Cosmology with supernova Encore in the strong lensing cluster MACS J0138−2155

We present a comprehensive spectroscopic analysis of MACS J0138-2155, at z = 0:336. It is the first galaxy cluster known to host two strongly lensed supernovae (SNe), Requiem and Encore, and thus provides us with a chance to obtain a reliable H0 measurement from the time delays between multiple images. We took advantage of new spectroscopic data from the Multi Unit Spectroscopic Explorer (MUSE) on the Very Large Telescope, complemented with archival imaging from the Hubble Space Telescope. The MUSE data cover a central 1 arcmin2 of the lensing cluster, for a total depth of 3.7 hours, including 2.9 hours recently obtained by our Target of Opportunity programme. Based on these observations, we release a new spectroscopic catalogue containing reliable redshifts for 107 objects. This includes 50 galaxy cluster members with secure redshift values in the range 0:324 < z < 0:349, and 13 lensed multiple images from four background sources between 0:767 ≤ z ≤ 3:420, including four images of the host galaxy of the two SNe. We exploited the MUSE data to study the stellar kinematics of 14 bright cluster members and two background galaxies, obtaining reliable measurements of their line-of-sight velocity dispersion. Finally, we combined these results with measurements of the total magnitude of the cluster members in the Hubble Space Telescope F160W band to calibrate the Faber-Jackson relation between luminosity and stellar velocity dispersion (L ∝ σ1/α v ) for the early-type cluster member galaxies, measuring a slope α = 0:25+0:05 -0:05 and a scatter of 25+6 -4 km s-1 about the relation. We compared the calibrated relation with six other strong lensing clusters in the redshift range 0:31 ≤ z ≤ 0:59, finding a general agreement and no clear sign of evolution with redshift. A pure and complete sample of cluster member galaxies and a reliable characterisation of their total mass structure are key to building accurate total mass maps of the cluster and mitigating the impact of parametric degeneracies, which is necessary for inferring the value of H0 from the measured time delays between the lensed images of the two SNe.