the potential for the chaotic inflation: $$\begin{aligned}
\frac{\kappa}{3}V(\phi )=H_{\rm R}^2\left(\frac{\phi}{\phi_{\rm R}}\right)^{2}
\exp\biggl[\frac{\phi^{2}-\phi_{\rm R}^2}{\phi_{\rm R}^{2}}\biggr]\,.
\label{eq:chaotic-type}\end{aligned}$$ The slow-roll parameter is given by $$\begin{aligned}
\epsilon (\phi )=\frac{2}{\kappa\phi_{\rm R}^2}
\left(\frac{\phi}{\phi_{\rm R}}+\frac{\phi_{\rm R}}{\phi}\right)^{2}
\,.
\label{slow-roll}\end{aligned}$$ Note that we will impose $\kappa\phi_{\rm R}^2>8$ for the slow-roll inflation to occur at $\phi =\phi_{\rm R}$. As with the exponential-type potential, the potential of a similar type also often appears in supergravity and superstring theory. As a concrete example, we will consider the case with $H_{\rm R}^2=10^{-40}M_{\rm pl}^2$ and $\phi_{\rm R}=7.5M_{\rm pl}$. In Fig. \[fig:chaotic\_evol\], the evolution of the energy densities for $\phi_{*}=9\phi_{\rm R}$, $7\phi_{\rm R}$, $5\phi_{\rm R}$, and $3\phi_{\rm R}$ is shown. Since the field varies slowly near the nucleation point, the chaotic-type potential Eq.  can be approximated there by the exponential-type. Therefore one can see that the tracking solution appears after $t\approx \epsilon_{*}^{-1/2}H_*^{-1}$ even in this chaotic-type potential.

CMB anisotropy in the landscape
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Tensor power spectrum {#sec: general power spectrum}
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We begin with reviewing how to compute the CMB power spectrum due to tensor perturbation for single-field models of one-bubble open inflation, following Refs. [@Garriga:1998he; @Garriga:1997wz]. Assuming that the observer is at the center of the spherical coordinates, the relevant tensor perturbation is the 