the RD epoch reads $$\begin{aligned}
h_k^{\prime\prime}+\frac{2}{\tilde{\tau}_r}h_k^{\prime}+k^2h_k=0\,,\label{eq:eomRD}\end{aligned}$$ where this time $'$ denotes derivatives with respect to $\tilde\tau_r$ (again simply because $d\tilde\tau_r = d\tau$). The solution during the RD era to (\[eq:eomRD\]) $\tau_{\text{RD}}\leq \tau \ll\tau_{\text{eq}}$, is $$h_k^{\text{(rad)}}(\tau)=\frac{1}{\sqrt{y}}\left[A(k) J_{1\over2}(y)+B(k) Y_{1\over2}(y)\right]\,, ~~~~~~y \equiv {k\over aH} = k\tilde\tau_{r}(\tau)\,, \label{radsol}$$ where $J_{1/2}(y), Y_{1/2}(y)$ are Bessel functions of the first and second kind and the superscript $^{(\text{rad})}$ indicates that the solution applies during RD. At $\tau=\tau_{\text{RD}}$ this solution must match with solution (\[stiffsol\]) \[and hence simultaneously with $h_k^{\text{inf}}$ if the mode is superhorizon\]. Continuity of the tensor modes and their derivatives requires $$\begin{aligned}
h_k^{\text{(stiff)}}(\tau_{\text{RD}}) = h_k^{\text{(rad)}}(\tau_{\text{RD}})\,,~~~~
h_k^{\prime \text{(stiff)}}(\tau_{\text{RD}}) = h_k^{\prime \text{(rad)}}(\tau_{\text{RD}})\end{aligned}$$ from which we get $$\begin{aligned}
\left\lbrace \begin{array}{c}
A(k)\vspace*{0.3cm}\\B(k)          
\end{array}
\right\rbrace = {1\over\sqrt{2}}\left(\frac{2}{\alpha_{\rm s}}\right)^{\alpha_{\rm s}}\Gamma\left(\alpha_{\rm s}+\frac{1}{2}\right)\hspace*{-1mm}~\kappa^{1-\alpha}~h_k^{\text{inf}}\times \left\lbrace \begin{array}{c}
a(\kappa)\vspace*{0.3cm}\\b(\kappa)          
\end{array}\right\rbrace\,,\label{ab}\end{aligned}$$ where $$\begin{aligned}
\kappa \equiv {k\over k_{\rm RD}}\,,\end{aligned}$$ with $k_{\rm RD} \equiv a_{\rm RD}H_{\rm RD}$, and $$\begin{aligned}
a(\kappa) =\sqrt{\alpha_{\rm s}}\left(\dfrac{J_{\alpha_{\rm s}-{1\over2}}(\alpha_{\rm s}\kappa)Y_{3\over2}(\kappa)-J_{\alpha_{\rm s}+{1\over2}}(\alpha_{\rm s}\kappa)Y_{1\over2}(\kappa)}{J_{1\over2}(\kappa)Y_{3\over2}(\kappa)-J_{3\over2}(\kappa)Y_{1\over2}(\kappa)}\right)\,,\\
b(\kappa) =\sqrt{\alpha_{\rm s}}\left(\dfrac{J_{\alpha_{\rm s}+{1\over2}}(\alpha_{\rm s}\kappa)J_{1\over2}(\kappa)-J_{\alpha_{\rm s}-{1\over2}}(\alpha_{\rm s}\kappa)J_{3\over2}(\kappa)}{J_{1\over2}(\kappa)Y_{3\over2}(\kappa)-J_{3\over2}(\kappa)Y_{1\over2}(\kappa)}\right)\,. \label{abtilde}\end{aligned}$$ The sub-horizon $k\tau\gg 1$ and $\tau\gg \tau_{\rm RD}$ limit of is $$h_k^{\text{(rad)}}(\tau)=\sqrt{\frac{2}{\pi y^2}}\,\left[\,A(k)\sin y-B(k)\cos y\,\right]\,,\label{eq:TensorRDsubHubble}$$ where we used the large argument expansion of Bessel functions $J_{\nu}(x \gg 1) \simeq \sqrt{2\over\pi x}\sin(x-\delta_\nu)$, $Y_{\nu}(x \gg 1) \simeq \sqrt{2\over\pi x}\cos(x-\delta_\nu)$ and $\tilde{\tau}\approx\tau$. Substituting into , squaring, and averaging over 