ner_tags: O O O O O O O O B-Formula I-Formula I-Formula O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O B-CelestialObject O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O B-CelestialObject O O O O B-Citation I-Citation I-Citation I-Citation O O B-CelestialObject O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Wavelength O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation O O B-Observatory O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: (SLAC) lenses for a substructure mass function slope α = 0.90 () (Vegetti et al. 2014 ). Thus, as long as baryonic physics does not significantly affect the result, it is reasonable to conclude that CDM substructures are not the entire reason for the flux anomaly problem. On the other hand, our result for line-of-sight lensing significantly differs from the result that was obtained using R cusp and R fold (Xu et al. 2012 ). The difference may come from the typical source redshift z S = 2 assumed in Xu et al. ( 2012 ). In fact, our result indicates that for z S 2, the ratio of the strength of the convergence perturbation from subhaloes to that of the total value is 30 to 40 per cent. For higher source redshifts, however, the ratio becomes much smaller. Moreover, the two lens systems B2045+265 and B1555+375 used in Xu et al. ( 2012 ) (the latter was excluded in our analysis), which have a relatively large R cusp = 0.4–0.5, show another source of anomaly: B2045+265 has a secondary lens (McKean et al. 2007 ) and B1555+375 has an edge-on disc component that can account for the anomaly (Hsueh et al. 2016 ). Excluding these complex lenses would surely weaken the discrepancy with our result. Complex lens environments, such as secondary lenses and disc components and neighbouring groups and clusters, should be taken into account properly. Our result indicates that a secondary lens can reside in the intergalactic space and is not necessarily a companion of the primary lens. Furthermore, the previously claimed subhalo with a mass of (1.9 ± 0.1) × 10 8 M ⊙ discovered from the residual image of an Einstein ring in B1938+666 (Koopmans 2005 ; Vegetti et al. 2012 ) may not be the subhalo of the primary lensing galaxy halo. It may be isolated from any haloes or belong to another distant one. To measure the redshift of a secondary or that of a third lens, we can use the distortion pattern of an extended source and future submillimetre observations would be able to detect such small signals (Inoue Chiba 2005a , b ). Using ALMA, the detection of other anomalous anomalies in lensed quadruple SMGs could be achieved by increasing the number of anomalous targets to ≳10, which would test our findings. In our analysis, we assumed that primary lenses are described by SIEs as indicated by the surface brightness of the lensing galaxies. This assumption may be too simple. We did not take into account the slight smooth distortion of the lens potential, such as low-order multipoles m 3 and m 4, or a deviation from an isothermal profile. As long as these complexities are related with the fluctuations of the first derivatives on scales comparable to or larger than the Einstein radius of the primary lens, we expect that our result would not change significantly. This is because the anomaly in the flux ratios is caused by a perturbation of the second derivatives of the gravitational potential rather than the first derivatives. However, to obtain more accurate results, we do