## 4. Systematic errorsOur lensing rate calculation relies on the assumption that the HDF is a reasonable sample of the distance universe and that it can be applied to the whole sky. In the case of optical arcs, we have included an additional number of faint sources to the photometric redshift catalog, and by doing so, may have introduced a systematic bias in our calculation. However, unless these sources are at either low or high redshift, we do not expect such sources to make a large change in the lensing rate. Also, there is a possibility that certain multiple sources, which we have counted as separate objects, may in fact represent star-forming regions within individual galaxies (e.g., Colley et al. 1997). If this is true, we may have overestimated the number of sources by as much as 40%, and may have caused a systematic increase in the lensing rate. Another possible systematic error is involved with the determination of the parameter. We have used the PS theory normalized to local cluster abundance and relations between velocity dispersion, cluster temperature, mass and luminosity to calculate . The used scaling relations, as well as parameters in the PS function, have in some cases large uncertainties. It is likely that our predicted numbers may be accurate to within 40% to 50%. Other than statistical errors, there may also be systematic uncertainties. For example, the relation may have additional dependences on the cosmological parameters (see, e.g., Voit & Donahue 1998), which we have not fully considered. Since was inferred based on PS function normalized to observations, and since these observables depend on the assumed cosmology, will also depend on it. The dependence of the inferred on cosmology also depends on the scaling relations, as well as the lower limit of the luminosity used in the PS calculation, which varies with cosmology. As suggested earlier, for the most part, we can ignore such small changes due to the choice of cosmological model in our scaling relations and other observables; there are much larger statistical and systematic errors in our calculation involving the normalization of the PS function etc. Even though we have used the PS theory to account for redshift evolution, it is possible that we have only partially accounted for evolutionary effects. For example, we have not taken into account the effects of on cluster formation, where clusters are expected to be less compact in a universe with than in a universe with . Thus, our analytical calculation is different from the numerical study of Bartelmann et al. (1997), where effects of on cluster structures are accounted based on numerical simulations. Our lens model is too simple to allow such effects, and by ignoring this important fact, we have included an additional simplification in our present analysis. Bartelmann et al. (1997) found that the observed number of arcs can be explained in an open universe, while with a cosmological constant the number predicted is smaller than the observed statistics. However, we note that the results from Bartelmann et al. (1997) may be in conflict with present estimates of cosmological parameters based on other methods, which suggest a flat universe with non-zero . In comparison, we find that the number predicted in an open universe is not enough to fully account for the observed statistics, unless . Also, we find a considerably large number of arcs with a dominated universe (), which has been considered in the past to account for lensed arcs statistics (e.g., Wu & Mao 1996). We have assumed that clusters can be described by singular isothermal spheres. However, for high amplification events such as arcs, substructures within clusters are important; substructures are also responsible for aspherical potentials. Such potentials have been considered important for lensing studies of individual clusters (e.g., Bézecourt 1998), where it has been shown that the true lensing rate can be as high as a factor of 2 from spherical potentials. However, it is likely that such biases may only exist for a small number of clusters, and thus, for overall statistics of lensed arcs, complex potentials can be ignored; a conclusion also supported by the numerical simulations of clusters and lensed arcs. However, for certain clusters, especially the ones that have been systematically studied in detail due to the large number of arcs, which includes A2218, A370 and A1689, complex potentials are important to model the individual arc distribution. © European Southern Observatory (ESO) 1999 Online publication: December 16, 1998 |