A Tentative Detection of Molecular Hydrogen (H$_2$) Emission Lines at Cosmic Dawn

We present a theoretical framework for interpreting far-ultraviolet (FUV) fluorescent emission from molecular hydrogen (H$_2$) in high-redshift galaxies, motivated by the unique capabilities of the James Webb Space Telescope (JWST) to probe the rest frame FUV at cosmic dawn. Using the Meudon photodissociation region (PDR) code, we model the H$_2$ fluorescence spectrum under extreme interstellar medium (ISM) conditions in terms of high pressure ($10^{11}~\mathrm{K~ cm^{-3}}$), high radiation field ($10^6$ $G_0$) combined with low metallicity ($Z = 0.1~Z_\odot$) and high cosmic ionization rate ($\zeta = 10^{-14}~\mathrm{s}^{-1}$), characteristic of early galaxies. As a case study, we apply this framework to stacked NIRSpec spectra from the JWST Advanced Deep Extragalactic Survey (JADES) for galaxies at redshifts $z\geq7$. The stacked spectrum exhibits emission features consistent in profile and wavelength with the predicted H$_2$ fluorescence lines, including a blue shift suggestive of an outflow of molecular gas. Although individual features remain below robust detection thresholds, this demonstration illustrates the feasibility of using FUV fluorescence modeling to guide and interpret JWST spectroscopy of the molecular ISM at high redshift. Our framework provides a foundation for future searches for molecular hydrogen emission and the study of galactic feedback processes in the early universe.