  We find new and universal relations for the properties of dark matter
particles consistent with standard relic abundances. Analysis is based on first
characterizing the $s$-channel resonant annihilation process in great detail,
keeping track of all velocity-dependence, the presence of multiple scales and
treating each physical regime above, below, and close to thresholds separately.
The resonant regime as well as extension to include non-resonant processes are
then reduced to analytic formulas and inequalities that describe the full range
of multi-dimensional numerical work. These results eliminate the need to
recompute relic abundance model by model, and reduce calculations to verifying
certain scale and parameter combinations are consistent. Remarkably simple
formulas describe the relation between the total width of an $s$-channel
intermediate particle, the masses and the couplings involved. Eliminating the
width in terms of the mass produces new consistency relations between dark
matter masses and the intermediate masses. The formulas are general enough to
test directly whether new particles can be identified as dark matter. Resonance
mass and total width are quantities directly observable at accelerators such as
the LHC, and will be sufficient to establish whether new discoveries are
consistent with the cosmological bounds on dark matter.
