Statistical strong lensing. II. Cosmology and galaxy structure with time-delay lenses

Context. Time delay lensing is a powerful tool to measure the Hubble constant $H_0$. In order to obtain an accurate estimate of $H_0$ from a sample of time delay strong lenses, however, it is necessary to have a very good knowledge of the mass structure of the lens galaxies. Strong lensing data on their own are not sufficient to break the degeneracy between $H_0$ and the lens model parameters, on a single object basis. Aims. The goal of this study is to determine whether it is possible to break the $H_0$-lens structure degeneracy with the statistical combination of a large sample of time-delay lenses, relying purely on strong lensing data (that is, with no stellar kinematics information). Methods. I simulated a set of 100 lenses with doubly imaged quasars and related time delay measurements. I fitted these data with a Bayesian hierarchical method and a flexible model for the lens population, emulating the lens modelling step. Results. The sample of 100 lenses, on its own, provides a measurement of $H_0$ with $3\%$ precision, but with a $-4\%$ bias. However, the addition of prior information on the lens structural parameters from a large sample of lenses with no time delays, such as that considered in Paper I, allows for a $1\%$-level inference. Moreover, the 100 lenses allow for a $0.03$~dex calibration of galaxy stellar masses, regardless of the level of prior knowledge of the Hubble constant. Conclusions. Breaking the $H_0$-lens model degeneracy with lensing data alone is possible, but $1\%$ measurements of $H_0$ require either a number of time delay lenses much larger than 100, or the knowledge of the structural parameter distribution of the lens population from a separate sample of lenses.