Warm H2 as a probe of massive accretion and feedback through shocks and turbulence across cosmic time

Galaxy formation depends on a complex interplay between gravitational collapse, gas accretion, merging, and feedback processes. Yet, after many decades of investigation, these concepts are poorly understood. This paper presents the argument that warm H$_2$ can be used as a tool to unlock some of these mysteries. Turbulence, shocks and outflows, driven by star formation, AGN activity or inflows, may prevent the rapid buildup of star formation in galaxies. Central to our understanding of how gas is converted into stars is the process by which gas can dissipate its mechanical energy through turbulence and shocks in order to cool. H$_2$ lines provide direct quantitative measurements of kinetic energy dissipation in molecular gas in galaxies throughout the Universe. Based on the detection of very powerful H$_2$ lines from z = 2 galaxies and proto-clusters at the detection limits of {\it Spitzer}, we are confident that future far-IR and UV H$_2$ observations will provide a wealth of new information and insight into galaxy evolution to high-z. Finally, at the very earliest epoch of star and galaxy formation, warm H$_2$ 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 first galaxies. Such measurements are beyond the reach of existing and planned observatories.