SIMBAD references

We investigated 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 a mass model whose innermost density profile was parameterized as r ∝ r^–α, we derived a series of mass models that can fit the observational data, and then compute the probability distribution functions of time delays. We find that larger α has longer time delays, longer tails at the higher end of the probability distribution, and larger model uncertainties. The ratios of the time delays slightly depend on the slope, α. 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 α is well fitted by a linear form, which reflects the well-known degeneracy between the mass profile and time delays. We performed a Monte Carlo simulation to illustrate how well the inner slope can be constrained from measurements of time delays. We found that measurements of more than one time delay resulted in reasonably tight constraints on the inner slope (s_α \lesssim 0.25), while only one time delay could not 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.