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

We investigate the dependence of the time delays for the large-separation gravitationally lensed quasar SDSS J1004+4112 on the inner mass profile of the lensing cluster. Adopting the mass model whose innermost density profile is parameterized as \rho \propto r^{-\alpha}, we derive a series of mass models which can fit observational data and then compute the probability distribution functions of time delays. We find that larger \alpha has longer time delays, longer tails at the higher end of the probability distribution, and larger model uncertainties. The ratios of time delays slightly depend on the slope \alpha. Among others, time delays between images C and A (or B) have little dependence on the inner slope, particularly when the time delays are short. The dependence of time delays on \alpha is well fitted by a linear form, which reflects well-known degeneracy between the mass profile and time delays. We perform a Monte-Carlo simulation to illustrate how well the inner slope can be constrained from measurements of time delays. We find that measurements of more than one time delays result in reasonably tight constraints on the inner slope (\sigma_{\alpha} \lesssim 0.25), while only one time delay cannot determine the inner slope very well. Our result indicates that time delays indeed serve as a powerful tool to determine the mass profile, despite the complexity of the lensing cluster.