Abstract
Using the upper bound on the inelastic reaction cross-section implied by S-matrix unitarity, we derive the thermally averaged maximum dark matter (DM) annihilation rate for general k → 2 number-changing reactions, with k ≥ 2, taking place either entirely within the dark sector, or involving standard model fields. This translates to a maximum mass of the particle saturating the observed DM abundance, which, for dominantly s-wave annihilations, is obtained to be around 130 TeV, 1 GeV, 7 MeV and 110 keV, for k = 2, 3, 4 and 5, respectively, in a radiation dominated Universe, for a real or complex scalar DM stabilized by a minimal symmetry. For modified thermal histories in the pre-big bang nucleosynthesis era, with an intermediate period of matter domination, values of reheating temperature higher than mathcal{O}(200) GeV for k ≥ 4, mathcal{O}(1) TeV for k = 3 and mathcal{O}(50) TeV for k = 2 are strongly disfavoured by the combined requirements of unitarity and DM relic abundance, for DM freeze-out before reheating.
Highlights
Number density of dark matter (DM), and the maximum mass that saturates the present density
This translates to a maximum mass of the particle saturating the observed DM abundance, which, for dominantly s-wave annihilations, is obtained to be around 130 TeV, 1 GeV, 7 MeV and 110 keV, for k = 2, 3, 4 and 5, respectively, in a radiation dominated Universe, for a real or complex scalar DM stabilized by a minimal symmetry
A natural question is what are the implications of S-matrix unitarity in such scenarios, where the dominant DM number changing interaction is of the type k → n, with k ≥ n? A completely general formulation of this problem is challenging since the partialwave decomposition of a k−body initial state, for k ≥ 3, is rather involved
Summary
We recall some of the basic results on the implications of S-matrix unitarity, following the treatment of Weinberg in ref. [17]. Our results for the maximum value of the inelastic scattering cross-section for identical initial state particles does not agree with the corresponding comment in ref. We can perform the thermal average integral in eq (3.6) for the 2 → k reactions with the maximum value of the inelastic cross-section as in eq (2.9) as input, and obtain, with all k + 2 particles having the same mass, σ2→kvrel max =. For identical particles in the initial state, we found in eq (2.14) that the maximum inelastic cross-section for a 2 → k reaction is a factor of two larger than the non-identical case. The thermal averaging integral in σ2→kvrel will in this case have a symmetry factor of 1/2 for the two identical particles in the initial state. For identical initial state particles, σ2→kvrel max as shown in eq (3.9) remains valid, and eq (3.11) remains the same as well
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