even parity one. We introduce the variable $U_p$ to expand the spatial metric perturbation $$\begin{aligned}
\delta g_{ij}=a_{\rm C}^2(\eta_{\rm C})
\sum\hat{a}^{({\rm T})}_{p\ell m}
U_p(\eta_{\rm C})Y^{(+)p\ell m}_{ij}(\chi_{\rm C} ,\Omega )
+{\rm h.c.}\,,\end{aligned}$$ where $\hat{a}^{({\rm T})}_{p\ell m}$ and $Y^{(+)p\ell m}_{ij}$ denote the annihilation operator and the analytic continuation of the even parity tensor harmonics on the 3-hyperboloid [@Tomita:1982], respectively. The function $Y^{(+)p\ell m}_{ij}$ is more explicitly expressed using the ordinary spherical harmonics $Y_{lm}$. In the Sachs-Wolfe formula we only need the $\chi\chi$ component, which is given by $$Y^{(+)p\ell m}_{\chi\chi}(\chi_{\rm C},\Omega )=
\frac{1}{\cosh^{2}\chi_{\rm C}}f^{p\ell}(\chi_{\rm C})Y_{\ell m}(\Omega)\,.$$ Then, the time evolution equation for $f^{p\ell}(\chi_{\rm C})$ is in a model-independent manner given by $$\begin{aligned}
&&\Biggl[ -\frac{1}{\cosh^{2}\chi_{\rm C}}\frac{{{\rm d}}}{{{\rm d}}\chi_{\rm C}}
\cosh^{2}\chi_{\rm C}\frac{{{\rm d}}}{{{\rm d}}\chi_{\rm C}}
-\frac{\ell (\ell +1)}{\cosh^{2}\chi_{\rm C}}\Biggr]
f^{p\ell}
\nonumber\\
&&\qquad \qquad \qquad \qquad \qquad \ \ \ \ \ \ \ \
=\left( p^2+1\right) f^{p\ell}\,.\end{aligned}$$

As a natural vacuum state after tunneling, we adopt the Euclidean vacuum, which is specified by requiring that the positive frequency functions are all regular at $\chi_{\rm C}=0$. Then, we have $$\begin{aligned}
&&f^{p\ell}(\chi_{\rm C})
=\sqrt{\frac{\Gamma (ip+\ell +1)\Gamma (-ip+\ell +1)}{\Gamma (ip)\Gamma (-ip)}}
\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ 
\times
\frac{1}{\sqrt{\cosh\chi_{\rm C}}}P^{-\ell -1/2}_{ip-1/2}\left( 