Astroparticle Physics

10.5 Synthesis of Light Nuclei 221is around 10 −9 . One can estimate roughly when deuteriumproduction can begin to proceed by finding the temperaturewhere the number of photons with energies greater than2.2 MeV is equal to the number of nucleons. For a nucleonto-photonratio of 10 −9 , this occurs at T = 0.086 MeV,which is reached at a time of t ≈ 3 minutes.Over the next several minutes, essentially all of the neutrons,except those that decay, are processed into 4 He. Theabundance of 4 He is usually quoted by giving its mass fraction,4 He mass fractionY P =mass of 4 Hemass of all nuclei = m Hen Hem N (n n + n p ) , (10.23)where the neutron and proton masses have both been approximatedby the nucleon mass, m N ≈ m n ≈ m p ≈0.94 GeV. There are four nucleons in 4 He, so, neglecting thebinding energy, one has m He ≈ 4m N . Furthermore, there aretwo neutrons per 4 He nucleus, so if one assumes that all ofthe neutrons end up in 4 He, one has n He = n n /2. This willturn out to be a good approximation, as the next most commonnucleus, deuterium, ends up with an abundance four tofive orders of magnitude smaller than that of hydrogen. The4 He mass fraction is thereforeY P = 4m N(n n /2)m N (n n + n p ) = 2(n n/n p )1 + n n /n p≈ 2 × 0.13 ≈ 0.23 . (10.24)1 + 0.13This rough estimate turns out to agree quite well with moredetailed calculations. 23% 4 He mass fraction means that thenumber fraction of 4 He is about 6% with respect to hydrogen.The fact that the universe finally ends up containingaround one quarter 4 He by mass can be seen as the result ofa number of rather remarkable coincidences. For example,mean lifetimes for weak decays can vary over many ordersof magnitude. The fact that the mean neutron lifetime turnsout to be 885.7 s is a complicated consequence of the ratherclose neutron and proton masses combined with strong andweak interaction physics. If it had turned out that τ n were,say, only a few seconds or less, then essentially all of theneutrons would decay before they could be bound up intodeuterium, and the chain of nuclear reactions could not havestarted.4 He number fractiontuning of parameters