i\sinh\chi_{\rm C}\right)\,,~~~~\end{aligned}$$ where $P^{\mu}_{\nu}$ is the associated Legendre function of the first kind. We have fixed the normalization constant so that the analytic continuation of $Y^{p\ell m}(\chi_{\rm C},\Omega )\equiv 
f^{p\ell} (\chi_{\rm C})Y_{\ell m}(\Omega )$ to region R or L behaves as a spatial harmonic function properly normalized on a unit 3-hyperboloid.

The equation for $U_p$ is the same as the one for the massless scalar field [@Tanaka:1997kq]: $$\begin{aligned}
\Biggl[\frac{1}{a_{\rm C}^3}\frac{{{\rm d}}}{{{\rm d}}t_{\rm C}}a_{\rm C}^3
\frac{{{\rm d}}}{{{\rm d}}t_{\rm C}}+\frac{p^2+1}{a_{\rm C}^2}\Biggr] 
U_p=0\,.
\label{eq:U_p eq}\end{aligned}$$ It is also convenient to introduce another variable ${\bm w}^p$, which is related to $U_p$ by [@Garriga:1998he; @Garriga:1997wz] $$\begin{aligned}
U_p=-\sqrt{\frac{\kappa}{p(1+p^2)\sinh\pi p}}\frac{1}{a_{\rm R}}
\frac{{{\rm d}}}{{{\rm d}}t_{\rm R}}\left( a_{\rm R}{\bm w}^p\right)\,.\end{aligned}$$ Then, the spatial eigenfunction ${\bm w}^p$ satisfies a simple equation $$\begin{aligned}
\Biggl[ -\frac{{{\rm d}}^{2}}{{{\rm d}}\eta_{\rm C}^2}
+U_{\rm T}(\eta_{\rm C})\Biggr]{\bm w}^p
=p^2{\bm w}^p\,,
\label{eq:w^p eq}\end{aligned}$$ with $$\begin{aligned}
U_{\rm T}(\eta_{\rm C})=\frac{\kappa}{2}{\phi'}^2(\eta_{\rm C})\,,
\label{eq:effective potential for mode func}\end{aligned}$$ where a prime represents differentiation with respect to the conformal time. Since the effective potential $U_{\rm T}$ is clearly positive definite, there is no supercurvature mode (bound state) in the tensor-type perturbation.

From the boundary conditions for the background solution, we find that the potential $U_T$ vanishes at both boundaries of the region C. 