Scalar perturbation spectra in an emergent cosmological island

The possibility that the Universe may have a fundamental and positive cosmological constant has motivated an interesting cosmological model, in which initially the Universe is in a cosmological constant sea, then the local quantum fluctuations violating the null energy condition create some islands with matter and radiation, which under certain conditions might correspond to our observable universe. We in this paper study the perturbation spectra of scalar fields not affecting the evolution of background during the fluctuation. We will examine whether they can be interesting, and responsible for the structure formation of the observable universe