The JWST-PRIMAL Legacy Survey. A JWST/NIRSpec reference sample for the physical properties and Lyman-$\alpha$ absorption and emission of $\sim 500$ galaxies at $z=5.5-13.4$

One of the surprising early findings with JWST has been the discovery of a strong "roll-over" or a softening of the absorption edge of Ly$\alpha$ in a large number of galaxies at ($z\gtrsim 6$), in addition to systematic offsets from photometric redshift estimates and fundamental galaxy scaling relations. This has been interpreted as damped Ly$\alpha$ absorption (DLA) wings from high column densities of neutral atomic hydrogen (HI), signifying major gas accretion events in the formation of these galaxies. To explore this new phenomenon systematically, we assemble the JWST/NIRSpec PRImordial gas Mass AssembLy (PRIMAL) legacy survey of 494 galaxies at $z=5.5-13.4$. We characterize this benchmark sample in full and spectroscopically derive the galaxy redshifts, metallicities, star-formation rates, and ultraviolet slopes. We define a new diagnostic, the Ly$\alpha$ damping parameter $D_{\rm Ly\alpha}$ to measure and quantify the Ly$\alpha$ emission strength, HI fraction in the IGM, or local HI column density for each source. The JWST-PRIMAL survey is based on the spectroscopic DAWN JWST Archive (DJA-Spec). All the software, reduced spectra, and spectroscopically derived quantities and catalogs are made publicly available in dedicated repositories. The fraction of strong galaxy DLAs are found to be in the range $65-95\%$ at $z>5.5$. The fraction of strong Ly$\alpha$ emitters (LAEs) is found to increase with decreasing redshift, in qualitative agreement with previous observational results, and are predominantly associated with low-metallicity and UV faint galaxies. By contrast, strong DLAs are observed in galaxies with a variety of intrinsic physical properties. Our results indicate that strong DLAs likely reflect a particular early assembly phase of reionization-era galaxies, at which point they are largely dominated by pristine HI gas accretion. [abridged]