Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 55 / 3 / a / Solution

The new neutrino species undergoes thermal decoupling in cosmology when its interaction rate becomes comparable to the Hubble parameter. Equating the given rates givesWith $g_{\ast }\simeq 100$, $G_A=10^{-12}{\mathrm{GeV}}^{-2}$ and $m_{\mathrm{Pl}}\simeq 10^{19}\mathrm{GeV}$,This is an order-of-magnitude electroweak-scale estimate under the given approximation. The mass in this formula is the unreduced Planck mass, not the reduced Planck mass.