Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 2 / i / c / Solution

A thermally populated extra light neutrino species is relativistic during nucleosynthesis; its $0.001\mathrm{eV}$ mass is negligible at MeV temperatures. One neutrino-antineutrino species adds $(7/8)\times 2=7/4$ to $g_{\ast }$. Thus the expansion rate increases by $\sqrt{12.5/10.75}$ and the freeze-out temperature by $(12.5/10.75{)}^{1/6}$. Earlier freeze-out leaves more neutrons, while faster cooling leaves less time for decay. The helium-4 abundance increases. This assumes the new species was populated at the standard neutrino temperature; an unpopulated species would not contribute this extra density.