Cosmic Reionization On Computers III. The Clumping Factor
  We use fully self-consistent numerical simulations of cosmic reionization,
completed under the Cosmic Reionization On Computers (CROC) project, to explore
how well the recombinations in the ionized IGM can be quantified by the
effective "clumping factor." The density distribution in the simulations (and,
presumably, in a real universe) is highly inhomogeneous and more-or-less
smoothly varying in space. However, even in highly complex and dynamic
environments the concept of the IGM remains reasonably well-defined; the
largest ambiguity comes from the unvirialized regions around galaxies that are
over-ionized by the local enhancement in the radiation field ("proximity
zones"). That ambiguity precludes computing the IGM clumping factor to better
than about 20\%. We also discuss a "local clumping factor," defined over a
particular spatial scale, and quantify its scatter on a given scale and its
variation as a function of scale.
