxies. The search for such galaxies will be aided by the amplification due to gravitational lensing, allowing detections of faint sources, below the flux limits of regular surveys. It is likely that a careful analysis of lensed µJy sources will allow the study of star formation at moderate to high redshift galaxies. Also, the low redshift µJy sources, associated with spiral galaxies are not likely to be found through clusters, due to the low lensing rate. Based on our predictions and the detection of lensed sources towards A370, we strongly recommend that deep radio observations of lensing clusters be carried out to find lensed sources and that such detections be followed up at other wavelengths.

5.3. Lensed sub-mm sources

We have predicted [FORMULA] lensed sub-mm sources with flux densities greater than 2 mJy at 850 µm, and with amplifications greater than 2, for a cosmology with [FORMULA] and [FORMULA]. The number with [FORMULA] for the same cosmology is [FORMULA] 3100, while the number with [FORMULA] is [FORMULA] 500. We predict a lensing rate of [FORMULA] 4 sources per cluster with amplifications greater than 2 down to a flux limit of 2 mJy.

We compare our predicted number of lensed sources to the observed number towards a sample of galaxy clusters imaged with the SCUBA by Smail et al. (1997, 1998). This sample contains 7 clusters with redshifts in the range [FORMULA] 0.2 to 0.4. All of these clusters are well known lensing clusters in the optical wavelengths. Unfortunately, this sample is incomplete either in terms of X-ray luminosity or total mass. This incompleteness doesn't allow us to perform a direct comparison between the predicted and observed numbers. Out of the 7 clusters, 3 clusters have X-ray luminosities greater than the lower limit imposed in our calculation. Towards these three clusters, A370, A2390 & A1835, there are 8 sub-mm sources, all of which may be gravitationally lensed. This implies a total of [FORMULA] lensed sub-mm sources on the whole sky. Based on our lensing rate, we expect [FORMULA] 6 lensed sources towards 3 clusters; this exact number is strongly sensitive to the cosmological parameters. Here, we have assumed a spatially-flat cosmological model with [FORMULA] and [FORMULA]. The predicted and observed numbers seem to be in agreement with each other for low [FORMULA] values in a flat universe ([FORMULA]).

However, we cannot use the present observational data to derive cosmological parameters for several reasons. These reasons include source contamination in the lensed source sample and systematic biases in the foreground cluster sample. For example, it is likely that the lensed source sample presented by Smail et al. (1998) contain foreground and cluster-member sources. Since the foreground or cluster-member sources are less bright than the background lensed sources, this contamination is likely to be small (see, Blain 1997). An additional systematic bias comes from the selection effects associated with the foreground cluster sample. Since the observed clusters are well known lensing clusters with high lensing rates at optical wavelengths, it is likely that there may be more lensed sub-mm sources towards these clusters than generally expected. Therefore, it is likely that the Smail et al. (1998) sample is biased towards a higher number of lensed sub-mm sources.

In order to constrain cosmological parameters based on statistics of lensed sub-mm sources, results from a complete sample of galaxy clusters, preferably from a large area survey, are needed. Further SCUBA observations of galaxy clusters, perhaps the same cluster sample as the Le Fèvre et al. (1994) sample, would be helpful in this regard. However, such a survey will require a considerable amount of observing time, suggesting that current instruments may not be able to obtain the necessary statistics. However, in the near future there will be two opportunities to perform a large area sub-mm survey of galaxy clusters: the Planck Surveyor and the South Pole 10-m sub-mm telescope.

5.3.1. Survey opportunities

South Pole 10 m sub-mm telescope -The planned South Pole (SP) 10-m sub-mm telescope 1 is expected to begin observations around year 2003 (see, Stark et al. 1998). At 850 µm, it is expected that within [FORMULA] 90 hours a square degree area will be surveyed down to a flux limit of 1 mJy. Given the resolution and flux sensitivity, it is likely that the SP telescope would be an ideal instrument to survey either a sample of clusters or random areas to obtain lensed source statistics down to few mJy. To obtain reliable values of the cosmological parameters based on the sub-mm lensed source statistics, a survey of several hundred square degrees down to few [FORMULA] 1 mJy will be needed. A more direct approach within a reasonable amount of observing time would be to survey a carefully selected sample of galaxy clusters, either based on X-ray luminosity or total mass, from which lensed source statistics can easily be derived.

Planck Surveyor -Considering the amplification distribution for SIS lens model, and the number counts defined by Scott & White (1998), we find that roughly 100 lensed sub-mm sources may be detected with the Planck Surveyor towards galaxy clusters 2. In Table 1, we list the number expected as a function of the cosmological parameters and assuming that the Planck data will allow detection of sources down to 50 mJy. However, given the limited observational data on source counts at 850 µm, we note that the predicted numbers may have large errors. We also note that the Planck data will be highly confused, as the beam size of Planck is [FORMULA] few arcmins at 850 µm; even with [FORMULA] 2 arcmin physical pixels for high signal-to-noise data, most of the sources down to 50 mJy would be separated by only one or two pixels. Assuming pixel sizes of the order beam size, the probability of finding two sources with flux densities greater than 50 mJy in one Planck pixel would be [FORMULA] 0.2 to 0.3. Thus, it is more likely that the Planck data will allow clear detection of sources down to [FORMULA] 100 mJy, but with additional information, such as from other frequency channels and filtering techniques (see, e.g., Tegmark & de Oliviera-Costa 1998), it may be possible to lower this flux limit.

Also, it is likely that the lensed background sources will contaminate the detection of Sunyaev-Zel'dovich (SZ) effect in galaxy clusters (see, Aghanim et al. 1997; Blain 1998). Given the source confusion and contamination, it is likely that that Planck data would not readily allow an adequate determination of lensed sub-mm source statistics to constrain cosmological parameters. It is more likely that the lensed sub-mm source catalog from Planck would be an important tool to study the star-formation history at high redshifts; since lensing brightens sources, such a lensed source catalog will contain sub-mm sources fainter than the current limit predicted to be observable with Planck for unlensed sources.

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© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998