over oscillations, the amplitude of the $plateau$ characterizing the energy density spectrum today is $$\begin{aligned}
\label{eq:InfGWtodayRD}
 \Omega_{\rm GW}^{(0)}{\Big |}_{\rm plateau}  &\simeq& ~~\mathcal{G}_k { \Omega_{\rm rad}^{(0)}\over12\pi^2}\left(H_{\rm inf}\over m_p\right)^2  \simeq ~1\cdot10^{-16}\left(H_{\rm inf}\over H_{\rm max}\right)^2\,,\end{aligned}$$ where we have used $k = a_kH_k$ and introduced the RD transfer function [@Boyle:2005se] $$\begin{aligned}
\label{eq:TransferAndGk}
 T_h(k) \simeq {1\over2}\left({a_k\over a_0}\right)^2 \simeq {1\over2}\mathcal{G}_k\Omega_{\rm rad}^{(0)}\left({a_0H_0\over a_k H_k}\right)^2\,,~~~\mathcal{G}_k \equiv \left(g_{*,k}\over g_{*,0}\right)\left(g_{s,0}\over g_{s,k}\right)^{4/3}\,.\end{aligned}$$ In the $rhs$ of Eq. (\[eq:InfGWtodayRD\]) we have used $\Omega_{\rm rad}^{(0)} \simeq 9\cdot 10^{-5}$, $g_{s,0} \simeq 3.91$, $g_{*,0} = 3.36$, and $g_{s,k} \simeq g_{*,k} \simeq 106.75$ (so that $\mathcal{G}_k \simeq 0.39$). For simplicity we have considered $g_{s,k}, g_{*,k}$ equal to the Standard Model (SM) degrees of freedom ($dof$) before the electroweak symmetry breaking, and hence independent of $k$. In reality the number of SM relativistic $dof$ change with the scale, but we postpone the discussion of this spectral distortion to Section \[s:changesindof\]. For the time being we simply consider an identical suppression for all the modes as $\mathcal{G}_k \sim 0.39$.

Eq. (\[eq:InfGWtodayRD\]) describes the amplitude of the $plateau$ of the inflationary GW (quasi-)scale invariant energy density spectrum today, corresponding to the modes that crossed the horizon during RD. However, if after inflation there is a transient period of evolution 