inflation.

In this paper we will concentrate on the investigation of the models with the CDL tunneling and investigate the CMB temperature fluctuations. We will focus on tensor perturbations since tensor perturbations are more sensitive to the value of the Hubble parameter and properties of tunneling than scalar perturbations. At the end of the paper we will briefly discuss scalar perturbations and their cosmological implications.

This paper is organized as follows. In Sec. II we describe the background evolution of the models of open inflation in the context of the string landscape scenario. There we introduce models with a rapid-roll phase after the bubble nucleation. In Sec. III we first review the method to evaluate the tensor perturbation after the false vacuum decay, and apply it to our models to identify the typical signature in the CMB spectrum. Section IV is devoted to summary.

Background evolution of the open inflation models in the landscape {#sec:open inflation}
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Bubble nucleation {#sec:existence of rolling down phase}
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![Schematic picture of the effective potential for inflaton in one-bubble open inflation scenario.[]{data-label="fig:open_potential"}](open_potential.eps){width="80mm"}

The system which we are going to consider consists of a minimally coupled scalar field, $\phi$ with Einstein gravity. As we mentioned 