The LCDM model has emerged as the concordance model of cosmology for its ability to explain a variety of observations, ranging from the anisotropies of the Cosmic Microwave Background (CMB) to the accelerated expansion of the Universe. However, with the ever-increasing precision of the measurements, tensions have recently emerged between the latest CMB observations and other cosmological probes. The most prominent one concerns the expansion rate of the Universe, that is the Hubble constant, H0, with a statistically significant discrepancy between local and early measurements. If not arising from unaccounted systematic errors, this discrepancy is an exciting opportunity to explore missing physics beyond the standard LCDM model.

Time-delay cosmography has emerged as a competitive and single-step method to measure locally the Hubble constant, providing an alternative to the cosmic distance ladder. The two techniques do not share any known source of errors, allowing us to obtain truly independent estimates. This technique is based on the strong lensing phenomenon, occurring when the light coming from a background source is split into multiple images by a massive foreground galaxy. In asymmetric configurations, the travel time is then slightly different for each of the lensed images, leading to measurable time delays in the signals received from the background object. The Hubble constant can then be directly inferred from these time delays, provided that an accurate reconstruction of the mass distribution of the lens galaxy is available.

The first part of my PhD work focuses on the precise determination of the time delays in strongly lensed quasars, an essential ingredient of the method. The COSMOGRAIL collaboration has been monitoring lensed quasars for two decades to record enough quasar variations that can be unambiguously matched in all light curves. I compiled these data obtained from the Euler 1.2m Swiss telescope in La Silla, Chile, and measured the time delays in 18 systems, more than doubling the current sample of lensed quasars with known delays. This work required the development and automation of curve-shifting algorithms presented in a dedicated chapter. As the sample of known lensed quasars is growing rapidly, I also present a novel monitoring strategy to obtain the time delays efficiently, in only one or two monitoring seasons.

Thanks to this long-term monitoring effort, the TDCOSMO collaboration published the most precise determination of the Hubble constant obtained with time-delay cosmography from a sample of seven lensed quasars. We found H0 = 73.7+/- 1.5 km/s/Mpc, at 2% precision. This result confirmed the existing tension, now reaching a high significance when combined with the distance ladder results. "As extraordinary claims require extraordinary evidence", the second part of my work focused on the verification of the assumptions made to obtain this result. In the last chapter of this thesis, I present my work on the search for unaccounted systematic errors in every step of the analysis. I also show the results of our participation to a blind data challenge designed to test lens modelling codes and the assumptions used to reconstruct the mass distribution of the galaxies. I conclude this thesis by presenting the new approach proposed by the TDCOSMO collaboration to relax all assumptions about the density profile of massive elliptical galaxies and adopt a parametrisation entirely constrained by stellar kinematic.