Gravitational lens time delays depend on the Hubble constant , the observed image positions , and the surface mass density of the lens in the annulus between the images .
Simple time delay lenses like PG1115+080 , SBS1520+530 , B1600+434 , PKS1830–211 and HE2149–2745 have H _ { 0 } = A ( 1 - \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle ) + B \langle \kappa \vphantom { R _ % { 1 } ^ { 2 } } \rangle ( \eta - 1 ) where the two coefficients A \simeq 90 km/s Mpc and B \simeq 10 km/s Mpc depend on the measured delays and the observed image positions , \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle is the mean surface density in the annulus between the images , and there is a small correction from the logarithmic slope \eta \simeq 2 of the surface density profile , \kappa \propto R ^ { 1 - \eta } , in the annulus .
These 5 systems are very homogeneous , since for fixed H _ { 0 } = 100 h km/s Mpc they must have the same surface density , \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle = 1.11 - 1.22 h \pm 0.04 , with an upper bound of \sigma _ { \kappa } < 0.07 on any dispersion in \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle beyond those due to the measurement errors .
If the lenses have their expected dark halos , \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle \simeq 0.5 and H _ { 0 } \simeq 51 \pm 5 km/s Mpc , while if they have constant mass-to-light ratios , \langle \kappa \vphantom { R _ { 1 } ^ { 2 } } \rangle \simeq 0.1 – 0.2 and H _ { 0 } \simeq 73 \pm 8 km/s Mpc .
More complicated lenses with multiple components or strong perturbations from nearby clusters , like RXJ0911+0551 and Q0957+561 , are easily recognized because they have significantly different coefficients .