The surface and flux densities of gravitational lenses in the millimetre/submillimetre wavebands can be predicted by combining models of the population of distant dusty star-forming galaxies (Blain & Longair (1996)) with a model of the magnification distribution due to lensing as a function of redshift (Peacock (1982), Blain (1996b)). The magnification distribution can be derived from the mass distribution of galaxies, and takes the form if is large. A range of estimates of are presented in Fig.1; these are calculated for four different world models and for both evolving and non-evolving models of the distribution of lensing masses. In the evolving model the mass distribution of the population of lensing galaxies is derived using the Press--Schechter formalism for structure formation by hierarchical clustering (Press & Schechter (1974)), in which galaxies typically become smaller and more numerous as redshift increases. The probability of lensing is predicted to be smaller in the evolving model as compared with the non-evolving model, and to increase in flat world models as the size of the cosmological constant increases. The form of assumed in this letter is normalised to match a particular set of earlier predictions (Pei (1993)).
Figure 1: The probability of galaxy--galaxy lensing as a function of redshift. The probability can change significantly if either the cosmological constant or the mass distribution of the population of lensing galaxies evolves. All world models are flat, with . The optical depth to lensing . The full references are listed elsewhere (Blain (1996b)).
Investigations of the evolution of galaxies and active galactic nuclei (AGN) in many different wavebands (for example Hewett et al. (1993), Dunlop & Peacock (1990), Oliver et al. (1993), Lilly et al. (1996)) indicate that the observed evolution of both the global star-formation rate and the luminosity density of AGN is consistent with pure luminosity evolution (PLE) of the form out to . All three models of galaxy evolution discussed here are normalised to match the - luminosity function of IRAS galaxies at small redshifts (Saunders et al. (1990)) and undergo this form of PLE out to a redshift . However, at larger redshifts each involves a different form of evolution. In modelA -- that is there are no galaxies at . In model B , but there is no further PLE at redshifts between and . In model C and galaxies undergo negative PLE of the form at redshifts ; is the cosmic epoch at redshift .
The counts predicted in all three models of galaxy evolution are compared in Fig.2(a), assuming both an evolving and a non-evolving distribution of lenses. The surface density of lensed images is predicted to be smaller in the evolving model. The counts predicted in different world models are compared in Fig.2(b). A non-zero cosmological constant both increases the predicted surface density of lensed galaxies -- due to an increased probability of lensing -- and decreases the counts of unlensed galaxies -- due to a smaller volume element at large redshifts as compared with an Einstein--de Sitter model. The wavelength dependence of both the lensed and unlensed counts is presented in Figs2(c)&(d). The principal feature is the large increase in the ratio of the surface densities of lensed and unlensed galaxies in the submillimetre waveband at flux densities of about , which correspond to the onset of the steep rise in the unlensed counts above the Euclidean slope at each wavelength. This increase is by two orders of magnitude as compared with the - counts, which have a similar form to the counts expected in the optical or radio wavebands. Another notable feature is the predicted increase in the surface density of unlensed 0.1-Jy galaxies as the observing wavelength in the submillimetre waveband decreases. The surface density of unlensed 0.1-Jy galaxies is predicted to be larger by a factor of about 40 at a wavelength of as compared with a wavelength of . The counts of lensed 0.1-Jy galaxies are also expected to increase as the observing wavelength in the submillimetre waveband decreases >from to about ; however, at wavelengths shorter than about this effect is reversed, and the counts are predicted to decrease as the observing wavelength decreases.
Figure 2: The counts of lensed and unlensed field galaxies expected in a range of galaxy evolution scenarios (a) and world models (b). The wavelength dependence of the predicted counts is presented in (c) and (d).
The general form of the lensed and unlensed counts at flux densities of a few tenths of mJy is consistent with the expectation of an increasing number of lenses, but with lenses contributing a smaller fraction of the total number of detected sources, as the observing wavelength decreases in the submillimetre waveband.