Strong gravitational lensing time delay statistics and the density profile of dark halos

The distribution of differential time delays Δt between images produced by strong gravitational lensing contains information on the mass distributions in the lensing objects as well as on cosmological parameters such as H₀. We derive an explicit expression for the conditional probability distribution function of time delays P(Δt | θ), given an image separation between multiple images θ and related statistics. We consider lensing halos described by the singular isothermal sphere (SIS) approximation and by its generalization as proposed by Navarro, Frenk, & White (NFW), which has a density profile ρ ∝ r^-α in the innermost region. The time delay distribution is very sensitive to these profiles; steeper inner slopes tend to produce larger time delays. For example, if H₀ = 70 km s^-1 Mpc^-1, a Λ-dominated cosmology and a source redshift z_S = 1.27 are assumed, lenses with θ = 5″ produce a time delay of Δt = 1.5_-0.9 ^+1.7, 0.39_-0.22^+0.37, 0.15_-0.09 ^+0.11, and 0.071_-0.038^+0.054 yr (50% confidence interval) for SIS, generalized NFW with α = 1.5, α = 1.0, and α = 0.5, respectively. At a fixed image separation, the time delay is determined by the difference in the lensing potential between the position of the two images, which typically occur at different impact parameters. Although the values of Δt are proportional to the inverse of H ₀, P(Δt | θ) is rather insensitive to all other cosmological model parameters, source redshifts, magnification biases, and so on. A knowledge of P(Δt | θ) will also be useful in designing the observing program of future large-scale synoptic variability surveys and for evaluating possible selection biases operating against large splitting lens systems.