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Galaxy formation depends on a complex interplay between gravitationalcollapse, gas accretion, merging, and feedback processes. Yet, after manydecades of investigation, these concepts are poorly understood. This paperpresents the argument that warm H$_2$ can be used as a tool to unlock some ofthese mysteries. Turbulence, shocks and outflows, driven by star formation, AGNactivity or inflows, may prevent the rapid buildup of star formation ingalaxies. Central to our understanding of how gas is converted into stars isthe process by which gas can dissipate its mechanical energy through turbulenceand shocks in order to cool. H$_2$ lines provide direct quantitativemeasurements of kinetic energy dissipation in molecular gas in galaxiesthroughout the Universe. Based on the detection of very powerful H$_2$ linesfrom z = 2 galaxies and proto-clusters at the detection limits of {\itSpitzer}, we are confident that future far-IR and UV H$_2$ observations willprovide a wealth of new information and insight into galaxy evolution tohigh-z. Finally, at the very earliest epoch of star and galaxy formation, warmH$_2$ may also provide a unique glimpse of molecular gas collapse at 7 $<$ z$<$ 12 in massive dark matter (DM) halos on their way to forming the very firstgalaxies. Such measurements are beyond the reach of existing and plannedobservatories.

Warm H$_2$ as a probe of massive accretion and feedback through shocks and turbulence across cosmic time