VLT/MUSE observations of SDSS J1029+2623: towards a high-precision
  strong lensing model
  We present a strong lensing analysis of the galaxy cluster SDSS J1029+2623 at
$z=0.588$, one of the few currently known lens clusters with multiple images of
a background ($z=2.1992$) quasar with a measured time delay. We use archival
Hubble Space Telescope multi-band imaging and new Multi Unit Spectroscopic
Explorer follow-up spectroscopy to build an accurate lens mass model, a crucial
step towards future cosmological applications. The spectroscopic data enable
the secure identification of 57 cluster members and of two nearby perturbers
along the line-of-sight. We estimate the inner kinematics of a sub-set of 20
cluster galaxies to calibrate the scaling relations parametrizing the sub-halo
mass component. We also reliably determine the redshift of 4 multiply imaged
sources, provide a tentative measurement for one system, and report the
discovery of a new four-image system. The final catalog comprises 26 multiple
images from 7 background sources, spanning a wide redshift range, from 1.02 to
5.06. We present two parametric lens models, with slightly different cluster
mass parametrizations. The observed positions of the multiple images are
accurately reproduced within approximately $0''.2$, the three image positions
of the quasar within only $\sim0''.1$. We estimate a cluster projected total
mass of $M(<300~ {\rm kpc}) \sim 2.1 \times 10^{14}~ M_{\odot}$, with a
statistical uncertainty of a few percent. Both models, that include a small
galaxy close to one of the quasar images, predict magnitude differences and
time delays between the quasar images that are consistent with the
observations.
