= Solution
The new <neutrino> species undergoes <thermal decoupling in cosmology> when its interaction rate becomes comparable to the <Hubble parameter>. Equating the given rates gives
$$
G_A^2T_D^5=\frac{1.66\sqrt{g_*}\,T_D^2}{m_{\mathrm{Pl}}},\qquad
T_D=\left(\frac{1.66\sqrt{g_*}}{G_A^2m_{\mathrm{Pl}}}\right)^{1/3}.
$$
With $g_*\simeq100$, $G_A=10^{-12}\,\mathrm{GeV}^{-2}$ and $m_{\mathrm{Pl}}\simeq10^{19}\,\mathrm{GeV}$,
$$
\boxed{T_D\simeq(1.66\times10^6)^{1/3}\,\mathrm{GeV}\simeq1.2\times10^2\,\mathrm{GeV}.}
$$
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>.
Back to article page
