Junction conditions of Friedmann-Robertson-Walker space-times.

We complete a classification of junctions of two Friedmann-RobertsonWalker space-times bounded by a spherical thin wall. Our analysis covers super-horizon bubbles and thus complements the previous work of Berezin, Kuzumin and Tkachev. Contrary to sub-horizon bubbles, various topology types for super-horizon bubbles are possible, regardless of the sign of the extrinsic curvature. We also derive a formula for the peculiar velocity of a domain wall for all types of junction. PACS number(s): 04.20.-q ∗ electronic mail: [email protected] ∗∗ electronic mail: [email protected] Using the thin-wall formalism first devised by Israel [1], many authors studied bubble dynamics [2]-[7]. In the formalism, a (2+1) dimensional hypersurface of a trajectory of the bubble wall is embedded in two different homogeneous space-times: One describes the inside of the bubble and the other does the outside. Two spacetimes are pasted on the hypersurface with the junction conditions for metrics and extrinsic curvature tensors. Berezin, Kuzumin and Tkachev [2] studied the junction conditions of two different Friedmann-Robertson-Walker (FRW) space-times bounded by a spherical thin wall. They considered only the case when bubbles are smaller than the horizon scale. From this restriction, they discussed the constraints on decaying-vacuum parameters, which are related to the size of a nucleated bubble, in an open or flat universe. Although Berezin et al. mentioned the possibility of bubbles which are larger than the horizon scale, they did not show the junction conditions explicitly. As long as the self-gravity of a bubble is ignored, a bubble is in fact nucleated within the horizon scale. However, in some inflationary models there may exist bubbles whose self-gravity is dominant [5] and, therefore, whose size could be larger than the horizon scale [6]. Further, vacuum parameters should be determined from microscopic physics but not from a macroscopic constraint such as the condition that a bubble should be smaller than the horizon size. In this paper, we re-examine junctions of FRW space-times and classify all possible spherical bubbles. Our study is a complement to that of Berezin et al., and will also be useful for analyzing super-horizon voids [8]. To begin, we define a unit space-like vector N which is orthogonal to the world hypersurface Σ denoting a trajectory of a bubble wall and which points from V − (inside of the bubble) to V + (outside). For convenience, we introduce a Gaussian normal coordinate system (n, τ, θ, φ), where n = 0 corresponds to Σ and τ is the proper time of the wall. Then the angular component of the extrinsic curvature