Consistent cosmic bubble embeddings

The Raychaudhuri equation for null rays is a powerful tool for finding consistent embeddings of cosmological bubbles in a background spacetime in a way that is largely independent of the matter content. We find that spatially flat or positively curved thin wall bubbles surrounded by a cosmological background must have a Hubble expansion that is either contracting or expanding slower than the background, which is a more stringent constraint than those obtained by the usual Israel thin-wall formalism. Similarly, a cosmological bubble surrounded by Schwarzschild space, occasionally used as a simple “swiss cheese” model of inhomogenities in an expanding universe, must be contracting (for spatially flat and positively curved bubbles) and bounded in size by the apparent horizon.

Figure 1

We will be considering radially ingoing null rays as they cross the boundary $\mathrm{\Sigma }$ from a spherically symmetric background into an embedded cosmological bubble.

Figure 2

A null ray traveling across a thick wall boundary between a cosmological background $H_{+}$ and bubble $H_{-}$ leaves the background at $\tilde{r}=R+\delta$ and enters the bubble at $\tilde{r}=R-\delta$.

Figure 3

A FLRW bubble surrounded by a Schwarzschild background can serve as a model for a “vacuole” or swiss cheese inhomogeneity embedded in a larger expanding background. We study the behavior of radial null rays as they traverse from the Schwarzschild background into the bubble.

Figure 4

One way to evade the constraints on bubble embeddings from the Raychaudhuri equation is for the bubble to become disconnected from the background spacetime by the creation of a wormhole, as shown in this Penrose diagram adapted from [16]. Radially ingoing null rays starting in the background either see the bubble contracting or are casually disconnected from the bubble, while the bubble continues to expand in a “baby universe.”