SPT0311-58 is the most massive infrared luminous system discovered so far during the Epoch of Reionization (EoR). In this paper, we present a detailed analysis of the molecular interstellar medium at z = 6.9, through high-resolution observations of the CO(6-5), CO(7-6), CO(10-9), [CI](2-1), and p-H2O(211-202) lines and dust continuum emission with the Atacama Large Millimeter/submillimeter Array (ALMA). The system consists of a pair of intensely star-forming gravitationally lensed galaxies (labelled West and East). The intrinsic far-infrared luminosity is (16 $pm$ 4)$timesrm 10^{12} rm L_{odot}$ in West and (27 $pm$ 4)$timesrm 10^{11} rm L_{odot}$ in East. We model the dust, CO, and [CI] using non-local thermodynamic equilibrium radiative transfer models and estimate the intrinsic gas mass to be (5.4 $pm$ 3.4)$timesrm 10^{11} rm M_{odot}$ in West and (3.1 $pm$ 2.7)$timesrm 10^{10} rm M_{odot}$ in East. We find that the CO spectral line energy distribution in West and East are typical of high-redshift sub-millimeter galaxies (SMGs). The CO-to-H2 conversion factor ($alpha_{CO}$) and the gas depletion time scales estimated from the model are consistent with the high-redshift SMGs in the literature within the uncertainties. We find no evidence of evolution of depletion time with redshift in SMGs at z > 3. This is the most detailed study of molecular gas content of a galaxy in the EoR to-date, with the most distant detection of H2O in a galaxy without any evidence for active galactic nuclei in the literature.
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