A Mass Model for the Lensing Cluster SDSS J1004+4112: Constraints From the Third Time Delay

We have built a new model for the lens system SDSS J1004+4112 including the recently measured time delay of the fourth quasar image. This time delay has a strong influence on the inner mass distribution of the lensing cluster ($\rho \propto r^{-\alpha }$) allowing us to determine $\alpha =1.18_{-0.03(-0.18)}^{+0.02(+0.11)}$ at the 68% (95%) confidence level in agreement with hydrodynamical simulations of massive galaxy clusters. We find an offset between the brightest cluster galaxy (BCG) and the dark matter halo of $3.8_{-0.7(-1.3)}^{+0.6(+1.4)}$ kpc at 68% (95%) confidence which is compatible with other galaxy cluster measurements. As an observational challenge, the estimated time delay between the leading image C and the faint (I=24.7) fifth image E is roughly 8 yr.