tokens: surface mass density; D L , D S and D LS are the angular diameter distances from the observer to the lens, to the source and from the lens to the source, respectively. We can get some simple but important results about the lensing efficiency for the DC14 model even before calculating the lensing probabilities. We notice that, from equation ( 2 ), NFW profile is a specific form that has (, , )(1, 3, 1), and SIS profile is similar to (, , )(2, 2, 2), where in the latter case, can be any non-zero number and we let 2 for definiteness. Therefore, it would be helpful to plot the parameters , and versus M halo from 10 11 to 10 14.9 M , as shown in Fig. 1 . We find that the inner slope has 1, an NFW like value, only in a narrow range of halo mass around 10 12 M , and it flattens for lower and higher mass ranges. This would result in a much lower lensing efficiency compared with that of NFW model and even further lower compared with SIS model. Figure 1. The parameters of , and as functions of M halo . Another parameter that helps us to understand the lensing efficiency is s in lensing equation ( 10 ). According to strong lensing theory, multiple images can occur only for sufficiently large values of s . Fig. 2 shows how s changes with M halo for both DC14 and NFW models, with a fixed typical value 0.45 of the lens redshift and the average value 1.56 of the source redshift (Inada etal. 2012 ). We find that the values of s for NFW model uniformly surpass that of DC14 in the whole range of halo mass of (10 11 , 10 14.9 M ), and the difference increases markedly for M halo 10 12 M . This would also mean an obviously lower lensing efficiency for the DC14 model compared with the NFW model. Figure 2. The parameter s as a function of M halo for NFW (dotted line) and DC14 (solid line). In both cases, the redshifts of source and lens are set to be 1.56 and 0.45, respectively. Figure 3. The relation between c and M 200 . Figure 4. The relation between i and M 200 . Figure 5. The relation between r s and M 200 . 2.2 <xref ref-type="bibr" rid="bib77">Schaller15</xref> model The Schaller15 model is derived from EAGLE simulations; the total (baryons plus DM) density profile consists of two terms (Schaller etal. 2015b ): where the first term is the NFW profile and the second term is NFW-like in that it shares the same asymptotic behaviour at small and large radii and has a slope of 2 at its scaleradius, r r i . We write the surface mass densities corresponding to the two terms as where and with For later calculations, we need to know the characteristic densities c and i , and characteristic radii r s and r i as functions of the total mass M 200 , which reads (Schaller etal. 2015b ) In practice, however, it is impossible to derive so many parameters from the only equation ( 16 ), we thus fit c etc. to M 200 with the data given by Schaller etal. ( 2015b ). The fitted results are displayed in Figs 3 5 . The fitting formulae are where X log10( M 200 ), and ca 0.06, cb
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