The quadruple quasar H1413+117 ( z _ { s } = 2.56 ) has been monitored with the 2.0 m Liverpool Telescope in the r Sloan band from 2008 February to July .
This optical follow–up leads to accurate light curves of the four quasar images ( A–D ) , which are defined by 33 epochs of observation and an average photometric error of \sim 15 mmag .
We then use the observed ( intrinsic ) variations of \sim 50 - 100 mmag to measure the three time delays for the lens system for the first time ( 1 \sigma confidence intervals ) : \Delta \tau _ { AB } = - 17 \pm 3 , \Delta \tau _ { AC } = - 20 \pm 4 , and \Delta \tau _ { AD } = 23 \pm 4 days ( \Delta \tau _ { ij } = \tau _ { j } - \tau _ { i } ; B and C are leading , while D is trailing ) .
Although time delays for lens systems are often used to obtain the Hubble constant ( H _ { 0 } ) , the unavailability of the spectroscopic lens redshift ( z _ { l } ) in the system H1413+117 prevents a determination of H _ { 0 } from the measured delays .
In this paper , the new time delay constraints and a concordance expansion rate ( H _ { 0 } = 70 km s ^ { -1 } Mpc ^ { -1 } ) allow us to improve the lens model and to estimate the previously unknown z _ { l } .
Our 1 \sigma estimate z _ { l } = 1.88 ^ { +0.09 } _ { -0.11 } is an example of how to infer the redshift of very distant galaxies via gravitational lensing .