The distribution of differential time delays \Delta t between images producedby strong gravitational lensing contains information on the mass distributionsin the lensing objects as well as on cosmological parameters such as H_0. Wederive an explicit expression for the conditional probability distributionfunction of time delays P(\Delta t|\theta), given an image separation betweenmultiple images \theta, and related statistics. We consider lensing halosdescribed by the singular isothermal sphere (SIS) approximation and by itsgeneralization as proposed by Navarro, Frenk, & White (NFW) which has a densityprofile \rho \propto r^{-\alpha} in the innermost region. The time delaydistribution is very sensitive to these profiles; steeper inner slopes tend toproduce larger time delays. For example, if H_0=70 km/s/Mpc, a\Lambda-dominated cosmology and a source redshift z_S=1.27 are assumed, lenseswith \theta=5'' produce a time delay of \Delta t[yr]=1.5^{+1.7}_{-0.9},0.39^{+0.37}_{-0.22}, 0.15^{+0.11}_{-0.09}, and 0.071^{+0.054}_{-0.038} (50%confidence interval), for SIS, generalized NFW with \alpha=1.5, \alpha=1.0, and\alpha=0.5, respectively. At a fixed image separation, the time delay isdetermined by the difference in the lensing potential between the position ofthe two images, which typically occur at different impact parameters. Althoughthe values of \Delta t are proportional to the inverse of H_0, P(\Deltat|\theta) is rather insensitive to all other cosmological model parameters,source redshifts, magnification biases and so on. A knowledge of P(\Deltat|\theta) will also be useful in designing the observing program of futurelarge scale synoptic variability surveys and for evaluating possible selectionbiases operating against large splitting lens systems.
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