Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 2 / c / Solution

After the real scalar decouples at $T_d$, its temperature redshifts as $a^{-1}$. The other particles continue sharing entropy. Just before standard-neutrino decoupling their entropy degrees of freedom areThe temperature of a decoupled relativistic relic therefore obeysA real scalar has one bosonic degree of freedom, whereas one effective neutrino species has energy weight $(7/8)\times 2=7/4$. Hence the contribution of a decoupled real scalar to Neff isHere $g_{\ast s}(T_d)$ counts the other particles still coupled to the thermal bath, as specified in the question.