Astroparticle Physics

218 10 Big Bang NucleosynthesisIgnoring for the moment the effect of neutron decay, theneutron-to-proton ratio at the freeze-out temperature isn n= e −(m n−m p )/T f≈ e −1.3/0.7 ≈ 0.16 . (10.14)n pneutron-to-proton ratioAccording to (10.4), this temperature is reached at a timet ≈ 1.5 s. By the end of the next five minutes, essentiallyall of the neutrons become bound into 4 He, and thus theneutron-to-proton ratio at the freeze-out temperature is thedominant factor in determining the amount of helium produced.Before proceeding to predict the helium abundance,a few questions will be addressed in somewhat more detail,namely, neutrino decoupling, positron annihilation, and neutrondecay.10.4 Neutrino Decoupling,Positron Annihilation, and Neutron Decay“Neutrinos, they are very small.They have no charge and have no massand do not interact at all.”John Updikeeffective number of degreesof freedomFor an estimate of the neutron freeze-out temperature ofT f ≈ 0.7 MeV the value g ∗ = 10.75 had been used for theeffective number of degrees of freedom. This corresponds tophotons, e + , e − , and three families of ν and ¯ν as relativisticparticles. The timeline of BBN is complicated slightly bythe fact that g ∗ changes from 10.75 to 3.36 as the temperaturedrops below the electron mass, m e ≈ 0.511 MeV, see(10.16). This means that somewhat more time elapses beforethe synthesis of deuterium can begin and, as a result, moreneutrons have a chance to decay than would be the case withthe higher value of g ∗ .Neutrinos are held in equilibrium with e + and e −through the reaction e + e − ↔ ν ¯ν. At a temperature around1 MeV, however, the rate of this reaction drops below the expansionrate. Muon- and tau-type neutrinos decouple com-pletely from the rest of the particles while ν e continues tointeract for a short while further through ν e n ↔ pe − .Afterthe neutron freeze-out at around 0.7 MeV, all neutrinoflavours are decoupled from other particles. As they are stable,however, their number densities continue to decreasein proportion to 1/R 3 , just like other relativistic particles,neutrino decoupling