Time Delays in the Gravitationally Lensed Quasar H1413+117 (Cloverleaf)
  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 +/- 3, \Delta
\tau_{AC} = -20 +/- 4, and \Delta \tau_{AD} = 23 +/- 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.
