Measuring the value of the Hubble constant "\`a la Refsdal"

Realizing Refsdal's original idea from 1964, we present estimates of the Hubble constant that are complementary to and potentially competitive with those of other cosmological probes. We use the observed positions of 89 multiple images, with extensive spectroscopic information, from 28 background sources and the measured time delays between the images S1-S4 and SX of supernova "Refsdal" ($z = 1.489$), which were obtained thanks to Hubble Space Telescope (HST) deep imaging and Multi Unit Spectroscopic Explorer (MUSE) data. We extend the strong lensing modeling of the Hubble Frontier Fields (HFF) galaxy cluster MACS J1149.5$+$2223 ($z = 0.542$), published by Grillo et al. (2016), and explore different $\Lambda$CDM models. Taking advantage of the lensing information associated to the presence of very close pairs of multiple images at various redshifts and to the extended surface brightness distribution of the SN Refsdal host, we can reconstruct the total mass density profile of the cluster very precisely. The combined dependence of the multiple image positions and time delays on the cosmological parameters allows us to infer the values of $H_{0}$ and $\Omega_{\rm m}$ with relative (1$\sigma$) statistical errors of, respectively, 6% (7%) and 31% (26%) in flat (general) cosmological models, assuming a conservative 3% uncertainty on the final time delay of image SX and, remarkably, no priors from other cosmological experiments. Our best estimate of $H_{0}$, based on the model described in this work, will be presented when the final time-delay measurement becomes available. Our results show that it is possible to utilize time delays in lens galaxy clusters as an important alternative tool for measuring the expansion rate and the geometry of the Universe.