Astroparticle Physics

216 10 Big Bang Nucleosynthesis10.3 The Neutron-to-Proton Ratio“The most serious uncertainty affectingthe ultimate fate of the universe isthe question whether the proton is absolutelystable against decay into lighterparticles. If the proton is unstable, allmatter is transitory and must dissolveinto radiation.”Freeman J. DysonFig. 10.1Feynman diagram for the reactionnν e ↔ pe −neutron-to-proton ratioAlthough the total baryon number is conserved, protons andneutrons can be transformed through reactions like nν e ↔pe − and ne + ↔ p ¯ν e . A typical Feynman diagram is shownin Fig. 10.1.The crucial question is whether these reactions proceedfaster than the expansion rate so that thermal equilibrium ismaintained. If this is the case, then the ratio of neutron-toprotonnumber densities is given by( )n 3/2 n mn= e −(m n−m p )/T ≈ e −m/T , (10.7)n p m pwhere m p = 938.272 MeV, m n = 939.565 MeV, and m =m n − m p = 1.293 MeV. To find out whether equilibriumis maintained, one needs to compare the expansion rate Hfrom (10.1) to the reaction rate Γ . In Sect. 9.3 this rate wasfound to be given by (9.43)Γ = n〈σv〉 . (10.8)reaction ratesweak cross sectionΓ is the the reaction rate per neutron for nν e ↔ pe − ,wherethe brackets denote an average of σv over a thermal distributionof velocities. The number density n in (10.8) refers tothe target particles, i.e., neutrinos, which is therefore givenby (10.5).The cross section for the reaction ν e n ↔ e − p can bepredicted using the Standard Model of electroweak interactions.An exact calculation is difficult but to a good approximationone finds for the thermally averaged speed timescross section〈σv〉≈G 2 F T 2 . (10.9)Here G F = 1.166 × 10 −5 GeV −2 is the Fermi constant,which characterizes the strength of weak interactions.