Astroparticle Physics

220 10 Big Bang Nucleosynthesisquickly absorbed to form deuterium and then helium. 2 So,from freeze-out to the start of deuterium production, the neutron-to-protonratio isFig. 10.3The ratio n n /n p as a function ofthe temperatureneutron-to-proton ration n= e −(m n−m p )/T fe −t/τ n, (10.18)n pwhere the mean neutron lifetime is τ n = 885.7 s. Figure 10.3shows n n /n p as a function of the temperature. The freezeouttemperature is at T f = 0.7 MeV, below which the ratiois almost constant, falling slightly because of neutron decay.At a time t = 180 s (T = 0.086 MeV), a value ofn n≈ 0.13 . (10.19)n pis found.10.5 Synthesis of Light Nuclei“Give me matter and I will construct aworld out of it.”Immanuel Kantdeuterium productionbaryon-to-photon ratioThe synthesis of 4 He proceeds through a chain of reactionswhich includes, for example,pn→ dγ , (10.20)dp→ 3 He γ, (10.21)d 3 He → 4 He p. (10.22)The binding energy of deuterium is E bind = 2.2MeV,so ifthe temperature is so high that there are many photons withenergies higher than this, then the deuterium will be brokenapart as soon as it is produced. One might naïvely expect thatthe reaction (10.20) would begin to be effective as soon asthe temperature drops to around 2.2 MeV. In fact this doesnot happen until a considerably lower temperature. This isbecause there are so many more photons than baryons, andthe photon energy distribution, i.e., the Planck distribution,has a long tail towards high energies.The nucleon-to-photon ratio is at this point essentiallythe same as the baryon-to-photon ratio η = n b /n γ ,which2 The calculations are based on average values of thermodynamicdistributions. The approximations therefore may showdiscontinuities which, however, would disappear if Maxwell–Boltzmann or Planck distributions, respectively, were used.