= Solution
Above the transition,
$$
\frac{\Gamma_{\rm int}}H=\frac{\alpha_Y^2m_{\rm pl}}{1.66\sqrt{g_*}\,T}=\frac{T_{\rm eq}}T,\qquad\boxed{T_{\rm eq}\sim7\times10^{11}\ {\rm GeV}}
$$
for $g_*\sim100$. The ratio increases during cooling, rather than at higher <temperature>. A GUT transition at $T_c\sim10^{15}$–$10^{16}$ GeV occurs well above this scale, so the interaction is too slow to establish or maintain <thermal equilibrium> throughout $T>T_c$. In fact its integrated interaction probability from an arbitrarily high <temperature> down to $T_c$ is at most of order $T_{\rm eq}/T_c\ll1$, since $dt=-dT/(HT)$.
An initial <Fermi-Dirac distribution> with zero <chemical potential> therefore cannot be assumed from this high-temperature rate alone. If imposed initially and collisionlessly redshifted while the particles are massless, its thermal-looking form can persist, but this is not dynamically maintained <thermal equilibrium>. Other initial states can remain nonthermal. This is the <thermalization temperature for a massless gauge mediator>.
The below-transition rate stipulated in the model has $\Gamma/H=(T/T_D)^3$. If $T_c\gg T_D$ and it includes efficient number-changing reactions, it can restore <chemical equilibrium> after the transition and erase the initial-state memory, making the estimates in a and b applicable. If it only equilibrates momenta, or if <reheating> never populates the relevant range, the <thermal relic> calculation must be reconsidered.
Back to article page
