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

Before electron-positron annihilation, photons, electrons, and positrons share one temperature. Their effective entropy degrees of freedom areThe neutrinos have already undergone thermal decoupling in cosmology, so $T_{\nu }a$ remains constant and they receive none of the electron-positron entropy. In the still-coupled electromagnetic plasma, cosmological entropy conservation givesThe decoupled neutrino temperature at the same later time is $T_{\nu }=T_{\mathrm{before}}{a}_{\mathrm{before}}/a$. Dividing the two relations gives the Cosmic neutrino background temperatureThis instantaneous-decoupling calculation neglects the small reheating correction from non-instantaneous neutrino decoupling.

Photons are bosons with two polarizations and temperature $T$, so ${\rho }_{\gamma }=({\pi }^2/30)2T^4$. Each relativistic neutrino species includes a neutrino and antineutrino helicity state, is fermionic, and has temperature $T_{\nu }=(4/11{)}^{1/3}T$. ConsequentlyAdding the two contributions gives the defining expression for the effective number of neutrino species:

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.

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.