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 can 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 .