Following the epoch of primordial element formation (Ex. 4.10), the universe continued to expand and cool. Eventually when the temperature of the photons was ∼3,000 K, the free electrons and protons combined to form atomic hydrogen; this was the epoch of recombination. Later, when the photon temperature had fallen to ∼30 K, some hot stars and quasars formed and their ultraviolet radiation dissociated the hydrogen; this was the epoch of reionization. The details are poorly understood. (Making the simplifying assumption that reionization happened rapidly and homogeneously, show that the increase in the entropy per baryon was ∼60k_B, depending weakly on the temperature of the atomic hydrogen, which you can assume to be ∼100 K.

Data from Exercises 4.10

(a) The early universe was ∼200 s old, its principal constituents were photons, protons, neutrons, electrons, positrons, and (thermodynamically isolated) neutrinos and gravitons. The photon temperature was ∼9 × 10⁸ K, and the baryon density was ∼0.02 kgm⁻³. The photons, protons, electrons, and positrons were undergoing rapid electromagnetic interactions that kept them in thermodynamic equilibrium. Use the neutron-proton mass difference of 1.3MeV to argue that, if the neutrons had also been in thermodynamic equilibrium, then their density would have been negligible at this time.

(b) However, the universe expanded too rapidly for weak interactions to keep the neutrons in equilibrium with the other particles, and so the fraction of the baryons at 200 s, in the form of neutrons that did not subsequently decay, was ∼0.14. At this time the neutrons began rapidly combining with protons to form alpha particles through a short chain of reactions, of which the first step—the one that was hardest to make go—was n + p → d + 2.22MeV, with the 2.22MeV going into heat. Only about 10−5 of the baryons needed to go into deuterium in order to start and then maintain the full reaction chain. Show, using entropy per particle and the first law of thermodynamics, that at time t ∼ 200 s after the big bang, this reaction was just barely entropically favorable.

(c) During this epoch, roughly how do you expect the baryon density to have decreased as a function of decreasing photon temperature? (For this you can neglect the heat being released by the nuclear burning and the role of positrons.) Show that before t ∼ 200 s, the deuterium-formation reaction could not take place, and after t ∼ 200 s, it rapidly became strongly entropically favorable.

(d) The nuclear reactions shut down when the neutrons had all cycled through deuterium and almost all had been incoroporated into α particles, leaving only ∼10⁻⁵ of them in deuterium. About what fraction of all the baryons wound up in α particles (Helium4)?