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

Relic Abundance Predicts Universal Mass-Width Relations for Dark Matter Interactions