Measuring the distances to quasars at high redshifts with strong lensing

Strongly lensed quasars with time-delay measurements are well known to provide the "time-delay distances" $D_{\Delta t}=(1+z_L)D_LD_S/D_{LS}$ and the angular diameter distances to lens galaxies $D_L$. These two kinds of distances give stringent constraints on cosmological parameters. In this work, we explore a different use of time-delay observables: Under the assumption of a flat Universe, strong lensing observations can accurately measure the angular diameter distances to sources $D_S$. The corresponding redshifts of quasars may be up to $z_S\sim4$ according to the forecast. The high-redshift distances would sample the Hubble diagram between SNe Ia and CMB, cosmological-model-independently providing direct information on the evolution of the nature of our Universe, for example, the dark energy Equation-of-State parameter $w(z)$. We apply our method to the existing lensing system SDSS 1206+4332 and get $D_S=2388_{-978}^{+2632}Mpc$ at $z_S=1.789$. We also make a forecast for the era of LSST. The uncertainty of $D_S$ depends on the redshifts of lens and source, the uncertainties of $D_{\Delta t}$ and $D_L$, and the correlation between $D_{\Delta t}$ and $D_L$ as well. Larger correlation would result in tighter $D_S$ determination.