2020ApJ...897..127W

With the distance sum rule in the Friedmann-Lemaitre-Robertson-Walker metric, model-independent constraints on both the Hubble constant H₀ and spatial curvature Ω_K can be obtained using strong lensing time-delay data and Type Ia supernovae (SNe Ia) luminosity distances. This method is limited by the relatively low redshifts of SNe Ia, however. Here, we propose using quasars as distance indicators, extending the coverage to encompass the redshift range of strong lensing systems. We provide a novel and improved method of determining H₀ and Ω_K simultaneously. By applying this technique to the time-delay measurements of seven strong lensing systems and the known ultraviolet versus X-ray luminosity correlation of quasars, we constrain the possible values of both H₀ and Ω_K, and find that H₀=75.3_–2.9^+3.0 km s^–1Mpc^–1 and Ω_K=-0.01_–0.17^+0.18. The measured Ω_K is consistent with zero spatial curvature, indicating that there is no significant deviation from a flat universe. If we use flatness as a prior, we infer that H₀=75.3_–1.9^+1.9 km s^–1Mpc^–1, representing a precision of 2.5%. If we further combine these data with the 1048 current Pantheon SNe Ia, our model-independent constraints can be further improved to H₀=75.3_–2.9^+3.0 km s^–1Mpc^–1 and Ω_K=0.05_–0.14^+0.16. In every case, we find that the Hubble constant measured with this technique is strongly consistent with the value (∼74 km s^–1Mpc^–1) measured using the local distance ladder, as opposed to the value optimized by Planck.