6.1. Magnification bias
Because surface brightness is conserved by all gravitational lenses, the net effect of magnifying a population of background galaxies depends on the slope of the counts dN( > S) / dS, where N( > S) is the number of galaxies per unit area on the sky brighter than S (Schneider et al., 1992). Subject to a magnification µ, the count becomes [1 / µ2]dN[ > (S / µ)] / dS. For a power-law count with dN( > S) / dS S, a value of < - 2 corresponds to an increase in surface density if µ > 1. This threshold value corresponds to a slope of -1 for the integral counts N( > S) shown in Fig. 9. Note that for a uniform non-evolving population of galaxies = - 2.5.
As shown in Fig. 9, submm-wave counts are expected to be steep, and to change slope sharply at mJy flux density levels, as compared with deep optical or radio counts. The significant changes in the count slope are particularly unusual, and not found in any other waveband. As a result, the magnification bias can be large, increasing the number of detectable bright galaxies (Blain, 1996, 1997), and providing a way to investigate very faint counts by comparing lensed and unlensed fields; for example in the innermost regions of clusters of galaxies (Blain, 2002).
That a significant magnification bias can be exploited using relatively weak lensing by clusters of galaxies can be seen by comparing the number of 10-20-mJy 850-µm galaxies detected in the SCUBA Lens Survey (Smail et al., 1997, 2002), and in the larger field of the unlensed 8-mJy survey (Scott et al., 2002). The ratio is about 3:1, showing a clear positive magnification bias, and indicating that if the 850-µm counts at flux densities greater than 10 mJy are represented by a power-law, then the index < - 2.