Astroparticle Physics

226 10 Big Bang Nucleosynthesis10.7 Constraints on the Numberof Neutrino Families“If there were many neutrino generations,we would not be here to countthem, because the whole universe wouldbe made of helium and no life could develop.”AnonymousN ν from cosmologyhow many neutrino families?In this section it will be shown how the comparison of themeasured and predicted 4 He mass fractions can result inconstraints on the particle content of the universe at BBNtemperatures. For example, the Standard Model has N ν = 3,but one can ask whether additional families exist. It will beseen that BBN was able to constrain N ν to be quite closeto three – a number of years earlier than accelerator experimentswere able to determine the same quantity to high precisionusing electron–positron collisions at energies near theZ resonance.Once the parameter η has been determined, the predicted4 He mass fraction is fixed to a narrow range of values closeto Y P = 0.24. As was noted earlier, this prediction is ingood agreement with the measured abundance. The predictiondepended, however, on the effective number of degreesof freedom,g ∗ = 2 + 7 8 (4 + 2N ν), (10.32)where N ν is the number of neutrino families. Earlier theStandard Model value N ν = 3 was used, which gave g ∗ =10.75. This then determines the expansion rate throughFig. 10.6The reaction rate Γ(ν e n ↔ e − p)and the expansion rate H forN ν = 2, 3, and 4 as a function oftemperatureH = 1.66 √ T 2g ∗ . (10.33)m PlThe effective number of degrees of freedom therefore has animpact on the freeze-out temperature, where one has H =Γ(nν e ↔ pe − ), see (10.11). This can be seen in Fig. 10.6,which shows the reaction rate Γ(nν e ↔ pe + ) ≈ G 2 F T 5 andthe expansion rate H versus temperature. The expansion rateis shown using three different values of g ∗ , corresponding toN ν = 2, 3, and 4.From Fig. 10.6 one can see that the freeze-out temperatureT f is higher for larger values of g ∗ , i.e., for higher N ν .At T f the neutron-to-proton ratio freezes out to n n /n p =