Astroparticle Physics

10.2 Start of the BBN Era 215these contribute together g ν = 2N ν . From (9.18) one thereforeobtainsg ∗ = 2 + 7 8 (4 + 2N ν). (10.3)For N ν = 3 one has g ∗ = 10.75. Using this value, (10.2) canbe written in a form convenient for description of the BBNera,tT 2 ≈ 0.74 s MeV 2 . (10.4)10.2 Start of the BBN Era“By the word of the Lord were the heavensmade. For he spoke, and it came tobe; he commanded, and it stood firm.”The Bible; Psalm 33:6,9From (10.4) a temperature of T = 10 MeV is reached at atime t ≈ 0.007 s. At this temperature, all of the relativisticparticles – γ , e − , ν e , ν µ , ν τ , and their antiparticles – are inthermal equilibrium through reactions of the type e + e − ↔ν ¯ν, e + e − ↔ γγ, etc. The number density of the neutrinos,for example, is given by the equilibrium formula appropriatefor relativistic fermions, see (9.11),n ν = 3 ζ(3)4 π 2 g νT 3 , (10.5)with a similar formula holding for the electron density.Already at temperatures around 20 MeV, essentially allof the antiprotons and antineutrons annihilated. The baryonto-photonratio is a number that one could, in principle, predict,if a complete theory of baryogenesis were available.Since this is not the case, however, the baryon density hasto be treated as a free parameter. Since one does not expectany more baryon-number-violating processes at temperaturesnear the BBN era, the total number of protons andneutrons in a comoving volume remains constant. That is,even though protons and neutrons are no longer relativistic,baryon-number conservation requires that the sum of theirnumber densities followsequilibriumof relativistic particlesbaryogenesisbaryon-number conservationn n + n p ∼ 1 R 3 ∼ T 3 . (10.6)At temperatures much greater than the neutron–proton massdifference, m = m n − m p ≈ 1.3 MeV, one has n n ≈ n p .