  We estimate the rate of dark matter scattering in collapsed structures
throughout the history of the Universe. If the scattering cross-section is
velocity-independent, then the canonical picture is correct that scatterings
occur mainly at late times. The scattering rate peaks slightly at redshift z~6,
and remains significant today. Half the scatterings occur after z~1, in
structures more massive than 10^12 M_sun. Within a factor of two, these numbers
are robust to changes in the assumed astrophysics, and the scatterings would be
captured in cosmological simulations. However, for particle physics models with
a velocity-dependent cross-section (as for Yukawa potential interactions via a
massive mediator), the scattering rate peaks before z~20, in objects with mass
less than 10^4 M_sun. These precise values are sensitive to the
redshift-dependent mass-concentration relation and the small-scale cutoff in
the matter power spectrum. In extreme cases, the qualitative effect of early
interactions may be reminiscent of warm dark matter and strongly affect the
subsequent growth of structure. However, these scatterings are being missed in
existing cosmological simulations with limited mass resolution.
