units in which $M_{\rm pl}$ is set to unity.) This amplitude of fluctuation has steep $\ell$ dependence and is much larger than the ordinary slow-roll power spectrum determined by the Hubble rate during the open inflation unless $1-\Omega_0$ is extremely small. Therefore we need to introduce some complications even in two field models, in order to make them compatible with observations [@Sasaki:1996qn].

Originally, the models of open inflation were proposed in order to describe an inflationary universe with $\Omega_{0} \sim 0.3$. The possibility that $\Omega_{0} \sim 0.3$ was eventually ruled out from the CMB observations, but CMB observations do not rule out the possibility that inflation after tunneling is sufficiently long and $\Omega_{0}$ is sufficiently close to $1$. This fact gives a revived interest in the generation of perturbation in these models from the point of view of testing some of the predictions of the string landscape scenario. Indeed, we will find that some models based on CDL tunneling are consistent with the existing observational data for $1-\Omega_{0} \sim 10^{-3}$, but may be ruled out for $1-\Omega_{0} \sim
10^{-2}$. Furthermore, models of quasiopen inflation, as well as the models leading to the Hawking-Moss tunneling [@Hawking:1981fz; @b60; @Linde:1991sk], may lead to 