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

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.