Warm H$_2$ 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.
