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Warm molecular Hydrogen at high redshift with the James Webb Space Telescope

The build-up of galaxies is regulated by a complex interplay between gravitational collapse, galaxy merging and feedback related to AGN and star formation. The energy released by these processes has to dissipate for gas to cool, condense, and form stars. How gas cools is thus a key to understand galaxy formation. \textit{Spitzer Space Telescope} infrared spectroscopy revealed a population of galaxies with weak star formation and unusually powerful H$_2$ line emission. This is a signature of turbulent dissipation, sustained by large-scale mechanical energy injection. The cooling of the multiphase interstellar medium is associated with emission in the H$_2$ lines. These results have profound consequences on our understanding of regulation of star formation, feedback and energetics of galaxy formation in general. The fact that H$_2$ lines can be strongly enhanced in high-redshift turbulent galaxies will be of great importance for the \textit{James Webb Space Telescope} observations which will unveil the role that H$_2$ plays as a cooling agent in the era of galaxy assembly.