Question B2 The number density of a non-relativistic particle of mass m in equilibrium is given by nx (mkT)3/2e -mc²/kT At early times in the universe the temperature of radiation evolves as a) 1/2 T(1) ≈ 8.07 x 10° K t (1) Neutrino decoupling happens when kT ~0.8 MeV. Determine the time in seconds when this occurs. Discuss the weak reactions which take place before this and after this time. b) State the implication that neutrino decoupling has for the subsequent number of neutrons. The neutron fraction is defined by c) nn Xn = nn + np Show that before deuterium formation starts, Xn(t) ~ 1/ 2 6 where Tn 881 s is the mean lifetime of a free neutron. Show that for the formation of deuterium, the temperature T given by the solution to 3/2 kT mpc² eBD/KT ~ 1 n must be reached. Here, BD is the deuterium binding energy, and n = n₁/n, is the baryon- to-photon ratio. You may use the fact that the number of photons no, ∞ (kT)³. d) If the solution to the equation in part (c) is kT ~0.06 MeV, show that the mass fraction of He to Hydrogen is approximately 1/4. Explain how the Helium abundance would be affected if the number of photons compared to baryons were reduced significantly.

Question B2 The number density of a non-relativistic particle of mass m in equilibrium is given by nx (mkT)3/2e -mc²/kT At early times in the universe the temperature of radiation evolves as a) 1/2 T(1) ≈ 8.07 x 10° K t (1) Neutrino decoupling happens when kT ~0.8 MeV. Determine the time in seconds when this occurs. Discuss the weak reactions which take place before this and after this time. b) State the implication that neutrino decoupling has for the subsequent number of neutrons. The neutron fraction is defined by c) nn Xn = nn + np Show that before deuterium formation starts, Xn(t) ~ 1/ 2 6 where Tn 881 s is the mean lifetime of a free neutron. Show that for the formation of deuterium, the temperature T given by the solution to 3/2 kT mpc² eBD/KT ~ 1 n must be reached. Here, BD is the deuterium binding energy, and n = n₁/n, is the baryon- to-photon ratio. You may use the fact that the number of photons no, ∞ (kT)³. d) If the solution to the equation in part (c) is kT ~0.06 MeV, show that the mass fraction of He to Hydrogen is approximately 1/4. Explain how the Helium abundance would be affected if the number of photons compared to baryons were reduced significantly.