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We investigate the dependence of the time delays for the large-separationgravitationally lensed quasar SDSS J1004+4112 on the inner mass profile of thelensing cluster. Adopting the mass model whose innermost density profile isparameterized as \rho \propto r^{-\alpha}, we derive a series of mass modelswhich can fit observational data and then compute the probability distributionfunctions of time delays. We find that larger \alpha has longer time delays,longer tails at the higher end of the probability distribution, and largermodel uncertainties. The ratios of time delays slightly depend on the slope\alpha. Among others, time delays between images C and A (or B) have littledependence on the inner slope, particularly when the time delays are short. Thedependence of time delays on \alpha is well fitted by a linear form, whichreflects well-known degeneracy between the mass profile and time delays. Weperform a Monte-Carlo simulation to illustrate how well the inner slope can beconstrained from measurements of time delays. We find that measurements of morethan one time delays result in reasonably tight constraints on the inner slope(\sigma_{\alpha} \lesssim 0.25), while only one time delay cannot determine theinner slope very well. Our result indicates that time delays indeed serve as apowerful tool to determine the mass profile, despite the complexity of thelensing cluster.Comment: 21 pages, 8 figures, accepted for publication in PAS

Time Delays for the Giant Quadruple Lensed Quasar SDSS J1004+4112: Prospects for Determining the Density Profile of the Lensing Cluster