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

The measurement cannot definitively exclude the model. If only Standard Model particles supplied entropy at scalar decoupling, the largest available value $g_{\ast s}=106.75$ would givewhich a $0.1\%$ measurement around the Standard Model value would clearly detect.

In the proposed model, however, the many additional relativistic species are also in equilibrium when the scalar decouples. They increase $g_{\ast s}(T_d)$, and their later disappearance transfers entropy to the coupled bath but not to the scalar. Since $\Delta N_{\mathrm{eff}}\propto g_{\ast s}(T_d{)}^{-4/3}$, a sufficiently large hidden particle content can dilute the scalar signal below any stated finite precision; a value above roughly $g_{\ast s}(T_d)\sim 550$ already pushes it below about $0.003$. The null measurement constrains the combination of decoupling time and total entropy degrees of freedom, but does not rule out the entire new-physics model.