  Cosmological simulations show that, at the present time, a substantial
fraction of the gas in the intergalactic medium (IGM) has been shock-heated to
T>10^5 K. Here we develop an analytic model to describe the fraction of
shocked, moderately overdense gas in the IGM. The model is an extension of the
Press & Schechter (1974) description for the mass function of halos: we assume
that large-scale structure shocks occur at a fixed overdensity during nonlinear
collapse. This in turn allows us to compute the fraction of gas at a given
redshift that has been shock-heated to a specified temperature. We show that,
if strong shocks occur at turnaround, our model provides a reasonable
description of the temperature distribution seen in cosmological simulations at
z~0, although it does overestimate the importance of weak shocks. We then apply
our model to shocks at high redshifts. We show that, before reionization, the
thermal energy of the IGM is dominated by large-scale structure shocks (rather
than virialized objects). These shocks can have a variety of effects, including
stripping ~10% of the gas from dark matter minihalos, accelerating cosmic rays,
and creating a diffuse radiation background from inverse Compton and cooling
radiation. This radiation background develops before the first stars form and
could have measurable effects on molecular hydrogen formation and the spin
temperature of the 21 cm transition of neutral hydrogen. Finally, we show that
shock-heating will also be directly detectable by redshifted 21 cm measurements
of the neutral IGM in the young universe.
