Redshift Leverage in Time-Delay Distances: What the Lens and Source Redshifts Buy and What They Cost

Time-delay distances are computed across a grid of lens and source redshifts in a flat reference cosmology, and the sensitivity of each configuration to the matter density is tabulated as the hidden cost of the leverage. The distance grows with source redshift, but so does model dependence: at the highest-redshift configuration tabulated, a five per cent change in the matter density moves the distance by -0.21 per cent, while the lowest-redshift configuration moves by only -0.0 per cent. Systems like the benchmark sample's, at lens redshifts near one half and source redshifts near 1.7, sit in an intermediate band where the matter-density sensitivity is of order one per cent for a five per cent parameter shift. Redshift choice is thus a trade between statistical reach and cosmological-model exposure, both arms of which are priced here. Computed in a stated reference cosmology; not measured.