Large-scale inhomogeneity of dark energy produced in the ancestor vacuum

We investigate large-scale inhomogeneity of dark energy in the bubble nucleation scenario of the universe. In this scenario, the present universe was created by a bubble nucleation due to quantum tunneling from a metastable ancestor vacuum, followed by a primordial inflationary era. During the bubble nucleation, supercurvature modes of some kind of a scalar field are produced, and remain until present without decaying; thus they can play a role of the dark energy, if the mass of the scalar field is sufficiently light in the present universe. The supercurvature modes fluctuate at a very large spatial scale, much longer than the Hubble length in the present universe. Thus they create large-scale inhomogeneities of the dark energy, and generate large-scale anisotropies in the cosmic microwave background (CMB) fluctuations. This is a notable feature of this scenario, where quantum fluctuations of a scalar field are responsible for the dark energy. In this paper, we calculate imprints of the scenario on the CMB anisotropies through the integrated Sachs-Wolfe (ISW) effect, and give observational constraints on the curvature parameter $\Omega_K$ and on an additional parameter $\epsilon$ describing some properties of the ancestor vacuum.