Measurements of $H_0$ in modified gravity theories: The role of lensed quasars in the late-time Universe

In this work, we obtain measurements of the Hubble constant in the context of modified gravity theories. We set up our theoretical framework by considering viable cosmological $f(R)$ and $f(T)$ models, and we analyzed them through the use of geometrical data sets obtained in a model-independent way, namely, gravitationally lensed quasars with measured time delays, standard clocks from cosmic chronometers, and standard candles from the Pantheon Supernovae Ia sample. We find $H_0=(72.4\pm 1.4)$ km s$^{-1}$ Mpc$^{-1}$ and $H_0=(71.5\pm 1.3)$ km s$^{-1}$ Mpc$^{-1}$ for the $f(R)$ and $f(T)$ models, respectively. Our results represent 1.9\% and 1.8\% measurements of the Hubble constant, which are fully consistent with the local estimate of $H_0$ by the Hubble Space Telescope. We do not find significant departures from general relativity, as our study shows that the characteristic parameters of the extensions of gravity beyond general relativity are compatible with the $\Lambda$CDM cosmology. Moreover, within the standard cosmological framework, our full joint analysis suggests that it is possible to measure the dark energy equation of state parameter at 1.2\% accuracy, although we find no statistical evidence for deviations from the cosmological constant case.