A Simple Way of Calculating Cosmological Relic Density

A simple procedure is presented which leads to a dramatic simplification in the calculation of the relic density of stable particles in the Universe. 1 E-mail address: [email protected] Any stable species χ contributes to the total mass-energy density in the Universe. If its number density cannot be reduced efficiently enough before it decouples from the thermal equilibrium, its relic abundance Ωχh 2 0 can be sizeable and it can affect the evolution and the age of the Universe. A conservative estimate that the Universe is at least 10 billion years old requires Ωχh 2 0 < ΩTOTh 2 0 < 1 [1]. Furthermore, there is growing evidence for dark matter at both galactic and larger length scales [1] which would most likely require the existence of some type of exotic neutral particles. Since such particles are often present in many theories beyond the Standard Model, it is important to develop a simple practical procedure for calculating their relic abundance with enough precision. Two groups [2, 3] have developed equivalent frameworks for properly calculating the relic density of χ’s, including relativistic corrections. The method of Ref. [2] is in practice applicable away from poles and new final-state thresholds, which is most often the case. In Ref. [3] also the vicinity of poles and thresholds has been carefully studied. Essentially, one needs to calculate the thermally averaged product of the χχ̄ annihilation cross section and their relative velocity 〈σvrel〉. In practice, one expands 〈σvrel〉 = a+ bx+O(x2) in powers of x ≡ T/mχ = O(1/20) in order to avoid difficult numerical integrations and approximates 〈σvrel〉 by a and b. Both techniques give equivalent results [3, 4] in the overlapping region (away from poles and thresholds). Unfortunately, in practice the actual calculation of even the first two terms of the expansion is typically very complicated and tedious. In this Letter I report on a dramatic technical simplification in practical applications of the method of Ref. [2]. Consider an annihilation of particles χ, χ̄ into a two-body final state. Furthermore, in many cases of interest, the final state particles have equal mass mF . (A general case of unequal final state masses will be presented elsewhere [5].) Let the momenta of the two initial states χ, χ̄, and the two final states F , F̄ , be p1, p2 and k1, k2, respectively. Srednicki, et al., introduce the function w(s) defined as w(s) ≡ 1 4 ∫ dLISP |M| = E1E2 σvrel, (1) where dLISP in this case takes the form dLISP = (2π)δ(p1 + p2 − k1 − k2) dk1 (2π)2E1 dk2 (2π)2E2 , (2) where E1,2 = ~k 2 1,2 +m 2 F , and |M|2 is the square of the reduced matrix element for the annihilation process χ̄χ → F̄F summed over the spins of the final-state particles and averaged over the spins of the initial particles [2]. Denote f ≡ |M|. (3)