p
+{\rm Re}\left(\rho_{+}^p\Bigl(\overline{\tilde{\bm w}^p}\Bigr)^{2}\right)
\biggr]
\,.\end{aligned}$$

After the horizon crossing, $a^2H^2\gg p^2+1$, Eq.  implies that the amplitude of $U_p$ freezes. In terms of ${\bm w}^p$, two independent solutions after the horizon crossing are given by $$\begin{aligned}
{\bm w}^p_1
\propto a_{\rm R}^{-1}\,,\ \ \ 
{\bm w}^p_2\propto 
a_{\rm R}^{-1}\int^{\eta}a_{\rm R}^2(\eta' ){{\rm d}}\eta'
\,.\end{aligned}$$ Neglecting the decaying mode ${\bm w}^p_1$, we denote the asymptotic behavior of $\tilde {\bm w}^p$ as $$\begin{aligned}
\tilde{\bm w}^p\rightarrow -{\mathbb H}_p
e^{ip\eta_{\rm T}^p}
a_{\rm R}^{-1}\int^{\eta}a_{\rm R}^2(\eta' ){{\rm d}}\eta'
\,,
\label{eq:asymptotic behavior of w^p}\end{aligned}$$ where ${\mathbb H}_p$ and $\eta_{\rm T}^p$ are the amplitude and phase, respectively. We note that $p\eta_{\rm T}^p$ vanishes in the limit $p\rightarrow 0$ because $\overline{\tilde{\bm w}^p}=\tilde{\bm w}^{-p}$.

Using ${\mathbb H}_p$ and $\eta_{\rm T}^p$, the even parity tensor power spectrum can be expressed as [@Garriga:1998he; @Garriga:1997wz] $$\begin{aligned}
&&\frac{p^3}{2\pi^{2}}P_{\rm T}(p)
\equiv
\frac{p^3}{2\pi^{2}}
\left(
\bigl| U_{p(+)}\bigl|^{2}
+\bigl| U_{p(-)}\bigl|^{2}
\right)
\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ 
=4\kappa\left(\frac{{\mathbb H}_p}{2\pi}\right)^{2}\frac{p^2\coth\pi p}{1+p^2}
\Bigl( 1-y_{\rm T}^p\Bigr)\,,
\label{eq:general power spectrum}\end{aligned}$$ where $$\begin{aligned}
&&y_{\rm T}^p=-\frac{1}{\cosh\pi p}{\rm Re}
\left(\rho_{+}^pe^{-2ip\eta_{\rm T}^p}\right)
\,,\end{aligned}$$ essentially represents the effects of the bubble wall.

To summarize, in order to find the power spectrum for tensor-type perturbations, one has to solve the scattering problem 