Low-redshift measurement of the sound horizon through gravitational time-delays

The matter sound horizon can be inferred from the cosmic microwave background within the Standard Model. Independent direct measurements of the sound horizon are then a probe of possible deviations from the Standard Model. We aim at measuring the sound horizon $r_s$ from low-redshift indicators, which are completely independent of CMB inference. We used the measured product $H(z)r_s$ from baryon acoustic oscillations (BAO) together with supernovae~\textsc{I}a to constrain $H(z)/H_{0}$ and time-delay lenses analysed by the H0LiCOW collaboration to anchor cosmological distances ($\propto H_{0}^{-1}$). {Additionally, we investigated the influence of adding a sample of quasars with higher redshift with standardisable UV-Xray luminosity distances. We adopted polynomial expansions in $H(z)$ or in comoving distances} so that our inference was completely independent of any cosmological model on which the expansion history might be based. Our measurements are independent of Cepheids and systematics from peculiar motions {to within percent-level accuracy.} The inferred sound horizon $r_s$ varies between $(133 \pm 8)$~Mpc and $(138 \pm 5)$~Mpc across different models. The discrepancy with CMB measurements is robust against model choice. Statistical uncertainties are comparable to systematics. The combination of time-delay lenses, supernovae, and BAO yields a distance ladder that is independent of cosmology (and of Cepheid calibration) and a measurement of $r_s $ that is independent of the CMB. These cosmographic measurements are then a competitive test of the Standard Model, regardless of the hypotheses on which the cosmology is based.