  The rate of merging of dark-matter halos is an absolutely essential
ingredient for studies of both structure and galaxy formation. Remarkably,
however, our quantitative understanding of the halo merger rate is still quite
limited, and current analytic descriptions based upon the extended
Press-Schechter formalism are fundamentally flawed. We show that a
mathematically self-consistent merger rate must be consistent with the
evolution of the halo abundance in the following sense: The merger rate must,
when inserted into the Smoluchowski coagulation equation, yield the correct
evolution of the halo abundance. We then describe a numerical technique to find
merger rates that are consistent with this evolution. We present results from a
preliminary study in which we find merger rates that reproduce the evolution of
the halo abundance according to Press-Schechter for power-law power spectra. We
discuss the limitations of the current approach and outline the questions that
must still be answered before we have a fully consistent and correct theory of
halo merger rates.
