Cosmology with supernova Encore in the strong lensing cluster MACS J0138-2155. Lens model comparison and H_0 measurement

Robust mass modeling of strong-lensing galaxy clusters is crucial for studying cosmology and galaxy evolution. We present and compare seven mass models of the galaxy cluster constructed using six independent modeling software programs, including parametric and free-form approaches. By conducting a blind analysis where all the mass-modeling teams constructed their models independently without exchanging results, we quantified uncertainties arising from modeling software and assumptions. is unique as the only cluster found to strongly lens two supernovae (SNe), Requiem and Encore, from the same host galaxy at a redshift of z=1.949, providing an excellent probe of cosmology through time delays between their multiple images. Through the Hubble Space Telescope, James Webb Space Telescope, and Multi Unit Spectroscopic Explorer observations, we assembled high-quality data products, including eight sets of ``gold" lensed-image systems consisting of 23 multiple images with secure spectroscopic redshifts. We further identified one ``silver" lensed-image system with a likely but nonsecure redshift measurement. By restricting ourselves to high-quality gold images, we obtain overall good consistency in the model predictions of the positions, magnifications, and time delays of the multiple images of SN Encore and SN Requiem -- especially from the teams whose models fit the observed image positions with ̧hiimsq łeq 25. We predict the next images of SNe Encore and Requiem to reappear with time delays ≳3000,days and ∼3700 to 4000,days, respectively, based on a fiducial cosmological model with H_0=70,̨msMpc and Om = 1- = 0.3. By considering a range of hypothetical time-delay values with the same Om = 1- = 0.3, we obtain relations between H_0 and the time delays of SN Encore and SN Requiem. In particular, for H_0=73,̨msMpc, the four lowest ̧hiimsq models forecast the next image of SN Requiem to appear approximately April-December 2026; for H_0=67,̨msMpc, they predict it to appear approximately March-November 2027 (1σ uncertainties). Using the newly measured time delay between the two detected multiple images of SN Encore by Pierel et al. (2026) and our mass modeling, we infer H_0 = where the uncertainty is dominated by that of the short time delay between the existing pair of images. The long time delays of the next-appearing SN Requiem and SN Encore images provide excellent opportunities to measure H_0 with 2-3% uncertainty. Our mass models form the basis for cosmological inference from this unique lens cluster with two strongly lensed SNe.